LMFDB - Embedded newform 3332.2.a.u.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3332,2,Mod(1,3332)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3332, 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("3332.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3332 = 2^{2} \cdot 7^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3332.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:	$26.6061539535$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 8x^{6} + 36x^{5} + 17x^{4} - 76x^{3} - 20x^{2} + 44x + 17 $ x^8 - 4x^7 - 8x^6 + 36x^5 + 17x^4 - 76x^3 - 20x^2 + 44*x + 17
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.43314$ of defining polynomial
Character	$\chi$	$=$	3332.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.43314 q^{3} -0.700514 q^{5} +2.92017 q^{9} -6.25232 q^{11} +3.33822 q^{13} +1.70445 q^{15} -1.00000 q^{17} +2.78435 q^{19} -5.97696 q^{23} -4.50928 q^{25} +0.194229 q^{27} -3.96405 q^{29} +1.55695 q^{31} +15.2128 q^{33} -2.53597 q^{37} -8.12236 q^{39} +5.62122 q^{41} -2.29290 q^{43} -2.04562 q^{45} -11.9664 q^{47} +2.43314 q^{51} +6.55500 q^{53} +4.37983 q^{55} -6.77471 q^{57} -8.78005 q^{59} +4.95327 q^{61} -2.33847 q^{65} +13.1327 q^{67} +14.5428 q^{69} -14.2118 q^{71} -8.40017 q^{73} +10.9717 q^{75} +3.03170 q^{79} -9.23311 q^{81} -4.93476 q^{83} +0.700514 q^{85} +9.64510 q^{87} +14.7216 q^{89} -3.78828 q^{93} -1.95047 q^{95} +0.898333 q^{97} -18.2578 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{3} + 4 q^{5} + 8 q^{9} - 4 q^{11} + 20 q^{13} + 12 q^{15} - 8 q^{17} + 8 q^{19} + 4 q^{23} + 8 q^{25} + 28 q^{27} - 16 q^{29} - 8 q^{31} + 16 q^{33} + 8 q^{37} + 20 q^{39} + 12 q^{41} - 4 q^{43}+ \cdots - 8 q^{99}+O(q^{100}) $ 8 * q + 4 * q^3 + 4 * q^5 + 8 * q^9 - 4 * q^11 + 20 * q^13 + 12 * q^15 - 8 * q^17 + 8 * q^19 + 4 * q^23 + 8 * q^25 + 28 * q^27 - 16 * q^29 - 8 * q^31 + 16 * q^33 + 8 * q^37 + 20 * q^39 + 12 * q^41 - 4 * q^43 + 20 * q^45 + 4 * q^47 - 4 * q^51 + 12 * q^55 - 16 * q^59 + 32 * q^61 + 44 * q^69 + 24 * q^73 + 24 * q^75 + 4 * q^79 + 36 * q^81 + 28 * q^83 - 4 * q^85 - 40 * q^87 + 20 * q^89 - 16 * q^93 - 20 * q^95 + 56 * q^97 - 8 * 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
$3$	−2.43314	−1.40477	−0.702387	−	0.711795i	$-0.747882\pi$
$3$	−2.43314	−1.40477	−0.702387	+	0.711795i	$0.747882\pi$
$4$	0	0
$5$	−0.700514	−0.313279	−0.156640	−	0.987656i	$-0.550066\pi$
$5$	−0.700514	−0.313279	−0.156640	+	0.987656i	$0.550066\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	2.92017	0.973391
$10$	0	0
$11$	−6.25232	−1.88514	−0.942572	−	0.334003i	$-0.891601\pi$
$11$	−6.25232	−1.88514	−0.942572	+	0.334003i	$0.891601\pi$
$12$	0	0
$13$	3.33822	0.925855	0.462928	−	0.886396i	$-0.346799\pi$
$13$	3.33822	0.925855	0.462928	+	0.886396i	$0.346799\pi$
$14$	0	0
$15$	1.70445	0.440087
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	2.78435	0.638773	0.319387	−	0.947625i	$-0.396523\pi$
$19$	2.78435	0.638773	0.319387	+	0.947625i	$0.396523\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−5.97696	−1.24628	−0.623141	−	0.782110i	$-0.714144\pi$
$23$	−5.97696	−1.24628	−0.623141	+	0.782110i	$0.714144\pi$
$24$	0	0
$25$	−4.50928	−0.901856
$26$	0	0
$27$	0.194229	0.0373793
$28$	0	0
$29$	−3.96405	−0.736106	−0.368053	−	0.929805i	$-0.619976\pi$
$29$	−3.96405	−0.736106	−0.368053	+	0.929805i	$0.619976\pi$
$30$	0	0
$31$	1.55695	0.279637	0.139818	−	0.990177i	$-0.455348\pi$
$31$	1.55695	0.279637	0.139818	+	0.990177i	$0.455348\pi$
$32$	0	0
$33$	15.2128	2.64820
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−2.53597	−0.416912	−0.208456	−	0.978032i	$-0.566844\pi$
$37$	−2.53597	−0.416912	−0.208456	+	0.978032i	$0.566844\pi$
$38$	0	0
$39$	−8.12236	−1.30062
$40$	0	0
$41$	5.62122	0.877887	0.438944	−	0.898515i	$-0.355353\pi$
$41$	5.62122	0.877887	0.438944	+	0.898515i	$0.355353\pi$
$42$	0	0
$43$	−2.29290	−0.349664	−0.174832	−	0.984598i	$-0.555938\pi$
$43$	−2.29290	−0.349664	−0.174832	+	0.984598i	$0.555938\pi$
$44$	0	0
$45$	−2.04562	−0.304943
$46$	0	0
$47$	−11.9664	−1.74548	−0.872742	−	0.488181i	$-0.837661\pi$
$47$	−11.9664	−1.74548	−0.872742	+	0.488181i	$0.837661\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	2.43314	0.340708
$52$	0	0
$53$	6.55500	0.900399	0.450200	−	0.892928i	$-0.351353\pi$
$53$	6.55500	0.900399	0.450200	+	0.892928i	$0.351353\pi$
$54$	0	0
$55$	4.37983	0.590577
$56$	0	0
$57$	−6.77471	−0.897332
$58$	0	0
$59$	−8.78005	−1.14307	−0.571533	−	0.820579i	$-0.693651\pi$
$59$	−8.78005	−1.14307	−0.571533	+	0.820579i	$0.693651\pi$
$60$	0	0
$61$	4.95327	0.634201	0.317101	−	0.948392i	$-0.397291\pi$
$61$	4.95327	0.634201	0.317101	+	0.948392i	$0.397291\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−2.33847	−0.290051
$66$	0	0
$67$	13.1327	1.60442	0.802210	−	0.597042i	$-0.203657\pi$
$67$	13.1327	1.60442	0.802210	+	0.597042i	$0.203657\pi$
$68$	0	0
$69$	14.5428	1.75074
$70$	0	0
$71$	−14.2118	−1.68663	−0.843317	−	0.537417i	$-0.819400\pi$
$71$	−14.2118	−1.68663	−0.843317	+	0.537417i	$0.819400\pi$
$72$	0	0
$73$	−8.40017	−0.983165	−0.491583	−	0.870831i	$-0.663581\pi$
$73$	−8.40017	−0.983165	−0.491583	+	0.870831i	$0.663581\pi$
$74$	0	0
$75$	10.9717	1.26690
$76$	0	0
$77$	0	0
$78$	0	0
$79$	3.03170	0.341093	0.170546	−	0.985350i	$-0.445447\pi$
$79$	3.03170	0.341093	0.170546	+	0.985350i	$0.445447\pi$
$80$	0	0
$81$	−9.23311	−1.02590
$82$	0	0
$83$	−4.93476	−0.541660	−0.270830	−	0.962627i	$-0.587298\pi$
$83$	−4.93476	−0.541660	−0.270830	+	0.962627i	$0.587298\pi$
$84$	0	0
$85$	0.700514	0.0759814
$86$	0	0
$87$	9.64510	1.03406
$88$	0	0
$89$	14.7216	1.56049	0.780244	−	0.625475i	$-0.215095\pi$
$89$	14.7216	1.56049	0.780244	+	0.625475i	$0.215095\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−3.78828	−0.392826
$94$	0	0
$95$	−1.95047	−0.200114
$96$	0	0
$97$	0.898333	0.0912119	0.0456060	−	0.998960i	$-0.485478\pi$
$97$	0.898333	0.0912119	0.0456060	+	0.998960i	$0.485478\pi$
$98$	0	0
$99$	−18.2578	−1.83498
$100$	0	0
$101$	−6.75291	−0.671940	−0.335970	−	0.941873i	$-0.609064\pi$
$101$	−6.75291	−0.671940	−0.335970	+	0.941873i	$0.609064\pi$
$102$	0	0
$103$	−3.74321	−0.368830	−0.184415	−	0.982849i	$-0.559039\pi$
$103$	−3.74321	−0.368830	−0.184415	+	0.982849i	$0.559039\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	12.7611	1.23367	0.616833	−	0.787094i	$-0.288415\pi$
$107$	12.7611	1.23367	0.616833	+	0.787094i	$0.288415\pi$
$108$	0	0
$109$	−9.26268	−0.887204	−0.443602	−	0.896224i	$-0.646300\pi$
$109$	−9.26268	−0.887204	−0.443602	+	0.896224i	$0.646300\pi$
$110$	0	0
$111$	6.17038	0.585667
$112$	0	0
$113$	2.64156	0.248497	0.124248	−	0.992251i	$-0.460348\pi$
$113$	2.64156	0.248497	0.124248	+	0.992251i	$0.460348\pi$
$114$	0	0
$115$	4.18694	0.390434
$116$	0	0
$117$	9.74818	0.901220
$118$	0	0
$119$	0	0
$120$	0	0
$121$	28.0915	2.55377
$122$	0	0
$123$	−13.6772	−1.23323
$124$	0	0
$125$	6.66138	0.595812
$126$	0	0
$127$	6.94328	0.616117	0.308058	−	0.951367i	$-0.400321\pi$
$127$	6.94328	0.616117	0.308058	+	0.951367i	$0.400321\pi$
$128$	0	0
$129$	5.57895	0.491199
$130$	0	0
$131$	−8.88103	−0.775939	−0.387969	−	0.921672i	$-0.626823\pi$
$131$	−8.88103	−0.775939	−0.387969	+	0.921672i	$0.626823\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−0.136060	−0.0117102
$136$	0	0
$137$	4.33823	0.370640	0.185320	−	0.982678i	$-0.440668\pi$
$137$	4.33823	0.370640	0.185320	+	0.982678i	$0.440668\pi$
$138$	0	0
$139$	18.0420	1.53030	0.765150	−	0.643852i	$-0.222665\pi$
$139$	18.0420	1.53030	0.765150	+	0.643852i	$0.222665\pi$
$140$	0	0
$141$	29.1160	2.45201
$142$	0	0
$143$	−20.8716	−1.74537
$144$	0	0
$145$	2.77687	0.230607
$146$	0	0
$147$	0	0
$148$	0	0
$149$	17.6645	1.44713	0.723567	−	0.690255i	$-0.242501\pi$
$149$	17.6645	1.44713	0.723567	+	0.690255i	$0.242501\pi$
$150$	0	0
$151$	11.6266	0.946159	0.473080	−	0.881020i	$-0.343142\pi$
$151$	11.6266	0.946159	0.473080	+	0.881020i	$0.343142\pi$
$152$	0	0
$153$	−2.92017	−0.236082
$154$	0	0
$155$	−1.09067	−0.0876043
$156$	0	0
$157$	18.8397	1.50357	0.751784	−	0.659410i	$-0.229194\pi$
$157$	18.8397	1.50357	0.751784	+	0.659410i	$0.229194\pi$
$158$	0	0
$159$	−15.9492	−1.26486
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−10.2038	−0.799221	−0.399611	−	0.916685i	$-0.630855\pi$
$163$	−10.2038	−0.799221	−0.399611	+	0.916685i	$0.630855\pi$
$164$	0	0
$165$	−10.6568	−0.829627
$166$	0	0
$167$	2.04752	0.158442	0.0792210	−	0.996857i	$-0.474757\pi$
$167$	2.04752	0.158442	0.0792210	+	0.996857i	$0.474757\pi$
$168$	0	0
$169$	−1.85629	−0.142792
$170$	0	0
$171$	8.13078	0.621776
$172$	0	0
$173$	3.51778	0.267452	0.133726	−	0.991018i	$-0.457306\pi$
$173$	3.51778	0.267452	0.133726	+	0.991018i	$0.457306\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	21.3631	1.60575
$178$	0	0
$179$	4.01805	0.300323	0.150162	−	0.988661i	$-0.452021\pi$
$179$	4.01805	0.300323	0.150162	+	0.988661i	$0.452021\pi$
$180$	0	0
$181$	3.00576	0.223416	0.111708	−	0.993741i	$-0.464368\pi$
$181$	3.00576	0.223416	0.111708	+	0.993741i	$0.464368\pi$
$182$	0	0
$183$	−12.0520	−0.890910
$184$	0	0
$185$	1.77648	0.130610
$186$	0	0
$187$	6.25232	0.457215
$188$	0	0
$189$	0	0
$190$	0	0
$191$	13.7743	0.996670	0.498335	−	0.866985i	$-0.333945\pi$
$191$	13.7743	0.996670	0.498335	+	0.866985i	$0.333945\pi$
$192$	0	0
$193$	−8.29653	−0.597197	−0.298599	−	0.954379i	$-0.596519\pi$
$193$	−8.29653	−0.597197	−0.298599	+	0.954379i	$0.596519\pi$
$194$	0	0
$195$	5.68982	0.407457
$196$	0	0
$197$	10.6075	0.755755	0.377878	−	0.925856i	$-0.376654\pi$
$197$	10.6075	0.755755	0.377878	+	0.925856i	$0.376654\pi$
$198$	0	0
$199$	17.8444	1.26495	0.632477	−	0.774579i	$-0.282038\pi$
$199$	17.8444	1.26495	0.632477	+	0.774579i	$0.282038\pi$
$200$	0	0
$201$	−31.9538	−2.25385
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−3.93774	−0.275024
$206$	0	0
$207$	−17.4537	−1.21312
$208$	0	0
$209$	−17.4086	−1.20418
$210$	0	0
$211$	−4.90813	−0.337889	−0.168945	−	0.985626i	$-0.554036\pi$
$211$	−4.90813	−0.337889	−0.168945	+	0.985626i	$0.554036\pi$
$212$	0	0
$213$	34.5794	2.36934
$214$	0	0
$215$	1.60621	0.109542
$216$	0	0
$217$	0	0
$218$	0	0
$219$	20.4388	1.38113
$220$	0	0
$221$	−3.33822	−0.224553
$222$	0	0
$223$	−10.7501	−0.719879	−0.359940	−	0.932976i	$-0.617203\pi$
$223$	−10.7501	−0.719879	−0.359940	+	0.932976i	$0.617203\pi$
$224$	0	0
$225$	−13.1679	−0.877859
$226$	0	0
$227$	−6.86801	−0.455846	−0.227923	−	0.973679i	$-0.573193\pi$
$227$	−6.86801	−0.455846	−0.227923	+	0.973679i	$0.573193\pi$
$228$	0	0
$229$	14.3348	0.947268	0.473634	−	0.880722i	$-0.342942\pi$
$229$	14.3348	0.947268	0.473634	+	0.880722i	$0.342942\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−9.26722	−0.607116	−0.303558	−	0.952813i	$-0.598175\pi$
$233$	−9.26722	−0.607116	−0.303558	+	0.952813i	$0.598175\pi$
$234$	0	0
$235$	8.38266	0.546824
$236$	0	0
$237$	−7.37655	−0.479158
$238$	0	0
$239$	−22.9940	−1.48736	−0.743680	−	0.668536i	$-0.766921\pi$
$239$	−22.9940	−1.48736	−0.743680	+	0.668536i	$0.766921\pi$
$240$	0	0
$241$	−5.18312	−0.333874	−0.166937	−	0.985968i	$-0.553388\pi$
$241$	−5.18312	−0.333874	−0.166937	+	0.985968i	$0.553388\pi$
$242$	0	0
$243$	21.8828	1.40378
$244$	0	0
$245$	0	0
$246$	0	0
$247$	9.29476	0.591412
$248$	0	0
$249$	12.0070	0.760910
$250$	0	0
$251$	27.0836	1.70950	0.854752	−	0.519037i	$-0.173709\pi$
$251$	27.0836	1.70950	0.854752	+	0.519037i	$0.173709\pi$
$252$	0	0
$253$	37.3698	2.34942
$254$	0	0
$255$	−1.70445	−0.106737
$256$	0	0
$257$	−0.655631	−0.0408972	−0.0204486	−	0.999791i	$-0.506509\pi$
$257$	−0.655631	−0.0408972	−0.0204486	+	0.999791i	$0.506509\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−11.5757	−0.716520
$262$	0	0
$263$	22.1584	1.36635	0.683173	−	0.730257i	$-0.260600\pi$
$263$	22.1584	1.36635	0.683173	+	0.730257i	$0.260600\pi$
$264$	0	0
$265$	−4.59187	−0.282076
$266$	0	0
$267$	−35.8198	−2.19213
$268$	0	0
$269$	7.04549	0.429571	0.214786	−	0.976661i	$-0.431095\pi$
$269$	7.04549	0.429571	0.214786	+	0.976661i	$0.431095\pi$
$270$	0	0
$271$	−25.4854	−1.54813	−0.774064	−	0.633107i	$-0.781779\pi$
$271$	−25.4854	−1.54813	−0.774064	+	0.633107i	$0.781779\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	28.1934	1.70013
$276$	0	0
$277$	32.4113	1.94741	0.973704	−	0.227815i	$-0.0731582\pi$
$277$	32.4113	1.94741	0.973704	+	0.227815i	$0.0731582\pi$
$278$	0	0
$279$	4.54657	0.272196
$280$	0	0
$281$	−1.29635	−0.0773339	−0.0386669	−	0.999252i	$-0.512311\pi$
$281$	−1.29635	−0.0773339	−0.0386669	+	0.999252i	$0.512311\pi$
$282$	0	0
$283$	25.0050	1.48639	0.743197	−	0.669072i	$-0.233308\pi$
$283$	25.0050	1.48639	0.743197	+	0.669072i	$0.233308\pi$
$284$	0	0
$285$	4.74578	0.281116
$286$	0	0
$287$	0	0
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−2.18577	−0.128132
$292$	0	0
$293$	25.4747	1.48825	0.744125	−	0.668041i	$-0.232867\pi$
$293$	25.4747	1.48825	0.744125	+	0.668041i	$0.232867\pi$
$294$	0	0
$295$	6.15055	0.358099
$296$	0	0
$297$	−1.21438	−0.0704654
$298$	0	0
$299$	−19.9524	−1.15388
$300$	0	0
$301$	0	0
$302$	0	0
$303$	16.4308	0.943924
$304$	0	0
$305$	−3.46983	−0.198682
$306$	0	0
$307$	33.5628	1.91553	0.957764	−	0.287555i	$-0.0928422\pi$
$307$	33.5628	1.91553	0.957764	+	0.287555i	$0.0928422\pi$
$308$	0	0
$309$	9.10776	0.518122
$310$	0	0
$311$	23.2347	1.31752	0.658760	−	0.752353i	$-0.271081\pi$
$311$	23.2347	1.31752	0.658760	+	0.752353i	$0.271081\pi$
$312$	0	0
$313$	30.2386	1.70919	0.854593	−	0.519299i	$-0.173807\pi$
$313$	30.2386	1.70919	0.854593	+	0.519299i	$0.173807\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−19.4442	−1.09210	−0.546049	−	0.837753i	$-0.683869\pi$
$317$	−19.4442	−1.09210	−0.546049	+	0.837753i	$0.683869\pi$
$318$	0	0
$319$	24.7845	1.38767
$320$	0	0
$321$	−31.0496	−1.73302
$322$	0	0
$323$	−2.78435	−0.154925
$324$	0	0
$325$	−15.0530	−0.834988
$326$	0	0
$327$	22.5374	1.24632
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−27.4473	−1.50864	−0.754321	−	0.656506i	$-0.772034\pi$
$331$	−27.4473	−1.50864	−0.754321	+	0.656506i	$0.772034\pi$
$332$	0	0
$333$	−7.40549	−0.405818
$334$	0	0
$335$	−9.19967	−0.502631
$336$	0	0
$337$	−13.4504	−0.732688	−0.366344	−	0.930479i	$-0.619391\pi$
$337$	−13.4504	−0.732688	−0.366344	+	0.930479i	$0.619391\pi$
$338$	0	0
$339$	−6.42728	−0.349082
$340$	0	0
$341$	−9.73455	−0.527155
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−10.1874	−0.548472
$346$	0	0
$347$	−1.62909	−0.0874544	−0.0437272	−	0.999044i	$-0.513923\pi$
$347$	−1.62909	−0.0874544	−0.0437272	+	0.999044i	$0.513923\pi$
$348$	0	0
$349$	−7.60959	−0.407332	−0.203666	−	0.979040i	$-0.565286\pi$
$349$	−7.60959	−0.407332	−0.203666	+	0.979040i	$0.565286\pi$
$350$	0	0
$351$	0.648378	0.0346079
$352$	0	0
$353$	27.3499	1.45569	0.727843	−	0.685744i	$-0.240523\pi$
$353$	27.3499	1.45569	0.727843	+	0.685744i	$0.240523\pi$
$354$	0	0
$355$	9.95558	0.528387
$356$	0	0
$357$	0	0
$358$	0	0
$359$	27.5849	1.45588	0.727939	−	0.685642i	$-0.240479\pi$
$359$	27.5849	1.45588	0.727939	+	0.685642i	$0.240479\pi$
$360$	0	0
$361$	−11.2474	−0.591969
$362$	0	0
$363$	−68.3505	−3.58747
$364$	0	0
$365$	5.88443	0.308005
$366$	0	0
$367$	−14.7116	−0.767938	−0.383969	−	0.923346i	$-0.625443\pi$
$367$	−14.7116	−0.767938	−0.383969	+	0.923346i	$0.625443\pi$
$368$	0	0
$369$	16.4149	0.854528
$370$	0	0
$371$	0	0
$372$	0	0
$373$	21.0714	1.09104	0.545518	−	0.838099i	$-0.316333\pi$
$373$	21.0714	1.09104	0.545518	+	0.838099i	$0.316333\pi$
$374$	0	0
$375$	−16.2081	−0.836982
$376$	0	0
$377$	−13.2329	−0.681528
$378$	0	0
$379$	−30.5227	−1.56784	−0.783922	−	0.620859i	$-0.786784\pi$
$379$	−30.5227	−1.56784	−0.783922	+	0.620859i	$0.786784\pi$
$380$	0	0
$381$	−16.8940	−0.865505
$382$	0	0
$383$	−13.5517	−0.692458	−0.346229	−	0.938150i	$-0.612538\pi$
$383$	−13.5517	−0.692458	−0.346229	+	0.938150i	$0.612538\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−6.69567	−0.340360
$388$	0	0
$389$	−7.02769	−0.356318	−0.178159	−	0.984002i	$-0.557014\pi$
$389$	−7.02769	−0.356318	−0.178159	+	0.984002i	$0.557014\pi$
$390$	0	0
$391$	5.97696	0.302268
$392$	0	0
$393$	21.6088	1.09002
$394$	0	0
$395$	−2.12375	−0.106857
$396$	0	0
$397$	−18.2330	−0.915088	−0.457544	−	0.889187i	$-0.651271\pi$
$397$	−18.2330	−0.915088	−0.457544	+	0.889187i	$0.651271\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−17.8638	−0.892077	−0.446038	−	0.895014i	$-0.647166\pi$
$401$	−17.8638	−0.892077	−0.446038	+	0.895014i	$0.647166\pi$
$402$	0	0
$403$	5.19744	0.258903
$404$	0	0
$405$	6.46792	0.321393
$406$	0	0
$407$	15.8557	0.785939
$408$	0	0
$409$	−10.8428	−0.536143	−0.268072	−	0.963399i	$-0.586386\pi$
$409$	−10.8428	−0.536143	−0.268072	+	0.963399i	$0.586386\pi$
$410$	0	0
$411$	−10.5555	−0.520665
$412$	0	0
$413$	0	0
$414$	0	0
$415$	3.45686	0.169691
$416$	0	0
$417$	−43.8987	−2.14973
$418$	0	0
$419$	23.8813	1.16668	0.583340	−	0.812228i	$-0.301745\pi$
$419$	23.8813	1.16668	0.583340	+	0.812228i	$0.301745\pi$
$420$	0	0
$421$	−6.82824	−0.332788	−0.166394	−	0.986059i	$-0.553212\pi$
$421$	−6.82824	−0.332788	−0.166394	+	0.986059i	$0.553212\pi$
$422$	0	0
$423$	−34.9441	−1.69904
$424$	0	0
$425$	4.50928	0.218732
$426$	0	0
$427$	0	0
$428$	0	0
$429$	50.7835	2.45185
$430$	0	0
$431$	−10.9429	−0.527101	−0.263551	−	0.964646i	$-0.584894\pi$
$431$	−10.9429	−0.527101	−0.263551	+	0.964646i	$0.584894\pi$
$432$	0	0
$433$	−18.8182	−0.904344	−0.452172	−	0.891931i	$-0.649351\pi$
$433$	−18.8182	−0.904344	−0.452172	+	0.891931i	$0.649351\pi$
$434$	0	0
$435$	−6.75653	−0.323951
$436$	0	0
$437$	−16.6419	−0.796091
$438$	0	0
$439$	−4.35808	−0.208000	−0.104000	−	0.994577i	$-0.533164\pi$
$439$	−4.35808	−0.208000	−0.104000	+	0.994577i	$0.533164\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−8.16946	−0.388143	−0.194071	−	0.980987i	$-0.562169\pi$
$443$	−8.16946	−0.388143	−0.194071	+	0.980987i	$0.562169\pi$
$444$	0	0
$445$	−10.3127	−0.488869
$446$	0	0
$447$	−42.9802	−2.03290
$448$	0	0
$449$	15.0508	0.710292	0.355146	−	0.934811i	$-0.384431\pi$
$449$	15.0508	0.710292	0.355146	+	0.934811i	$0.384431\pi$
$450$	0	0
$451$	−35.1456	−1.65494
$452$	0	0
$453$	−28.2891	−1.32914
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−37.2417	−1.74209	−0.871047	−	0.491199i	$-0.836559\pi$
$457$	−37.2417	−1.74209	−0.871047	+	0.491199i	$0.836559\pi$
$458$	0	0
$459$	−0.194229	−0.00906582
$460$	0	0
$461$	33.6939	1.56928	0.784642	−	0.619949i	$-0.212847\pi$
$461$	33.6939	1.56928	0.784642	+	0.619949i	$0.212847\pi$
$462$	0	0
$463$	−7.90942	−0.367582	−0.183791	−	0.982965i	$-0.558837\pi$
$463$	−7.90942	−0.367582	−0.183791	+	0.982965i	$0.558837\pi$
$464$	0	0
$465$	2.65374	0.123064
$466$	0	0
$467$	−12.4018	−0.573887	−0.286944	−	0.957947i	$-0.592639\pi$
$467$	−12.4018	−0.573887	−0.286944	+	0.957947i	$0.592639\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−45.8395	−2.11217
$472$	0	0
$473$	14.3359	0.659167
$474$	0	0
$475$	−12.5554	−0.576082
$476$	0	0
$477$	19.1418	0.876441
$478$	0	0
$479$	−0.175096	−0.00800033	−0.00400016	−	0.999992i	$-0.501273\pi$
$479$	−0.175096	−0.00800033	−0.00400016	+	0.999992i	$0.501273\pi$
$480$	0	0
$481$	−8.46564	−0.386000
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−0.629295	−0.0285748
$486$	0	0
$487$	37.8139	1.71351	0.856755	−	0.515724i	$-0.172477\pi$
$487$	37.8139	1.71351	0.856755	+	0.515724i	$0.172477\pi$
$488$	0	0
$489$	24.8272	1.12273
$490$	0	0
$491$	36.7418	1.65814	0.829068	−	0.559148i	$-0.188872\pi$
$491$	36.7418	1.65814	0.829068	+	0.559148i	$0.188872\pi$
$492$	0	0
$493$	3.96405	0.178532
$494$	0	0
$495$	12.7899	0.574862
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−32.8512	−1.47062	−0.735311	−	0.677729i	$-0.762964\pi$
$499$	−32.8512	−1.47062	−0.735311	+	0.677729i	$0.762964\pi$
$500$	0	0
$501$	−4.98191	−0.222575
$502$	0	0
$503$	−7.33672	−0.327128	−0.163564	−	0.986533i	$-0.552299\pi$
$503$	−7.33672	−0.327128	−0.163564	+	0.986533i	$0.552299\pi$
$504$	0	0
$505$	4.73051	0.210505
$506$	0	0
$507$	4.51662	0.200590
$508$	0	0
$509$	14.8523	0.658318	0.329159	−	0.944275i	$-0.393235\pi$
$509$	14.8523	0.658318	0.329159	+	0.944275i	$0.393235\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0.540800	0.0238769
$514$	0	0
$515$	2.62217	0.115547
$516$	0	0
$517$	74.8180	3.29049
$518$	0	0
$519$	−8.55925	−0.375709
$520$	0	0
$521$	0.695446	0.0304680	0.0152340	−	0.999884i	$-0.495151\pi$
$521$	0.695446	0.0304680	0.0152340	+	0.999884i	$0.495151\pi$
$522$	0	0
$523$	6.79974	0.297332	0.148666	−	0.988887i	$-0.452502\pi$
$523$	6.79974	0.297332	0.148666	+	0.988887i	$0.452502\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1.55695	−0.0678218
$528$	0	0
$529$	12.7240	0.553217
$530$	0	0
$531$	−25.6393	−1.11265
$532$	0	0
$533$	18.7649	0.812797
$534$	0	0
$535$	−8.93935	−0.386482
$536$	0	0
$537$	−9.77648	−0.421886
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−43.5784	−1.87358	−0.936792	−	0.349887i	$-0.886220\pi$
$541$	−43.5784	−1.87358	−0.936792	+	0.349887i	$0.886220\pi$
$542$	0	0
$543$	−7.31343	−0.313849
$544$	0	0
$545$	6.48863	0.277943
$546$	0	0
$547$	−26.6669	−1.14019	−0.570097	−	0.821577i	$-0.693094\pi$
$547$	−26.6669	−1.14019	−0.570097	+	0.821577i	$0.693094\pi$
$548$	0	0
$549$	14.4644	0.617326
$550$	0	0
$551$	−11.0373	−0.470205
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−4.32244	−0.183477
$556$	0	0
$557$	−18.5815	−0.787322	−0.393661	−	0.919256i	$-0.628792\pi$
$557$	−18.5815	−0.787322	−0.393661	+	0.919256i	$0.628792\pi$
$558$	0	0
$559$	−7.65420	−0.323738
$560$	0	0
$561$	−15.2128	−0.642283
$562$	0	0
$563$	−9.61461	−0.405208	−0.202604	−	0.979261i	$-0.564940\pi$
$563$	−9.61461	−0.405208	−0.202604	+	0.979261i	$0.564940\pi$
$564$	0	0
$565$	−1.85045	−0.0778489
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−21.8304	−0.915179	−0.457590	−	0.889163i	$-0.651287\pi$
$569$	−21.8304	−0.915179	−0.457590	+	0.889163i	$0.651287\pi$
$570$	0	0
$571$	−14.1272	−0.591207	−0.295603	−	0.955311i	$-0.595521\pi$
$571$	−14.1272	−0.591207	−0.295603	+	0.955311i	$0.595521\pi$
$572$	0	0
$573$	−33.5147	−1.40010
$574$	0	0
$575$	26.9518	1.12397
$576$	0	0
$577$	24.2960	1.01146	0.505729	−	0.862693i	$-0.331224\pi$
$577$	24.2960	1.01146	0.505729	+	0.862693i	$0.331224\pi$
$578$	0	0
$579$	20.1866	0.838928
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−40.9840	−1.69738
$584$	0	0
$585$	−6.82873	−0.282333
$586$	0	0
$587$	3.98164	0.164340	0.0821700	−	0.996618i	$-0.473815\pi$
$587$	3.98164	0.164340	0.0821700	+	0.996618i	$0.473815\pi$
$588$	0	0
$589$	4.33509	0.178624
$590$	0	0
$591$	−25.8096	−1.06167
$592$	0	0
$593$	−32.4429	−1.33227	−0.666135	−	0.745831i	$-0.732053\pi$
$593$	−32.4429	−1.33227	−0.666135	+	0.745831i	$0.732053\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−43.4178	−1.77697
$598$	0	0
$599$	−48.4024	−1.97767	−0.988835	−	0.149016i	$-0.952389\pi$
$599$	−48.4024	−1.97767	−0.988835	+	0.149016i	$0.952389\pi$
$600$	0	0
$601$	14.7447	0.601448	0.300724	−	0.953711i	$-0.402772\pi$
$601$	14.7447	0.601448	0.300724	+	0.953711i	$0.402772\pi$
$602$	0	0
$603$	38.3499	1.56173
$604$	0	0
$605$	−19.6785	−0.800043
$606$	0	0
$607$	19.2021	0.779390	0.389695	−	0.920944i	$-0.372580\pi$
$607$	19.2021	0.779390	0.389695	+	0.920944i	$0.372580\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−39.9466	−1.61607
$612$	0	0
$613$	32.5312	1.31392	0.656961	−	0.753924i	$-0.271841\pi$
$613$	32.5312	1.31392	0.656961	+	0.753924i	$0.271841\pi$
$614$	0	0
$615$	9.58108	0.386346
$616$	0	0
$617$	27.4908	1.10674	0.553369	−	0.832937i	$-0.313342\pi$
$617$	27.4908	1.10674	0.553369	+	0.832937i	$0.313342\pi$
$618$	0	0
$619$	−2.33700	−0.0939321	−0.0469661	−	0.998896i	$-0.514955\pi$
$619$	−2.33700	−0.0939321	−0.0469661	+	0.998896i	$0.514955\pi$
$620$	0	0
$621$	−1.16090	−0.0465852
$622$	0	0
$623$	0	0
$624$	0	0
$625$	17.8800	0.715201
$626$	0	0
$627$	42.3576	1.69160
$628$	0	0
$629$	2.53597	0.101116
$630$	0	0
$631$	−34.5715	−1.37627	−0.688135	−	0.725583i	$-0.741570\pi$
$631$	−34.5715	−1.37627	−0.688135	+	0.725583i	$0.741570\pi$
$632$	0	0
$633$	11.9422	0.474658
$634$	0	0
$635$	−4.86387	−0.193017
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−41.5010	−1.64175
$640$	0	0
$641$	−23.9277	−0.945086	−0.472543	−	0.881308i	$-0.656664\pi$
$641$	−23.9277	−0.945086	−0.472543	+	0.881308i	$0.656664\pi$
$642$	0	0
$643$	−18.3241	−0.722632	−0.361316	−	0.932443i	$-0.617672\pi$
$643$	−18.3241	−0.722632	−0.361316	+	0.932443i	$0.617672\pi$
$644$	0	0
$645$	−3.90813	−0.153882
$646$	0	0
$647$	−40.3569	−1.58659	−0.793297	−	0.608834i	$-0.791637\pi$
$647$	−40.3569	−1.58659	−0.793297	+	0.608834i	$0.791637\pi$
$648$	0	0
$649$	54.8957	2.15484
$650$	0	0
$651$	0	0
$652$	0	0
$653$	16.7302	0.654705	0.327353	−	0.944902i	$-0.393844\pi$
$653$	16.7302	0.654705	0.327353	+	0.944902i	$0.393844\pi$
$654$	0	0
$655$	6.22128	0.243086
$656$	0	0
$657$	−24.5299	−0.957004
$658$	0	0
$659$	−5.34497	−0.208211	−0.104105	−	0.994566i	$-0.533198\pi$
$659$	−5.34497	−0.208211	−0.104105	+	0.994566i	$0.533198\pi$
$660$	0	0
$661$	−17.9109	−0.696654	−0.348327	−	0.937373i	$-0.613250\pi$
$661$	−17.9109	−0.696654	−0.348327	+	0.937373i	$0.613250\pi$
$662$	0	0
$663$	8.12236	0.315446
$664$	0	0
$665$	0	0
$666$	0	0
$667$	23.6930	0.917396
$668$	0	0
$669$	26.1565	1.01127
$670$	0	0
$671$	−30.9694	−1.19556
$672$	0	0
$673$	9.72661	0.374933	0.187467	−	0.982271i	$-0.439972\pi$
$673$	9.72661	0.374933	0.187467	+	0.982271i	$0.439972\pi$
$674$	0	0
$675$	−0.875832	−0.0337108
$676$	0	0
$677$	1.35298	0.0519993	0.0259997	−	0.999662i	$-0.491723\pi$
$677$	1.35298	0.0519993	0.0259997	+	0.999662i	$0.491723\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	16.7108	0.640361
$682$	0	0
$683$	30.2177	1.15625	0.578125	−	0.815948i	$-0.303785\pi$
$683$	30.2177	1.15625	0.578125	+	0.815948i	$0.303785\pi$
$684$	0	0
$685$	−3.03899	−0.116114
$686$	0	0
$687$	−34.8785	−1.33070
$688$	0	0
$689$	21.8820	0.833639
$690$	0	0
$691$	14.9535	0.568859	0.284430	−	0.958697i	$-0.408196\pi$
$691$	14.9535	0.568859	0.284430	+	0.958697i	$0.408196\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−12.6386	−0.479411
$696$	0	0
$697$	−5.62122	−0.212919
$698$	0	0
$699$	22.5485	0.852861
$700$	0	0
$701$	−26.6974	−1.00835	−0.504173	−	0.863603i	$-0.668202\pi$
$701$	−26.6974	−1.00835	−0.504173	+	0.863603i	$0.668202\pi$
$702$	0	0
$703$	−7.06104	−0.266312
$704$	0	0
$705$	−20.3962	−0.768165
$706$	0	0
$707$	0	0
$708$	0	0
$709$	1.43807	0.0540077	0.0270038	−	0.999635i	$-0.491403\pi$
$709$	1.43807	0.0540077	0.0270038	+	0.999635i	$0.491403\pi$
$710$	0	0
$711$	8.85309	0.332017
$712$	0	0
$713$	−9.30582	−0.348506
$714$	0	0
$715$	14.6208	0.546789
$716$	0	0
$717$	55.9477	2.08940
$718$	0	0
$719$	−36.5231	−1.36208	−0.681041	−	0.732245i	$-0.738473\pi$
$719$	−36.5231	−1.36208	−0.681041	+	0.732245i	$0.738473\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	12.6113	0.469018
$724$	0	0
$725$	17.8750	0.663862
$726$	0	0
$727$	−3.65584	−0.135588	−0.0677938	−	0.997699i	$-0.521596\pi$
$727$	−3.65584	−0.135588	−0.0677938	+	0.997699i	$0.521596\pi$
$728$	0	0
$729$	−25.5445	−0.946093
$730$	0	0
$731$	2.29290	0.0848060
$732$	0	0
$733$	27.5214	1.01652	0.508262	−	0.861202i	$-0.330288\pi$
$733$	27.5214	1.01652	0.508262	+	0.861202i	$0.330288\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−82.1101	−3.02456
$738$	0	0
$739$	38.0727	1.40053	0.700264	−	0.713884i	$-0.253066\pi$
$739$	38.0727	1.40053	0.700264	+	0.713884i	$0.253066\pi$
$740$	0	0
$741$	−22.6155	−0.830800
$742$	0	0
$743$	52.8685	1.93956	0.969778	−	0.243990i	$-0.0784564\pi$
$743$	52.8685	1.93956	0.969778	+	0.243990i	$0.0784564\pi$
$744$	0	0
$745$	−12.3742	−0.453357
$746$	0	0
$747$	−14.4103	−0.527247
$748$	0	0
$749$	0	0
$750$	0	0
$751$	10.4052	0.379691	0.189846	−	0.981814i	$-0.439201\pi$
$751$	10.4052	0.379691	0.189846	+	0.981814i	$0.439201\pi$
$752$	0	0
$753$	−65.8983	−2.40147
$754$	0	0
$755$	−8.14459	−0.296412
$756$	0	0
$757$	−24.5697	−0.893000	−0.446500	−	0.894784i	$-0.647330\pi$
$757$	−24.5697	−0.893000	−0.446500	+	0.894784i	$0.647330\pi$
$758$	0	0
$759$	−90.9260	−3.30041
$760$	0	0
$761$	3.52282	0.127702	0.0638511	−	0.997959i	$-0.479662\pi$
$761$	3.52282	0.127702	0.0638511	+	0.997959i	$0.479662\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	2.04562	0.0739596
$766$	0	0
$767$	−29.3097	−1.05831
$768$	0	0
$769$	25.0896	0.904753	0.452377	−	0.891827i	$-0.350576\pi$
$769$	25.0896	0.904753	0.452377	+	0.891827i	$0.350576\pi$
$770$	0	0
$771$	1.59524	0.0574513
$772$	0	0
$773$	−32.0298	−1.15203	−0.576016	−	0.817438i	$-0.695394\pi$
$773$	−32.0298	−1.15203	−0.576016	+	0.817438i	$0.695394\pi$
$774$	0	0
$775$	−7.02073	−0.252192
$776$	0	0
$777$	0	0
$778$	0	0
$779$	15.6514	0.560771
$780$	0	0
$781$	88.8569	3.17955
$782$	0	0
$783$	−0.769933	−0.0275152
$784$	0	0
$785$	−13.1974	−0.471037
$786$	0	0
$787$	−12.4077	−0.442288	−0.221144	−	0.975241i	$-0.570979\pi$
$787$	−12.4077	−0.442288	−0.221144	+	0.975241i	$0.570979\pi$
$788$	0	0
$789$	−53.9145	−1.91941
$790$	0	0
$791$	0	0
$792$	0	0
$793$	16.5351	0.587179
$794$	0	0
$795$	11.1727	0.396254
$796$	0	0
$797$	34.2865	1.21449	0.607245	−	0.794515i	$-0.292275\pi$
$797$	34.2865	1.21449	0.607245	+	0.794515i	$0.292275\pi$
$798$	0	0
$799$	11.9664	0.423342
$800$	0	0
$801$	42.9897	1.51897
$802$	0	0
$803$	52.5205	1.85341
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−17.1427	−0.603451
$808$	0	0
$809$	50.3540	1.77035	0.885176	−	0.465256i	$-0.154038\pi$
$809$	50.3540	1.77035	0.885176	+	0.465256i	$0.154038\pi$
$810$	0	0
$811$	−49.5402	−1.73959	−0.869796	−	0.493412i	$-0.835750\pi$
$811$	−49.5402	−1.73959	−0.869796	+	0.493412i	$0.835750\pi$
$812$	0	0
$813$	62.0096	2.17477
$814$	0	0
$815$	7.14789	0.250379
$816$	0	0
$817$	−6.38423	−0.223356
$818$	0	0
$819$	0	0
$820$	0	0
$821$	12.0853	0.421781	0.210890	−	0.977510i	$-0.432364\pi$
$821$	12.0853	0.421781	0.210890	+	0.977510i	$0.432364\pi$
$822$	0	0
$823$	−10.2751	−0.358166	−0.179083	−	0.983834i	$-0.557313\pi$
$823$	−10.2751	−0.358166	−0.179083	+	0.983834i	$0.557313\pi$
$824$	0	0
$825$	−68.5986	−2.38830
$826$	0	0
$827$	51.7650	1.80005	0.900023	−	0.435843i	$-0.143550\pi$
$827$	51.7650	1.80005	0.900023	+	0.435843i	$0.143550\pi$
$828$	0	0
$829$	−47.9968	−1.66700	−0.833500	−	0.552520i	$-0.813666\pi$
$829$	−47.9968	−1.66700	−0.833500	+	0.552520i	$0.813666\pi$
$830$	0	0
$831$	−78.8614	−2.73567
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−1.43432	−0.0496366
$836$	0	0
$837$	0.302405	0.0104526
$838$	0	0
$839$	−52.5562	−1.81444	−0.907220	−	0.420656i	$-0.861800\pi$
$839$	−52.5562	−1.81444	−0.907220	+	0.420656i	$0.861800\pi$
$840$	0	0
$841$	−13.2863	−0.458147
$842$	0	0
$843$	3.15421	0.108637
$844$	0	0
$845$	1.30036	0.0447337
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−60.8408	−2.08805
$850$	0	0
$851$	15.1574	0.519589
$852$	0	0
$853$	−7.65942	−0.262254	−0.131127	−	0.991366i	$-0.541859\pi$
$853$	−7.65942	−0.262254	−0.131127	+	0.991366i	$0.541859\pi$
$854$	0	0
$855$	−5.69572	−0.194790
$856$	0	0
$857$	48.9114	1.67078	0.835391	−	0.549656i	$-0.185241\pi$
$857$	48.9114	1.67078	0.835391	+	0.549656i	$0.185241\pi$
$858$	0	0
$859$	16.7120	0.570206	0.285103	−	0.958497i	$-0.407972\pi$
$859$	16.7120	0.570206	0.285103	+	0.958497i	$0.407972\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	45.7856	1.55856	0.779279	−	0.626677i	$-0.215585\pi$
$863$	45.7856	1.55856	0.779279	+	0.626677i	$0.215585\pi$
$864$	0	0
$865$	−2.46425	−0.0837871
$866$	0	0
$867$	−2.43314	−0.0826338
$868$	0	0
$869$	−18.9551	−0.643009
$870$	0	0
$871$	43.8400	1.48546
$872$	0	0
$873$	2.62329	0.0887849
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−26.1183	−0.881952	−0.440976	−	0.897519i	$-0.645368\pi$
$877$	−26.1183	−0.881952	−0.440976	+	0.897519i	$0.645368\pi$
$878$	0	0
$879$	−61.9836	−2.09065
$880$	0	0
$881$	23.3437	0.786469	0.393234	−	0.919438i	$-0.371356\pi$
$881$	23.3437	0.786469	0.393234	+	0.919438i	$0.371356\pi$
$882$	0	0
$883$	31.0746	1.04574	0.522872	−	0.852411i	$-0.324861\pi$
$883$	31.0746	1.04574	0.522872	+	0.852411i	$0.324861\pi$
$884$	0	0
$885$	−14.9651	−0.503048
$886$	0	0
$887$	−25.0640	−0.841566	−0.420783	−	0.907161i	$-0.638245\pi$
$887$	−25.0640	−0.841566	−0.420783	+	0.907161i	$0.638245\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	57.7283	1.93397
$892$	0	0
$893$	−33.3187	−1.11497
$894$	0	0
$895$	−2.81470	−0.0940850
$896$	0	0
$897$	48.5470	1.62094
$898$	0	0
$899$	−6.17184	−0.205842
$900$	0	0
$901$	−6.55500	−0.218379
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−2.10557	−0.0699916
$906$	0	0
$907$	56.7568	1.88458	0.942290	−	0.334798i	$-0.108668\pi$
$907$	56.7568	1.88458	0.942290	+	0.334798i	$0.108668\pi$
$908$	0	0
$909$	−19.7197	−0.654060
$910$	0	0
$911$	−6.93958	−0.229919	−0.114959	−	0.993370i	$-0.536674\pi$
$911$	−6.93958	−0.229919	−0.114959	+	0.993370i	$0.536674\pi$
$912$	0	0
$913$	30.8537	1.02111
$914$	0	0
$915$	8.44259	0.279103
$916$	0	0
$917$	0	0
$918$	0	0
$919$	42.2004	1.39206	0.696031	−	0.718012i	$-0.254948\pi$
$919$	42.2004	1.39206	0.696031	+	0.718012i	$0.254948\pi$
$920$	0	0
$921$	−81.6629	−2.69089
$922$	0	0
$923$	−47.4422	−1.56158
$924$	0	0
$925$	11.4354	0.375994
$926$	0	0
$927$	−10.9308	−0.359015
$928$	0	0
$929$	−34.3374	−1.12657	−0.563286	−	0.826262i	$-0.690463\pi$
$929$	−34.3374	−1.12657	−0.563286	+	0.826262i	$0.690463\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−56.5334	−1.85082
$934$	0	0
$935$	−4.37983	−0.143236
$936$	0	0
$937$	−16.5267	−0.539903	−0.269952	−	0.962874i	$-0.587008\pi$
$937$	−16.5267	−0.539903	−0.269952	+	0.962874i	$0.587008\pi$
$938$	0	0
$939$	−73.5747	−2.40102
$940$	0	0
$941$	37.6133	1.22616	0.613079	−	0.790022i	$-0.289931\pi$
$941$	37.6133	1.22616	0.613079	+	0.790022i	$0.289931\pi$
$942$	0	0
$943$	−33.5978	−1.09409
$944$	0	0
$945$	0	0
$946$	0	0
$947$	30.7256	0.998448	0.499224	−	0.866473i	$-0.333618\pi$
$947$	30.7256	0.998448	0.499224	+	0.866473i	$0.333618\pi$
$948$	0	0
$949$	−28.0416	−0.910269
$950$	0	0
$951$	47.3106	1.53415
$952$	0	0
$953$	10.1120	0.327558	0.163779	−	0.986497i	$-0.447632\pi$
$953$	10.1120	0.327558	0.163779	+	0.986497i	$0.447632\pi$
$954$	0	0
$955$	−9.64905	−0.312236
$956$	0	0
$957$	−60.3042	−1.94936
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−28.5759	−0.921803
$962$	0	0
$963$	37.2647	1.20084
$964$	0	0
$965$	5.81183	0.187090
$966$	0	0
$967$	−26.0882	−0.838940	−0.419470	−	0.907769i	$-0.637784\pi$
$967$	−26.0882	−0.838940	−0.419470	+	0.907769i	$0.637784\pi$
$968$	0	0
$969$	6.77471	0.217635
$970$	0	0
$971$	−11.8527	−0.380372	−0.190186	−	0.981748i	$-0.560909\pi$
$971$	−11.8527	−0.380372	−0.190186	+	0.981748i	$0.560909\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	36.6260	1.17297
$976$	0	0
$977$	−13.1874	−0.421902	−0.210951	−	0.977497i	$-0.567656\pi$
$977$	−13.1874	−0.421902	−0.210951	+	0.977497i	$0.567656\pi$
$978$	0	0
$979$	−92.0442	−2.94175
$980$	0	0
$981$	−27.0486	−0.863596
$982$	0	0
$983$	−19.3555	−0.617344	−0.308672	−	0.951169i	$-0.599885\pi$
$983$	−19.3555	−0.617344	−0.308672	+	0.951169i	$0.599885\pi$
$984$	0	0
$985$	−7.43072	−0.236762
$986$	0	0
$987$	0	0
$988$	0	0
$989$	13.7046	0.435780
$990$	0	0
$991$	−5.67221	−0.180184	−0.0900918	−	0.995933i	$-0.528716\pi$
$991$	−5.67221	−0.180184	−0.0900918	+	0.995933i	$0.528716\pi$
$992$	0	0
$993$	66.7832	2.11930
$994$	0	0
$995$	−12.5002	−0.396284
$996$	0	0
$997$	31.7031	1.00405	0.502023	−	0.864854i	$-0.332589\pi$
$997$	31.7031	1.00405	0.502023	+	0.864854i	$0.332589\pi$
$998$	0	0
$999$	−0.492559	−0.0155839

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3332.2.a.u.1.1	yes	8
7.6	odd	2		3332.2.a.t.1.8	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3332.2.a.t.1.8	✓	8	7.6	odd	2
3332.2.a.u.1.1	yes	8	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 3332 = 2^{2} \cdot 7^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3332.a (trivial)

\(f(q)\)	\(=\)	\(q-2.43314 q^{3} -0.700514 q^{5} +2.92017 q^{9} -6.25232 q^{11} +3.33822 q^{13} +1.70445 q^{15} -1.00000 q^{17} +2.78435 q^{19} -5.97696 q^{23} -4.50928 q^{25} +0.194229 q^{27} -3.96405 q^{29} +1.55695 q^{31} +15.2128 q^{33} -2.53597 q^{37} -8.12236 q^{39} +5.62122 q^{41} -2.29290 q^{43} -2.04562 q^{45} -11.9664 q^{47} +2.43314 q^{51} +6.55500 q^{53} +4.37983 q^{55} -6.77471 q^{57} -8.78005 q^{59} +4.95327 q^{61} -2.33847 q^{65} +13.1327 q^{67} +14.5428 q^{69} -14.2118 q^{71} -8.40017 q^{73} +10.9717 q^{75} +3.03170 q^{79} -9.23311 q^{81} -4.93476 q^{83} +0.700514 q^{85} +9.64510 q^{87} +14.7216 q^{89} -3.78828 q^{93} -1.95047 q^{95} +0.898333 q^{97} -18.2578 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{3} + 4 q^{5} + 8 q^{9} - 4 q^{11} + 20 q^{13} + 12 q^{15} - 8 q^{17} + 8 q^{19} + 4 q^{23} + 8 q^{25} + 28 q^{27} - 16 q^{29} - 8 q^{31} + 16 q^{33} + 8 q^{37} + 20 q^{39} + 12 q^{41} - 4 q^{43}+ \cdots - 8 q^{99}+O(q^{100}) \) 8 * q + 4 * q^3 + 4 * q^5 + 8 * q^9 - 4 * q^11 + 20 * q^13 + 12 * q^15 - 8 * q^17 + 8 * q^19 + 4 * q^23 + 8 * q^25 + 28 * q^27 - 16 * q^29 - 8 * q^31 + 16 * q^33 + 8 * q^37 + 20 * q^39 + 12 * q^41 - 4 * q^43 + 20 * q^45 + 4 * q^47 - 4 * q^51 + 12 * q^55 - 16 * q^59 + 32 * q^61 + 44 * q^69 + 24 * q^73 + 24 * q^75 + 4 * q^79 + 36 * q^81 + 28 * q^83 - 4 * q^85 - 40 * q^87 + 20 * q^89 - 16 * q^93 - 20 * q^95 + 56 * q^97 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more