LMFDB - Embedded newform 4232.2.a.ba.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4232,2,Mod(1,4232)]

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

lf = mfeigenbasis(mf)

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")

//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);

Level:	$ N $	$=$	$ 4232 = 2^{3} \cdot 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4232.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:	$33.7926901354$
Analytic rank:	$1$
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			1.10
Root			$1.44536$ 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)

$f(q)$	$=$	$q+1.44536 q^{3} +2.02221 q^{5} -4.89726 q^{7} -0.910921 q^{9} +2.40830 q^{11} -1.19222 q^{13} +2.92283 q^{15} +1.22201 q^{17} -3.46894 q^{19} -7.07833 q^{21} -0.910658 q^{25} -5.65271 q^{27} +8.90576 q^{29} +6.51675 q^{31} +3.48087 q^{33} -9.90331 q^{35} -9.45159 q^{37} -1.72319 q^{39} -7.19755 q^{41} -3.89976 q^{43} -1.84208 q^{45} -3.20546 q^{47} +16.9832 q^{49} +1.76625 q^{51} -5.15861 q^{53} +4.87009 q^{55} -5.01389 q^{57} -5.88545 q^{59} -4.13077 q^{61} +4.46102 q^{63} -2.41092 q^{65} -2.40517 q^{67} -7.84660 q^{71} +10.9919 q^{73} -1.31623 q^{75} -11.7941 q^{77} -15.8288 q^{79} -5.43746 q^{81} -7.19108 q^{83} +2.47116 q^{85} +12.8721 q^{87} -3.16847 q^{89} +5.83860 q^{91} +9.41908 q^{93} -7.01494 q^{95} -12.5790 q^{97} -2.19377 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	1.44536	0.834482	0.417241	−	0.908796i	$-0.362997\pi$
$3$	1.44536	0.834482	0.417241	+	0.908796i	$0.362997\pi$
$4$	0	0
$5$	2.02221	0.904361	0.452180	−	0.891927i	$-0.350646\pi$
$5$	2.02221	0.904361	0.452180	+	0.891927i	$0.350646\pi$
$6$	0	0
$7$	−4.89726	−1.85099	−0.925496	−	0.378757i	$-0.876352\pi$
$7$	−4.89726	−1.85099	−0.925496	+	0.378757i	$0.876352\pi$
$8$	0	0
$9$	−0.910921	−0.303640
$10$	0	0
$11$	2.40830	0.726130	0.363065	−	0.931764i	$-0.381730\pi$
$11$	2.40830	0.726130	0.363065	+	0.931764i	$0.381730\pi$
$12$	0	0
$13$	−1.19222	−0.330662	−0.165331	−	0.986238i	$-0.552869\pi$
$13$	−1.19222	−0.330662	−0.165331	+	0.986238i	$0.552869\pi$
$14$	0	0
$15$	2.92283	0.754672
$16$	0	0
$17$	1.22201	0.296381	0.148190	−	0.988959i	$-0.452655\pi$
$17$	1.22201	0.296381	0.148190	+	0.988959i	$0.452655\pi$
$18$	0	0
$19$	−3.46894	−0.795830	−0.397915	−	0.917422i	$-0.630266\pi$
$19$	−3.46894	−0.795830	−0.397915	+	0.917422i	$0.630266\pi$
$20$	0	0
$21$	−7.07833	−1.54462
$22$	0	0
$23$	0	0
$23$	0	0
$24$	0	0
$25$	−0.910658	−0.182132
$26$	0	0
$27$	−5.65271	−1.08786
$28$	0	0
$29$	8.90576	1.65376	0.826879	−	0.562380i	$-0.190114\pi$
$29$	8.90576	1.65376	0.826879	+	0.562380i	$0.190114\pi$
$30$	0	0
$31$	6.51675	1.17044	0.585221	−	0.810874i	$-0.301008\pi$
$31$	6.51675	1.17044	0.585221	+	0.810874i	$0.301008\pi$
$32$	0	0
$33$	3.48087	0.605942
$34$	0	0
$35$	−9.90331	−1.67396
$36$	0	0
$37$	−9.45159	−1.55383	−0.776916	−	0.629604i	$-0.783217\pi$
$37$	−9.45159	−1.55383	−0.776916	+	0.629604i	$0.783217\pi$
$38$	0	0
$39$	−1.72319	−0.275931
$40$	0	0
$41$	−7.19755	−1.12407	−0.562034	−	0.827114i	$-0.689981\pi$
$41$	−7.19755	−1.12407	−0.562034	+	0.827114i	$0.689981\pi$
$42$	0	0
$43$	−3.89976	−0.594707	−0.297354	−	0.954767i	$-0.596104\pi$
$43$	−3.89976	−0.594707	−0.297354	+	0.954767i	$0.596104\pi$
$44$	0	0
$45$	−1.84208	−0.274601
$46$	0	0
$47$	−3.20546	−0.467564	−0.233782	−	0.972289i	$-0.575110\pi$
$47$	−3.20546	−0.467564	−0.233782	+	0.972289i	$0.575110\pi$
$48$	0	0
$49$	16.9832	2.42617
$50$	0	0
$51$	1.76625	0.247324
$52$	0	0
$53$	−5.15861	−0.708590	−0.354295	−	0.935134i	$-0.615279\pi$
$53$	−5.15861	−0.708590	−0.354295	+	0.935134i	$0.615279\pi$
$54$	0	0
$55$	4.87009	0.656683
$56$	0	0
$57$	−5.01389	−0.664105
$58$	0	0
$59$	−5.88545	−0.766220	−0.383110	−	0.923703i	$-0.625147\pi$
$59$	−5.88545	−0.766220	−0.383110	+	0.923703i	$0.625147\pi$
$60$	0	0
$61$	−4.13077	−0.528891	−0.264446	−	0.964401i	$-0.585189\pi$
$61$	−4.13077	−0.528891	−0.264446	+	0.964401i	$0.585189\pi$
$62$	0	0
$63$	4.46102	0.562036
$64$	0	0
$65$	−2.41092	−0.299037
$66$	0	0
$67$	−2.40517	−0.293839	−0.146919	−	0.989148i	$-0.546936\pi$
$67$	−2.40517	−0.293839	−0.146919	+	0.989148i	$0.546936\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−7.84660	−0.931220	−0.465610	−	0.884990i	$-0.654165\pi$
$71$	−7.84660	−0.931220	−0.465610	+	0.884990i	$0.654165\pi$
$72$	0	0
$73$	10.9919	1.28651	0.643253	−	0.765654i	$-0.277585\pi$
$73$	10.9919	1.28651	0.643253	+	0.765654i	$0.277585\pi$
$74$	0	0
$75$	−1.31623	−0.151986
$76$	0	0
$77$	−11.7941	−1.34406
$78$	0	0
$79$	−15.8288	−1.78088	−0.890441	−	0.455099i	$-0.849604\pi$
$79$	−15.8288	−1.78088	−0.890441	+	0.455099i	$0.849604\pi$
$80$	0	0
$81$	−5.43746	−0.604162
$82$	0	0
$83$	−7.19108	−0.789324	−0.394662	−	0.918826i	$-0.629138\pi$
$83$	−7.19108	−0.789324	−0.394662	+	0.918826i	$0.629138\pi$
$84$	0	0
$85$	2.47116	0.268035
$86$	0	0
$87$	12.8721	1.38003
$88$	0	0
$89$	−3.16847	−0.335858	−0.167929	−	0.985799i	$-0.553708\pi$
$89$	−3.16847	−0.335858	−0.167929	+	0.985799i	$0.553708\pi$
$90$	0	0
$91$	5.83860	0.612052
$92$	0	0
$93$	9.41908	0.976713
$94$	0	0
$95$	−7.01494	−0.719717
$96$	0	0
$97$	−12.5790	−1.27720	−0.638601	−	0.769538i	$-0.720487\pi$
$97$	−12.5790	−1.27720	−0.638601	+	0.769538i	$0.720487\pi$
$98$	0	0
$99$	−2.19377	−0.220482
$100$	0	0
$101$	14.4595	1.43877	0.719387	−	0.694609i	$-0.244423\pi$
$101$	14.4595	1.43877	0.719387	+	0.694609i	$0.244423\pi$
$102$	0	0
$103$	1.42796	0.140701	0.0703503	−	0.997522i	$-0.477588\pi$
$103$	1.42796	0.140701	0.0703503	+	0.997522i	$0.477588\pi$
$104$	0	0
$105$	−14.3139	−1.39689
$106$	0	0
$107$	−12.1845	−1.17792	−0.588960	−	0.808162i	$-0.700463\pi$
$107$	−12.1845	−1.17792	−0.588960	+	0.808162i	$0.700463\pi$
$108$	0	0
$109$	16.1124	1.54329	0.771644	−	0.636055i	$-0.219435\pi$
$109$	16.1124	1.54329	0.771644	+	0.636055i	$0.219435\pi$
$110$	0	0
$111$	−13.6610	−1.29664
$112$	0	0
$113$	6.15964	0.579450	0.289725	−	0.957110i	$-0.406436\pi$
$113$	6.15964	0.579450	0.289725	+	0.957110i	$0.406436\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	1.08602	0.100402
$118$	0	0
$119$	−5.98450	−0.548598
$120$	0	0
$121$	−5.20009	−0.472736
$122$	0	0
$123$	−10.4031	−0.938015
$124$	0	0
$125$	−11.9526	−1.06907
$126$	0	0
$127$	−16.6996	−1.48185	−0.740923	−	0.671589i	$-0.765612\pi$
$127$	−16.6996	−1.48185	−0.740923	+	0.671589i	$0.765612\pi$
$128$	0	0
$129$	−5.63657	−0.496272
$130$	0	0
$131$	−2.29064	−0.200135	−0.100067	−	0.994981i	$-0.531906\pi$
$131$	−2.29064	−0.200135	−0.100067	+	0.994981i	$0.531906\pi$
$132$	0	0
$133$	16.9883	1.47307
$134$	0	0
$135$	−11.4310	−0.983821
$136$	0	0
$137$	−20.7171	−1.76998	−0.884992	−	0.465607i	$-0.845836\pi$
$137$	−20.7171	−1.76998	−0.884992	+	0.465607i	$0.845836\pi$
$138$	0	0
$139$	0.114710	0.00972957	0.00486479	−	0.999988i	$-0.498451\pi$
$139$	0.114710	0.00972957	0.00486479	+	0.999988i	$0.498451\pi$
$140$	0	0
$141$	−4.63306	−0.390174
$142$	0	0
$143$	−2.87122	−0.240103
$144$	0	0
$145$	18.0093	1.49559
$146$	0	0
$147$	24.5469	2.02460
$148$	0	0
$149$	−12.9237	−1.05875	−0.529374	−	0.848389i	$-0.677573\pi$
$149$	−12.9237	−1.05875	−0.529374	+	0.848389i	$0.677573\pi$
$150$	0	0
$151$	2.68072	0.218154	0.109077	−	0.994033i	$-0.465210\pi$
$151$	2.68072	0.218154	0.109077	+	0.994033i	$0.465210\pi$
$152$	0	0
$153$	−1.11315	−0.0899932
$154$	0	0
$155$	13.1782	1.05850
$156$	0	0
$157$	18.3460	1.46417	0.732085	−	0.681214i	$-0.238547\pi$
$157$	18.3460	1.46417	0.732085	+	0.681214i	$0.238547\pi$
$158$	0	0
$159$	−7.45607	−0.591305
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−9.78039	−0.766059	−0.383030	−	0.923736i	$-0.625119\pi$
$163$	−9.78039	−0.766059	−0.383030	+	0.923736i	$0.625119\pi$
$164$	0	0
$165$	7.03906	0.547990
$166$	0	0
$167$	−15.5748	−1.20521	−0.602606	−	0.798039i	$-0.705871\pi$
$167$	−15.5748	−1.20521	−0.602606	+	0.798039i	$0.705871\pi$
$168$	0	0
$169$	−11.5786	−0.890663
$170$	0	0
$171$	3.15993	0.241646
$172$	0	0
$173$	−2.97497	−0.226183	−0.113092	−	0.993585i	$-0.536075\pi$
$173$	−2.97497	−0.226183	−0.113092	+	0.993585i	$0.536075\pi$
$174$	0	0
$175$	4.45974	0.337124
$176$	0	0
$177$	−8.50662	−0.639397
$178$	0	0
$179$	1.07242	0.0801568	0.0400784	−	0.999197i	$-0.487239\pi$
$179$	1.07242	0.0801568	0.0400784	+	0.999197i	$0.487239\pi$
$180$	0	0
$181$	12.9077	0.959424	0.479712	−	0.877426i	$-0.340741\pi$
$181$	12.9077	0.959424	0.479712	+	0.877426i	$0.340741\pi$
$182$	0	0
$183$	−5.97047	−0.441350
$184$	0	0
$185$	−19.1131	−1.40523
$186$	0	0
$187$	2.94296	0.215211
$188$	0	0
$189$	27.6828	2.01363
$190$	0	0
$191$	22.1315	1.60138	0.800690	−	0.599079i	$-0.204466\pi$
$191$	22.1315	1.60138	0.800690	+	0.599079i	$0.204466\pi$
$192$	0	0
$193$	3.34666	0.240898	0.120449	−	0.992720i	$-0.461567\pi$
$193$	3.34666	0.240898	0.120449	+	0.992720i	$0.461567\pi$
$194$	0	0
$195$	−3.48465	−0.249541
$196$	0	0
$197$	−4.75407	−0.338714	−0.169357	−	0.985555i	$-0.554169\pi$
$197$	−4.75407	−0.338714	−0.169357	+	0.985555i	$0.554169\pi$
$198$	0	0
$199$	−9.29236	−0.658718	−0.329359	−	0.944205i	$-0.606833\pi$
$199$	−9.29236	−0.658718	−0.329359	+	0.944205i	$0.606833\pi$
$200$	0	0
$201$	−3.47635	−0.245203
$202$	0	0
$203$	−43.6138	−3.06109
$204$	0	0
$205$	−14.5550	−1.01656
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−8.35425	−0.577876
$210$	0	0
$211$	20.9031	1.43903	0.719514	−	0.694478i	$-0.244365\pi$
$211$	20.9031	1.43903	0.719514	+	0.694478i	$0.244365\pi$
$212$	0	0
$213$	−11.3412	−0.777086
$214$	0	0
$215$	−7.88613	−0.537830
$216$	0	0
$217$	−31.9142	−2.16648
$218$	0	0
$219$	15.8873	1.07356
$220$	0	0
$221$	−1.45690	−0.0980017
$222$	0	0
$223$	5.57661	0.373437	0.186719	−	0.982413i	$-0.440215\pi$
$223$	5.57661	0.373437	0.186719	+	0.982413i	$0.440215\pi$
$224$	0	0
$225$	0.829538	0.0553026
$226$	0	0
$227$	−6.54438	−0.434366	−0.217183	−	0.976131i	$-0.569687\pi$
$227$	−6.54438	−0.434366	−0.217183	+	0.976131i	$0.569687\pi$
$228$	0	0
$229$	11.3038	0.746975	0.373487	−	0.927635i	$-0.378162\pi$
$229$	11.3038	0.746975	0.373487	+	0.927635i	$0.378162\pi$
$230$	0	0
$231$	−17.0467	−1.12159
$232$	0	0
$233$	18.4650	1.20968	0.604841	−	0.796346i	$-0.293237\pi$
$233$	18.4650	1.20968	0.604841	+	0.796346i	$0.293237\pi$
$234$	0	0
$235$	−6.48212	−0.422847
$236$	0	0
$237$	−22.8784	−1.48611
$238$	0	0
$239$	−2.15943	−0.139682	−0.0698409	−	0.997558i	$-0.522249\pi$
$239$	−2.15943	−0.139682	−0.0698409	+	0.997558i	$0.522249\pi$
$240$	0	0
$241$	3.75026	0.241575	0.120788	−	0.992678i	$-0.461458\pi$
$241$	3.75026	0.241575	0.120788	+	0.992678i	$0.461458\pi$
$242$	0	0
$243$	9.09901	0.583702
$244$	0	0
$245$	34.3436	2.19413
$246$	0	0
$247$	4.13573	0.263150
$248$	0	0
$249$	−10.3937	−0.658676
$250$	0	0
$251$	−19.7829	−1.24869	−0.624343	−	0.781151i	$-0.714633\pi$
$251$	−19.7829	−1.24869	−0.624343	+	0.781151i	$0.714633\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	3.57173	0.223670
$256$	0	0
$257$	−11.5620	−0.721219	−0.360610	−	0.932717i	$-0.617431\pi$
$257$	−11.5620	−0.721219	−0.360610	+	0.932717i	$0.617431\pi$
$258$	0	0
$259$	46.2870	2.87613
$260$	0	0
$261$	−8.11244	−0.502148
$262$	0	0
$263$	20.9772	1.29351	0.646753	−	0.762699i	$-0.276126\pi$
$263$	20.9772	1.29351	0.646753	+	0.762699i	$0.276126\pi$
$264$	0	0
$265$	−10.4318	−0.640821
$266$	0	0
$267$	−4.57960	−0.280267
$268$	0	0
$269$	26.2685	1.60162	0.800809	−	0.598920i	$-0.204403\pi$
$269$	26.2685	1.60162	0.800809	+	0.598920i	$0.204403\pi$
$270$	0	0
$271$	27.8245	1.69022	0.845109	−	0.534595i	$-0.179536\pi$
$271$	27.8245	1.69022	0.845109	+	0.534595i	$0.179536\pi$
$272$	0	0
$273$	8.43891	0.510746
$274$	0	0
$275$	−2.19314	−0.132251
$276$	0	0
$277$	15.0981	0.907160	0.453580	−	0.891216i	$-0.350147\pi$
$277$	15.0981	0.907160	0.453580	+	0.891216i	$0.350147\pi$
$278$	0	0
$279$	−5.93625	−0.355394
$280$	0	0
$281$	7.53422	0.449454	0.224727	−	0.974422i	$-0.427851\pi$
$281$	7.53422	0.449454	0.224727	+	0.974422i	$0.427851\pi$
$282$	0	0
$283$	3.61800	0.215068	0.107534	−	0.994201i	$-0.465705\pi$
$283$	3.61800	0.215068	0.107534	+	0.994201i	$0.465705\pi$
$284$	0	0
$285$	−10.1391	−0.600591
$286$	0	0
$287$	35.2483	2.08064
$288$	0	0
$289$	−15.5067	−0.912159
$290$	0	0
$291$	−18.1812	−1.06580
$292$	0	0
$293$	8.77390	0.512577	0.256288	−	0.966600i	$-0.417500\pi$
$293$	8.77390	0.512577	0.256288	+	0.966600i	$0.417500\pi$
$294$	0	0
$295$	−11.9016	−0.692939
$296$	0	0
$297$	−13.6134	−0.789930
$298$	0	0
$299$	0	0
$300$	0	0
$301$	19.0981	1.10080
$302$	0	0
$303$	20.8993	1.20063
$304$	0	0
$305$	−8.35330	−0.478309
$306$	0	0
$307$	29.0039	1.65534	0.827669	−	0.561216i	$-0.189666\pi$
$307$	29.0039	1.65534	0.827669	+	0.561216i	$0.189666\pi$
$308$	0	0
$309$	2.06392	0.117412
$310$	0	0
$311$	9.91599	0.562284	0.281142	−	0.959666i	$-0.409287\pi$
$311$	9.91599	0.562284	0.281142	+	0.959666i	$0.409287\pi$
$312$	0	0
$313$	−1.74566	−0.0986707	−0.0493354	−	0.998782i	$-0.515710\pi$
$313$	−1.74566	−0.0986707	−0.0493354	+	0.998782i	$0.515710\pi$
$314$	0	0
$315$	9.02114	0.508283
$316$	0	0
$317$	10.6371	0.597441	0.298721	−	0.954341i	$-0.403440\pi$
$317$	10.6371	0.597441	0.298721	+	0.954341i	$0.403440\pi$
$318$	0	0
$319$	21.4477	1.20084
$320$	0	0
$321$	−17.6111	−0.982953
$322$	0	0
$323$	−4.23908	−0.235869
$324$	0	0
$325$	1.08570	0.0602239
$326$	0	0
$327$	23.2883	1.28784
$328$	0	0
$329$	15.6980	0.865458
$330$	0	0
$331$	−10.6274	−0.584132	−0.292066	−	0.956398i	$-0.594343\pi$
$331$	−10.6274	−0.584132	−0.292066	+	0.956398i	$0.594343\pi$
$332$	0	0
$333$	8.60966	0.471806
$334$	0	0
$335$	−4.86377	−0.265736
$336$	0	0
$337$	26.9262	1.46676	0.733381	−	0.679818i	$-0.237941\pi$
$337$	26.9262	1.46676	0.733381	+	0.679818i	$0.237941\pi$
$338$	0	0
$339$	8.90293	0.483541
$340$	0	0
$341$	15.6943	0.849893
$342$	0	0
$343$	−48.8904	−2.63983
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−2.12167	−0.113897	−0.0569486	−	0.998377i	$-0.518137\pi$
$347$	−2.12167	−0.113897	−0.0569486	+	0.998377i	$0.518137\pi$
$348$	0	0
$349$	33.2453	1.77958	0.889790	−	0.456370i	$-0.150851\pi$
$349$	33.2453	1.77958	0.889790	+	0.456370i	$0.150851\pi$
$350$	0	0
$351$	6.73925	0.359715
$352$	0	0
$353$	16.1739	0.860850	0.430425	−	0.902626i	$-0.358364\pi$
$353$	16.1739	0.860850	0.430425	+	0.902626i	$0.358364\pi$
$354$	0	0
$355$	−15.8675	−0.842159
$356$	0	0
$357$	−8.64978	−0.457795
$358$	0	0
$359$	15.5224	0.819241	0.409621	−	0.912256i	$-0.365661\pi$
$359$	15.5224	0.819241	0.409621	+	0.912256i	$0.365661\pi$
$360$	0	0
$361$	−6.96644	−0.366655
$362$	0	0
$363$	−7.51603	−0.394489
$364$	0	0
$365$	22.2280	1.16346
$366$	0	0
$367$	6.66147	0.347726	0.173863	−	0.984770i	$-0.444375\pi$
$367$	6.66147	0.347726	0.173863	+	0.984770i	$0.444375\pi$
$368$	0	0
$369$	6.55640	0.341313
$370$	0	0
$371$	25.2631	1.31159
$372$	0	0
$373$	−12.4917	−0.646795	−0.323397	−	0.946263i	$-0.604825\pi$
$373$	−12.4917	−0.646795	−0.323397	+	0.946263i	$0.604825\pi$
$374$	0	0
$375$	−17.2759	−0.892122
$376$	0	0
$377$	−10.6176	−0.546834
$378$	0	0
$379$	−6.85759	−0.352251	−0.176125	−	0.984368i	$-0.556356\pi$
$379$	−6.85759	−0.352251	−0.176125	+	0.984368i	$0.556356\pi$
$380$	0	0
$381$	−24.1370	−1.23657
$382$	0	0
$383$	−14.3183	−0.731630	−0.365815	−	0.930688i	$-0.619210\pi$
$383$	−14.3183	−0.731630	−0.365815	+	0.930688i	$0.619210\pi$
$384$	0	0
$385$	−23.8501	−1.21552
$386$	0	0
$387$	3.55237	0.180577
$388$	0	0
$389$	−2.84243	−0.144117	−0.0720585	−	0.997400i	$-0.522957\pi$
$389$	−2.84243	−0.144117	−0.0720585	+	0.997400i	$0.522957\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−3.31082	−0.167009
$394$	0	0
$395$	−32.0092	−1.61056
$396$	0	0
$397$	−26.0574	−1.30778	−0.653891	−	0.756589i	$-0.726864\pi$
$397$	−26.0574	−1.30778	−0.653891	+	0.756589i	$0.726864\pi$
$398$	0	0
$399$	24.5543	1.22925
$400$	0	0
$401$	5.43728	0.271525	0.135762	−	0.990741i	$-0.456652\pi$
$401$	5.43728	0.271525	0.135762	+	0.990741i	$0.456652\pi$
$402$	0	0
$403$	−7.76938	−0.387020
$404$	0	0
$405$	−10.9957	−0.546380
$406$	0	0
$407$	−22.7623	−1.12828
$408$	0	0
$409$	−36.3042	−1.79512	−0.897562	−	0.440887i	$-0.854664\pi$
$409$	−36.3042	−1.79512	−0.897562	+	0.440887i	$0.854664\pi$
$410$	0	0
$411$	−29.9438	−1.47702
$412$	0	0
$413$	28.8226	1.41827
$414$	0	0
$415$	−14.5419	−0.713833
$416$	0	0
$417$	0.165798	0.00811915
$418$	0	0
$419$	−18.5793	−0.907658	−0.453829	−	0.891089i	$-0.649942\pi$
$419$	−18.5793	−0.907658	−0.453829	+	0.891089i	$0.649942\pi$
$420$	0	0
$421$	2.92780	0.142692	0.0713461	−	0.997452i	$-0.477271\pi$
$421$	2.92780	0.142692	0.0713461	+	0.997452i	$0.477271\pi$
$422$	0	0
$423$	2.91992	0.141972
$424$	0	0
$425$	−1.11283	−0.0539803
$426$	0	0
$427$	20.2295	0.978974
$428$	0	0
$429$	−4.14995	−0.200362
$430$	0	0
$431$	−23.0798	−1.11171	−0.555857	−	0.831278i	$-0.687610\pi$
$431$	−23.0798	−1.11171	−0.555857	+	0.831278i	$0.687610\pi$
$432$	0	0
$433$	−13.3765	−0.642832	−0.321416	−	0.946938i	$-0.604159\pi$
$433$	−13.3765	−0.642832	−0.321416	+	0.946938i	$0.604159\pi$
$434$	0	0
$435$	26.0300	1.24804
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−18.4769	−0.881856	−0.440928	−	0.897543i	$-0.645351\pi$
$439$	−18.4769	−0.881856	−0.440928	+	0.897543i	$0.645351\pi$
$440$	0	0
$441$	−15.4704	−0.736684
$442$	0	0
$443$	5.74649	0.273024	0.136512	−	0.990638i	$-0.456411\pi$
$443$	5.74649	0.273024	0.136512	+	0.990638i	$0.456411\pi$
$444$	0	0
$445$	−6.40733	−0.303736
$446$	0	0
$447$	−18.6794	−0.883505
$448$	0	0
$449$	−6.35693	−0.300002	−0.150001	−	0.988686i	$-0.547928\pi$
$449$	−6.35693	−0.300002	−0.150001	+	0.988686i	$0.547928\pi$
$450$	0	0
$451$	−17.3339	−0.816220
$452$	0	0
$453$	3.87462	0.182045
$454$	0	0
$455$	11.8069	0.553516
$456$	0	0
$457$	−5.60464	−0.262174	−0.131087	−	0.991371i	$-0.541847\pi$
$457$	−5.60464	−0.262174	−0.131087	+	0.991371i	$0.541847\pi$
$458$	0	0
$459$	−6.90766	−0.322422
$460$	0	0
$461$	−13.7185	−0.638935	−0.319467	−	0.947597i	$-0.603504\pi$
$461$	−13.7185	−0.638935	−0.319467	+	0.947597i	$0.603504\pi$
$462$	0	0
$463$	−31.6222	−1.46961	−0.734804	−	0.678280i	$-0.762726\pi$
$463$	−31.6222	−1.46961	−0.734804	+	0.678280i	$0.762726\pi$
$464$	0	0
$465$	19.0474	0.883301
$466$	0	0
$467$	−8.08857	−0.374294	−0.187147	−	0.982332i	$-0.559924\pi$
$467$	−8.08857	−0.374294	−0.187147	+	0.982332i	$0.559924\pi$
$468$	0	0
$469$	11.7788	0.543893
$470$	0	0
$471$	26.5166	1.22182
$472$	0	0
$473$	−9.39178	−0.431835
$474$	0	0
$475$	3.15902	0.144946
$476$	0	0
$477$	4.69909	0.215157
$478$	0	0
$479$	20.7341	0.947363	0.473682	−	0.880696i	$-0.342925\pi$
$479$	20.7341	0.947363	0.473682	+	0.880696i	$0.342925\pi$
$480$	0	0
$481$	11.2684	0.513793
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−25.4374	−1.15505
$486$	0	0
$487$	−4.64883	−0.210658	−0.105329	−	0.994437i	$-0.533590\pi$
$487$	−4.64883	−0.210658	−0.105329	+	0.994437i	$0.533590\pi$
$488$	0	0
$489$	−14.1362	−0.639262
$490$	0	0
$491$	−31.0974	−1.40341	−0.701704	−	0.712469i	$-0.747577\pi$
$491$	−31.0974	−1.40341	−0.701704	+	0.712469i	$0.747577\pi$
$492$	0	0
$493$	10.8829	0.490142
$494$	0	0
$495$	−4.43627	−0.199396
$496$	0	0
$497$	38.4269	1.72368
$498$	0	0
$499$	−6.58689	−0.294869	−0.147435	−	0.989072i	$-0.547102\pi$
$499$	−6.58689	−0.294869	−0.147435	+	0.989072i	$0.547102\pi$
$500$	0	0
$501$	−22.5112	−1.00573
$502$	0	0
$503$	−30.2889	−1.35052	−0.675259	−	0.737581i	$-0.735968\pi$
$503$	−30.2889	−1.35052	−0.675259	+	0.737581i	$0.735968\pi$
$504$	0	0
$505$	29.2402	1.30117
$506$	0	0
$507$	−16.7353	−0.743242
$508$	0	0
$509$	−2.64250	−0.117127	−0.0585634	−	0.998284i	$-0.518652\pi$
$509$	−2.64250	−0.117127	−0.0585634	+	0.998284i	$0.518652\pi$
$510$	0	0
$511$	−53.8303	−2.38131
$512$	0	0
$513$	19.6089	0.865755
$514$	0	0
$515$	2.88763	0.127244
$516$	0	0
$517$	−7.71971	−0.339512
$518$	0	0
$519$	−4.29992	−0.188746
$520$	0	0
$521$	−15.2126	−0.666478	−0.333239	−	0.942842i	$-0.608142\pi$
$521$	−15.2126	−0.666478	−0.333239	+	0.942842i	$0.608142\pi$
$522$	0	0
$523$	23.2078	1.01481	0.507404	−	0.861708i	$-0.330605\pi$
$523$	23.2078	1.01481	0.507404	+	0.861708i	$0.330605\pi$
$524$	0	0
$525$	6.44594	0.281324
$526$	0	0
$527$	7.96352	0.346897
$528$	0	0
$529$	0	0
$530$	0	0
$531$	5.36118	0.232655
$532$	0	0
$533$	8.58104	0.371686
$534$	0	0
$535$	−24.6397	−1.06527
$536$	0	0
$537$	1.55004	0.0668893
$538$	0	0
$539$	40.9006	1.76172
$540$	0	0
$541$	12.7454	0.547968	0.273984	−	0.961734i	$-0.411658\pi$
$541$	12.7454	0.547968	0.273984	+	0.961734i	$0.411658\pi$
$542$	0	0
$543$	18.6564	0.800622
$544$	0	0
$545$	32.5827	1.39569
$546$	0	0
$547$	20.9251	0.894693	0.447347	−	0.894361i	$-0.352369\pi$
$547$	20.9251	0.894693	0.447347	+	0.894361i	$0.352369\pi$
$548$	0	0
$549$	3.76281	0.160593
$550$	0	0
$551$	−30.8935	−1.31611
$552$	0	0
$553$	77.5179	3.29640
$554$	0	0
$555$	−27.6254	−1.17263
$556$	0	0
$557$	12.8732	0.545457	0.272728	−	0.962091i	$-0.412074\pi$
$557$	12.8732	0.545457	0.272728	+	0.962091i	$0.412074\pi$
$558$	0	0
$559$	4.64936	0.196647
$560$	0	0
$561$	4.25365	0.179589
$562$	0	0
$563$	−4.84746	−0.204296	−0.102148	−	0.994769i	$-0.532572\pi$
$563$	−4.84746	−0.204296	−0.102148	+	0.994769i	$0.532572\pi$
$564$	0	0
$565$	12.4561	0.524032
$566$	0	0
$567$	26.6287	1.11830
$568$	0	0
$569$	43.4330	1.82081	0.910404	−	0.413721i	$-0.135771\pi$
$569$	43.4330	1.82081	0.910404	+	0.413721i	$0.135771\pi$
$570$	0	0
$571$	25.9544	1.08616	0.543079	−	0.839681i	$-0.317258\pi$
$571$	25.9544	1.08616	0.543079	+	0.839681i	$0.317258\pi$
$572$	0	0
$573$	31.9881	1.33632
$574$	0	0
$575$	0	0
$576$	0	0
$577$	3.06969	0.127793	0.0638964	−	0.997957i	$-0.479647\pi$
$577$	3.06969	0.127793	0.0638964	+	0.997957i	$0.479647\pi$
$578$	0	0
$579$	4.83715	0.201025
$580$	0	0
$581$	35.2166	1.46103
$582$	0	0
$583$	−12.4235	−0.514528
$584$	0	0
$585$	2.19616	0.0907998
$586$	0	0
$587$	−23.8413	−0.984037	−0.492019	−	0.870585i	$-0.663741\pi$
$587$	−23.8413	−0.984037	−0.492019	+	0.870585i	$0.663741\pi$
$588$	0	0
$589$	−22.6062	−0.931473
$590$	0	0
$591$	−6.87137	−0.282650
$592$	0	0
$593$	−32.9927	−1.35485	−0.677424	−	0.735593i	$-0.736904\pi$
$593$	−32.9927	−1.35485	−0.677424	+	0.735593i	$0.736904\pi$
$594$	0	0
$595$	−12.1019	−0.496131
$596$	0	0
$597$	−13.4309	−0.549688
$598$	0	0
$599$	10.2143	0.417343	0.208672	−	0.977986i	$-0.433086\pi$
$599$	10.2143	0.417343	0.208672	+	0.977986i	$0.433086\pi$
$600$	0	0
$601$	−16.1059	−0.656972	−0.328486	−	0.944509i	$-0.606538\pi$
$601$	−16.1059	−0.656972	−0.328486	+	0.944509i	$0.606538\pi$
$602$	0	0
$603$	2.19093	0.0892214
$604$	0	0
$605$	−10.5157	−0.427524
$606$	0	0
$607$	−29.9050	−1.21381	−0.606904	−	0.794775i	$-0.707589\pi$
$607$	−29.9050	−1.21381	−0.606904	+	0.794775i	$0.707589\pi$
$608$	0	0
$609$	−63.0379	−2.55442
$610$	0	0
$611$	3.82161	0.154606
$612$	0	0
$613$	−23.7162	−0.957890	−0.478945	−	0.877845i	$-0.658981\pi$
$613$	−23.7162	−0.957890	−0.478945	+	0.877845i	$0.658981\pi$
$614$	0	0
$615$	−21.0372	−0.848304
$616$	0	0
$617$	33.0802	1.33176	0.665879	−	0.746060i	$-0.268057\pi$
$617$	33.0802	1.33176	0.665879	+	0.746060i	$0.268057\pi$
$618$	0	0
$619$	33.5280	1.34760	0.673802	−	0.738912i	$-0.264660\pi$
$619$	33.5280	1.34760	0.673802	+	0.738912i	$0.264660\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	15.5169	0.621670
$624$	0	0
$625$	−19.6174	−0.784696
$626$	0	0
$627$	−12.0749	−0.482227
$628$	0	0
$629$	−11.5499	−0.460526
$630$	0	0
$631$	−13.4819	−0.536706	−0.268353	−	0.963321i	$-0.586479\pi$
$631$	−13.4819	−0.536706	−0.268353	+	0.963321i	$0.586479\pi$
$632$	0	0
$633$	30.2126	1.20084
$634$	0	0
$635$	−33.7701	−1.34012
$636$	0	0
$637$	−20.2477	−0.802242
$638$	0	0
$639$	7.14764	0.282756
$640$	0	0
$641$	−1.21395	−0.0479481	−0.0239741	−	0.999713i	$-0.507632\pi$
$641$	−1.21395	−0.0479481	−0.0239741	+	0.999713i	$0.507632\pi$
$642$	0	0
$643$	1.08705	0.0428692	0.0214346	−	0.999770i	$-0.493177\pi$
$643$	1.08705	0.0428692	0.0214346	+	0.999770i	$0.493177\pi$
$644$	0	0
$645$	−11.3983	−0.448809
$646$	0	0
$647$	−26.0340	−1.02350	−0.511750	−	0.859134i	$-0.671003\pi$
$647$	−26.0340	−1.02350	−0.511750	+	0.859134i	$0.671003\pi$
$648$	0	0
$649$	−14.1739	−0.556375
$650$	0	0
$651$	−46.1277	−1.80789
$652$	0	0
$653$	35.0930	1.37329	0.686647	−	0.726991i	$-0.259082\pi$
$653$	35.0930	1.37329	0.686647	+	0.726991i	$0.259082\pi$
$654$	0	0
$655$	−4.63217	−0.180994
$656$	0	0
$657$	−10.0128	−0.390635
$658$	0	0
$659$	−36.5324	−1.42310	−0.711551	−	0.702635i	$-0.752007\pi$
$659$	−36.5324	−1.42310	−0.711551	+	0.702635i	$0.752007\pi$
$660$	0	0
$661$	6.70114	0.260644	0.130322	−	0.991472i	$-0.458399\pi$
$661$	6.70114	0.260644	0.130322	+	0.991472i	$0.458399\pi$
$662$	0	0
$663$	−2.10575	−0.0817806
$664$	0	0
$665$	34.3540	1.33219
$666$	0	0
$667$	0	0
$668$	0	0
$669$	8.06023	0.311626
$670$	0	0
$671$	−9.94814	−0.384044
$672$	0	0
$673$	1.05231	0.0405634	0.0202817	−	0.999794i	$-0.493544\pi$
$673$	1.05231	0.0405634	0.0202817	+	0.999794i	$0.493544\pi$
$674$	0	0
$675$	5.14769	0.198135
$676$	0	0
$677$	24.8621	0.955529	0.477765	−	0.878488i	$-0.341447\pi$
$677$	24.8621	0.955529	0.477765	+	0.878488i	$0.341447\pi$
$678$	0	0
$679$	61.6026	2.36409
$680$	0	0
$681$	−9.45902	−0.362470
$682$	0	0
$683$	−40.0210	−1.53136	−0.765680	−	0.643222i	$-0.777597\pi$
$683$	−40.0210	−1.53136	−0.765680	+	0.643222i	$0.777597\pi$
$684$	0	0
$685$	−41.8944	−1.60070
$686$	0	0
$687$	16.3381	0.623337
$688$	0	0
$689$	6.15018	0.234303
$690$	0	0
$691$	4.88876	0.185977	0.0929886	−	0.995667i	$-0.470358\pi$
$691$	4.88876	0.185977	0.0929886	+	0.995667i	$0.470358\pi$
$692$	0	0
$693$	10.7435	0.408111
$694$	0	0
$695$	0.231968	0.00879904
$696$	0	0
$697$	−8.79547	−0.333152
$698$	0	0
$699$	26.6887	1.00946
$700$	0	0
$701$	−6.61481	−0.249838	−0.124919	−	0.992167i	$-0.539867\pi$
$701$	−6.61481	−0.249838	−0.124919	+	0.992167i	$0.539867\pi$
$702$	0	0
$703$	32.7870	1.23659
$704$	0	0
$705$	−9.36903	−0.352858
$706$	0	0
$707$	−70.8120	−2.66316
$708$	0	0
$709$	−6.62998	−0.248994	−0.124497	−	0.992220i	$-0.539732\pi$
$709$	−6.62998	−0.248994	−0.124497	+	0.992220i	$0.539732\pi$
$710$	0	0
$711$	14.4188	0.540748
$712$	0	0
$713$	0	0
$714$	0	0
$715$	−5.80621	−0.217140
$716$	0	0
$717$	−3.12116	−0.116562
$718$	0	0
$719$	−0.956212	−0.0356607	−0.0178303	−	0.999841i	$-0.505676\pi$
$719$	−0.956212	−0.0356607	−0.0178303	+	0.999841i	$0.505676\pi$
$720$	0	0
$721$	−6.99308	−0.260436
$722$	0	0
$723$	5.42049	0.201590
$724$	0	0
$725$	−8.11010	−0.301202
$726$	0	0
$727$	18.3313	0.679871	0.339935	−	0.940449i	$-0.389595\pi$
$727$	18.3313	0.679871	0.339935	+	0.940449i	$0.389595\pi$
$728$	0	0
$729$	29.4638	1.09125
$730$	0	0
$731$	−4.76553	−0.176260
$732$	0	0
$733$	2.75172	0.101637	0.0508186	−	0.998708i	$-0.483817\pi$
$733$	2.75172	0.101637	0.0508186	+	0.998708i	$0.483817\pi$
$734$	0	0
$735$	49.6391	1.83096
$736$	0	0
$737$	−5.79238	−0.213365
$738$	0	0
$739$	23.2814	0.856421	0.428211	−	0.903679i	$-0.359144\pi$
$739$	23.2814	0.856421	0.428211	+	0.903679i	$0.359144\pi$
$740$	0	0
$741$	5.97764	0.219594
$742$	0	0
$743$	33.4868	1.22851	0.614257	−	0.789106i	$-0.289456\pi$
$743$	33.4868	1.22851	0.614257	+	0.789106i	$0.289456\pi$
$744$	0	0
$745$	−26.1344	−0.957490
$746$	0	0
$747$	6.55051	0.239671
$748$	0	0
$749$	59.6708	2.18032
$750$	0	0
$751$	12.6341	0.461023	0.230512	−	0.973070i	$-0.425960\pi$
$751$	12.6341	0.461023	0.230512	+	0.973070i	$0.425960\pi$
$752$	0	0
$753$	−28.5935	−1.04200
$754$	0	0
$755$	5.42099	0.197290
$756$	0	0
$757$	12.4802	0.453602	0.226801	−	0.973941i	$-0.427173\pi$
$757$	12.4802	0.453602	0.226801	+	0.973941i	$0.427173\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	40.2487	1.45901	0.729507	−	0.683973i	$-0.239749\pi$
$761$	40.2487	1.45901	0.729507	+	0.683973i	$0.239749\pi$
$762$	0	0
$763$	−78.9066	−2.85661
$764$	0	0
$765$	−2.25103	−0.0813863
$766$	0	0
$767$	7.01673	0.253360
$768$	0	0
$769$	9.95061	0.358828	0.179414	−	0.983774i	$-0.442580\pi$
$769$	9.95061	0.358828	0.179414	+	0.983774i	$0.442580\pi$
$770$	0	0
$771$	−16.7113	−0.601844
$772$	0	0
$773$	50.1107	1.80236	0.901178	−	0.433449i	$-0.142704\pi$
$773$	50.1107	1.80236	0.901178	+	0.433449i	$0.142704\pi$
$774$	0	0
$775$	−5.93453	−0.213175
$776$	0	0
$777$	66.9015	2.40008
$778$	0	0
$779$	24.9679	0.894567
$780$	0	0
$781$	−18.8970	−0.676187
$782$	0	0
$783$	−50.3416	−1.79906
$784$	0	0
$785$	37.0995	1.32414
$786$	0	0
$787$	−11.8773	−0.423380	−0.211690	−	0.977337i	$-0.567897\pi$
$787$	−11.8773	−0.423380	−0.211690	+	0.977337i	$0.567897\pi$
$788$	0	0
$789$	30.3196	1.07941
$790$	0	0
$791$	−30.1654	−1.07256
$792$	0	0
$793$	4.92478	0.174884
$794$	0	0
$795$	−15.0778	−0.534753
$796$	0	0
$797$	15.5473	0.550715	0.275358	−	0.961342i	$-0.411204\pi$
$797$	15.5473	0.550715	0.275358	+	0.961342i	$0.411204\pi$
$798$	0	0
$799$	−3.91710	−0.138577
$800$	0	0
$801$	2.88623	0.101980
$802$	0	0
$803$	26.4718	0.934170
$804$	0	0
$805$	0	0
$806$	0	0
$807$	37.9675	1.33652
$808$	0	0
$809$	33.5677	1.18018	0.590089	−	0.807338i	$-0.299093\pi$
$809$	33.5677	1.18018	0.590089	+	0.807338i	$0.299093\pi$
$810$	0	0
$811$	46.6891	1.63948	0.819738	−	0.572739i	$-0.194119\pi$
$811$	46.6891	1.63948	0.819738	+	0.572739i	$0.194119\pi$
$812$	0	0
$813$	40.2165	1.41046
$814$	0	0
$815$	−19.7780	−0.692794
$816$	0	0
$817$	13.5280	0.473286
$818$	0	0
$819$	−5.31851	−0.185844
$820$	0	0
$821$	−24.6315	−0.859646	−0.429823	−	0.902913i	$-0.641424\pi$
$821$	−24.6315	−0.859646	−0.429823	+	0.902913i	$0.641424\pi$
$822$	0	0
$823$	−27.2657	−0.950422	−0.475211	−	0.879872i	$-0.657628\pi$
$823$	−27.2657	−0.950422	−0.475211	+	0.879872i	$0.657628\pi$
$824$	0	0
$825$	−3.16988	−0.110361
$826$	0	0
$827$	−35.1129	−1.22100	−0.610498	−	0.792018i	$-0.709031\pi$
$827$	−35.1129	−1.22100	−0.610498	+	0.792018i	$0.709031\pi$
$828$	0	0
$829$	4.43108	0.153898	0.0769488	−	0.997035i	$-0.475482\pi$
$829$	4.43108	0.153898	0.0769488	+	0.997035i	$0.475482\pi$
$830$	0	0
$831$	21.8223	0.757008
$832$	0	0
$833$	20.7536	0.719070
$834$	0	0
$835$	−31.4955	−1.08995
$836$	0	0
$837$	−36.8373	−1.27328
$838$	0	0
$839$	5.82003	0.200930	0.100465	−	0.994941i	$-0.467967\pi$
$839$	5.82003	0.200930	0.100465	+	0.994941i	$0.467967\pi$
$840$	0	0
$841$	50.3125	1.73491
$842$	0	0
$843$	10.8897	0.375061
$844$	0	0
$845$	−23.4144	−0.805481
$846$	0	0
$847$	25.4662	0.875030
$848$	0	0
$849$	5.22934	0.179470
$850$	0	0
$851$	0	0
$852$	0	0
$853$	−39.4660	−1.35129	−0.675645	−	0.737227i	$-0.736135\pi$
$853$	−39.4660	−1.35129	−0.675645	+	0.737227i	$0.736135\pi$
$854$	0	0
$855$	6.39006	0.218535
$856$	0	0
$857$	10.9397	0.373691	0.186846	−	0.982389i	$-0.440174\pi$
$857$	10.9397	0.373691	0.186846	+	0.982389i	$0.440174\pi$
$858$	0	0
$859$	−18.9069	−0.645094	−0.322547	−	0.946553i	$-0.604539\pi$
$859$	−18.9069	−0.645094	−0.322547	+	0.946553i	$0.604539\pi$
$860$	0	0
$861$	50.9467	1.73626
$862$	0	0
$863$	−14.2965	−0.486659	−0.243329	−	0.969944i	$-0.578240\pi$
$863$	−14.2965	−0.486659	−0.243329	+	0.969944i	$0.578240\pi$
$864$	0	0
$865$	−6.01603	−0.204551
$866$	0	0
$867$	−22.4128	−0.761179
$868$	0	0
$869$	−38.1206	−1.29315
$870$	0	0
$871$	2.86749	0.0971612
$872$	0	0
$873$	11.4585	0.387811
$874$	0	0
$875$	58.5351	1.97885
$876$	0	0
$877$	13.1813	0.445099	0.222550	−	0.974921i	$-0.428562\pi$
$877$	13.1813	0.445099	0.222550	+	0.974921i	$0.428562\pi$
$878$	0	0
$879$	12.6815	0.427736
$880$	0	0
$881$	−38.0168	−1.28082	−0.640409	−	0.768034i	$-0.721235\pi$
$881$	−38.0168	−1.28082	−0.640409	+	0.768034i	$0.721235\pi$
$882$	0	0
$883$	−31.0587	−1.04521	−0.522604	−	0.852575i	$-0.675040\pi$
$883$	−31.0587	−1.04521	−0.522604	+	0.852575i	$0.675040\pi$
$884$	0	0
$885$	−17.2022	−0.578245
$886$	0	0
$887$	10.0983	0.339067	0.169534	−	0.985524i	$-0.445774\pi$
$887$	10.0983	0.339067	0.169534	+	0.985524i	$0.445774\pi$
$888$	0	0
$889$	81.7822	2.74289
$890$	0	0
$891$	−13.0950	−0.438700
$892$	0	0
$893$	11.1196	0.372102
$894$	0	0
$895$	2.16867	0.0724906
$896$	0	0
$897$	0	0
$898$	0	0
$899$	58.0366	1.93563
$900$	0	0
$901$	−6.30387	−0.210012
$902$	0	0
$903$	27.6038	0.918596
$904$	0	0
$905$	26.1022	0.867665
$906$	0	0
$907$	10.6476	0.353546	0.176773	−	0.984252i	$-0.443434\pi$
$907$	10.6476	0.353546	0.176773	+	0.984252i	$0.443434\pi$
$908$	0	0
$909$	−13.1715	−0.436870
$910$	0	0
$911$	4.33299	0.143558	0.0717791	−	0.997421i	$-0.477132\pi$
$911$	4.33299	0.143558	0.0717791	+	0.997421i	$0.477132\pi$
$912$	0	0
$913$	−17.3183	−0.573151
$914$	0	0
$915$	−12.0736	−0.399140
$916$	0	0
$917$	11.2179	0.370447
$918$	0	0
$919$	−21.0208	−0.693411	−0.346705	−	0.937974i	$-0.612700\pi$
$919$	−21.0208	−0.693411	−0.346705	+	0.937974i	$0.612700\pi$
$920$	0	0
$921$	41.9212	1.38135
$922$	0	0
$923$	9.35485	0.307919
$924$	0	0
$925$	8.60717	0.283002
$926$	0	0
$927$	−1.30076	−0.0427224
$928$	0	0
$929$	−46.5849	−1.52840	−0.764200	−	0.644979i	$-0.776866\pi$
$929$	−46.5849	−1.52840	−0.764200	+	0.644979i	$0.776866\pi$
$930$	0	0
$931$	−58.9138	−1.93082
$932$	0	0
$933$	14.3322	0.469216
$934$	0	0
$935$	5.95130	0.194628
$936$	0	0
$937$	3.18673	0.104106	0.0520530	−	0.998644i	$-0.483424\pi$
$937$	3.18673	0.104106	0.0520530	+	0.998644i	$0.483424\pi$
$938$	0	0
$939$	−2.52312	−0.0823389
$940$	0	0
$941$	29.3290	0.956097	0.478049	−	0.878333i	$-0.341344\pi$
$941$	29.3290	0.956097	0.478049	+	0.878333i	$0.341344\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	55.9805	1.82105
$946$	0	0
$947$	−9.43418	−0.306570	−0.153285	−	0.988182i	$-0.548985\pi$
$947$	−9.43418	−0.306570	−0.153285	+	0.988182i	$0.548985\pi$
$948$	0	0
$949$	−13.1047	−0.425398
$950$	0	0
$951$	15.3745	0.498554
$952$	0	0
$953$	−19.6830	−0.637596	−0.318798	−	0.947823i	$-0.603279\pi$
$953$	−19.6830	−0.637596	−0.318798	+	0.947823i	$0.603279\pi$
$954$	0	0
$955$	44.7546	1.44823
$956$	0	0
$957$	30.9998	1.00208
$958$	0	0
$959$	101.457	3.27622
$960$	0	0
$961$	11.4680	0.369936
$962$	0	0
$963$	11.0991	0.357664
$964$	0	0
$965$	6.76766	0.217859
$966$	0	0
$967$	−15.0388	−0.483616	−0.241808	−	0.970324i	$-0.577740\pi$
$967$	−15.0388	−0.483616	−0.241808	+	0.970324i	$0.577740\pi$
$968$	0	0
$969$	−6.12701	−0.196828
$970$	0	0
$971$	5.75409	0.184658	0.0923288	−	0.995729i	$-0.470569\pi$
$971$	5.75409	0.184658	0.0923288	+	0.995729i	$0.470569\pi$
$972$	0	0
$973$	−0.561765	−0.0180094
$974$	0	0
$975$	1.56924	0.0502558
$976$	0	0
$977$	50.3570	1.61106	0.805531	−	0.592554i	$-0.201880\pi$
$977$	50.3570	1.61106	0.805531	+	0.592554i	$0.201880\pi$
$978$	0	0
$979$	−7.63063	−0.243876
$980$	0	0
$981$	−14.6771	−0.468604
$982$	0	0
$983$	9.40619	0.300011	0.150005	−	0.988685i	$-0.452071\pi$
$983$	9.40619	0.300011	0.150005	+	0.988685i	$0.452071\pi$
$984$	0	0
$985$	−9.61375	−0.306319
$986$	0	0
$987$	22.6893	0.722209
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−21.6672	−0.688281	−0.344140	−	0.938918i	$-0.611830\pi$
$991$	−21.6672	−0.688281	−0.344140	+	0.938918i	$0.611830\pi$
$992$	0	0
$993$	−15.3604	−0.487448
$994$	0	0
$995$	−18.7911	−0.595719
$996$	0	0
$997$	7.76043	0.245775	0.122888	−	0.992421i	$-0.460785\pi$
$997$	7.76043	0.245775	0.122888	+	0.992421i	$0.460785\pi$
$998$	0	0
$999$	53.4271	1.69036

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4232.2.a.ba.1.10		15
4.3	odd	2		8464.2.a.ch.1.6		15
23.15	odd	22		184.2.i.b.41.2	yes	30
23.20	odd	22		184.2.i.b.9.2	✓	30
23.22	odd	2		4232.2.a.bb.1.10		15
92.15	even	22		368.2.m.e.225.2		30
92.43	even	22		368.2.m.e.193.2		30
92.91	even	2		8464.2.a.cg.1.6		15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
184.2.i.b.9.2	✓	30	23.20	odd	22
184.2.i.b.41.2	yes	30	23.15	odd	22
368.2.m.e.193.2		30	92.43	even	22
368.2.m.e.225.2		30	92.15	even	22
4232.2.a.ba.1.10		15	1.1	even	1	trivial
4232.2.a.bb.1.10		15	23.22	odd	2
8464.2.a.cg.1.6		15	92.91	even	2
8464.2.a.ch.1.6		15	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:	\( N \)	\(=\)	\( 4232 = 2^{3} \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4232.a (trivial)

\(f(q)\)	\(=\)	\(q+1.44536 q^{3} +2.02221 q^{5} -4.89726 q^{7} -0.910921 q^{9} +2.40830 q^{11} -1.19222 q^{13} +2.92283 q^{15} +1.22201 q^{17} -3.46894 q^{19} -7.07833 q^{21} -0.910658 q^{25} -5.65271 q^{27} +8.90576 q^{29} +6.51675 q^{31} +3.48087 q^{33} -9.90331 q^{35} -9.45159 q^{37} -1.72319 q^{39} -7.19755 q^{41} -3.89976 q^{43} -1.84208 q^{45} -3.20546 q^{47} +16.9832 q^{49} +1.76625 q^{51} -5.15861 q^{53} +4.87009 q^{55} -5.01389 q^{57} -5.88545 q^{59} -4.13077 q^{61} +4.46102 q^{63} -2.41092 q^{65} -2.40517 q^{67} -7.84660 q^{71} +10.9919 q^{73} -1.31623 q^{75} -11.7941 q^{77} -15.8288 q^{79} -5.43746 q^{81} -7.19108 q^{83} +2.47116 q^{85} +12.8721 q^{87} -3.16847 q^{89} +5.83860 q^{91} +9.41908 q^{93} -7.01494 q^{95} -12.5790 q^{97} -2.19377 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + q^{3} - 10 q^{7} + 16 q^{9} - 23 q^{11} - 10 q^{15} - 29 q^{19} - q^{21} + 23 q^{25} + q^{27} - 2 q^{29} + 20 q^{31} - 18 q^{33} - 18 q^{35} - 24 q^{37} - 19 q^{39} + 9 q^{41} - 48 q^{43} - 4 q^{45}+ \cdots - 63 q^{99}+O(q^{100}) \) 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

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