LMFDB - Embedded newform 4232.2.a.bb.1.9

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4232,2,Mod(1,4232)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4232.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4232, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4232 = 2^{3} \cdot 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4232.a (trivial)

Newform invariants

comment:select newform

sage:traces = [15,0,1,0,0,0,10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$33.7926901354$
Analytic rank:	$0$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - x^{14} - 30 x^{13} + 28 x^{12} + 354 x^{11} - 302 x^{10} - 2111 x^{9} + 1596 x^{8} + 6777 x^{7} + \cdots - 419 $ x^15 - x^14 - 30x^13 + 28x^12 + 354x^11 - 302x^10 - 2111x^9 + 1596x^8 + 6777x^7 - 4292x^6 - 11506x^5 + 5297x^4 + 9480x^3 - 1825x^2 - 3136*x - 419
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 184)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$1.07060$ of defining polynomial
Character	$\chi$	$=$	4232.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.07060 q^{3} -3.54589 q^{5} +3.82947 q^{7} -1.85382 q^{9} +4.38386 q^{11} +5.21152 q^{13} -3.79622 q^{15} -0.517206 q^{17} +3.99525 q^{19} +4.09982 q^{21} +7.57336 q^{25} -5.19649 q^{27} -9.23921 q^{29} -3.51763 q^{31} +4.69334 q^{33} -13.5789 q^{35} +3.44042 q^{37} +5.57944 q^{39} +10.8104 q^{41} +0.391013 q^{43} +6.57346 q^{45} +1.50879 q^{47} +7.66482 q^{49} -0.553720 q^{51} +0.746394 q^{53} -15.5447 q^{55} +4.27730 q^{57} -5.69987 q^{59} +5.94097 q^{61} -7.09915 q^{63} -18.4795 q^{65} +2.72082 q^{67} -3.85262 q^{71} -5.78124 q^{73} +8.10802 q^{75} +16.7878 q^{77} +2.75488 q^{79} -0.00187825 q^{81} +2.73234 q^{83} +1.83396 q^{85} -9.89147 q^{87} +7.82654 q^{89} +19.9574 q^{91} -3.76596 q^{93} -14.1667 q^{95} -1.93011 q^{97} -8.12689 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q + q^{3} + 10 q^{7} + 16 q^{9} + 23 q^{11} + 10 q^{15} + 29 q^{19} + q^{21} + 23 q^{25} + q^{27} - 2 q^{29} + 20 q^{31} + 18 q^{33} - 18 q^{35} + 24 q^{37} - 19 q^{39} + 9 q^{41} + 48 q^{43} + 4 q^{45}+ \cdots + 63 q^{99}+O(q^{100}) $ 15 * q + q^3 + 10 * q^7 + 16 * q^9 + 23 * q^11 + 10 * q^15 + 29 * q^19 + q^21 + 23 * q^25 + q^27 - 2 * q^29 + 20 * q^31 + 18 * q^33 - 18 * q^35 + 24 * q^37 - 19 * q^39 + 9 * q^41 + 48 * q^43 + 4 * q^45 - 36 * q^47 + 25 * q^49 + 35 * q^51 - 5 * q^53 - 10 * q^55 + 23 * q^57 - 22 * q^59 + 12 * q^61 + 35 * q^63 - 26 * q^65 + 58 * q^67 + 2 * q^71 + 5 * q^73 - 17 * q^75 + 26 * q^77 + 26 * q^79 - 21 * q^81 + 68 * q^83 - 72 * q^85 + 19 * q^87 - 6 * q^89 + 71 * q^91 - 55 * q^93 - 12 * q^95 + 40 * q^97 + 63 * 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
$3$	1.07060	0.618109	0.309055	−	0.951044i	$-0.399987\pi$
$3$	1.07060	0.618109	0.309055	+	0.951044i	$0.399987\pi$
$4$	0	0
$5$	−3.54589	−1.58577	−0.792886	−	0.609370i	$-0.791422\pi$
$5$	−3.54589	−1.58577	−0.792886	+	0.609370i	$0.791422\pi$
$6$	0	0
$7$	3.82947	1.44740	0.723701	−	0.690113i	$-0.242439\pi$
$7$	3.82947	1.44740	0.723701	+	0.690113i	$0.242439\pi$
$8$	0	0
$9$	−1.85382	−0.617941
$10$	0	0
$11$	4.38386	1.32178	0.660891	−	0.750482i	$-0.270178\pi$
$11$	4.38386	1.32178	0.660891	+	0.750482i	$0.270178\pi$
$12$	0	0
$13$	5.21152	1.44542	0.722708	−	0.691153i	$-0.242897\pi$
$13$	5.21152	1.44542	0.722708	+	0.691153i	$0.242897\pi$
$14$	0	0
$15$	−3.79622	−0.980181
$16$	0	0
$17$	−0.517206	−0.125441	−0.0627205	−	0.998031i	$-0.519978\pi$
$17$	−0.517206	−0.125441	−0.0627205	+	0.998031i	$0.519978\pi$
$18$	0	0
$19$	3.99525	0.916572	0.458286	−	0.888805i	$-0.348464\pi$
$19$	3.99525	0.916572	0.458286	+	0.888805i	$0.348464\pi$
$20$	0	0
$21$	4.09982	0.894653
$22$	0	0
$23$	0	0
$23$	0	0
$24$	0	0
$25$	7.57336	1.51467
$26$	0	0
$27$	−5.19649	−1.00006
$28$	0	0
$29$	−9.23921	−1.71568	−0.857839	−	0.513918i	$-0.828194\pi$
$29$	−9.23921	−1.71568	−0.857839	+	0.513918i	$0.828194\pi$
$30$	0	0
$31$	−3.51763	−0.631785	−0.315892	−	0.948795i	$-0.602304\pi$
$31$	−3.51763	−0.631785	−0.315892	+	0.948795i	$0.602304\pi$
$32$	0	0
$33$	4.69334	0.817006
$34$	0	0
$35$	−13.5789	−2.29525
$36$	0	0
$37$	3.44042	0.565601	0.282801	−	0.959179i	$-0.408737\pi$
$37$	3.44042	0.565601	0.282801	+	0.959179i	$0.408737\pi$
$38$	0	0
$39$	5.57944	0.893426
$40$	0	0
$41$	10.8104	1.68830	0.844149	−	0.536108i	$-0.180106\pi$
$41$	10.8104	1.68830	0.844149	+	0.536108i	$0.180106\pi$
$42$	0	0
$43$	0.391013	0.0596289	0.0298145	−	0.999555i	$-0.490508\pi$
$43$	0.391013	0.0596289	0.0298145	+	0.999555i	$0.490508\pi$
$44$	0	0
$45$	6.57346	0.979913
$46$	0	0
$47$	1.50879	0.220080	0.110040	−	0.993927i	$-0.464902\pi$
$47$	1.50879	0.220080	0.110040	+	0.993927i	$0.464902\pi$
$48$	0	0
$49$	7.66482	1.09497
$50$	0	0
$51$	−0.553720	−0.0775363
$52$	0	0
$53$	0.746394	0.102525	0.0512626	−	0.998685i	$-0.483675\pi$
$53$	0.746394	0.102525	0.0512626	+	0.998685i	$0.483675\pi$
$54$	0	0
$55$	−15.5447	−2.09605
$56$	0	0
$57$	4.27730	0.566542
$58$	0	0
$59$	−5.69987	−0.742060	−0.371030	−	0.928621i	$-0.620995\pi$
$59$	−5.69987	−0.742060	−0.371030	+	0.928621i	$0.620995\pi$
$60$	0	0
$61$	5.94097	0.760663	0.380332	−	0.924850i	$-0.375810\pi$
$61$	5.94097	0.760663	0.380332	+	0.924850i	$0.375810\pi$
$62$	0	0
$63$	−7.09915	−0.894409
$64$	0	0
$65$	−18.4795	−2.29210
$66$	0	0
$67$	2.72082	0.332401	0.166201	−	0.986092i	$-0.446850\pi$
$67$	2.72082	0.332401	0.166201	+	0.986092i	$0.446850\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−3.85262	−0.457221	−0.228611	−	0.973518i	$-0.573418\pi$
$71$	−3.85262	−0.457221	−0.228611	+	0.973518i	$0.573418\pi$
$72$	0	0
$73$	−5.78124	−0.676642	−0.338321	−	0.941031i	$-0.609859\pi$
$73$	−5.78124	−0.676642	−0.338321	+	0.941031i	$0.609859\pi$
$74$	0	0
$75$	8.10802	0.936233
$76$	0	0
$77$	16.7878	1.91315
$78$	0	0
$79$	2.75488	0.309948	0.154974	−	0.987919i	$-0.450471\pi$
$79$	2.75488	0.309948	0.154974	+	0.987919i	$0.450471\pi$
$80$	0	0
$81$	−0.00187825	−0.000208695 0
$82$	0	0
$83$	2.73234	0.299913	0.149957	−	0.988693i	$-0.452087\pi$
$83$	2.73234	0.299913	0.149957	+	0.988693i	$0.452087\pi$
$84$	0	0
$85$	1.83396	0.198921
$86$	0	0
$87$	−9.89147	−1.06048
$88$	0	0
$89$	7.82654	0.829611	0.414806	−	0.909910i	$-0.363850\pi$
$89$	7.82654	0.829611	0.414806	+	0.909910i	$0.363850\pi$
$90$	0	0
$91$	19.9574	2.09210
$92$	0	0
$93$	−3.76596	−0.390512
$94$	0	0
$95$	−14.1667	−1.45347
$96$	0	0
$97$	−1.93011	−0.195973	−0.0979867	−	0.995188i	$-0.531240\pi$
$97$	−1.93011	−0.195973	−0.0979867	+	0.995188i	$0.531240\pi$
$98$	0	0
$99$	−8.12689	−0.816783
$100$	0	0
$101$	−9.28145	−0.923539	−0.461770	−	0.887000i	$-0.652785\pi$
$101$	−9.28145	−0.923539	−0.461770	+	0.887000i	$0.652785\pi$
$102$	0	0
$103$	−0.511115	−0.0503616	−0.0251808	−	0.999683i	$-0.508016\pi$
$103$	−0.511115	−0.0503616	−0.0251808	+	0.999683i	$0.508016\pi$
$104$	0	0
$105$	−14.5375	−1.41872
$106$	0	0
$107$	−1.89569	−0.183263	−0.0916314	−	0.995793i	$-0.529208\pi$
$107$	−1.89569	−0.183263	−0.0916314	+	0.995793i	$0.529208\pi$
$108$	0	0
$109$	18.0810	1.73184	0.865922	−	0.500178i	$-0.166732\pi$
$109$	18.0810	1.73184	0.865922	+	0.500178i	$0.166732\pi$
$110$	0	0
$111$	3.68330	0.349603
$112$	0	0
$113$	10.3834	0.976787	0.488394	−	0.872623i	$-0.337583\pi$
$113$	10.3834	0.976787	0.488394	+	0.872623i	$0.337583\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−9.66124	−0.893182
$118$	0	0
$119$	−1.98063	−0.181564
$120$	0	0
$121$	8.21819	0.747109
$122$	0	0
$123$	11.5736	1.04355
$124$	0	0
$125$	−9.12486	−0.816153
$126$	0	0
$127$	19.4144	1.72275	0.861376	−	0.507967i	$-0.169603\pi$
$127$	19.4144	1.72275	0.861376	+	0.507967i	$0.169603\pi$
$128$	0	0
$129$	0.418617	0.0368572
$130$	0	0
$131$	6.82961	0.596706	0.298353	−	0.954456i	$-0.403563\pi$
$131$	6.82961	0.596706	0.298353	+	0.954456i	$0.403563\pi$
$132$	0	0
$133$	15.2997	1.32665
$134$	0	0
$135$	18.4262	1.58587
$136$	0	0
$137$	−2.24390	−0.191709	−0.0958546	−	0.995395i	$-0.530558\pi$
$137$	−2.24390	−0.191709	−0.0958546	+	0.995395i	$0.530558\pi$
$138$	0	0
$139$	−11.5961	−0.983566	−0.491783	−	0.870718i	$-0.663655\pi$
$139$	−11.5961	−0.983566	−0.491783	+	0.870718i	$0.663655\pi$
$140$	0	0
$141$	1.61531	0.136034
$142$	0	0
$143$	22.8466	1.91053
$144$	0	0
$145$	32.7613	2.72067
$146$	0	0
$147$	8.20594	0.676814
$148$	0	0
$149$	−9.77690	−0.800955	−0.400478	−	0.916307i	$-0.631156\pi$
$149$	−9.77690	−0.800955	−0.400478	+	0.916307i	$0.631156\pi$
$150$	0	0
$151$	−20.3172	−1.65339	−0.826693	−	0.562653i	$-0.809781\pi$
$151$	−20.3172	−1.65339	−0.826693	+	0.562653i	$0.809781\pi$
$152$	0	0
$153$	0.958809	0.0775151
$154$	0	0
$155$	12.4731	1.00187
$156$	0	0
$157$	12.6266	1.00771	0.503855	−	0.863788i	$-0.331915\pi$
$157$	12.6266	1.00771	0.503855	+	0.863788i	$0.331915\pi$
$158$	0	0
$159$	0.799088	0.0633718
$160$	0	0
$161$	0	0
$162$	0	0
$163$	10.9449	0.857271	0.428635	−	0.903478i	$-0.358995\pi$
$163$	10.9449	0.857271	0.428635	+	0.903478i	$0.358995\pi$
$164$	0	0
$165$	−16.6421	−1.29559
$166$	0	0
$167$	−4.67905	−0.362076	−0.181038	−	0.983476i	$-0.557946\pi$
$167$	−4.67905	−0.362076	−0.181038	+	0.983476i	$0.557946\pi$
$168$	0	0
$169$	14.1600	1.08923
$170$	0	0
$171$	−7.40647	−0.566387
$172$	0	0
$173$	16.9273	1.28696	0.643478	−	0.765464i	$-0.277491\pi$
$173$	16.9273	1.28696	0.643478	+	0.765464i	$0.277491\pi$
$174$	0	0
$175$	29.0019	2.19234
$176$	0	0
$177$	−6.10227	−0.458674
$178$	0	0
$179$	12.1143	0.905463	0.452731	−	0.891647i	$-0.350450\pi$
$179$	12.1143	0.905463	0.452731	+	0.891647i	$0.350450\pi$
$180$	0	0
$181$	6.87963	0.511359	0.255680	−	0.966762i	$-0.417701\pi$
$181$	6.87963	0.511359	0.255680	+	0.966762i	$0.417701\pi$
$182$	0	0
$183$	6.36039	0.470173
$184$	0	0
$185$	−12.1994	−0.896914
$186$	0	0
$187$	−2.26736	−0.165806
$188$	0	0
$189$	−19.8998	−1.44750
$190$	0	0
$191$	12.5817	0.910380	0.455190	−	0.890394i	$-0.349571\pi$
$191$	12.5817	0.910380	0.455190	+	0.890394i	$0.349571\pi$
$192$	0	0
$193$	8.40550	0.605041	0.302520	−	0.953143i	$-0.402172\pi$
$193$	8.40550	0.605041	0.302520	+	0.953143i	$0.402172\pi$
$194$	0	0
$195$	−19.7841	−1.41677
$196$	0	0
$197$	−20.1389	−1.43484	−0.717420	−	0.696641i	$-0.754677\pi$
$197$	−20.1389	−1.43484	−0.717420	+	0.696641i	$0.754677\pi$
$198$	0	0
$199$	−26.1425	−1.85319	−0.926597	−	0.376056i	$-0.877280\pi$
$199$	−26.1425	−1.85319	−0.926597	+	0.376056i	$0.877280\pi$
$200$	0	0
$201$	2.91290	0.205460
$202$	0	0
$203$	−35.3813	−2.48328
$204$	0	0
$205$	−38.3325	−2.67726
$206$	0	0
$207$	0	0
$208$	0	0
$209$	17.5146	1.21151
$210$	0	0
$211$	27.0002	1.85877	0.929385	−	0.369111i	$-0.120338\pi$
$211$	27.0002	1.85877	0.929385	+	0.369111i	$0.120338\pi$
$212$	0	0
$213$	−4.12460	−0.282613
$214$	0	0
$215$	−1.38649	−0.0945578
$216$	0	0
$217$	−13.4706	−0.914447
$218$	0	0
$219$	−6.18937	−0.418239
$220$	0	0
$221$	−2.69543	−0.181314
$222$	0	0
$223$	14.6913	0.983802	0.491901	−	0.870651i	$-0.336302\pi$
$223$	14.6913	0.983802	0.491901	+	0.870651i	$0.336302\pi$
$224$	0	0
$225$	−14.0397	−0.935977
$226$	0	0
$227$	8.02183	0.532427	0.266214	−	0.963914i	$-0.414227\pi$
$227$	8.02183	0.532427	0.266214	+	0.963914i	$0.414227\pi$
$228$	0	0
$229$	14.7699	0.976022	0.488011	−	0.872837i	$-0.337723\pi$
$229$	14.7699	0.976022	0.488011	+	0.872837i	$0.337723\pi$
$230$	0	0
$231$	17.9730	1.18254
$232$	0	0
$233$	−1.53936	−0.100847	−0.0504233	−	0.998728i	$-0.516057\pi$
$233$	−1.53936	−0.100847	−0.0504233	+	0.998728i	$0.516057\pi$
$234$	0	0
$235$	−5.35002	−0.348997
$236$	0	0
$237$	2.94937	0.191582
$238$	0	0
$239$	14.8698	0.961846	0.480923	−	0.876763i	$-0.340302\pi$
$239$	14.8698	0.961846	0.480923	+	0.876763i	$0.340302\pi$
$240$	0	0
$241$	10.5569	0.680031	0.340016	−	0.940420i	$-0.389568\pi$
$241$	10.5569	0.680031	0.340016	+	0.940420i	$0.389568\pi$
$242$	0	0
$243$	15.5875	0.999935
$244$	0	0
$245$	−27.1786	−1.73638
$246$	0	0
$247$	20.8213	1.32483
$248$	0	0
$249$	2.92524	0.185379
$250$	0	0
$251$	29.9932	1.89315	0.946576	−	0.322482i	$-0.104517\pi$
$251$	29.9932	1.89315	0.946576	+	0.322482i	$0.104517\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	1.96343	0.122955
$256$	0	0
$257$	1.86747	0.116489	0.0582447	−	0.998302i	$-0.481450\pi$
$257$	1.86747	0.116489	0.0582447	+	0.998302i	$0.481450\pi$
$258$	0	0
$259$	13.1750	0.818652
$260$	0	0
$261$	17.1279	1.06019
$262$	0	0
$263$	−3.40882	−0.210197	−0.105098	−	0.994462i	$-0.533516\pi$
$263$	−3.40882	−0.210197	−0.105098	+	0.994462i	$0.533516\pi$
$264$	0	0
$265$	−2.64664	−0.162582
$266$	0	0
$267$	8.37907	0.512790
$268$	0	0
$269$	−7.29587	−0.444837	−0.222418	−	0.974951i	$-0.571395\pi$
$269$	−7.29587	−0.444837	−0.222418	+	0.974951i	$0.571395\pi$
$270$	0	0
$271$	9.73925	0.591617	0.295809	−	0.955247i	$-0.404411\pi$
$271$	9.73925	0.591617	0.295809	+	0.955247i	$0.404411\pi$
$272$	0	0
$273$	21.3663	1.29315
$274$	0	0
$275$	33.2005	2.00207
$276$	0	0
$277$	20.9394	1.25813	0.629064	−	0.777354i	$-0.283439\pi$
$277$	20.9394	1.25813	0.629064	+	0.777354i	$0.283439\pi$
$278$	0	0
$279$	6.52106	0.390405
$280$	0	0
$281$	18.2292	1.08747	0.543733	−	0.839259i	$-0.317011\pi$
$281$	18.2292	1.08747	0.543733	+	0.839259i	$0.317011\pi$
$282$	0	0
$283$	−32.2173	−1.91512	−0.957559	−	0.288238i	$-0.906930\pi$
$283$	−32.2173	−1.91512	−0.957559	+	0.288238i	$0.906930\pi$
$284$	0	0
$285$	−15.1668	−0.898406
$286$	0	0
$287$	41.3980	2.44365
$288$	0	0
$289$	−16.7325	−0.984265
$290$	0	0
$291$	−2.06638	−0.121133
$292$	0	0
$293$	−29.0502	−1.69713	−0.848564	−	0.529093i	$-0.822532\pi$
$293$	−29.0502	−1.69713	−0.848564	+	0.529093i	$0.822532\pi$
$294$	0	0
$295$	20.2111	1.17674
$296$	0	0
$297$	−22.7807	−1.32187
$298$	0	0
$299$	0	0
$300$	0	0
$301$	1.49737	0.0863071
$302$	0	0
$303$	−9.93670	−0.570848
$304$	0	0
$305$	−21.0661	−1.20624
$306$	0	0
$307$	5.94318	0.339195	0.169597	−	0.985513i	$-0.445753\pi$
$307$	5.94318	0.339195	0.169597	+	0.985513i	$0.445753\pi$
$308$	0	0
$309$	−0.547198	−0.0311290
$310$	0	0
$311$	−4.20872	−0.238654	−0.119327	−	0.992855i	$-0.538074\pi$
$311$	−4.20872	−0.238654	−0.119327	+	0.992855i	$0.538074\pi$
$312$	0	0
$313$	0.981845	0.0554972	0.0277486	−	0.999615i	$-0.491166\pi$
$313$	0.981845	0.0554972	0.0277486	+	0.999615i	$0.491166\pi$
$314$	0	0
$315$	25.1728	1.41833
$316$	0	0
$317$	−1.91691	−0.107665	−0.0538323	−	0.998550i	$-0.517144\pi$
$317$	−1.91691	−0.107665	−0.0538323	+	0.998550i	$0.517144\pi$
$318$	0	0
$319$	−40.5034	−2.26775
$320$	0	0
$321$	−2.02952	−0.113277
$322$	0	0
$323$	−2.06637	−0.114976
$324$	0	0
$325$	39.4688	2.18933
$326$	0	0
$327$	19.3575	1.07047
$328$	0	0
$329$	5.77788	0.318545
$330$	0	0
$331$	1.87565	0.103095	0.0515476	−	0.998671i	$-0.483585\pi$
$331$	1.87565	0.103095	0.0515476	+	0.998671i	$0.483585\pi$
$332$	0	0
$333$	−6.37792	−0.349508
$334$	0	0
$335$	−9.64775	−0.527113
$336$	0	0
$337$	−6.35295	−0.346067	−0.173034	−	0.984916i	$-0.555357\pi$
$337$	−6.35295	−0.346067	−0.173034	+	0.984916i	$0.555357\pi$
$338$	0	0
$339$	11.1164	0.603762
$340$	0	0
$341$	−15.4208	−0.835082
$342$	0	0
$343$	2.54592	0.137467
$344$	0	0
$345$	0	0
$346$	0	0
$347$	6.14644	0.329958	0.164979	−	0.986297i	$-0.447244\pi$
$347$	6.14644	0.329958	0.164979	+	0.986297i	$0.447244\pi$
$348$	0	0
$349$	−23.5878	−1.26263	−0.631314	−	0.775527i	$-0.717484\pi$
$349$	−23.5878	−1.26263	−0.631314	+	0.775527i	$0.717484\pi$
$350$	0	0
$351$	−27.0816	−1.44551
$352$	0	0
$353$	−3.10697	−0.165368	−0.0826838	−	0.996576i	$-0.526349\pi$
$353$	−3.10697	−0.165368	−0.0826838	+	0.996576i	$0.526349\pi$
$354$	0	0
$355$	13.6610	0.725049
$356$	0	0
$357$	−2.12045	−0.112226
$358$	0	0
$359$	−17.4020	−0.918444	−0.459222	−	0.888322i	$-0.651872\pi$
$359$	−17.4020	−0.918444	−0.459222	+	0.888322i	$0.651872\pi$
$360$	0	0
$361$	−3.03802	−0.159896
$362$	0	0
$363$	8.79837	0.461795
$364$	0	0
$365$	20.4996	1.07300
$366$	0	0
$367$	−33.2650	−1.73642	−0.868210	−	0.496197i	$-0.834730\pi$
$367$	−33.2650	−1.73642	−0.868210	+	0.496197i	$0.834730\pi$
$368$	0	0
$369$	−20.0405	−1.04327
$370$	0	0
$371$	2.85829	0.148395
$372$	0	0
$373$	−11.5445	−0.597752	−0.298876	−	0.954292i	$-0.596612\pi$
$373$	−11.5445	−0.597752	−0.298876	+	0.954292i	$0.596612\pi$
$374$	0	0
$375$	−9.76905	−0.504472
$376$	0	0
$377$	−48.1504	−2.47987
$378$	0	0
$379$	19.4924	1.00126	0.500628	−	0.865663i	$-0.333103\pi$
$379$	19.4924	1.00126	0.500628	+	0.865663i	$0.333103\pi$
$380$	0	0
$381$	20.7850	1.06485
$382$	0	0
$383$	−24.6027	−1.25714	−0.628568	−	0.777754i	$-0.716359\pi$
$383$	−24.6027	−1.25714	−0.628568	+	0.777754i	$0.716359\pi$
$384$	0	0
$385$	−59.5279	−3.03382
$386$	0	0
$387$	−0.724868	−0.0368471
$388$	0	0
$389$	0.544240	0.0275941	0.0137970	−	0.999905i	$-0.495608\pi$
$389$	0.544240	0.0275941	0.0137970	+	0.999905i	$0.495608\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	7.31177	0.368830
$394$	0	0
$395$	−9.76852	−0.491507
$396$	0	0
$397$	2.18896	0.109861	0.0549303	−	0.998490i	$-0.482506\pi$
$397$	2.18896	0.109861	0.0549303	+	0.998490i	$0.482506\pi$
$398$	0	0
$399$	16.3798	0.820014
$400$	0	0
$401$	−3.96429	−0.197967	−0.0989837	−	0.995089i	$-0.531559\pi$
$401$	−3.96429	−0.197967	−0.0989837	+	0.995089i	$0.531559\pi$
$402$	0	0
$403$	−18.3322	−0.913192
$404$	0	0
$405$	0.00666009	0.000330943 0
$406$	0	0
$407$	15.0823	0.747601
$408$	0	0
$409$	2.80613	0.138754	0.0693770	−	0.997591i	$-0.477899\pi$
$409$	2.80613	0.138754	0.0693770	+	0.997591i	$0.477899\pi$
$410$	0	0
$411$	−2.40231	−0.118497
$412$	0	0
$413$	−21.8275	−1.07406
$414$	0	0
$415$	−9.68859	−0.475594
$416$	0	0
$417$	−12.4147	−0.607952
$418$	0	0
$419$	−1.29682	−0.0633540	−0.0316770	−	0.999498i	$-0.510085\pi$
$419$	−1.29682	−0.0633540	−0.0316770	+	0.999498i	$0.510085\pi$
$420$	0	0
$421$	23.1196	1.12678	0.563391	−	0.826190i	$-0.309496\pi$
$421$	23.1196	1.12678	0.563391	+	0.826190i	$0.309496\pi$
$422$	0	0
$423$	−2.79704	−0.135997
$424$	0	0
$425$	−3.91699	−0.190002
$426$	0	0
$427$	22.7508	1.10099
$428$	0	0
$429$	24.4595	1.18091
$430$	0	0
$431$	23.3800	1.12618	0.563088	−	0.826397i	$-0.309613\pi$
$431$	23.3800	1.12618	0.563088	+	0.826397i	$0.309613\pi$
$432$	0	0
$433$	−12.5035	−0.600881	−0.300441	−	0.953801i	$-0.597134\pi$
$433$	−12.5035	−0.600881	−0.300441	+	0.953801i	$0.597134\pi$
$434$	0	0
$435$	35.0741	1.68168
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−21.1780	−1.01077	−0.505385	−	0.862894i	$-0.668649\pi$
$439$	−21.1780	−1.01077	−0.505385	+	0.862894i	$0.668649\pi$
$440$	0	0
$441$	−14.2092	−0.676629
$442$	0	0
$443$	−17.1827	−0.816373	−0.408186	−	0.912899i	$-0.633839\pi$
$443$	−17.1827	−0.816373	−0.408186	+	0.912899i	$0.633839\pi$
$444$	0	0
$445$	−27.7521	−1.31557
$446$	0	0
$447$	−10.4671	−0.495078
$448$	0	0
$449$	−23.2984	−1.09952	−0.549760	−	0.835323i	$-0.685281\pi$
$449$	−23.2984	−1.09952	−0.549760	+	0.835323i	$0.685281\pi$
$450$	0	0
$451$	47.3912	2.23156
$452$	0	0
$453$	−21.7515	−1.02197
$454$	0	0
$455$	−70.7667	−3.31759
$456$	0	0
$457$	−38.4740	−1.79974	−0.899870	−	0.436159i	$-0.856339\pi$
$457$	−38.4740	−1.79974	−0.899870	+	0.436159i	$0.856339\pi$
$458$	0	0
$459$	2.68766	0.125449
$460$	0	0
$461$	−10.0360	−0.467424	−0.233712	−	0.972306i	$-0.575087\pi$
$461$	−10.0360	−0.467424	−0.233712	+	0.972306i	$0.575087\pi$
$462$	0	0
$463$	−23.0965	−1.07338	−0.536692	−	0.843778i	$-0.680326\pi$
$463$	−23.0965	−1.07338	−0.536692	+	0.843778i	$0.680326\pi$
$464$	0	0
$465$	13.3537	0.619263
$466$	0	0
$467$	29.6966	1.37420	0.687098	−	0.726565i	$-0.258884\pi$
$467$	29.6966	1.37420	0.687098	+	0.726565i	$0.258884\pi$
$468$	0	0
$469$	10.4193	0.481119
$470$	0	0
$471$	13.5180	0.622875
$472$	0	0
$473$	1.71414	0.0788164
$474$	0	0
$475$	30.2574	1.38831
$476$	0	0
$477$	−1.38368	−0.0633545
$478$	0	0
$479$	−26.5299	−1.21218	−0.606090	−	0.795396i	$-0.707263\pi$
$479$	−26.5299	−1.21218	−0.606090	+	0.795396i	$0.707263\pi$
$480$	0	0
$481$	17.9298	0.817529
$482$	0	0
$483$	0	0
$484$	0	0
$485$	6.84398	0.310769
$486$	0	0
$487$	−35.1111	−1.59104	−0.795519	−	0.605929i	$-0.792802\pi$
$487$	−35.1111	−1.59104	−0.795519	+	0.605929i	$0.792802\pi$
$488$	0	0
$489$	11.7176	0.529887
$490$	0	0
$491$	−33.2586	−1.50094	−0.750470	−	0.660905i	$-0.770173\pi$
$491$	−33.2586	−1.50094	−0.750470	+	0.660905i	$0.770173\pi$
$492$	0	0
$493$	4.77858	0.215216
$494$	0	0
$495$	28.8171	1.29523
$496$	0	0
$497$	−14.7535	−0.661784
$498$	0	0
$499$	−12.7124	−0.569085	−0.284542	−	0.958663i	$-0.591842\pi$
$499$	−12.7124	−0.569085	−0.284542	+	0.958663i	$0.591842\pi$
$500$	0	0
$501$	−5.00938	−0.223803
$502$	0	0
$503$	−14.1849	−0.632471	−0.316236	−	0.948681i	$-0.602419\pi$
$503$	−14.1849	−0.632471	−0.316236	+	0.948681i	$0.602419\pi$
$504$	0	0
$505$	32.9110	1.46452
$506$	0	0
$507$	15.1596	0.673263
$508$	0	0
$509$	31.1700	1.38158	0.690792	−	0.723053i	$-0.257262\pi$
$509$	31.1700	1.38158	0.690792	+	0.723053i	$0.257262\pi$
$510$	0	0
$511$	−22.1391	−0.979374
$512$	0	0
$513$	−20.7612	−0.916631
$514$	0	0
$515$	1.81236	0.0798620
$516$	0	0
$517$	6.61434	0.290898
$518$	0	0
$519$	18.1223	0.795480
$520$	0	0
$521$	20.6181	0.903295	0.451648	−	0.892196i	$-0.350836\pi$
$521$	20.6181	0.903295	0.451648	+	0.892196i	$0.350836\pi$
$522$	0	0
$523$	32.3598	1.41500	0.707499	−	0.706715i	$-0.249824\pi$
$523$	32.3598	1.41500	0.707499	+	0.706715i	$0.249824\pi$
$524$	0	0
$525$	31.0494	1.35511
$526$	0	0
$527$	1.81934	0.0792517
$528$	0	0
$529$	0	0
$530$	0	0
$531$	10.5665	0.458549
$532$	0	0
$533$	56.3386	2.44029
$534$	0	0
$535$	6.72190	0.290613
$536$	0	0
$537$	12.9695	0.559675
$538$	0	0
$539$	33.6015	1.44732
$540$	0	0
$541$	14.8905	0.640193	0.320097	−	0.947385i	$-0.396285\pi$
$541$	14.8905	0.640193	0.320097	+	0.947385i	$0.396285\pi$
$542$	0	0
$543$	7.36531	0.316076
$544$	0	0
$545$	−64.1133	−2.74631
$546$	0	0
$547$	21.5597	0.921825	0.460912	−	0.887446i	$-0.347522\pi$
$547$	21.5597	0.921825	0.460912	+	0.887446i	$0.347522\pi$
$548$	0	0
$549$	−11.0135	−0.470045
$550$	0	0
$551$	−36.9129	−1.57254
$552$	0	0
$553$	10.5497	0.448620
$554$	0	0
$555$	−13.0606	−0.554391
$556$	0	0
$557$	−33.7378	−1.42952	−0.714758	−	0.699372i	$-0.753463\pi$
$557$	−33.7378	−1.42952	−0.714758	+	0.699372i	$0.753463\pi$
$558$	0	0
$559$	2.03777	0.0861886
$560$	0	0
$561$	−2.42743	−0.102486
$562$	0	0
$563$	−26.1916	−1.10384	−0.551922	−	0.833896i	$-0.686105\pi$
$563$	−26.1916	−1.10384	−0.551922	+	0.833896i	$0.686105\pi$
$564$	0	0
$565$	−36.8184	−1.54896
$566$	0	0
$567$	−0.00719271	−0.000302066 0
$568$	0	0
$569$	17.3204	0.726110	0.363055	−	0.931768i	$-0.381734\pi$
$569$	17.3204	0.726110	0.363055	+	0.931768i	$0.381734\pi$
$570$	0	0
$571$	25.7842	1.07903	0.539517	−	0.841975i	$-0.318607\pi$
$571$	25.7842	1.07903	0.539517	+	0.841975i	$0.318607\pi$
$572$	0	0
$573$	13.4699	0.562714
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−32.3714	−1.34764	−0.673820	−	0.738896i	$-0.735348\pi$
$577$	−32.3714	−1.34764	−0.673820	+	0.738896i	$0.735348\pi$
$578$	0	0
$579$	8.99890	0.373982
$580$	0	0
$581$	10.4634	0.434095
$582$	0	0
$583$	3.27209	0.135516
$584$	0	0
$585$	34.2577	1.41638
$586$	0	0
$587$	−1.09740	−0.0452946	−0.0226473	−	0.999744i	$-0.507209\pi$
$587$	−1.09740	−0.0452946	−0.0226473	+	0.999744i	$0.507209\pi$
$588$	0	0
$589$	−14.0538	−0.579076
$590$	0	0
$591$	−21.5607	−0.886889
$592$	0	0
$593$	36.4220	1.49567	0.747837	−	0.663883i	$-0.231093\pi$
$593$	36.4220	1.49567	0.747837	+	0.663883i	$0.231093\pi$
$594$	0	0
$595$	7.02309	0.287918
$596$	0	0
$597$	−27.9881	−1.14548
$598$	0	0
$599$	−30.3382	−1.23959	−0.619793	−	0.784765i	$-0.712783\pi$
$599$	−30.3382	−1.23959	−0.619793	+	0.784765i	$0.712783\pi$
$600$	0	0
$601$	34.1619	1.39349	0.696747	−	0.717317i	$-0.254630\pi$
$601$	34.1619	1.39349	0.696747	+	0.717317i	$0.254630\pi$
$602$	0	0
$603$	−5.04392	−0.205404
$604$	0	0
$605$	−29.1408	−1.18474
$606$	0	0
$607$	7.77776	0.315689	0.157845	−	0.987464i	$-0.449545\pi$
$607$	7.77776	0.315689	0.157845	+	0.987464i	$0.449545\pi$
$608$	0	0
$609$	−37.8791	−1.53494
$610$	0	0
$611$	7.86312	0.318108
$612$	0	0
$613$	23.2972	0.940965	0.470482	−	0.882409i	$-0.344080\pi$
$613$	23.2972	0.940965	0.470482	+	0.882409i	$0.344080\pi$
$614$	0	0
$615$	−41.0386	−1.65484
$616$	0	0
$617$	14.0407	0.565259	0.282629	−	0.959229i	$-0.408793\pi$
$617$	14.0407	0.565259	0.282629	+	0.959229i	$0.408793\pi$
$618$	0	0
$619$	5.94363	0.238895	0.119447	−	0.992841i	$-0.461888\pi$
$619$	5.94363	0.238895	0.119447	+	0.992841i	$0.461888\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	29.9715	1.20078
$624$	0	0
$625$	−5.51101	−0.220440
$626$	0	0
$627$	18.7511	0.748845
$628$	0	0
$629$	−1.77941	−0.0709495
$630$	0	0
$631$	−29.6088	−1.17871	−0.589353	−	0.807876i	$-0.700617\pi$
$631$	−29.6088	−1.17871	−0.589353	+	0.807876i	$0.700617\pi$
$632$	0	0
$633$	28.9063	1.14892
$634$	0	0
$635$	−68.8415	−2.73189
$636$	0	0
$637$	39.9454	1.58269
$638$	0	0
$639$	7.14206	0.282536
$640$	0	0
$641$	−20.7555	−0.819794	−0.409897	−	0.912132i	$-0.634435\pi$
$641$	−20.7555	−0.819794	−0.409897	+	0.912132i	$0.634435\pi$
$642$	0	0
$643$	10.9240	0.430801	0.215400	−	0.976526i	$-0.430894\pi$
$643$	10.9240	0.430801	0.215400	+	0.976526i	$0.430894\pi$
$644$	0	0
$645$	−1.48437	−0.0584471
$646$	0	0
$647$	−44.4615	−1.74796	−0.873981	−	0.485960i	$-0.838470\pi$
$647$	−44.4615	−1.74796	−0.873981	+	0.485960i	$0.838470\pi$
$648$	0	0
$649$	−24.9874	−0.980842
$650$	0	0
$651$	−14.4216	−0.565228
$652$	0	0
$653$	17.9012	0.700527	0.350264	−	0.936651i	$-0.386092\pi$
$653$	17.9012	0.700527	0.350264	+	0.936651i	$0.386092\pi$
$654$	0	0
$655$	−24.2171	−0.946240
$656$	0	0
$657$	10.7174	0.418125
$658$	0	0
$659$	29.9766	1.16772	0.583861	−	0.811853i	$-0.301541\pi$
$659$	29.9766	1.16772	0.583861	+	0.811853i	$0.301541\pi$
$660$	0	0
$661$	23.8555	0.927873	0.463936	−	0.885869i	$-0.346437\pi$
$661$	23.8555	0.927873	0.463936	+	0.885869i	$0.346437\pi$
$662$	0	0
$663$	−2.88572	−0.112072
$664$	0	0
$665$	−54.2510	−2.10376
$666$	0	0
$667$	0	0
$668$	0	0
$669$	15.7285	0.608097
$670$	0	0
$671$	26.0444	1.00543
$672$	0	0
$673$	42.7522	1.64798	0.823988	−	0.566608i	$-0.191745\pi$
$673$	42.7522	1.64798	0.823988	+	0.566608i	$0.191745\pi$
$674$	0	0
$675$	−39.3549	−1.51477
$676$	0	0
$677$	1.26531	0.0486297	0.0243149	−	0.999704i	$-0.492260\pi$
$677$	1.26531	0.0486297	0.0243149	+	0.999704i	$0.492260\pi$
$678$	0	0
$679$	−7.39131	−0.283653
$680$	0	0
$681$	8.58814	0.329098
$682$	0	0
$683$	38.9921	1.49199	0.745995	−	0.665951i	$-0.231974\pi$
$683$	38.9921	1.49199	0.745995	+	0.665951i	$0.231974\pi$
$684$	0	0
$685$	7.95662	0.304007
$686$	0	0
$687$	15.8126	0.603289
$688$	0	0
$689$	3.88985	0.148192
$690$	0	0
$691$	8.04344	0.305987	0.152993	−	0.988227i	$-0.451109\pi$
$691$	8.04344	0.305987	0.152993	+	0.988227i	$0.451109\pi$
$692$	0	0
$693$	−31.1217	−1.18221
$694$	0	0
$695$	41.1184	1.55971
$696$	0	0
$697$	−5.59120	−0.211782
$698$	0	0
$699$	−1.64803	−0.0623342
$700$	0	0
$701$	−38.0452	−1.43695	−0.718474	−	0.695553i	$-0.755159\pi$
$701$	−38.0452	−1.43695	−0.718474	+	0.695553i	$0.755159\pi$
$702$	0	0
$703$	13.7453	0.518414
$704$	0	0
$705$	−5.72772	−0.215718
$706$	0	0
$707$	−35.5430	−1.33673
$708$	0	0
$709$	−48.0465	−1.80443	−0.902213	−	0.431291i	$-0.858058\pi$
$709$	−48.0465	−1.80443	−0.902213	+	0.431291i	$0.858058\pi$
$710$	0	0
$711$	−5.10706	−0.191530
$712$	0	0
$713$	0	0
$714$	0	0
$715$	−81.0115	−3.02966
$716$	0	0
$717$	15.9195	0.594526
$718$	0	0
$719$	28.0329	1.04545	0.522725	−	0.852501i	$-0.324915\pi$
$719$	28.0329	1.04545	0.522725	+	0.852501i	$0.324915\pi$
$720$	0	0
$721$	−1.95730	−0.0728936
$722$	0	0
$723$	11.3022	0.420334
$724$	0	0
$725$	−69.9719	−2.59869
$726$	0	0
$727$	16.1018	0.597184	0.298592	−	0.954381i	$-0.403483\pi$
$727$	16.1018	0.597184	0.298592	+	0.954381i	$0.403483\pi$
$728$	0	0
$729$	16.6935	0.618278
$730$	0	0
$731$	−0.202234	−0.00747991
$732$	0	0
$733$	−16.5524	−0.611377	−0.305688	−	0.952132i	$-0.598887\pi$
$733$	−16.5524	−0.611377	−0.305688	+	0.952132i	$0.598887\pi$
$734$	0	0
$735$	−29.0974	−1.07327
$736$	0	0
$737$	11.9277	0.439362
$738$	0	0
$739$	−6.76652	−0.248910	−0.124455	−	0.992225i	$-0.539718\pi$
$739$	−6.76652	−0.248910	−0.124455	+	0.992225i	$0.539718\pi$
$740$	0	0
$741$	22.2912	0.818889
$742$	0	0
$743$	−30.5250	−1.11985	−0.559927	−	0.828542i	$-0.689171\pi$
$743$	−30.5250	−1.11985	−0.559927	+	0.828542i	$0.689171\pi$
$744$	0	0
$745$	34.6679	1.27013
$746$	0	0
$747$	−5.06527	−0.185329
$748$	0	0
$749$	−7.25947	−0.265255
$750$	0	0
$751$	−30.7731	−1.12293	−0.561464	−	0.827501i	$-0.689762\pi$
$751$	−30.7731	−1.12293	−0.561464	+	0.827501i	$0.689762\pi$
$752$	0	0
$753$	32.1106	1.17017
$754$	0	0
$755$	72.0425	2.62189
$756$	0	0
$757$	−0.712632	−0.0259011	−0.0129505	−	0.999916i	$-0.504122\pi$
$757$	−0.712632	−0.0259011	−0.0129505	+	0.999916i	$0.504122\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−36.8606	−1.33620	−0.668098	−	0.744073i	$-0.732891\pi$
$761$	−36.8606	−1.33620	−0.668098	+	0.744073i	$0.732891\pi$
$762$	0	0
$763$	69.2406	2.50668
$764$	0	0
$765$	−3.39983	−0.122921
$766$	0	0
$767$	−29.7050	−1.07259
$768$	0	0
$769$	−18.5714	−0.669701	−0.334850	−	0.942271i	$-0.608686\pi$
$769$	−18.5714	−0.669701	−0.334850	+	0.942271i	$0.608686\pi$
$770$	0	0
$771$	1.99931	0.0720033
$772$	0	0
$773$	−21.4543	−0.771658	−0.385829	−	0.922570i	$-0.626085\pi$
$773$	−21.4543	−0.771658	−0.385829	+	0.922570i	$0.626085\pi$
$774$	0	0
$775$	−26.6403	−0.956946
$776$	0	0
$777$	14.1051	0.506017
$778$	0	0
$779$	43.1901	1.54745
$780$	0	0
$781$	−16.8893	−0.604347
$782$	0	0
$783$	48.0115	1.71579
$784$	0	0
$785$	−44.7725	−1.59800
$786$	0	0
$787$	12.1365	0.432619	0.216310	−	0.976325i	$-0.430598\pi$
$787$	12.1365	0.432619	0.216310	+	0.976325i	$0.430598\pi$
$788$	0	0
$789$	−3.64947	−0.129925
$790$	0	0
$791$	39.7629	1.41380
$792$	0	0
$793$	30.9615	1.09948
$794$	0	0
$795$	−2.83348	−0.100493
$796$	0	0
$797$	−5.80025	−0.205456	−0.102728	−	0.994710i	$-0.532757\pi$
$797$	−5.80025	−0.205456	−0.102728	+	0.994710i	$0.532757\pi$
$798$	0	0
$799$	−0.780358	−0.0276071
$800$	0	0
$801$	−14.5090	−0.512650
$802$	0	0
$803$	−25.3441	−0.894374
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−7.81093	−0.274958
$808$	0	0
$809$	−18.8497	−0.662720	−0.331360	−	0.943504i	$-0.607507\pi$
$809$	−18.8497	−0.662720	−0.331360	+	0.943504i	$0.607507\pi$
$810$	0	0
$811$	−29.0930	−1.02159	−0.510796	−	0.859702i	$-0.670649\pi$
$811$	−29.0930	−1.02159	−0.510796	+	0.859702i	$0.670649\pi$
$812$	0	0
$813$	10.4268	0.365684
$814$	0	0
$815$	−38.8095	−1.35944
$816$	0	0
$817$	1.56219	0.0546542
$818$	0	0
$819$	−36.9974	−1.29279
$820$	0	0
$821$	9.92528	0.346395	0.173197	−	0.984887i	$-0.444590\pi$
$821$	9.92528	0.346395	0.173197	+	0.984887i	$0.444590\pi$
$822$	0	0
$823$	−52.6311	−1.83460	−0.917302	−	0.398192i	$-0.869638\pi$
$823$	−52.6311	−1.83460	−0.917302	+	0.398192i	$0.869638\pi$
$824$	0	0
$825$	35.5444	1.23750
$826$	0	0
$827$	10.9941	0.382302	0.191151	−	0.981561i	$-0.438778\pi$
$827$	10.9941	0.382302	0.191151	+	0.981561i	$0.438778\pi$
$828$	0	0
$829$	47.8423	1.66163	0.830815	−	0.556548i	$-0.187875\pi$
$829$	47.8423	1.66163	0.830815	+	0.556548i	$0.187875\pi$
$830$	0	0
$831$	22.4177	0.777660
$832$	0	0
$833$	−3.96429	−0.137355
$834$	0	0
$835$	16.5914	0.574170
$836$	0	0
$837$	18.2793	0.631825
$838$	0	0
$839$	−7.20814	−0.248852	−0.124426	−	0.992229i	$-0.539709\pi$
$839$	−7.20814	−0.248852	−0.124426	+	0.992229i	$0.539709\pi$
$840$	0	0
$841$	56.3631	1.94355
$842$	0	0
$843$	19.5162	0.672172
$844$	0	0
$845$	−50.2098	−1.72727
$846$	0	0
$847$	31.4713	1.08137
$848$	0	0
$849$	−34.4917	−1.18375
$850$	0	0
$851$	0	0
$852$	0	0
$853$	−16.6244	−0.569207	−0.284603	−	0.958645i	$-0.591862\pi$
$853$	−16.6244	−0.569207	−0.284603	+	0.958645i	$0.591862\pi$
$854$	0	0
$855$	26.2626	0.898161
$856$	0	0
$857$	−34.8561	−1.19066	−0.595332	−	0.803480i	$-0.702979\pi$
$857$	−34.8561	−1.19066	−0.595332	+	0.803480i	$0.702979\pi$
$858$	0	0
$859$	6.10354	0.208250	0.104125	−	0.994564i	$-0.466796\pi$
$859$	6.10354	0.208250	0.104125	+	0.994564i	$0.466796\pi$
$860$	0	0
$861$	44.3206	1.51044
$862$	0	0
$863$	27.5967	0.939402	0.469701	−	0.882825i	$-0.344362\pi$
$863$	27.5967	0.939402	0.469701	+	0.882825i	$0.344362\pi$
$864$	0	0
$865$	−60.0223	−2.04082
$866$	0	0
$867$	−17.9138	−0.608383
$868$	0	0
$869$	12.0770	0.409684
$870$	0	0
$871$	14.1796	0.480459
$872$	0	0
$873$	3.57809	0.121100
$874$	0	0
$875$	−34.9434	−1.18130
$876$	0	0
$877$	−24.1591	−0.815793	−0.407897	−	0.913028i	$-0.633738\pi$
$877$	−24.1591	−0.815793	−0.407897	+	0.913028i	$0.633738\pi$
$878$	0	0
$879$	−31.1010	−1.04901
$880$	0	0
$881$	−13.2428	−0.446160	−0.223080	−	0.974800i	$-0.571611\pi$
$881$	−13.2428	−0.446160	−0.223080	+	0.974800i	$0.571611\pi$
$882$	0	0
$883$	−31.1265	−1.04749	−0.523744	−	0.851875i	$-0.675465\pi$
$883$	−31.1265	−1.04749	−0.523744	+	0.851875i	$0.675465\pi$
$884$	0	0
$885$	21.6380	0.727353
$886$	0	0
$887$	25.3248	0.850325	0.425163	−	0.905117i	$-0.360217\pi$
$887$	25.3248	0.850325	0.425163	+	0.905117i	$0.360217\pi$
$888$	0	0
$889$	74.3470	2.49352
$890$	0	0
$891$	−0.00823400	−0.000275849 0
$892$	0	0
$893$	6.02800	0.201719
$894$	0	0
$895$	−42.9559	−1.43586
$896$	0	0
$897$	0	0
$898$	0	0
$899$	32.5001	1.08394
$900$	0	0
$901$	−0.386040	−0.0128609
$902$	0	0
$903$	1.60308	0.0533472
$904$	0	0
$905$	−24.3944	−0.810899
$906$	0	0
$907$	2.75421	0.0914520	0.0457260	−	0.998954i	$-0.485440\pi$
$907$	2.75421	0.0914520	0.0457260	+	0.998954i	$0.485440\pi$
$908$	0	0
$909$	17.2062	0.570692
$910$	0	0
$911$	14.5734	0.482839	0.241419	−	0.970421i	$-0.422387\pi$
$911$	14.5734	0.482839	0.241419	+	0.970421i	$0.422387\pi$
$912$	0	0
$913$	11.9782	0.396420
$914$	0	0
$915$	−22.5533	−0.745588
$916$	0	0
$917$	26.1538	0.863674
$918$	0	0
$919$	22.2003	0.732320	0.366160	−	0.930552i	$-0.380672\pi$
$919$	22.2003	0.732320	0.366160	+	0.930552i	$0.380672\pi$
$920$	0	0
$921$	6.36275	0.209660
$922$	0	0
$923$	−20.0780	−0.660876
$924$	0	0
$925$	26.0555	0.856700
$926$	0	0
$927$	0.947516	0.0311205
$928$	0	0
$929$	44.3607	1.45543	0.727713	−	0.685882i	$-0.240583\pi$
$929$	44.3607	1.45543	0.727713	+	0.685882i	$0.240583\pi$
$930$	0	0
$931$	30.6228	1.00362
$932$	0	0
$933$	−4.50584	−0.147515
$934$	0	0
$935$	8.03981	0.262930
$936$	0	0
$937$	21.6418	0.707006	0.353503	−	0.935433i	$-0.384990\pi$
$937$	21.6418	0.707006	0.353503	+	0.935433i	$0.384990\pi$
$938$	0	0
$939$	1.05116	0.0343033
$940$	0	0
$941$	−0.118996	−0.00387915	−0.00193957	−	0.999998i	$-0.500617\pi$
$941$	−0.118996	−0.00387915	−0.00193957	+	0.999998i	$0.500617\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	70.5625	2.29540
$946$	0	0
$947$	−49.6057	−1.61197	−0.805984	−	0.591937i	$-0.798363\pi$
$947$	−49.6057	−1.61197	−0.805984	+	0.591937i	$0.798363\pi$
$948$	0	0
$949$	−30.1290	−0.978030
$950$	0	0
$951$	−2.05224	−0.0665485
$952$	0	0
$953$	−2.99008	−0.0968583	−0.0484292	−	0.998827i	$-0.515422\pi$
$953$	−2.99008	−0.0968583	−0.0484292	+	0.998827i	$0.515422\pi$
$954$	0	0
$955$	−44.6134	−1.44365
$956$	0	0
$957$	−43.3628	−1.40172
$958$	0	0
$959$	−8.59294	−0.277480
$960$	0	0
$961$	−18.6263	−0.600848
$962$	0	0
$963$	3.51426	0.113246
$964$	0	0
$965$	−29.8050	−0.959457
$966$	0	0
$967$	−22.6066	−0.726980	−0.363490	−	0.931598i	$-0.618415\pi$
$967$	−22.6066	−0.726980	−0.363490	+	0.931598i	$0.618415\pi$
$968$	0	0
$969$	−2.21225	−0.0710676
$970$	0	0
$971$	−15.3138	−0.491443	−0.245721	−	0.969341i	$-0.579025\pi$
$971$	−15.3138	−0.491443	−0.245721	+	0.969341i	$0.579025\pi$
$972$	0	0
$973$	−44.4068	−1.42362
$974$	0	0
$975$	42.2551	1.35325
$976$	0	0
$977$	−3.46868	−0.110973	−0.0554864	−	0.998459i	$-0.517671\pi$
$977$	−3.46868	−0.110973	−0.0554864	+	0.998459i	$0.517671\pi$
$978$	0	0
$979$	34.3104	1.09657
$980$	0	0
$981$	−33.5189	−1.07018
$982$	0	0
$983$	−27.4595	−0.875821	−0.437911	−	0.899019i	$-0.644281\pi$
$983$	−27.4595	−0.875821	−0.437911	+	0.899019i	$0.644281\pi$
$984$	0	0
$985$	71.4106	2.27533
$986$	0	0
$987$	6.18578	0.196896
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−24.1034	−0.765670	−0.382835	−	0.923817i	$-0.625052\pi$
$991$	−24.1034	−0.765670	−0.382835	+	0.923817i	$0.625052\pi$
$992$	0	0
$993$	2.00807	0.0637241
$994$	0	0
$995$	92.6986	2.93874
$996$	0	0
$997$	−26.7269	−0.846450	−0.423225	−	0.906025i	$-0.639102\pi$
$997$	−26.7269	−0.846450	−0.423225	+	0.906025i	$0.639102\pi$
$998$	0	0
$999$	−17.8781	−0.565637

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4232.2.a.bb.1.9		15
4.3	odd	2		8464.2.a.cg.1.7		15
23.9	even	11		184.2.i.b.81.1	yes	30
23.18	even	11		184.2.i.b.25.1	✓	30
23.22	odd	2		4232.2.a.ba.1.9		15
92.55	odd	22		368.2.m.e.81.3		30
92.87	odd	22		368.2.m.e.209.3		30
92.91	even	2		8464.2.a.ch.1.7		15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
184.2.i.b.25.1	✓	30	23.18	even	11
184.2.i.b.81.1	yes	30	23.9	even	11
368.2.m.e.81.3		30	92.55	odd	22
368.2.m.e.209.3		30	92.87	odd	22
4232.2.a.ba.1.9		15	23.22	odd	2
4232.2.a.bb.1.9		15	1.1	even	1	trivial
8464.2.a.cg.1.7		15	4.3	odd	2
8464.2.a.ch.1.7		15	92.91	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4232 = 2^{3} \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4232.a (trivial)

\(f(q)\)	\(=\)	\(q+1.07060 q^{3} -3.54589 q^{5} +3.82947 q^{7} -1.85382 q^{9} +4.38386 q^{11} +5.21152 q^{13} -3.79622 q^{15} -0.517206 q^{17} +3.99525 q^{19} +4.09982 q^{21} +7.57336 q^{25} -5.19649 q^{27} -9.23921 q^{29} -3.51763 q^{31} +4.69334 q^{33} -13.5789 q^{35} +3.44042 q^{37} +5.57944 q^{39} +10.8104 q^{41} +0.391013 q^{43} +6.57346 q^{45} +1.50879 q^{47} +7.66482 q^{49} -0.553720 q^{51} +0.746394 q^{53} -15.5447 q^{55} +4.27730 q^{57} -5.69987 q^{59} +5.94097 q^{61} -7.09915 q^{63} -18.4795 q^{65} +2.72082 q^{67} -3.85262 q^{71} -5.78124 q^{73} +8.10802 q^{75} +16.7878 q^{77} +2.75488 q^{79} -0.00187825 q^{81} +2.73234 q^{83} +1.83396 q^{85} -9.89147 q^{87} +7.82654 q^{89} +19.9574 q^{91} -3.76596 q^{93} -14.1667 q^{95} -1.93011 q^{97} -8.12689 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + q^{3} + 10 q^{7} + 16 q^{9} + 23 q^{11} + 10 q^{15} + 29 q^{19} + q^{21} + 23 q^{25} + q^{27} - 2 q^{29} + 20 q^{31} + 18 q^{33} - 18 q^{35} + 24 q^{37} - 19 q^{39} + 9 q^{41} + 48 q^{43} + 4 q^{45}+ \cdots + 63 q^{99}+O(q^{100}) \) 15 * q + q^3 + 10 * q^7 + 16 * q^9 + 23 * q^11 + 10 * q^15 + 29 * q^19 + q^21 + 23 * q^25 + q^27 - 2 * q^29 + 20 * q^31 + 18 * q^33 - 18 * q^35 + 24 * q^37 - 19 * q^39 + 9 * q^41 + 48 * q^43 + 4 * q^45 - 36 * q^47 + 25 * q^49 + 35 * q^51 - 5 * q^53 - 10 * q^55 + 23 * q^57 - 22 * q^59 + 12 * q^61 + 35 * q^63 - 26 * q^65 + 58 * q^67 + 2 * q^71 + 5 * q^73 - 17 * q^75 + 26 * q^77 + 26 * q^79 - 21 * q^81 + 68 * q^83 - 72 * q^85 + 19 * q^87 - 6 * q^89 + 71 * q^91 - 55 * q^93 - 12 * q^95 + 40 * q^97 + 63 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more