LMFDB - Embedded newform 8325.2.a.ca.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8325,2,Mod(1,8325)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8325, 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("8325.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8325 = 3^{2} \cdot 5^{2} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8325.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:	$66.4754596827$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.528933.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 6x^{3} + 4x^{2} + 7x - 1 $ x^5 - 2x^4 - 6x^3 + 4x^2 + 7x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 2775)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$3.26942$ of defining polynomial
Character	$\chi$	$=$	8325.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.03520 q^{2} -0.928359 q^{4} -1.11914 q^{7} -3.03144 q^{8} -4.41971 q^{11} +3.45615 q^{13} -1.15853 q^{14} -1.28143 q^{16} -1.54385 q^{17} -1.52655 q^{19} -4.57529 q^{22} -2.69819 q^{23} +3.57781 q^{26} +1.03896 q^{28} -1.31411 q^{29} -2.78632 q^{31} +4.73634 q^{32} -1.59819 q^{34} +1.00000 q^{37} -1.58029 q^{38} -4.89192 q^{41} -0.770260 q^{43} +4.10308 q^{44} -2.79317 q^{46} +3.73758 q^{47} -5.74753 q^{49} -3.20855 q^{52} -3.73758 q^{53} +3.39260 q^{56} -1.36037 q^{58} +3.42095 q^{59} +2.86481 q^{61} -2.88440 q^{62} +7.46593 q^{64} +8.50365 q^{67} +1.43325 q^{68} +13.9777 q^{71} +0.0402008 q^{73} +1.03520 q^{74} +1.41719 q^{76} +4.94626 q^{77} -6.48635 q^{79} -5.06412 q^{82} -11.6567 q^{83} -0.797374 q^{86} +13.3981 q^{88} -5.31368 q^{89} -3.86791 q^{91} +2.50489 q^{92} +3.86915 q^{94} +14.3116 q^{97} -5.94985 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 2 q^{2} + 8 q^{4} - 3 q^{8} + q^{11} + 14 q^{13} + 10 q^{14} + 10 q^{16} - 11 q^{17} + 10 q^{19} - 14 q^{22} - 4 q^{23} - 13 q^{26} + 3 q^{28} - 5 q^{29} - 3 q^{31} - 23 q^{32} - 3 q^{34} + 5 q^{37}+ \cdots - 35 q^{98}+O(q^{100}) $ 5 * q - 2 * q^2 + 8 * q^4 - 3 * q^8 + q^11 + 14 * q^13 + 10 * q^14 + 10 * q^16 - 11 * q^17 + 10 * q^19 - 14 * q^22 - 4 * q^23 - 13 * q^26 + 3 * q^28 - 5 * q^29 - 3 * q^31 - 23 * q^32 - 3 * q^34 + 5 * q^37 - 31 * q^38 + 8 * q^41 + q^43 + 21 * q^44 - 13 * q^46 - q^47 + 21 * q^49 + 40 * q^52 + q^53 + 38 * q^56 + 40 * q^58 + 21 * q^59 + 26 * q^61 + 28 * q^62 + 45 * q^64 + 21 * q^67 + 17 * q^71 + 25 * q^73 - 2 * q^74 + 6 * q^76 - 16 * q^77 + 10 * q^79 - 28 * q^82 + 3 * q^83 + 45 * q^86 - 9 * q^88 - 33 * q^89 + 9 * q^91 + 18 * q^92 + 18 * q^94 + 16 * q^97 - 35 * q^98

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 $

$2$	1.03520	0.731997	0.365999	−	0.930615i	$-0.380727\pi$
$2$	1.03520	0.731997	0.365999	+	0.930615i	$0.380727\pi$
$3$	0	0
$3$	0	0
$4$	−0.928359	−0.464180
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−1.11914	−0.422994	−0.211497	−	0.977379i	$-0.567834\pi$
$7$	−1.11914	−0.422994	−0.211497	+	0.977379i	$0.567834\pi$
$8$	−3.03144	−1.07178
$9$	0	0
$10$	0	0
$11$	−4.41971	−1.33259	−0.666297	−	0.745687i	$-0.732122\pi$
$11$	−4.41971	−1.33259	−0.666297	+	0.745687i	$0.732122\pi$
$12$	0	0
$13$	3.45615	0.958564	0.479282	−	0.877661i	$-0.340897\pi$
$13$	3.45615	0.958564	0.479282	+	0.877661i	$0.340897\pi$
$14$	−1.15853	−0.309631
$15$	0	0
$16$	−1.28143	−0.320358
$17$	−1.54385	−0.374438	−0.187219	−	0.982318i	$-0.559947\pi$
$17$	−1.54385	−0.374438	−0.187219	+	0.982318i	$0.559947\pi$
$18$	0	0
$19$	−1.52655	−0.350215	−0.175108	−	0.984549i	$-0.556027\pi$
$19$	−1.52655	−0.350215	−0.175108	+	0.984549i	$0.556027\pi$
$20$	0	0
$21$	0	0
$22$	−4.57529	−0.975455
$23$	−2.69819	−0.562611	−0.281305	−	0.959618i	$-0.590767\pi$
$23$	−2.69819	−0.562611	−0.281305	+	0.959618i	$0.590767\pi$
$24$	0	0
$25$	0	0
$26$	3.57781	0.701666
$27$	0	0
$28$	1.03896	0.196345
$29$	−1.31411	−0.244024	−0.122012	−	0.992529i	$-0.538935\pi$
$29$	−1.31411	−0.244024	−0.122012	+	0.992529i	$0.538935\pi$
$30$	0	0
$31$	−2.78632	−0.500437	−0.250219	−	0.968189i	$-0.580503\pi$
$31$	−2.78632	−0.500437	−0.250219	+	0.968189i	$0.580503\pi$
$32$	4.73634	0.837275
$33$	0	0
$34$	−1.59819	−0.274088
$35$	0	0
$36$	0	0
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	−1.58029	−0.256357
$39$	0	0
$40$	0	0
$41$	−4.89192	−0.763989	−0.381995	−	0.924164i	$-0.624763\pi$
$41$	−4.89192	−0.763989	−0.381995	+	0.924164i	$0.624763\pi$
$42$	0	0
$43$	−0.770260	−0.117464	−0.0587318	−	0.998274i	$-0.518706\pi$
$43$	−0.770260	−0.117464	−0.0587318	+	0.998274i	$0.518706\pi$
$44$	4.10308	0.618563
$45$	0	0
$46$	−2.79317	−0.411830
$47$	3.73758	0.545182	0.272591	−	0.962130i	$-0.412119\pi$
$47$	3.73758	0.545182	0.272591	+	0.962130i	$0.412119\pi$
$48$	0	0
$49$	−5.74753	−0.821076
$50$	0	0
$51$	0	0
$52$	−3.20855	−0.444946
$53$	−3.73758	−0.513396	−0.256698	−	0.966492i	$-0.582635\pi$
$53$	−3.73758	−0.513396	−0.256698	+	0.966492i	$0.582635\pi$
$54$	0	0
$55$	0	0
$56$	3.39260	0.453355
$57$	0	0
$58$	−1.36037	−0.178625
$59$	3.42095	0.445370	0.222685	−	0.974890i	$-0.428518\pi$
$59$	3.42095	0.445370	0.222685	+	0.974890i	$0.428518\pi$
$60$	0	0
$61$	2.86481	0.366801	0.183400	−	0.983038i	$-0.441290\pi$
$61$	2.86481	0.366801	0.183400	+	0.983038i	$0.441290\pi$
$62$	−2.88440	−0.366319
$63$	0	0
$64$	7.46593	0.933241
$65$	0	0
$66$	0	0
$67$	8.50365	1.03889	0.519443	−	0.854505i	$-0.326139\pi$
$67$	8.50365	1.03889	0.519443	+	0.854505i	$0.326139\pi$
$68$	1.43325	0.173807
$69$	0	0
$70$	0	0
$71$	13.9777	1.65885	0.829424	−	0.558619i	$-0.188669\pi$
$71$	13.9777	1.65885	0.829424	+	0.558619i	$0.188669\pi$
$72$	0	0
$73$	0.0402008	0.00470514	0.00235257	−	0.999997i	$-0.499251\pi$
$73$	0.0402008	0.00470514	0.00235257	+	0.999997i	$0.499251\pi$
$74$	1.03520	0.120340
$75$	0	0
$76$	1.41719	0.162563
$77$	4.94626	0.563679
$78$	0	0
$79$	−6.48635	−0.729772	−0.364886	−	0.931052i	$-0.618892\pi$
$79$	−6.48635	−0.729772	−0.364886	+	0.931052i	$0.618892\pi$
$80$	0	0
$81$	0	0
$82$	−5.06412	−0.559238
$83$	−11.6567	−1.27949	−0.639747	−	0.768585i	$-0.720961\pi$
$83$	−11.6567	−1.27949	−0.639747	+	0.768585i	$0.720961\pi$
$84$	0	0
$85$	0	0
$86$	−0.797374	−0.0859831
$87$	0	0
$88$	13.3981	1.42824
$89$	−5.31368	−0.563249	−0.281624	−	0.959525i	$-0.590873\pi$
$89$	−5.31368	−0.563249	−0.281624	+	0.959525i	$0.590873\pi$
$90$	0	0
$91$	−3.86791	−0.405467
$92$	2.50489	0.261153
$93$	0	0
$94$	3.86915	0.399072
$95$	0	0
$96$	0	0
$97$	14.3116	1.45313	0.726563	−	0.687100i	$-0.241117\pi$
$97$	14.3116	1.45313	0.726563	+	0.687100i	$0.241117\pi$
$98$	−5.94985	−0.601026
$99$	0	0
$100$	0	0
$101$	12.2018	1.21413	0.607064	−	0.794653i	$-0.292347\pi$
$101$	12.2018	1.21413	0.607064	+	0.794653i	$0.292347\pi$
$102$	0	0
$103$	3.77168	0.371634	0.185817	−	0.982584i	$-0.440507\pi$
$103$	3.77168	0.371634	0.185817	+	0.982584i	$0.440507\pi$
$104$	−10.4771	−1.02737
$105$	0	0
$106$	−3.86915	−0.375805
$107$	−2.70238	−0.261249	−0.130624	−	0.991432i	$-0.541698\pi$
$107$	−2.70238	−0.261249	−0.130624	+	0.991432i	$0.541698\pi$
$108$	0	0
$109$	1.25853	0.120545	0.0602724	−	0.998182i	$-0.480803\pi$
$109$	1.25853	0.120545	0.0602724	+	0.998182i	$0.480803\pi$
$110$	0	0
$111$	0	0
$112$	1.43410	0.135509
$113$	−14.6838	−1.38134	−0.690670	−	0.723170i	$-0.742684\pi$
$113$	−14.6838	−1.38134	−0.690670	+	0.723170i	$0.742684\pi$
$114$	0	0
$115$	0	0
$116$	1.21997	0.113271
$117$	0	0
$118$	3.54137	0.326010
$119$	1.72778	0.158385
$120$	0	0
$121$	8.53385	0.775804
$122$	2.96565	0.268497
$123$	0	0
$124$	2.58670	0.232293
$125$	0	0
$126$	0	0
$127$	17.0214	1.51041	0.755204	−	0.655490i	$-0.227538\pi$
$127$	17.0214	1.51041	0.755204	+	0.655490i	$0.227538\pi$
$128$	−1.74395	−0.154145
$129$	0	0
$130$	0	0
$131$	−2.45572	−0.214557	−0.107279	−	0.994229i	$-0.534214\pi$
$131$	−2.45572	−0.214557	−0.107279	+	0.994229i	$0.534214\pi$
$132$	0	0
$133$	1.70842	0.148139
$134$	8.80298	0.760462
$135$	0	0
$136$	4.68008	0.401314
$137$	−3.89750	−0.332986	−0.166493	−	0.986043i	$-0.553244\pi$
$137$	−3.89750	−0.332986	−0.166493	+	0.986043i	$0.553244\pi$
$138$	0	0
$139$	5.79612	0.491620	0.245810	−	0.969318i	$-0.420946\pi$
$139$	5.79612	0.491620	0.245810	+	0.969318i	$0.420946\pi$
$140$	0	0
$141$	0	0
$142$	14.4697	1.21427
$143$	−15.2752	−1.27738
$144$	0	0
$145$	0	0
$146$	0.0416159	0.00344415
$147$	0	0
$148$	−0.928359	−0.0763107
$149$	−2.52279	−0.206675	−0.103338	−	0.994646i	$-0.532952\pi$
$149$	−2.52279	−0.206675	−0.103338	+	0.994646i	$0.532952\pi$
$150$	0	0
$151$	2.12351	0.172809	0.0864043	−	0.996260i	$-0.472462\pi$
$151$	2.12351	0.172809	0.0864043	+	0.996260i	$0.472462\pi$
$152$	4.62765	0.375352
$153$	0	0
$154$	5.12038	0.412612
$155$	0	0
$156$	0	0
$157$	6.00295	0.479088	0.239544	−	0.970886i	$-0.423002\pi$
$157$	6.00295	0.479088	0.239544	+	0.970886i	$0.423002\pi$
$158$	−6.71468	−0.534191
$159$	0	0
$160$	0	0
$161$	3.01964	0.237981
$162$	0	0
$163$	20.4393	1.60093	0.800465	−	0.599380i	$-0.204586\pi$
$163$	20.4393	1.60093	0.800465	+	0.599380i	$0.204586\pi$
$164$	4.54146	0.354628
$165$	0	0
$166$	−12.0671	−0.936587
$167$	21.0479	1.62873	0.814367	−	0.580350i	$-0.197084\pi$
$167$	21.0479	1.62873	0.814367	+	0.580350i	$0.197084\pi$
$168$	0	0
$169$	−1.05502	−0.0811553
$170$	0	0
$171$	0	0
$172$	0.715078	0.0545242
$173$	−3.05560	−0.232313	−0.116156	−	0.993231i	$-0.537057\pi$
$173$	−3.05560	−0.232313	−0.116156	+	0.993231i	$0.537057\pi$
$174$	0	0
$175$	0	0
$176$	5.66355	0.426906
$177$	0	0
$178$	−5.50072	−0.412297
$179$	11.7380	0.877340	0.438670	−	0.898648i	$-0.355450\pi$
$179$	11.7380	0.877340	0.438670	+	0.898648i	$0.355450\pi$
$180$	0	0
$181$	10.2545	0.762212	0.381106	−	0.924531i	$-0.375543\pi$
$181$	10.2545	0.762212	0.381106	+	0.924531i	$0.375543\pi$
$182$	−4.00406	−0.296801
$183$	0	0
$184$	8.17939	0.602993
$185$	0	0
$186$	0	0
$187$	6.82337	0.498974
$188$	−3.46982	−0.253063
$189$	0	0
$190$	0	0
$191$	17.3820	1.25772	0.628860	−	0.777519i	$-0.283522\pi$
$191$	17.3820	1.25772	0.628860	+	0.777519i	$0.283522\pi$
$192$	0	0
$193$	−20.9875	−1.51071	−0.755357	−	0.655313i	$-0.772537\pi$
$193$	−20.9875	−1.51071	−0.755357	+	0.655313i	$0.772537\pi$
$194$	14.8154	1.06368
$195$	0	0
$196$	5.33577	0.381127
$197$	−11.3766	−0.810546	−0.405273	−	0.914196i	$-0.632824\pi$
$197$	−11.3766	−0.810546	−0.405273	+	0.914196i	$0.632824\pi$
$198$	0	0
$199$	26.6589	1.88980	0.944899	−	0.327361i	$-0.106159\pi$
$199$	26.6589	1.88980	0.944899	+	0.327361i	$0.106159\pi$
$200$	0	0
$201$	0	0
$202$	12.6313	0.888739
$203$	1.47067	0.103221
$204$	0	0
$205$	0	0
$206$	3.90444	0.272035
$207$	0	0
$208$	−4.42882	−0.307083
$209$	6.74692	0.466694
$210$	0	0
$211$	4.21679	0.290295	0.145148	−	0.989410i	$-0.453634\pi$
$211$	4.21679	0.290295	0.145148	+	0.989410i	$0.453634\pi$
$212$	3.46982	0.238308
$213$	0	0
$214$	−2.79751	−0.191234
$215$	0	0
$216$	0	0
$217$	3.11827	0.211682
$218$	1.30283	0.0882385
$219$	0	0
$220$	0	0
$221$	−5.33577	−0.358923
$222$	0	0
$223$	−3.32515	−0.222669	−0.111334	−	0.993783i	$-0.535512\pi$
$223$	−3.32515	−0.222669	−0.111334	+	0.993783i	$0.535512\pi$
$224$	−5.30062	−0.354162
$225$	0	0
$226$	−15.2007	−1.01114
$227$	−3.11399	−0.206683	−0.103341	−	0.994646i	$-0.532953\pi$
$227$	−3.11399	−0.206683	−0.103341	+	0.994646i	$0.532953\pi$
$228$	0	0
$229$	7.74316	0.511683	0.255841	−	0.966719i	$-0.417648\pi$
$229$	7.74316	0.511683	0.255841	+	0.966719i	$0.417648\pi$
$230$	0	0
$231$	0	0
$232$	3.98364	0.261539
$233$	−3.33406	−0.218421	−0.109211	−	0.994019i	$-0.534832\pi$
$233$	−3.33406	−0.218421	−0.109211	+	0.994019i	$0.534832\pi$
$234$	0	0
$235$	0	0
$236$	−3.17587	−0.206732
$237$	0	0
$238$	1.78860	0.115938
$239$	−7.91336	−0.511873	−0.255936	−	0.966694i	$-0.582384\pi$
$239$	−7.91336	−0.511873	−0.255936	+	0.966694i	$0.582384\pi$
$240$	0	0
$241$	12.4081	0.799275	0.399638	−	0.916673i	$-0.369136\pi$
$241$	12.4081	0.799275	0.399638	+	0.916673i	$0.369136\pi$
$242$	8.83425	0.567887
$243$	0	0
$244$	−2.65957	−0.170261
$245$	0	0
$246$	0	0
$247$	−5.27600	−0.335704
$248$	8.44655	0.536357
$249$	0	0
$250$	0	0
$251$	−8.54489	−0.539349	−0.269674	−	0.962952i	$-0.586916\pi$
$251$	−8.54489	−0.539349	−0.269674	+	0.962952i	$0.586916\pi$
$252$	0	0
$253$	11.9252	0.749731
$254$	17.6206	1.10562
$255$	0	0
$256$	−16.7372	−1.04607
$257$	15.1499	0.945028	0.472514	−	0.881323i	$-0.343347\pi$
$257$	15.1499	0.945028	0.472514	+	0.881323i	$0.343347\pi$
$258$	0	0
$259$	−1.11914	−0.0695398
$260$	0	0
$261$	0	0
$262$	−2.54216	−0.157055
$263$	23.6047	1.45553	0.727764	−	0.685828i	$-0.240560\pi$
$263$	23.6047	1.45553	0.727764	+	0.685828i	$0.240560\pi$
$264$	0	0
$265$	0	0
$266$	1.76856	0.108437
$267$	0	0
$268$	−7.89444	−0.482230
$269$	3.92799	0.239494	0.119747	−	0.992804i	$-0.461792\pi$
$269$	3.92799	0.239494	0.119747	+	0.992804i	$0.461792\pi$
$270$	0	0
$271$	8.11060	0.492684	0.246342	−	0.969183i	$-0.420771\pi$
$271$	8.11060	0.492684	0.246342	+	0.969183i	$0.420771\pi$
$272$	1.97833	0.119954
$273$	0	0
$274$	−4.03469	−0.243745
$275$	0	0
$276$	0	0
$277$	5.48883	0.329792	0.164896	−	0.986311i	$-0.447271\pi$
$277$	5.48883	0.329792	0.164896	+	0.986311i	$0.447271\pi$
$278$	6.00015	0.359865
$279$	0	0
$280$	0	0
$281$	−0.976662	−0.0582628	−0.0291314	−	0.999576i	$-0.509274\pi$
$281$	−0.976662	−0.0582628	−0.0291314	+	0.999576i	$0.509274\pi$
$282$	0	0
$283$	−18.3901	−1.09318	−0.546590	−	0.837401i	$-0.684074\pi$
$283$	−18.3901	−1.09318	−0.546590	+	0.837401i	$0.684074\pi$
$284$	−12.9763	−0.770004
$285$	0	0
$286$	−15.8129	−0.935036
$287$	5.47473	0.323163
$288$	0	0
$289$	−14.6165	−0.859796
$290$	0	0
$291$	0	0
$292$	−0.0373208	−0.00218403
$293$	14.1756	0.828150	0.414075	−	0.910243i	$-0.364105\pi$
$293$	14.1756	0.828150	0.414075	+	0.910243i	$0.364105\pi$
$294$	0	0
$295$	0	0
$296$	−3.03144	−0.176199
$297$	0	0
$298$	−2.61160	−0.151286
$299$	−9.32534	−0.539298
$300$	0	0
$301$	0.862027	0.0496864
$302$	2.19826	0.126495
$303$	0	0
$304$	1.95617	0.112194
$305$	0	0
$306$	0	0
$307$	4.23790	0.241870	0.120935	−	0.992660i	$-0.461411\pi$
$307$	4.23790	0.241870	0.120935	+	0.992660i	$0.461411\pi$
$308$	−4.59191	−0.261648
$309$	0	0
$310$	0	0
$311$	−8.80673	−0.499384	−0.249692	−	0.968325i	$-0.580329\pi$
$311$	−8.80673	−0.499384	−0.249692	+	0.968325i	$0.580329\pi$
$312$	0	0
$313$	12.8888	0.728520	0.364260	−	0.931297i	$-0.381322\pi$
$313$	12.8888	0.728520	0.364260	+	0.931297i	$0.381322\pi$
$314$	6.21426	0.350691
$315$	0	0
$316$	6.02167	0.338745
$317$	−22.7425	−1.27735	−0.638673	−	0.769478i	$-0.720517\pi$
$317$	−22.7425	−1.27735	−0.638673	+	0.769478i	$0.720517\pi$
$318$	0	0
$319$	5.80798	0.325185
$320$	0	0
$321$	0	0
$322$	3.12594	0.174202
$323$	2.35677	0.131134
$324$	0	0
$325$	0	0
$326$	21.1588	1.17188
$327$	0	0
$328$	14.8296	0.818825
$329$	−4.18287	−0.230609
$330$	0	0
$331$	32.0593	1.76214	0.881070	−	0.472985i	$-0.156824\pi$
$331$	32.0593	1.76214	0.881070	+	0.472985i	$0.156824\pi$
$332$	10.8217	0.593915
$333$	0	0
$334$	21.7888	1.19223
$335$	0	0
$336$	0	0
$337$	29.9910	1.63372	0.816858	−	0.576839i	$-0.195714\pi$
$337$	29.9910	1.63372	0.816858	+	0.576839i	$0.195714\pi$
$338$	−1.09216	−0.0594055
$339$	0	0
$340$	0	0
$341$	12.3147	0.666879
$342$	0	0
$343$	14.2662	0.770304
$344$	2.33500	0.125895
$345$	0	0
$346$	−3.16316	−0.170052
$347$	19.1225	1.02655	0.513276	−	0.858224i	$-0.328432\pi$
$347$	19.1225	1.02655	0.513276	+	0.858224i	$0.328432\pi$
$348$	0	0
$349$	−1.56638	−0.0838464	−0.0419232	−	0.999121i	$-0.513348\pi$
$349$	−1.56638	−0.0838464	−0.0419232	+	0.999121i	$0.513348\pi$
$350$	0	0
$351$	0	0
$352$	−20.9333	−1.11575
$353$	−4.62110	−0.245956	−0.122978	−	0.992409i	$-0.539244\pi$
$353$	−4.62110	−0.245956	−0.122978	+	0.992409i	$0.539244\pi$
$354$	0	0
$355$	0	0
$356$	4.93300	0.261449
$357$	0	0
$358$	12.1512	0.642211
$359$	19.9663	1.05378	0.526890	−	0.849934i	$-0.323358\pi$
$359$	19.9663	1.05378	0.526890	+	0.849934i	$0.323358\pi$
$360$	0	0
$361$	−16.6696	−0.877349
$362$	10.6155	0.557937
$363$	0	0
$364$	3.59081	0.188210
$365$	0	0
$366$	0	0
$367$	1.70108	0.0887958	0.0443979	−	0.999014i	$-0.485863\pi$
$367$	1.70108	0.0887958	0.0443979	+	0.999014i	$0.485863\pi$
$368$	3.45754	0.180237
$369$	0	0
$370$	0	0
$371$	4.18287	0.217164
$372$	0	0
$373$	−4.60844	−0.238616	−0.119308	−	0.992857i	$-0.538068\pi$
$373$	−4.60844	−0.238616	−0.119308	+	0.992857i	$0.538068\pi$
$374$	7.06355	0.365248
$375$	0	0
$376$	−11.3303	−0.584313
$377$	−4.54176	−0.233913
$378$	0	0
$379$	−6.09375	−0.313015	−0.156508	−	0.987677i	$-0.550024\pi$
$379$	−6.09375	−0.313015	−0.156508	+	0.987677i	$0.550024\pi$
$380$	0	0
$381$	0	0
$382$	17.9939	0.920648
$383$	−10.6299	−0.543164	−0.271582	−	0.962415i	$-0.587547\pi$
$383$	−10.6299	−0.543164	−0.271582	+	0.962415i	$0.587547\pi$
$384$	0	0
$385$	0	0
$386$	−21.7263	−1.10584
$387$	0	0
$388$	−13.2863	−0.674512
$389$	−31.2014	−1.58197	−0.790987	−	0.611833i	$-0.790433\pi$
$389$	−31.2014	−1.58197	−0.790987	+	0.611833i	$0.790433\pi$
$390$	0	0
$391$	4.16559	0.210663
$392$	17.4233	0.880009
$393$	0	0
$394$	−11.7770	−0.593318
$395$	0	0
$396$	0	0
$397$	−14.0978	−0.707546	−0.353773	−	0.935331i	$-0.615101\pi$
$397$	−14.0978	−0.707546	−0.353773	+	0.935331i	$0.615101\pi$
$398$	27.5973	1.38333
$399$	0	0
$400$	0	0
$401$	17.4354	0.870682	0.435341	−	0.900266i	$-0.356628\pi$
$401$	17.4354	0.870682	0.435341	+	0.900266i	$0.356628\pi$
$402$	0	0
$403$	−9.62993	−0.479701
$404$	−11.3277	−0.563574
$405$	0	0
$406$	1.52244	0.0755573
$407$	−4.41971	−0.219077
$408$	0	0
$409$	−7.59745	−0.375670	−0.187835	−	0.982201i	$-0.560147\pi$
$409$	−7.59745	−0.375670	−0.187835	+	0.982201i	$0.560147\pi$
$410$	0	0
$411$	0	0
$412$	−3.50147	−0.172505
$413$	−3.82851	−0.188389
$414$	0	0
$415$	0	0
$416$	16.3695	0.802582
$417$	0	0
$418$	6.98442	0.341619
$419$	35.1905	1.71917	0.859583	−	0.510996i	$-0.170723\pi$
$419$	35.1905	1.71917	0.859583	+	0.510996i	$0.170723\pi$
$420$	0	0
$421$	24.1940	1.17914	0.589571	−	0.807716i	$-0.299297\pi$
$421$	24.1940	1.17914	0.589571	+	0.807716i	$0.299297\pi$
$422$	4.36522	0.212496
$423$	0	0
$424$	11.3303	0.550246
$425$	0	0
$426$	0	0
$427$	−3.20611	−0.155155
$428$	2.50878	0.121266
$429$	0	0
$430$	0	0
$431$	12.2546	0.590282	0.295141	−	0.955454i	$-0.404633\pi$
$431$	12.2546	0.590282	0.295141	+	0.955454i	$0.404633\pi$
$432$	0	0
$433$	−5.67574	−0.272759	−0.136379	−	0.990657i	$-0.543547\pi$
$433$	−5.67574	−0.272759	−0.136379	+	0.990657i	$0.543547\pi$
$434$	3.22804	0.154951
$435$	0	0
$436$	−1.16836	−0.0559545
$437$	4.11892	0.197035
$438$	0	0
$439$	29.2305	1.39510	0.697549	−	0.716537i	$-0.254274\pi$
$439$	29.2305	1.39510	0.697549	+	0.716537i	$0.254274\pi$
$440$	0	0
$441$	0	0
$442$	−5.52360	−0.262731
$443$	−15.9668	−0.758607	−0.379304	−	0.925272i	$-0.623836\pi$
$443$	−15.9668	−0.758607	−0.379304	+	0.925272i	$0.623836\pi$
$444$	0	0
$445$	0	0
$446$	−3.44220	−0.162993
$447$	0	0
$448$	−8.35540	−0.394755
$449$	13.3288	0.629023	0.314511	−	0.949254i	$-0.398159\pi$
$449$	13.3288	0.629023	0.314511	+	0.949254i	$0.398159\pi$
$450$	0	0
$451$	21.6209	1.01809
$452$	13.6319	0.641190
$453$	0	0
$454$	−3.22360	−0.151291
$455$	0	0
$456$	0	0
$457$	−13.1682	−0.615980	−0.307990	−	0.951390i	$-0.599656\pi$
$457$	−13.1682	−0.615980	−0.307990	+	0.951390i	$0.599656\pi$
$458$	8.01573	0.374550
$459$	0	0
$460$	0	0
$461$	−1.08658	−0.0506072	−0.0253036	−	0.999680i	$-0.508055\pi$
$461$	−1.08658	−0.0506072	−0.0253036	+	0.999680i	$0.508055\pi$
$462$	0	0
$463$	−25.3893	−1.17994	−0.589970	−	0.807425i	$-0.700861\pi$
$463$	−25.3893	−1.17994	−0.589970	+	0.807425i	$0.700861\pi$
$464$	1.68394	0.0781749
$465$	0	0
$466$	−3.45142	−0.159884
$467$	−24.2164	−1.12060	−0.560300	−	0.828290i	$-0.689314\pi$
$467$	−24.2164	−1.12060	−0.560300	+	0.828290i	$0.689314\pi$
$468$	0	0
$469$	−9.51675	−0.439443
$470$	0	0
$471$	0	0
$472$	−10.3704	−0.477337
$473$	3.40433	0.156531
$474$	0	0
$475$	0	0
$476$	−1.60400	−0.0735192
$477$	0	0
$478$	−8.19192	−0.374690
$479$	8.36493	0.382204	0.191102	−	0.981570i	$-0.438794\pi$
$479$	8.36493	0.382204	0.191102	+	0.981570i	$0.438794\pi$
$480$	0	0
$481$	3.45615	0.157587
$482$	12.8449	0.585068
$483$	0	0
$484$	−7.92248	−0.360113
$485$	0	0
$486$	0	0
$487$	13.7271	0.622032	0.311016	−	0.950405i	$-0.399331\pi$
$487$	13.7271	0.622032	0.311016	+	0.950405i	$0.399331\pi$
$488$	−8.68449	−0.393128
$489$	0	0
$490$	0	0
$491$	−33.0595	−1.49195	−0.745976	−	0.665972i	$-0.768017\pi$
$491$	−33.0595	−1.49195	−0.745976	+	0.665972i	$0.768017\pi$
$492$	0	0
$493$	2.02879	0.0913719
$494$	−5.46172	−0.245734
$495$	0	0
$496$	3.57047	0.160319
$497$	−15.6430	−0.701683
$498$	0	0
$499$	−20.3734	−0.912041	−0.456020	−	0.889969i	$-0.650726\pi$
$499$	−20.3734	−0.912041	−0.456020	+	0.889969i	$0.650726\pi$
$500$	0	0
$501$	0	0
$502$	−8.84568	−0.394802
$503$	−4.30818	−0.192092	−0.0960461	−	0.995377i	$-0.530620\pi$
$503$	−4.30818	−0.192092	−0.0960461	+	0.995377i	$0.530620\pi$
$504$	0	0
$505$	0	0
$506$	12.3450	0.548801
$507$	0	0
$508$	−15.8020	−0.701101
$509$	10.0595	0.445881	0.222941	−	0.974832i	$-0.428434\pi$
$509$	10.0595	0.445881	0.222941	+	0.974832i	$0.428434\pi$
$510$	0	0
$511$	−0.0449902	−0.00199025
$512$	−13.8384	−0.611579
$513$	0	0
$514$	15.6832	0.691758
$515$	0	0
$516$	0	0
$517$	−16.5190	−0.726506
$518$	−1.15853	−0.0509030
$519$	0	0
$520$	0	0
$521$	−30.1033	−1.31885	−0.659425	−	0.751770i	$-0.729200\pi$
$521$	−30.1033	−1.31885	−0.659425	+	0.751770i	$0.729200\pi$
$522$	0	0
$523$	−41.9861	−1.83592	−0.917962	−	0.396669i	$-0.870166\pi$
$523$	−41.9861	−1.83592	−0.917962	+	0.396669i	$0.870166\pi$
$524$	2.27979	0.0995931
$525$	0	0
$526$	24.4356	1.06544
$527$	4.30165	0.187383
$528$	0	0
$529$	−15.7198	−0.683469
$530$	0	0
$531$	0	0
$532$	−1.58603	−0.0687631
$533$	−16.9072	−0.732333
$534$	0	0
$535$	0	0
$536$	−25.7783	−1.11345
$537$	0	0
$538$	4.06625	0.175309
$539$	25.4024	1.09416
$540$	0	0
$541$	42.3908	1.82252	0.911262	−	0.411828i	$-0.135109\pi$
$541$	42.3908	1.82252	0.911262	+	0.411828i	$0.135109\pi$
$542$	8.39610	0.360643
$543$	0	0
$544$	−7.31220	−0.313508
$545$	0	0
$546$	0	0
$547$	5.04402	0.215667	0.107833	−	0.994169i	$-0.465609\pi$
$547$	5.04402	0.215667	0.107833	+	0.994169i	$0.465609\pi$
$548$	3.61828	0.154565
$549$	0	0
$550$	0	0
$551$	2.00606	0.0854609
$552$	0	0
$553$	7.25912	0.308689
$554$	5.68204	0.241407
$555$	0	0
$556$	−5.38088	−0.228200
$557$	−11.9490	−0.506296	−0.253148	−	0.967428i	$-0.581466\pi$
$557$	−11.9490	−0.506296	−0.253148	+	0.967428i	$0.581466\pi$
$558$	0	0
$559$	−2.66214	−0.112596
$560$	0	0
$561$	0	0
$562$	−1.01104	−0.0426482
$563$	6.06416	0.255574	0.127787	−	0.991802i	$-0.459213\pi$
$563$	6.06416	0.255574	0.127787	+	0.991802i	$0.459213\pi$
$564$	0	0
$565$	0	0
$566$	−19.0375	−0.800204
$567$	0	0
$568$	−42.3726	−1.77791
$569$	31.8955	1.33713	0.668564	−	0.743655i	$-0.266909\pi$
$569$	31.8955	1.33713	0.668564	+	0.743655i	$0.266909\pi$
$570$	0	0
$571$	40.0882	1.67764	0.838820	−	0.544408i	$-0.183246\pi$
$571$	40.0882	1.67764	0.838820	+	0.544408i	$0.183246\pi$
$572$	14.1809	0.592932
$573$	0	0
$574$	5.66744	0.236555
$575$	0	0
$576$	0	0
$577$	−39.4370	−1.64178	−0.820892	−	0.571084i	$-0.806523\pi$
$577$	−39.4370	−1.64178	−0.820892	+	0.571084i	$0.806523\pi$
$578$	−15.1310	−0.629368
$579$	0	0
$580$	0	0
$581$	13.0455	0.541219
$582$	0	0
$583$	16.5190	0.684148
$584$	−0.121866	−0.00504286
$585$	0	0
$586$	14.6746	0.606204
$587$	18.3058	0.755561	0.377781	−	0.925895i	$-0.376687\pi$
$587$	18.3058	0.755561	0.377781	+	0.925895i	$0.376687\pi$
$588$	0	0
$589$	4.25346	0.175261
$590$	0	0
$591$	0	0
$592$	−1.28143	−0.0526665
$593$	−42.2895	−1.73662	−0.868311	−	0.496020i	$-0.834794\pi$
$593$	−42.2895	−1.73662	−0.868311	+	0.496020i	$0.834794\pi$
$594$	0	0
$595$	0	0
$596$	2.34206	0.0959344
$597$	0	0
$598$	−9.65360	−0.394765
$599$	38.3964	1.56883	0.784417	−	0.620234i	$-0.212963\pi$
$599$	38.3964	1.56883	0.784417	+	0.620234i	$0.212963\pi$
$600$	0	0
$601$	−22.0508	−0.899470	−0.449735	−	0.893162i	$-0.648482\pi$
$601$	−22.0508	−0.899470	−0.449735	+	0.893162i	$0.648482\pi$
$602$	0.892371	0.0363703
$603$	0	0
$604$	−1.97138	−0.0802142
$605$	0	0
$606$	0	0
$607$	−28.8824	−1.17230	−0.586149	−	0.810203i	$-0.699357\pi$
$607$	−28.8824	−1.17230	−0.586149	+	0.810203i	$0.699357\pi$
$608$	−7.23028	−0.293226
$609$	0	0
$610$	0	0
$611$	12.9176	0.522592
$612$	0	0
$613$	−2.02571	−0.0818175	−0.0409088	−	0.999163i	$-0.513025\pi$
$613$	−2.02571	−0.0818175	−0.0409088	+	0.999163i	$0.513025\pi$
$614$	4.38708	0.177048
$615$	0	0
$616$	−14.9943	−0.604138
$617$	−23.6766	−0.953183	−0.476591	−	0.879125i	$-0.658128\pi$
$617$	−23.6766	−0.953183	−0.476591	+	0.879125i	$0.658128\pi$
$618$	0	0
$619$	−5.30766	−0.213333	−0.106666	−	0.994295i	$-0.534018\pi$
$619$	−5.30766	−0.213333	−0.106666	+	0.994295i	$0.534018\pi$
$620$	0	0
$621$	0	0
$622$	−9.11673	−0.365548
$623$	5.94673	0.238251
$624$	0	0
$625$	0	0
$626$	13.3425	0.533275
$627$	0	0
$628$	−5.57290	−0.222383
$629$	−1.54385	−0.0615573
$630$	0	0
$631$	45.6685	1.81803	0.909017	−	0.416760i	$-0.136834\pi$
$631$	45.6685	1.81803	0.909017	+	0.416760i	$0.136834\pi$
$632$	19.6630	0.782152
$633$	0	0
$634$	−23.5431	−0.935015
$635$	0	0
$636$	0	0
$637$	−19.8643	−0.787054
$638$	6.01243	0.238034
$639$	0	0
$640$	0	0
$641$	18.4272	0.727832	0.363916	−	0.931432i	$-0.381440\pi$
$641$	18.4272	0.727832	0.363916	+	0.931432i	$0.381440\pi$
$642$	0	0
$643$	−3.86585	−0.152454	−0.0762272	−	0.997090i	$-0.524287\pi$
$643$	−3.86585	−0.152454	−0.0762272	+	0.997090i	$0.524287\pi$
$644$	−2.80331	−0.110466
$645$	0	0
$646$	2.43973	0.0959898
$647$	1.12134	0.0440846	0.0220423	−	0.999757i	$-0.492983\pi$
$647$	1.12134	0.0440846	0.0220423	+	0.999757i	$0.492983\pi$
$648$	0	0
$649$	−15.1196	−0.593497
$650$	0	0
$651$	0	0
$652$	−18.9750	−0.743119
$653$	14.0497	0.549805	0.274903	−	0.961472i	$-0.411354\pi$
$653$	14.0497	0.549805	0.274903	+	0.961472i	$0.411354\pi$
$654$	0	0
$655$	0	0
$656$	6.26865	0.244750
$657$	0	0
$658$	−4.33011	−0.168805
$659$	32.2839	1.25760	0.628801	−	0.777566i	$-0.283546\pi$
$659$	32.2839	1.25760	0.628801	+	0.777566i	$0.283546\pi$
$660$	0	0
$661$	−45.7280	−1.77861	−0.889306	−	0.457312i	$-0.848812\pi$
$661$	−45.7280	−1.77861	−0.889306	+	0.457312i	$0.848812\pi$
$662$	33.1879	1.28988
$663$	0	0
$664$	35.3367	1.37133
$665$	0	0
$666$	0	0
$667$	3.54571	0.137291
$668$	−19.5400	−0.756025
$669$	0	0
$670$	0	0
$671$	−12.6616	−0.488796
$672$	0	0
$673$	6.76356	0.260716	0.130358	−	0.991467i	$-0.458387\pi$
$673$	6.76356	0.260716	0.130358	+	0.991467i	$0.458387\pi$
$674$	31.0467	1.19588
$675$	0	0
$676$	0.979436	0.0376706
$677$	3.64779	0.140196	0.0700981	−	0.997540i	$-0.477669\pi$
$677$	3.64779	0.140196	0.0700981	+	0.997540i	$0.477669\pi$
$678$	0	0
$679$	−16.0167	−0.614664
$680$	0	0
$681$	0	0
$682$	12.7482	0.488154
$683$	9.05590	0.346514	0.173257	−	0.984877i	$-0.444571\pi$
$683$	9.05590	0.346514	0.173257	+	0.984877i	$0.444571\pi$
$684$	0	0
$685$	0	0
$686$	14.7684	0.563861
$687$	0	0
$688$	0.987035	0.0376303
$689$	−12.9176	−0.492123
$690$	0	0
$691$	−19.7616	−0.751765	−0.375883	−	0.926667i	$-0.622660\pi$
$691$	−19.7616	−0.751765	−0.375883	+	0.926667i	$0.622660\pi$
$692$	2.83669	0.107835
$693$	0	0
$694$	19.7957	0.751433
$695$	0	0
$696$	0	0
$697$	7.55238	0.286067
$698$	−1.62152	−0.0613754
$699$	0	0
$700$	0	0
$701$	−12.3724	−0.467299	−0.233650	−	0.972321i	$-0.575067\pi$
$701$	−12.3724	−0.467299	−0.233650	+	0.972321i	$0.575067\pi$
$702$	0	0
$703$	−1.52655	−0.0575750
$704$	−32.9972	−1.24363
$705$	0	0
$706$	−4.78376	−0.180039
$707$	−13.6555	−0.513569
$708$	0	0
$709$	−33.1915	−1.24653	−0.623266	−	0.782010i	$-0.714194\pi$
$709$	−33.1915	−1.24653	−0.623266	+	0.782010i	$0.714194\pi$
$710$	0	0
$711$	0	0
$712$	16.1081	0.603676
$713$	7.51800	0.281551
$714$	0	0
$715$	0	0
$716$	−10.8971	−0.407244
$717$	0	0
$718$	20.6691	0.771364
$719$	52.0978	1.94292	0.971460	−	0.237205i	$-0.0762314\pi$
$719$	52.0978	1.94292	0.971460	+	0.237205i	$0.0762314\pi$
$720$	0	0
$721$	−4.22102	−0.157199
$722$	−17.2564	−0.642217
$723$	0	0
$724$	−9.51987	−0.353803
$725$	0	0
$726$	0	0
$727$	−32.8479	−1.21826	−0.609131	−	0.793069i	$-0.708482\pi$
$727$	−32.8479	−1.21826	−0.609131	+	0.793069i	$0.708482\pi$
$728$	11.7253	0.434570
$729$	0	0
$730$	0	0
$731$	1.18917	0.0439829
$732$	0	0
$733$	−43.3499	−1.60116	−0.800582	−	0.599223i	$-0.795476\pi$
$733$	−43.3499	−1.60116	−0.800582	+	0.599223i	$0.795476\pi$
$734$	1.76096	0.0649983
$735$	0	0
$736$	−12.7795	−0.471060
$737$	−37.5837	−1.38441
$738$	0	0
$739$	19.4215	0.714433	0.357216	−	0.934022i	$-0.383726\pi$
$739$	19.4215	0.714433	0.357216	+	0.934022i	$0.383726\pi$
$740$	0	0
$741$	0	0
$742$	4.33011	0.158963
$743$	19.7858	0.725872	0.362936	−	0.931814i	$-0.381774\pi$
$743$	19.7858	0.725872	0.362936	+	0.931814i	$0.381774\pi$
$744$	0	0
$745$	0	0
$746$	−4.77066	−0.174666
$747$	0	0
$748$	−6.33454	−0.231614
$749$	3.02434	0.110507
$750$	0	0
$751$	31.9047	1.16422	0.582109	−	0.813110i	$-0.302228\pi$
$751$	31.9047	1.16422	0.582109	+	0.813110i	$0.302228\pi$
$752$	−4.78945	−0.174653
$753$	0	0
$754$	−4.70163	−0.171223
$755$	0	0
$756$	0	0
$757$	12.9205	0.469602	0.234801	−	0.972043i	$-0.424556\pi$
$757$	12.9205	0.469602	0.234801	+	0.972043i	$0.424556\pi$
$758$	−6.30826	−0.229126
$759$	0	0
$760$	0	0
$761$	30.3662	1.10077	0.550386	−	0.834910i	$-0.314481\pi$
$761$	30.3662	1.10077	0.550386	+	0.834910i	$0.314481\pi$
$762$	0	0
$763$	−1.40846	−0.0509898
$764$	−16.1368	−0.583808
$765$	0	0
$766$	−11.0041	−0.397595
$767$	11.8233	0.426915
$768$	0	0
$769$	30.1604	1.08761	0.543806	−	0.839211i	$-0.316983\pi$
$769$	30.1604	1.08761	0.543806	+	0.839211i	$0.316983\pi$
$770$	0	0
$771$	0	0
$772$	19.4840	0.701243
$773$	−36.0839	−1.29785	−0.648923	−	0.760854i	$-0.724780\pi$
$773$	−36.0839	−1.29785	−0.648923	+	0.760854i	$0.724780\pi$
$774$	0	0
$775$	0	0
$776$	−43.3848	−1.55743
$777$	0	0
$778$	−32.2997	−1.15800
$779$	7.46777	0.267561
$780$	0	0
$781$	−61.7774	−2.21057
$782$	4.31222	0.154205
$783$	0	0
$784$	7.36506	0.263038
$785$	0	0
$786$	0	0
$787$	4.61177	0.164392	0.0821960	−	0.996616i	$-0.473807\pi$
$787$	4.61177	0.164392	0.0821960	+	0.996616i	$0.473807\pi$
$788$	10.5615	0.376239
$789$	0	0
$790$	0	0
$791$	16.4332	0.584299
$792$	0	0
$793$	9.90120	0.351602
$794$	−14.5940	−0.517922
$795$	0	0
$796$	−24.7490	−0.877206
$797$	−24.7200	−0.875627	−0.437814	−	0.899066i	$-0.644247\pi$
$797$	−24.7200	−0.875627	−0.437814	+	0.899066i	$0.644247\pi$
$798$	0	0
$799$	−5.77026	−0.204137
$800$	0	0
$801$	0	0
$802$	18.0491	0.637337
$803$	−0.177676	−0.00627004
$804$	0	0
$805$	0	0
$806$	−9.96891	−0.351140
$807$	0	0
$808$	−36.9891	−1.30127
$809$	15.0367	0.528663	0.264331	−	0.964432i	$-0.414849\pi$
$809$	15.0367	0.528663	0.264331	+	0.964432i	$0.414849\pi$
$810$	0	0
$811$	−5.05087	−0.177360	−0.0886800	−	0.996060i	$-0.528265\pi$
$811$	−5.05087	−0.177360	−0.0886800	+	0.996060i	$0.528265\pi$
$812$	−1.36531	−0.0479130
$813$	0	0
$814$	−4.57529	−0.160364
$815$	0	0
$816$	0	0
$817$	1.17584	0.0411375
$818$	−7.86489	−0.274989
$819$	0	0
$820$	0	0
$821$	35.5461	1.24057	0.620285	−	0.784377i	$-0.287017\pi$
$821$	35.5461	1.24057	0.620285	+	0.784377i	$0.287017\pi$
$822$	0	0
$823$	−53.3551	−1.85984	−0.929921	−	0.367760i	$-0.880125\pi$
$823$	−53.3551	−1.85984	−0.929921	+	0.367760i	$0.880125\pi$
$824$	−11.4336	−0.398309
$825$	0	0
$826$	−3.96328	−0.137900
$827$	35.6694	1.24035	0.620174	−	0.784464i	$-0.287062\pi$
$827$	35.6694	1.24035	0.620174	+	0.784464i	$0.287062\pi$
$828$	0	0
$829$	9.61754	0.334031	0.167016	−	0.985954i	$-0.446587\pi$
$829$	9.61754	0.334031	0.167016	+	0.985954i	$0.446587\pi$
$830$	0	0
$831$	0	0
$832$	25.8034	0.894571
$833$	8.87332	0.307442
$834$	0	0
$835$	0	0
$836$	−6.26357	−0.216630
$837$	0	0
$838$	36.4292	1.25843
$839$	1.25253	0.0432421	0.0216210	−	0.999766i	$-0.493117\pi$
$839$	1.25253	0.0432421	0.0216210	+	0.999766i	$0.493117\pi$
$840$	0	0
$841$	−27.2731	−0.940452
$842$	25.0456	0.863129
$843$	0	0
$844$	−3.91469	−0.134749
$845$	0	0
$846$	0	0
$847$	−9.55055	−0.328161
$848$	4.78945	0.164470
$849$	0	0
$850$	0	0
$851$	−2.69819	−0.0924927
$852$	0	0
$853$	16.4029	0.561625	0.280812	−	0.959763i	$-0.409396\pi$
$853$	16.4029	0.561625	0.280812	+	0.959763i	$0.409396\pi$
$854$	−3.31897	−0.113573
$855$	0	0
$856$	8.19210	0.280000
$857$	−30.3328	−1.03615	−0.518074	−	0.855336i	$-0.673351\pi$
$857$	−30.3328	−1.03615	−0.518074	+	0.855336i	$0.673351\pi$
$858$	0	0
$859$	0.581991	0.0198573	0.00992864	−	0.999951i	$-0.496840\pi$
$859$	0.581991	0.0198573	0.00992864	+	0.999951i	$0.496840\pi$
$860$	0	0
$861$	0	0
$862$	12.6859	0.432085
$863$	49.9890	1.70164	0.850822	−	0.525454i	$-0.176104\pi$
$863$	49.9890	1.70164	0.850822	+	0.525454i	$0.176104\pi$
$864$	0	0
$865$	0	0
$866$	−5.87553	−0.199659
$867$	0	0
$868$	−2.89488	−0.0982585
$869$	28.6678	0.972489
$870$	0	0
$871$	29.3899	0.995839
$872$	−3.81514	−0.129197
$873$	0	0
$874$	4.26391	0.144229
$875$	0	0
$876$	0	0
$877$	−15.5113	−0.523781	−0.261890	−	0.965098i	$-0.584346\pi$
$877$	−15.5113	−0.523781	−0.261890	+	0.965098i	$0.584346\pi$
$878$	30.2595	1.02121
$879$	0	0
$880$	0	0
$881$	−31.3544	−1.05636	−0.528178	−	0.849134i	$-0.677125\pi$
$881$	−31.3544	−1.05636	−0.528178	+	0.849134i	$0.677125\pi$
$882$	0	0
$883$	36.7810	1.23778	0.618890	−	0.785478i	$-0.287583\pi$
$883$	36.7810	1.23778	0.618890	+	0.785478i	$0.287583\pi$
$884$	4.95352	0.166605
$885$	0	0
$886$	−16.5289	−0.555299
$887$	39.8443	1.33784	0.668920	−	0.743335i	$-0.266757\pi$
$887$	39.8443	1.33784	0.668920	+	0.743335i	$0.266757\pi$
$888$	0	0
$889$	−19.0493	−0.638894
$890$	0	0
$891$	0	0
$892$	3.08693	0.103358
$893$	−5.70562	−0.190931
$894$	0	0
$895$	0	0
$896$	1.95172	0.0652025
$897$	0	0
$898$	13.7979	0.460443
$899$	3.66152	0.122119
$900$	0	0
$901$	5.77026	0.192235
$902$	22.3819	0.745237
$903$	0	0
$904$	44.5132	1.48049
$905$	0	0
$906$	0	0
$907$	7.19614	0.238944	0.119472	−	0.992838i	$-0.461880\pi$
$907$	7.19614	0.238944	0.119472	+	0.992838i	$0.461880\pi$
$908$	2.89090	0.0959379
$909$	0	0
$910$	0	0
$911$	−22.9749	−0.761193	−0.380596	−	0.924741i	$-0.624281\pi$
$911$	−22.9749	−0.761193	−0.380596	+	0.924741i	$0.624281\pi$
$912$	0	0
$913$	51.5195	1.70505
$914$	−13.6317	−0.450896
$915$	0	0
$916$	−7.18844	−0.237513
$917$	2.74829	0.0907564
$918$	0	0
$919$	12.4363	0.410237	0.205118	−	0.978737i	$-0.434242\pi$
$919$	12.4363	0.410237	0.205118	+	0.978737i	$0.434242\pi$
$920$	0	0
$921$	0	0
$922$	−1.12483	−0.0370443
$923$	48.3091	1.59011
$924$	0	0
$925$	0	0
$926$	−26.2830	−0.863714
$927$	0	0
$928$	−6.22407	−0.204315
$929$	−1.24689	−0.0409091	−0.0204545	−	0.999791i	$-0.506511\pi$
$929$	−1.24689	−0.0409091	−0.0204545	+	0.999791i	$0.506511\pi$
$930$	0	0
$931$	8.77391	0.287553
$932$	3.09521	0.101387
$933$	0	0
$934$	−25.0688	−0.820276
$935$	0	0
$936$	0	0
$937$	16.7786	0.548135	0.274067	−	0.961711i	$-0.411631\pi$
$937$	16.7786	0.548135	0.274067	+	0.961711i	$0.411631\pi$
$938$	−9.85175	−0.321671
$939$	0	0
$940$	0	0
$941$	33.4454	1.09029	0.545145	−	0.838342i	$-0.316475\pi$
$941$	33.4454	1.09029	0.545145	+	0.838342i	$0.316475\pi$
$942$	0	0
$943$	13.1993	0.429829
$944$	−4.38371	−0.142678
$945$	0	0
$946$	3.52416	0.114580
$947$	−23.5223	−0.764373	−0.382187	−	0.924085i	$-0.624829\pi$
$947$	−23.5223	−0.764373	−0.382187	+	0.924085i	$0.624829\pi$
$948$	0	0
$949$	0.138940	0.00451018
$950$	0	0
$951$	0	0
$952$	−5.23766	−0.169753
$953$	6.88378	0.222988	0.111494	−	0.993765i	$-0.464436\pi$
$953$	6.88378	0.222988	0.111494	+	0.993765i	$0.464436\pi$
$954$	0	0
$955$	0	0
$956$	7.34644	0.237601
$957$	0	0
$958$	8.65939	0.279772
$959$	4.36184	0.140851
$960$	0	0
$961$	−23.2364	−0.749562
$962$	3.57781	0.115353
$963$	0	0
$964$	−11.5192	−0.371007
$965$	0	0
$966$	0	0
$967$	6.71730	0.216014	0.108007	−	0.994150i	$-0.465553\pi$
$967$	6.71730	0.216014	0.108007	+	0.994150i	$0.465553\pi$
$968$	−25.8698	−0.831488
$969$	0	0
$970$	0	0
$971$	24.3441	0.781238	0.390619	−	0.920552i	$-0.372261\pi$
$971$	24.3441	0.781238	0.390619	+	0.920552i	$0.372261\pi$
$972$	0	0
$973$	−6.48665	−0.207953
$974$	14.2103	0.455326
$975$	0	0
$976$	−3.67105	−0.117507
$977$	−15.1279	−0.483984	−0.241992	−	0.970278i	$-0.577801\pi$
$977$	−15.1279	−0.483984	−0.241992	+	0.970278i	$0.577801\pi$
$978$	0	0
$979$	23.4849	0.750581
$980$	0	0
$981$	0	0
$982$	−34.2232	−1.09211
$983$	−29.6121	−0.944481	−0.472240	−	0.881470i	$-0.656555\pi$
$983$	−29.6121	−0.944481	−0.472240	+	0.881470i	$0.656555\pi$
$984$	0	0
$985$	0	0
$986$	2.10020	0.0668840
$987$	0	0
$988$	4.89802	0.155827
$989$	2.07831	0.0660863
$990$	0	0
$991$	−36.7988	−1.16895	−0.584476	−	0.811411i	$-0.698700\pi$
$991$	−36.7988	−1.16895	−0.584476	+	0.811411i	$0.698700\pi$
$992$	−13.1970	−0.419004
$993$	0	0
$994$	−16.1936	−0.513630
$995$	0	0
$996$	0	0
$997$	−38.0854	−1.20618	−0.603089	−	0.797674i	$-0.706064\pi$
$997$	−38.0854	−1.20618	−0.603089	+	0.797674i	$0.706064\pi$
$998$	−21.0906	−0.667612
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8325.2.a.ca.1.4		5
3.2	odd	2		2775.2.a.bc.1.2	yes	5
5.4	even	2		8325.2.a.cg.1.2		5
15.14	odd	2		2775.2.a.z.1.4	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2775.2.a.z.1.4	✓	5	15.14	odd	2
2775.2.a.bc.1.2	yes	5	3.2	odd	2
8325.2.a.ca.1.4		5	1.1	even	1	trivial
8325.2.a.cg.1.2		5	5.4	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 8325 = 3^{2} \cdot 5^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8325.a (trivial)

\(f(q)\)	\(=\)	\(q+1.03520 q^{2} -0.928359 q^{4} -1.11914 q^{7} -3.03144 q^{8} -4.41971 q^{11} +3.45615 q^{13} -1.15853 q^{14} -1.28143 q^{16} -1.54385 q^{17} -1.52655 q^{19} -4.57529 q^{22} -2.69819 q^{23} +3.57781 q^{26} +1.03896 q^{28} -1.31411 q^{29} -2.78632 q^{31} +4.73634 q^{32} -1.59819 q^{34} +1.00000 q^{37} -1.58029 q^{38} -4.89192 q^{41} -0.770260 q^{43} +4.10308 q^{44} -2.79317 q^{46} +3.73758 q^{47} -5.74753 q^{49} -3.20855 q^{52} -3.73758 q^{53} +3.39260 q^{56} -1.36037 q^{58} +3.42095 q^{59} +2.86481 q^{61} -2.88440 q^{62} +7.46593 q^{64} +8.50365 q^{67} +1.43325 q^{68} +13.9777 q^{71} +0.0402008 q^{73} +1.03520 q^{74} +1.41719 q^{76} +4.94626 q^{77} -6.48635 q^{79} -5.06412 q^{82} -11.6567 q^{83} -0.797374 q^{86} +13.3981 q^{88} -5.31368 q^{89} -3.86791 q^{91} +2.50489 q^{92} +3.86915 q^{94} +14.3116 q^{97} -5.94985 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 2 q^{2} + 8 q^{4} - 3 q^{8} + q^{11} + 14 q^{13} + 10 q^{14} + 10 q^{16} - 11 q^{17} + 10 q^{19} - 14 q^{22} - 4 q^{23} - 13 q^{26} + 3 q^{28} - 5 q^{29} - 3 q^{31} - 23 q^{32} - 3 q^{34} + 5 q^{37}+ \cdots - 35 q^{98}+O(q^{100}) \) 5 * q - 2 * q^2 + 8 * q^4 - 3 * q^8 + q^11 + 14 * q^13 + 10 * q^14 + 10 * q^16 - 11 * q^17 + 10 * q^19 - 14 * q^22 - 4 * q^23 - 13 * q^26 + 3 * q^28 - 5 * q^29 - 3 * q^31 - 23 * q^32 - 3 * q^34 + 5 * q^37 - 31 * q^38 + 8 * q^41 + q^43 + 21 * q^44 - 13 * q^46 - q^47 + 21 * q^49 + 40 * q^52 + q^53 + 38 * q^56 + 40 * q^58 + 21 * q^59 + 26 * q^61 + 28 * q^62 + 45 * q^64 + 21 * q^67 + 17 * q^71 + 25 * q^73 - 2 * q^74 + 6 * q^76 - 16 * q^77 + 10 * q^79 - 28 * q^82 + 3 * q^83 + 45 * q^86 - 9 * q^88 - 33 * q^89 + 9 * q^91 + 18 * q^92 + 18 * q^94 + 16 * q^97 - 35 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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