LMFDB - Embedded newform 3800.2.a.bb.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3800,2,Mod(1,3800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3800, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3800.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3800 = 2^{3} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3800.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:	$30.3431527681$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 12x^{4} + 16x^{3} + 33x^{2} - 4x - 7 $ x^6 - 2x^5 - 12x^4 + 16x^3 + 33x^2 - 4*x - 7
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-2.79951$ of defining polynomial
Character	$\chi$	$=$	3800.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.79951 q^{3} +0.127550 q^{7} +4.83726 q^{9} +O(q^{10})$	$q+2.79951 q^{3} +0.127550 q^{7} +4.83726 q^{9} +5.21326 q^{11} -0.515371 q^{13} +3.58010 q^{17} +1.00000 q^{19} +0.357078 q^{21} -6.50922 q^{23} +5.14343 q^{27} -3.05772 q^{29} +5.44243 q^{31} +14.5946 q^{33} -4.99696 q^{37} -1.44279 q^{39} +11.4394 q^{41} -2.06955 q^{43} +11.9078 q^{47} -6.98373 q^{49} +10.0225 q^{51} +2.25821 q^{53} +2.79951 q^{57} +1.89137 q^{59} -5.83111 q^{61} +0.616994 q^{63} +0.432668 q^{67} -18.2226 q^{69} +4.77748 q^{71} +9.98684 q^{73} +0.664953 q^{77} -3.28280 q^{79} -0.112689 q^{81} +0.496452 q^{83} -8.56013 q^{87} -12.4720 q^{89} -0.0657358 q^{91} +15.2361 q^{93} +7.00361 q^{97} +25.2179 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{3} - 2 q^{7} + 10 q^{9}+O(q^{10}) $ 6 * q - 2 * q^3 - 2 * q^7 + 10 * q^9	$ 6 q - 2 q^{3} - 2 q^{7} + 10 q^{9} + 3 q^{11} + 3 q^{13} + 2 q^{17} + 6 q^{19} + 11 q^{21} - 4 q^{23} - 20 q^{27} + 7 q^{29} + 5 q^{31} + 16 q^{33} + 8 q^{39} + 11 q^{41} + 7 q^{43} - 20 q^{47} - 2 q^{49} + 13 q^{51} + 7 q^{53} - 2 q^{57} - 4 q^{59} + 13 q^{61} + q^{63} - 25 q^{67} + 7 q^{69} + 29 q^{71} + 19 q^{73} - 24 q^{77} + 28 q^{79} + 38 q^{81} + 15 q^{83} - 57 q^{87} - 12 q^{89} + 27 q^{93} + 13 q^{97} + 10 q^{99}+O(q^{100}) $ 6 * q - 2 * q^3 - 2 * q^7 + 10 * q^9 + 3 * q^11 + 3 * q^13 + 2 * q^17 + 6 * q^19 + 11 * q^21 - 4 * q^23 - 20 * q^27 + 7 * q^29 + 5 * q^31 + 16 * q^33 + 8 * q^39 + 11 * q^41 + 7 * q^43 - 20 * q^47 - 2 * q^49 + 13 * q^51 + 7 * q^53 - 2 * q^57 - 4 * q^59 + 13 * q^61 + q^63 - 25 * q^67 + 7 * q^69 + 29 * q^71 + 19 * q^73 - 24 * q^77 + 28 * q^79 + 38 * q^81 + 15 * q^83 - 57 * q^87 - 12 * q^89 + 27 * q^93 + 13 * q^97 + 10 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	2.79951	1.61630	0.808149	−	0.588978i	$-0.200469\pi$
$3$	2.79951	1.61630	0.808149	+	0.588978i	$0.200469\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0.127550	0.0482095	0.0241047	−	0.999709i	$-0.492326\pi$
$7$	0.127550	0.0482095	0.0241047	+	0.999709i	$0.492326\pi$
$8$	0	0
$9$	4.83726	1.61242
$10$	0	0
$11$	5.21326	1.57186	0.785928	−	0.618318i	$-0.212185\pi$
$11$	5.21326	1.57186	0.785928	+	0.618318i	$0.212185\pi$
$12$	0	0
$13$	−0.515371	−0.142938	−0.0714691	−	0.997443i	$-0.522769\pi$
$13$	−0.515371	−0.142938	−0.0714691	+	0.997443i	$0.522769\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.58010	0.868302	0.434151	−	0.900840i	$-0.357048\pi$
$17$	3.58010	0.868302	0.434151	+	0.900840i	$0.357048\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0.357078	0.0779209
$22$	0	0
$23$	−6.50922	−1.35727	−0.678633	−	0.734477i	$-0.737427\pi$
$23$	−6.50922	−1.35727	−0.678633	+	0.734477i	$0.737427\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	5.14343	0.989854
$28$	0	0
$29$	−3.05772	−0.567805	−0.283902	−	0.958853i	$-0.591629\pi$
$29$	−3.05772	−0.567805	−0.283902	+	0.958853i	$0.591629\pi$
$30$	0	0
$31$	5.44243	0.977490	0.488745	−	0.872427i	$-0.337455\pi$
$31$	5.44243	0.977490	0.488745	+	0.872427i	$0.337455\pi$
$32$	0	0
$33$	14.5946	2.54059
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−4.99696	−0.821495	−0.410748	−	0.911749i	$-0.634732\pi$
$37$	−4.99696	−0.821495	−0.410748	+	0.911749i	$0.634732\pi$
$38$	0	0
$39$	−1.44279	−0.231031
$40$	0	0
$41$	11.4394	1.78653	0.893267	−	0.449527i	$-0.148408\pi$
$41$	11.4394	1.78653	0.893267	+	0.449527i	$0.148408\pi$
$42$	0	0
$43$	−2.06955	−0.315603	−0.157801	−	0.987471i	$-0.550441\pi$
$43$	−2.06955	−0.315603	−0.157801	+	0.987471i	$0.550441\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	11.9078	1.73692	0.868462	−	0.495755i	$-0.165109\pi$
$47$	11.9078	1.73692	0.868462	+	0.495755i	$0.165109\pi$
$48$	0	0
$49$	−6.98373	−0.997676
$50$	0	0
$51$	10.0225	1.40344
$52$	0	0
$53$	2.25821	0.310189	0.155095	−	0.987900i	$-0.450432\pi$
$53$	2.25821	0.310189	0.155095	+	0.987900i	$0.450432\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	2.79951	0.370804
$58$	0	0
$59$	1.89137	0.246235	0.123118	−	0.992392i	$-0.460711\pi$
$59$	1.89137	0.246235	0.123118	+	0.992392i	$0.460711\pi$
$60$	0	0
$61$	−5.83111	−0.746597	−0.373299	−	0.927711i	$-0.621773\pi$
$61$	−5.83111	−0.746597	−0.373299	+	0.927711i	$0.621773\pi$
$62$	0	0
$63$	0.616994	0.0777340
$64$	0	0
$65$	0	0
$66$	0	0
$67$	0.432668	0.0528588	0.0264294	−	0.999651i	$-0.491586\pi$
$67$	0.432668	0.0528588	0.0264294	+	0.999651i	$0.491586\pi$
$68$	0	0
$69$	−18.2226	−2.19375
$70$	0	0
$71$	4.77748	0.566983	0.283491	−	0.958975i	$-0.408507\pi$
$71$	4.77748	0.566983	0.283491	+	0.958975i	$0.408507\pi$
$72$	0	0
$73$	9.98684	1.16887	0.584436	−	0.811440i	$-0.301316\pi$
$73$	9.98684	1.16887	0.584436	+	0.811440i	$0.301316\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0.664953	0.0757784
$78$	0	0
$79$	−3.28280	−0.369344	−0.184672	−	0.982800i	$-0.559122\pi$
$79$	−3.28280	−0.369344	−0.184672	+	0.982800i	$0.559122\pi$
$80$	0	0
$81$	−0.112689	−0.0125211
$82$	0	0
$83$	0.496452	0.0544926	0.0272463	−	0.999629i	$-0.491326\pi$
$83$	0.496452	0.0544926	0.0272463	+	0.999629i	$0.491326\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−8.56013	−0.917742
$88$	0	0
$89$	−12.4720	−1.32203	−0.661013	−	0.750374i	$-0.729873\pi$
$89$	−12.4720	−1.32203	−0.661013	+	0.750374i	$0.729873\pi$
$90$	0	0
$91$	−0.0657358	−0.00689098
$92$	0	0
$93$	15.2361	1.57991
$94$	0	0
$95$	0	0
$96$	0	0
$97$	7.00361	0.711109	0.355555	−	0.934655i	$-0.384292\pi$
$97$	7.00361	0.711109	0.355555	+	0.934655i	$0.384292\pi$
$98$	0	0
$99$	25.2179	2.53449
$100$	0	0
$101$	−10.9573	−1.09029	−0.545147	−	0.838340i	$-0.683526\pi$
$101$	−10.9573	−1.09029	−0.545147	+	0.838340i	$0.683526\pi$
$102$	0	0
$103$	−11.0593	−1.08971	−0.544854	−	0.838531i	$-0.683415\pi$
$103$	−11.0593	−1.08971	−0.544854	+	0.838531i	$0.683415\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−13.0145	−1.25816	−0.629078	−	0.777342i	$-0.716568\pi$
$107$	−13.0145	−1.25816	−0.629078	+	0.777342i	$0.716568\pi$
$108$	0	0
$109$	7.65171	0.732901	0.366450	−	0.930438i	$-0.380573\pi$
$109$	7.65171	0.732901	0.366450	+	0.930438i	$0.380573\pi$
$110$	0	0
$111$	−13.9890	−1.32778
$112$	0	0
$113$	−9.93850	−0.934935	−0.467468	−	0.884010i	$-0.654834\pi$
$113$	−9.93850	−0.934935	−0.467468	+	0.884010i	$0.654834\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−2.49298	−0.230477
$118$	0	0
$119$	0.456643	0.0418604
$120$	0	0
$121$	16.1781	1.47073
$122$	0	0
$123$	32.0247	2.88757
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−8.54647	−0.758376	−0.379188	−	0.925320i	$-0.623797\pi$
$127$	−8.54647	−0.758376	−0.379188	+	0.925320i	$0.623797\pi$
$128$	0	0
$129$	−5.79372	−0.510108
$130$	0	0
$131$	17.0384	1.48865	0.744327	−	0.667816i	$-0.232771\pi$
$131$	17.0384	1.48865	0.744327	+	0.667816i	$0.232771\pi$
$132$	0	0
$133$	0.127550	0.0110600
$134$	0	0
$135$	0	0
$136$	0	0
$137$	6.28203	0.536710	0.268355	−	0.963320i	$-0.413520\pi$
$137$	6.28203	0.536710	0.268355	+	0.963320i	$0.413520\pi$
$138$	0	0
$139$	1.87860	0.159341	0.0796704	−	0.996821i	$-0.474613\pi$
$139$	1.87860	0.159341	0.0796704	+	0.996821i	$0.474613\pi$
$140$	0	0
$141$	33.3359	2.80739
$142$	0	0
$143$	−2.68676	−0.224678
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−19.5510	−1.61254
$148$	0	0
$149$	−1.98213	−0.162383	−0.0811914	−	0.996699i	$-0.525872\pi$
$149$	−1.98213	−0.162383	−0.0811914	+	0.996699i	$0.525872\pi$
$150$	0	0
$151$	1.05971	0.0862378	0.0431189	−	0.999070i	$-0.486271\pi$
$151$	1.05971	0.0862378	0.0431189	+	0.999070i	$0.486271\pi$
$152$	0	0
$153$	17.3179	1.40007
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−11.4832	−0.916458	−0.458229	−	0.888834i	$-0.651516\pi$
$157$	−11.4832	−0.916458	−0.458229	+	0.888834i	$0.651516\pi$
$158$	0	0
$159$	6.32189	0.501358
$160$	0	0
$161$	−0.830253	−0.0654331
$162$	0	0
$163$	0.595407	0.0466359	0.0233179	−	0.999728i	$-0.492577\pi$
$163$	0.595407	0.0466359	0.0233179	+	0.999728i	$0.492577\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	7.77000	0.601260	0.300630	−	0.953741i	$-0.402803\pi$
$167$	7.77000	0.601260	0.300630	+	0.953741i	$0.402803\pi$
$168$	0	0
$169$	−12.7344	−0.979569
$170$	0	0
$171$	4.83726	0.369915
$172$	0	0
$173$	−7.63268	−0.580302	−0.290151	−	0.956981i	$-0.593706\pi$
$173$	−7.63268	−0.580302	−0.290151	+	0.956981i	$0.593706\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	5.29491	0.397990
$178$	0	0
$179$	21.7950	1.62903	0.814516	−	0.580141i	$-0.197003\pi$
$179$	21.7950	1.62903	0.814516	+	0.580141i	$0.197003\pi$
$180$	0	0
$181$	25.0255	1.86013	0.930067	−	0.367390i	$-0.119749\pi$
$181$	25.0255	1.86013	0.930067	+	0.367390i	$0.119749\pi$
$182$	0	0
$183$	−16.3243	−1.20672
$184$	0	0
$185$	0	0
$186$	0	0
$187$	18.6640	1.36485
$188$	0	0
$189$	0.656046	0.0477204
$190$	0	0
$191$	−0.291147	−0.0210667	−0.0105333	−	0.999945i	$-0.503353\pi$
$191$	−0.291147	−0.0210667	−0.0105333	+	0.999945i	$0.503353\pi$
$192$	0	0
$193$	−5.23615	−0.376906	−0.188453	−	0.982082i	$-0.560347\pi$
$193$	−5.23615	−0.376906	−0.188453	+	0.982082i	$0.560347\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−21.5134	−1.53277	−0.766384	−	0.642383i	$-0.777946\pi$
$197$	−21.5134	−1.53277	−0.766384	+	0.642383i	$0.777946\pi$
$198$	0	0
$199$	22.4064	1.58834	0.794172	−	0.607693i	$-0.207905\pi$
$199$	22.4064	1.58834	0.794172	+	0.607693i	$0.207905\pi$
$200$	0	0
$201$	1.21126	0.0854356
$202$	0	0
$203$	−0.390014	−0.0273736
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−31.4868	−2.18848
$208$	0	0
$209$	5.21326	0.360609
$210$	0	0
$211$	−6.01277	−0.413936	−0.206968	−	0.978348i	$-0.566360\pi$
$211$	−6.01277	−0.413936	−0.206968	+	0.978348i	$0.566360\pi$
$212$	0	0
$213$	13.3746	0.916413
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0.694184	0.0471243
$218$	0	0
$219$	27.9583	1.88925
$220$	0	0
$221$	−1.84508	−0.124114
$222$	0	0
$223$	21.6542	1.45007	0.725035	−	0.688712i	$-0.241824\pi$
$223$	21.6542	1.45007	0.725035	+	0.688712i	$0.241824\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−15.8894	−1.05462	−0.527308	−	0.849674i	$-0.676799\pi$
$227$	−15.8894	−1.05462	−0.527308	+	0.849674i	$0.676799\pi$
$228$	0	0
$229$	9.32826	0.616429	0.308214	−	0.951317i	$-0.400269\pi$
$229$	9.32826	0.616429	0.308214	+	0.951317i	$0.400269\pi$
$230$	0	0
$231$	1.86154	0.122481
$232$	0	0
$233$	14.5960	0.956215	0.478107	−	0.878301i	$-0.341323\pi$
$233$	14.5960	0.956215	0.478107	+	0.878301i	$0.341323\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−9.19025	−0.596971
$238$	0	0
$239$	4.85092	0.313780	0.156890	−	0.987616i	$-0.449853\pi$
$239$	4.85092	0.313780	0.156890	+	0.987616i	$0.449853\pi$
$240$	0	0
$241$	13.7085	0.883043	0.441521	−	0.897251i	$-0.354439\pi$
$241$	13.7085	0.883043	0.441521	+	0.897251i	$0.354439\pi$
$242$	0	0
$243$	−15.7458	−1.01009
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−0.515371	−0.0327923
$248$	0	0
$249$	1.38982	0.0880764
$250$	0	0
$251$	−26.3986	−1.66626	−0.833131	−	0.553076i	$-0.813454\pi$
$251$	−26.3986	−1.66626	−0.833131	+	0.553076i	$0.813454\pi$
$252$	0	0
$253$	−33.9343	−2.13343
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−15.4439	−0.963363	−0.481681	−	0.876346i	$-0.659974\pi$
$257$	−15.4439	−0.963363	−0.481681	+	0.876346i	$0.659974\pi$
$258$	0	0
$259$	−0.637364	−0.0396039
$260$	0	0
$261$	−14.7910	−0.915540
$262$	0	0
$263$	−29.7108	−1.83204	−0.916022	−	0.401127i	$-0.868619\pi$
$263$	−29.7108	−1.83204	−0.916022	+	0.401127i	$0.868619\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−34.9154	−2.13679
$268$	0	0
$269$	27.1894	1.65776	0.828882	−	0.559423i	$-0.188977\pi$
$269$	27.1894	1.65776	0.828882	+	0.559423i	$0.188977\pi$
$270$	0	0
$271$	−28.1742	−1.71146	−0.855731	−	0.517421i	$-0.826892\pi$
$271$	−28.1742	−1.71146	−0.855731	+	0.517421i	$0.826892\pi$
$272$	0	0
$273$	−0.184028	−0.0111379
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−8.63783	−0.518997	−0.259498	−	0.965744i	$-0.583557\pi$
$277$	−8.63783	−0.518997	−0.259498	+	0.965744i	$0.583557\pi$
$278$	0	0
$279$	26.3265	1.57612
$280$	0	0
$281$	6.87156	0.409923	0.204961	−	0.978770i	$-0.434293\pi$
$281$	6.87156	0.409923	0.204961	+	0.978770i	$0.434293\pi$
$282$	0	0
$283$	9.90553	0.588823	0.294411	−	0.955679i	$-0.404876\pi$
$283$	9.90553	0.588823	0.294411	+	0.955679i	$0.404876\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1.45910	0.0861279
$288$	0	0
$289$	−4.18287	−0.246051
$290$	0	0
$291$	19.6067	1.14936
$292$	0	0
$293$	27.3706	1.59901	0.799504	−	0.600661i	$-0.205096\pi$
$293$	27.3706	1.59901	0.799504	+	0.600661i	$0.205096\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	26.8140	1.55591
$298$	0	0
$299$	3.35467	0.194005
$300$	0	0
$301$	−0.263971	−0.0152150
$302$	0	0
$303$	−30.6752	−1.76224
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−16.8240	−0.960194	−0.480097	−	0.877215i	$-0.659399\pi$
$307$	−16.8240	−0.960194	−0.480097	+	0.877215i	$0.659399\pi$
$308$	0	0
$309$	−30.9607	−1.76129
$310$	0	0
$311$	22.3539	1.26758	0.633788	−	0.773507i	$-0.281499\pi$
$311$	22.3539	1.26758	0.633788	+	0.773507i	$0.281499\pi$
$312$	0	0
$313$	−6.08636	−0.344021	−0.172011	−	0.985095i	$-0.555026\pi$
$313$	−6.08636	−0.344021	−0.172011	+	0.985095i	$0.555026\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	25.8523	1.45201	0.726005	−	0.687689i	$-0.241375\pi$
$317$	25.8523	1.45201	0.726005	+	0.687689i	$0.241375\pi$
$318$	0	0
$319$	−15.9407	−0.892508
$320$	0	0
$321$	−36.4342	−2.03356
$322$	0	0
$323$	3.58010	0.199202
$324$	0	0
$325$	0	0
$326$	0	0
$327$	21.4210	1.18459
$328$	0	0
$329$	1.51884	0.0837362
$330$	0	0
$331$	−23.1983	−1.27509	−0.637546	−	0.770412i	$-0.720050\pi$
$331$	−23.1983	−1.27509	−0.637546	+	0.770412i	$0.720050\pi$
$332$	0	0
$333$	−24.1716	−1.32460
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−19.1802	−1.04481	−0.522405	−	0.852698i	$-0.674965\pi$
$337$	−19.1802	−1.04481	−0.522405	+	0.852698i	$0.674965\pi$
$338$	0	0
$339$	−27.8229	−1.51113
$340$	0	0
$341$	28.3728	1.53647
$342$	0	0
$343$	−1.78363	−0.0963069
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−7.90306	−0.424259	−0.212129	−	0.977242i	$-0.568040\pi$
$347$	−7.90306	−0.424259	−0.212129	+	0.977242i	$0.568040\pi$
$348$	0	0
$349$	−0.888437	−0.0475570	−0.0237785	−	0.999717i	$-0.507570\pi$
$349$	−0.888437	−0.0475570	−0.0237785	+	0.999717i	$0.507570\pi$
$350$	0	0
$351$	−2.65078	−0.141488
$352$	0	0
$353$	23.0005	1.22419	0.612097	−	0.790783i	$-0.290326\pi$
$353$	23.0005	1.22419	0.612097	+	0.790783i	$0.290326\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	1.27838	0.0676589
$358$	0	0
$359$	−32.3510	−1.70742	−0.853709	−	0.520750i	$-0.825653\pi$
$359$	−32.3510	−1.70742	−0.853709	+	0.520750i	$0.825653\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	45.2907	2.37714
$364$	0	0
$365$	0	0
$366$	0	0
$367$	8.68117	0.453154	0.226577	−	0.973993i	$-0.427247\pi$
$367$	8.68117	0.453154	0.226577	+	0.973993i	$0.427247\pi$
$368$	0	0
$369$	55.3353	2.88064
$370$	0	0
$371$	0.288036	0.0149541
$372$	0	0
$373$	−28.0175	−1.45069	−0.725346	−	0.688384i	$-0.758320\pi$
$373$	−28.0175	−1.45069	−0.725346	+	0.688384i	$0.758320\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	1.57586	0.0811610
$378$	0	0
$379$	−3.74032	−0.192127	−0.0960637	−	0.995375i	$-0.530625\pi$
$379$	−3.74032	−0.192127	−0.0960637	+	0.995375i	$0.530625\pi$
$380$	0	0
$381$	−23.9259	−1.22576
$382$	0	0
$383$	2.88209	0.147268	0.0736340	−	0.997285i	$-0.476540\pi$
$383$	2.88209	0.147268	0.0736340	+	0.997285i	$0.476540\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−10.0109	−0.508884
$388$	0	0
$389$	−6.50836	−0.329987	−0.164994	−	0.986295i	$-0.552760\pi$
$389$	−6.50836	−0.329987	−0.164994	+	0.986295i	$0.552760\pi$
$390$	0	0
$391$	−23.3037	−1.17852
$392$	0	0
$393$	47.6992	2.40611
$394$	0	0
$395$	0	0
$396$	0	0
$397$	0.814506	0.0408789	0.0204394	−	0.999791i	$-0.493493\pi$
$397$	0.814506	0.0408789	0.0204394	+	0.999791i	$0.493493\pi$
$398$	0	0
$399$	0.357078	0.0178763
$400$	0	0
$401$	−20.4471	−1.02108	−0.510540	−	0.859854i	$-0.670555\pi$
$401$	−20.4471	−1.02108	−0.510540	+	0.859854i	$0.670555\pi$
$402$	0	0
$403$	−2.80487	−0.139721
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−26.0505	−1.29127
$408$	0	0
$409$	−2.25738	−0.111620	−0.0558102	−	0.998441i	$-0.517774\pi$
$409$	−2.25738	−0.111620	−0.0558102	+	0.998441i	$0.517774\pi$
$410$	0	0
$411$	17.5866	0.867484
$412$	0	0
$413$	0.241245	0.0118709
$414$	0	0
$415$	0	0
$416$	0	0
$417$	5.25916	0.257542
$418$	0	0
$419$	−37.5231	−1.83312	−0.916562	−	0.399893i	$-0.869047\pi$
$419$	−37.5231	−1.83312	−0.916562	+	0.399893i	$0.869047\pi$
$420$	0	0
$421$	26.0909	1.27159	0.635797	−	0.771856i	$-0.280672\pi$
$421$	26.0909	1.27159	0.635797	+	0.771856i	$0.280672\pi$
$422$	0	0
$423$	57.6009	2.80065
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−0.743760	−0.0359931
$428$	0	0
$429$	−7.52162	−0.363147
$430$	0	0
$431$	−31.1228	−1.49913	−0.749566	−	0.661930i	$-0.769738\pi$
$431$	−31.1228	−1.49913	−0.749566	+	0.661930i	$0.769738\pi$
$432$	0	0
$433$	−19.0781	−0.916835	−0.458417	−	0.888737i	$-0.651583\pi$
$433$	−19.0781	−0.916835	−0.458417	+	0.888737i	$0.651583\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−6.50922	−0.311378
$438$	0	0
$439$	−4.46337	−0.213025	−0.106513	−	0.994311i	$-0.533968\pi$
$439$	−4.46337	−0.213025	−0.106513	+	0.994311i	$0.533968\pi$
$440$	0	0
$441$	−33.7821	−1.60867
$442$	0	0
$443$	−29.8652	−1.41894	−0.709469	−	0.704736i	$-0.751065\pi$
$443$	−29.8652	−1.41894	−0.709469	+	0.704736i	$0.751065\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−5.54900	−0.262459
$448$	0	0
$449$	5.65190	0.266730	0.133365	−	0.991067i	$-0.457422\pi$
$449$	5.65190	0.266730	0.133365	+	0.991067i	$0.457422\pi$
$450$	0	0
$451$	59.6365	2.80817
$452$	0	0
$453$	2.96666	0.139386
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−4.36131	−0.204014	−0.102007	−	0.994784i	$-0.532526\pi$
$457$	−4.36131	−0.204014	−0.102007	+	0.994784i	$0.532526\pi$
$458$	0	0
$459$	18.4140	0.859492
$460$	0	0
$461$	−13.9282	−0.648699	−0.324349	−	0.945937i	$-0.605145\pi$
$461$	−13.9282	−0.648699	−0.324349	+	0.945937i	$0.605145\pi$
$462$	0	0
$463$	−39.1712	−1.82044	−0.910221	−	0.414124i	$-0.864088\pi$
$463$	−39.1712	−1.82044	−0.910221	+	0.414124i	$0.864088\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−28.8967	−1.33718	−0.668589	−	0.743632i	$-0.733101\pi$
$467$	−28.8967	−1.33718	−0.668589	+	0.743632i	$0.733101\pi$
$468$	0	0
$469$	0.0551869	0.00254830
$470$	0	0
$471$	−32.1473	−1.48127
$472$	0	0
$473$	−10.7891	−0.496082
$474$	0	0
$475$	0	0
$476$	0	0
$477$	10.9236	0.500156
$478$	0	0
$479$	1.24704	0.0569787	0.0284893	−	0.999594i	$-0.490930\pi$
$479$	1.24704	0.0569787	0.0284893	+	0.999594i	$0.490930\pi$
$480$	0	0
$481$	2.57529	0.117423
$482$	0	0
$483$	−2.32430	−0.105759
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−25.0138	−1.13348	−0.566742	−	0.823895i	$-0.691796\pi$
$487$	−25.0138	−1.13348	−0.566742	+	0.823895i	$0.691796\pi$
$488$	0	0
$489$	1.66685	0.0753775
$490$	0	0
$491$	10.3761	0.468267	0.234134	−	0.972204i	$-0.424775\pi$
$491$	10.3761	0.468267	0.234134	+	0.972204i	$0.424775\pi$
$492$	0	0
$493$	−10.9470	−0.493026
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0.609369	0.0273339
$498$	0	0
$499$	24.7623	1.10851	0.554256	−	0.832347i	$-0.313003\pi$
$499$	24.7623	1.10851	0.554256	+	0.832347i	$0.313003\pi$
$500$	0	0
$501$	21.7522	0.971816
$502$	0	0
$503$	−29.5651	−1.31824	−0.659122	−	0.752036i	$-0.729072\pi$
$503$	−29.5651	−1.31824	−0.659122	+	0.752036i	$0.729072\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−35.6501	−1.58328
$508$	0	0
$509$	−2.82819	−0.125358	−0.0626788	−	0.998034i	$-0.519964\pi$
$509$	−2.82819	−0.125358	−0.0626788	+	0.998034i	$0.519964\pi$
$510$	0	0
$511$	1.27382	0.0563507
$512$	0	0
$513$	5.14343	0.227088
$514$	0	0
$515$	0	0
$516$	0	0
$517$	62.0782	2.73020
$518$	0	0
$519$	−21.3678	−0.937941
$520$	0	0
$521$	21.5226	0.942921	0.471460	−	0.881887i	$-0.343727\pi$
$521$	21.5226	0.942921	0.471460	+	0.881887i	$0.343727\pi$
$522$	0	0
$523$	−27.1156	−1.18568	−0.592840	−	0.805320i	$-0.701993\pi$
$523$	−27.1156	−1.18568	−0.592840	+	0.805320i	$0.701993\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	19.4845	0.848757
$528$	0	0
$529$	19.3700	0.842172
$530$	0	0
$531$	9.14905	0.397035
$532$	0	0
$533$	−5.89553	−0.255364
$534$	0	0
$535$	0	0
$536$	0	0
$537$	61.0152	2.63300
$538$	0	0
$539$	−36.4080	−1.56820
$540$	0	0
$541$	32.0041	1.37596	0.687982	−	0.725728i	$-0.258497\pi$
$541$	32.0041	1.37596	0.687982	+	0.725728i	$0.258497\pi$
$542$	0	0
$543$	70.0593	3.00653
$544$	0	0
$545$	0	0
$546$	0	0
$547$	28.4212	1.21520	0.607601	−	0.794242i	$-0.292132\pi$
$547$	28.4212	1.21520	0.607601	+	0.794242i	$0.292132\pi$
$548$	0	0
$549$	−28.2066	−1.20383
$550$	0	0
$551$	−3.05772	−0.130263
$552$	0	0
$553$	−0.418723	−0.0178059
$554$	0	0
$555$	0	0
$556$	0	0
$557$	13.0041	0.551000	0.275500	−	0.961301i	$-0.411157\pi$
$557$	13.0041	0.551000	0.275500	+	0.961301i	$0.411157\pi$
$558$	0	0
$559$	1.06658	0.0451117
$560$	0	0
$561$	52.2501	2.20600
$562$	0	0
$563$	−6.28547	−0.264901	−0.132451	−	0.991190i	$-0.542285\pi$
$563$	−6.28547	−0.264901	−0.132451	+	0.991190i	$0.542285\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−0.0143736	−0.000603634 0
$568$	0	0
$569$	−9.12254	−0.382437	−0.191218	−	0.981548i	$-0.561244\pi$
$569$	−9.12254	−0.382437	−0.191218	+	0.981548i	$0.561244\pi$
$570$	0	0
$571$	12.7709	0.534444	0.267222	−	0.963635i	$-0.413894\pi$
$571$	12.7709	0.534444	0.267222	+	0.963635i	$0.413894\pi$
$572$	0	0
$573$	−0.815070	−0.0340500
$574$	0	0
$575$	0	0
$576$	0	0
$577$	13.4160	0.558514	0.279257	−	0.960216i	$-0.409912\pi$
$577$	13.4160	0.558514	0.279257	+	0.960216i	$0.409912\pi$
$578$	0	0
$579$	−14.6587	−0.609193
$580$	0	0
$581$	0.0633225	0.00262706
$582$	0	0
$583$	11.7726	0.487573
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−27.6937	−1.14304	−0.571521	−	0.820587i	$-0.693647\pi$
$587$	−27.6937	−1.14304	−0.571521	+	0.820587i	$0.693647\pi$
$588$	0	0
$589$	5.44243	0.224252
$590$	0	0
$591$	−60.2271	−2.47741
$592$	0	0
$593$	46.2283	1.89837	0.949185	−	0.314718i	$-0.101910\pi$
$593$	46.2283	1.89837	0.949185	+	0.314718i	$0.101910\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	62.7268	2.56724
$598$	0	0
$599$	−25.9606	−1.06072	−0.530360	−	0.847772i	$-0.677943\pi$
$599$	−25.9606	−1.06072	−0.530360	+	0.847772i	$0.677943\pi$
$600$	0	0
$601$	35.6322	1.45347	0.726735	−	0.686918i	$-0.241037\pi$
$601$	35.6322	1.45347	0.726735	+	0.686918i	$0.241037\pi$
$602$	0	0
$603$	2.09293	0.0852306
$604$	0	0
$605$	0	0
$606$	0	0
$607$	39.5302	1.60448	0.802242	−	0.597000i	$-0.203641\pi$
$607$	39.5302	1.60448	0.802242	+	0.597000i	$0.203641\pi$
$608$	0	0
$609$	−1.09185	−0.0442439
$610$	0	0
$611$	−6.13691	−0.248273
$612$	0	0
$613$	−22.2903	−0.900296	−0.450148	−	0.892954i	$-0.648629\pi$
$613$	−22.2903	−0.900296	−0.450148	+	0.892954i	$0.648629\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−31.2400	−1.25768	−0.628838	−	0.777537i	$-0.716469\pi$
$617$	−31.2400	−1.25768	−0.628838	+	0.777537i	$0.716469\pi$
$618$	0	0
$619$	35.0032	1.40690	0.703449	−	0.710746i	$-0.251642\pi$
$619$	35.0032	1.40690	0.703449	+	0.710746i	$0.251642\pi$
$620$	0	0
$621$	−33.4797	−1.34350
$622$	0	0
$623$	−1.59080	−0.0637342
$624$	0	0
$625$	0	0
$626$	0	0
$627$	14.5946	0.582851
$628$	0	0
$629$	−17.8896	−0.713306
$630$	0	0
$631$	−23.0271	−0.916695	−0.458347	−	0.888773i	$-0.651558\pi$
$631$	−23.0271	−0.916695	−0.458347	+	0.888773i	$0.651558\pi$
$632$	0	0
$633$	−16.8328	−0.669044
$634$	0	0
$635$	0	0
$636$	0	0
$637$	3.59921	0.142606
$638$	0	0
$639$	23.1099	0.914214
$640$	0	0
$641$	15.0359	0.593883	0.296942	−	0.954896i	$-0.404033\pi$
$641$	15.0359	0.593883	0.296942	+	0.954896i	$0.404033\pi$
$642$	0	0
$643$	−26.5794	−1.04819	−0.524095	−	0.851660i	$-0.675596\pi$
$643$	−26.5794	−1.04819	−0.524095	+	0.851660i	$0.675596\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	38.9210	1.53014	0.765071	−	0.643945i	$-0.222704\pi$
$647$	38.9210	1.53014	0.765071	+	0.643945i	$0.222704\pi$
$648$	0	0
$649$	9.86020	0.387047
$650$	0	0
$651$	1.94338	0.0761669
$652$	0	0
$653$	−18.7392	−0.733320	−0.366660	−	0.930355i	$-0.619499\pi$
$653$	−18.7392	−0.733320	−0.366660	+	0.930355i	$0.619499\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	48.3090	1.88471
$658$	0	0
$659$	−45.9031	−1.78813	−0.894066	−	0.447936i	$-0.852159\pi$
$659$	−45.9031	−1.78813	−0.894066	+	0.447936i	$0.852159\pi$
$660$	0	0
$661$	−11.3827	−0.442735	−0.221368	−	0.975190i	$-0.571052\pi$
$661$	−11.3827	−0.442735	−0.221368	+	0.975190i	$0.571052\pi$
$662$	0	0
$663$	−5.16533	−0.200605
$664$	0	0
$665$	0	0
$666$	0	0
$667$	19.9034	0.770663
$668$	0	0
$669$	60.6211	2.34375
$670$	0	0
$671$	−30.3991	−1.17354
$672$	0	0
$673$	33.0693	1.27473	0.637365	−	0.770562i	$-0.280024\pi$
$673$	33.0693	1.27473	0.637365	+	0.770562i	$0.280024\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	12.3049	0.472917	0.236458	−	0.971642i	$-0.424013\pi$
$677$	12.3049	0.472917	0.236458	+	0.971642i	$0.424013\pi$
$678$	0	0
$679$	0.893313	0.0342822
$680$	0	0
$681$	−44.4825	−1.70457
$682$	0	0
$683$	−22.4044	−0.857280	−0.428640	−	0.903475i	$-0.641007\pi$
$683$	−22.4044	−0.857280	−0.428640	+	0.903475i	$0.641007\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	26.1146	0.996333
$688$	0	0
$689$	−1.16382	−0.0443379
$690$	0	0
$691$	24.7072	0.939905	0.469952	−	0.882692i	$-0.344271\pi$
$691$	24.7072	0.939905	0.469952	+	0.882692i	$0.344271\pi$
$692$	0	0
$693$	3.21655	0.122187
$694$	0	0
$695$	0	0
$696$	0	0
$697$	40.9542	1.55125
$698$	0	0
$699$	40.8616	1.54553
$700$	0	0
$701$	48.4470	1.82982	0.914908	−	0.403662i	$-0.132263\pi$
$701$	48.4470	1.82982	0.914908	+	0.403662i	$0.132263\pi$
$702$	0	0
$703$	−4.99696	−0.188464
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−1.39761	−0.0525626
$708$	0	0
$709$	−17.3382	−0.651150	−0.325575	−	0.945516i	$-0.605558\pi$
$709$	−17.3382	−0.651150	−0.325575	+	0.945516i	$0.605558\pi$
$710$	0	0
$711$	−15.8798	−0.595538
$712$	0	0
$713$	−35.4260	−1.32671
$714$	0	0
$715$	0	0
$716$	0	0
$717$	13.5802	0.507162
$718$	0	0
$719$	−31.3919	−1.17072	−0.585360	−	0.810774i	$-0.699047\pi$
$719$	−31.3919	−1.17072	−0.585360	+	0.810774i	$0.699047\pi$
$720$	0	0
$721$	−1.41062	−0.0525342
$722$	0	0
$723$	38.3771	1.42726
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−19.6191	−0.727633	−0.363816	−	0.931471i	$-0.618526\pi$
$727$	−19.6191	−0.727633	−0.363816	+	0.931471i	$0.618526\pi$
$728$	0	0
$729$	−43.7424	−1.62009
$730$	0	0
$731$	−7.40919	−0.274039
$732$	0	0
$733$	6.22866	0.230061	0.115030	−	0.993362i	$-0.463303\pi$
$733$	6.22866	0.230061	0.115030	+	0.993362i	$0.463303\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	2.25561	0.0830865
$738$	0	0
$739$	−0.540267	−0.0198741	−0.00993703	−	0.999951i	$-0.503163\pi$
$739$	−0.540267	−0.0198741	−0.00993703	+	0.999951i	$0.503163\pi$
$740$	0	0
$741$	−1.44279	−0.0530021
$742$	0	0
$743$	13.6809	0.501904	0.250952	−	0.967999i	$-0.419256\pi$
$743$	13.6809	0.501904	0.250952	+	0.967999i	$0.419256\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	2.40147	0.0878650
$748$	0	0
$749$	−1.66000	−0.0606551
$750$	0	0
$751$	−27.2771	−0.995356	−0.497678	−	0.867362i	$-0.665814\pi$
$751$	−27.2771	−0.995356	−0.497678	+	0.867362i	$0.665814\pi$
$752$	0	0
$753$	−73.9031	−2.69318
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−38.4230	−1.39651	−0.698254	−	0.715850i	$-0.746039\pi$
$757$	−38.4230	−1.39651	−0.698254	+	0.715850i	$0.746039\pi$
$758$	0	0
$759$	−94.9993	−3.44826
$760$	0	0
$761$	−51.5643	−1.86920	−0.934602	−	0.355695i	$-0.884244\pi$
$761$	−51.5643	−1.86920	−0.934602	+	0.355695i	$0.884244\pi$
$762$	0	0
$763$	0.975978	0.0353328
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−0.974757	−0.0351964
$768$	0	0
$769$	52.1060	1.87899	0.939496	−	0.342561i	$-0.111294\pi$
$769$	52.1060	1.87899	0.939496	+	0.342561i	$0.111294\pi$
$770$	0	0
$771$	−43.2353	−1.55708
$772$	0	0
$773$	10.3653	0.372814	0.186407	−	0.982473i	$-0.440316\pi$
$773$	10.3653	0.372814	0.186407	+	0.982473i	$0.440316\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−1.78431	−0.0640117
$778$	0	0
$779$	11.4394	0.409859
$780$	0	0
$781$	24.9062	0.891215
$782$	0	0
$783$	−15.7272	−0.562044
$784$	0	0
$785$	0	0
$786$	0	0
$787$	29.7162	1.05927	0.529635	−	0.848226i	$-0.322329\pi$
$787$	29.7162	1.05927	0.529635	+	0.848226i	$0.322329\pi$
$788$	0	0
$789$	−83.1756	−2.96113
$790$	0	0
$791$	−1.26766	−0.0450727
$792$	0	0
$793$	3.00519	0.106717
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−39.2872	−1.39162	−0.695812	−	0.718224i	$-0.744955\pi$
$797$	−39.2872	−1.39162	−0.695812	+	0.718224i	$0.744955\pi$
$798$	0	0
$799$	42.6310	1.50818
$800$	0	0
$801$	−60.3302	−2.13166
$802$	0	0
$803$	52.0640	1.83730
$804$	0	0
$805$	0	0
$806$	0	0
$807$	76.1169	2.67944
$808$	0	0
$809$	−39.0027	−1.37126	−0.685630	−	0.727950i	$-0.740473\pi$
$809$	−39.0027	−1.37126	−0.685630	+	0.727950i	$0.740473\pi$
$810$	0	0
$811$	4.49460	0.157827	0.0789133	−	0.996881i	$-0.474855\pi$
$811$	4.49460	0.157827	0.0789133	+	0.996881i	$0.474855\pi$
$812$	0	0
$813$	−78.8740	−2.76623
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−2.06955	−0.0724042
$818$	0	0
$819$	−0.317981	−0.0111112
$820$	0	0
$821$	18.2356	0.636426	0.318213	−	0.948019i	$-0.396917\pi$
$821$	18.2356	0.636426	0.318213	+	0.948019i	$0.396917\pi$
$822$	0	0
$823$	−21.7245	−0.757268	−0.378634	−	0.925546i	$-0.623606\pi$
$823$	−21.7245	−0.757268	−0.378634	+	0.925546i	$0.623606\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−17.5265	−0.609455	−0.304727	−	0.952440i	$-0.598565\pi$
$827$	−17.5265	−0.609455	−0.304727	+	0.952440i	$0.598565\pi$
$828$	0	0
$829$	−33.5267	−1.16443	−0.582215	−	0.813035i	$-0.697814\pi$
$829$	−33.5267	−1.16443	−0.582215	+	0.813035i	$0.697814\pi$
$830$	0	0
$831$	−24.1817	−0.838853
$832$	0	0
$833$	−25.0025	−0.866284
$834$	0	0
$835$	0	0
$836$	0	0
$837$	27.9928	0.967572
$838$	0	0
$839$	−20.4096	−0.704617	−0.352309	−	0.935884i	$-0.614603\pi$
$839$	−20.4096	−0.704617	−0.352309	+	0.935884i	$0.614603\pi$
$840$	0	0
$841$	−19.6503	−0.677598
$842$	0	0
$843$	19.2370	0.662558
$844$	0	0
$845$	0	0
$846$	0	0
$847$	2.06352	0.0709033
$848$	0	0
$849$	27.7306	0.951713
$850$	0	0
$851$	32.5263	1.11499
$852$	0	0
$853$	−19.1968	−0.657287	−0.328644	−	0.944454i	$-0.606592\pi$
$853$	−19.1968	−0.657287	−0.328644	+	0.944454i	$0.606592\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−14.7861	−0.505083	−0.252542	−	0.967586i	$-0.581266\pi$
$857$	−14.7861	−0.505083	−0.252542	+	0.967586i	$0.581266\pi$
$858$	0	0
$859$	−6.83929	−0.233353	−0.116677	−	0.993170i	$-0.537224\pi$
$859$	−6.83929	−0.233353	−0.116677	+	0.993170i	$0.537224\pi$
$860$	0	0
$861$	4.08476	0.139208
$862$	0	0
$863$	−41.6367	−1.41733	−0.708666	−	0.705545i	$-0.750702\pi$
$863$	−41.6367	−1.41733	−0.708666	+	0.705545i	$0.750702\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−11.7100	−0.397692
$868$	0	0
$869$	−17.1141	−0.580557
$870$	0	0
$871$	−0.222985	−0.00755554
$872$	0	0
$873$	33.8783	1.14661
$874$	0	0
$875$	0	0
$876$	0	0
$877$	45.0143	1.52002	0.760012	−	0.649909i	$-0.225193\pi$
$877$	45.0143	1.52002	0.760012	+	0.649909i	$0.225193\pi$
$878$	0	0
$879$	76.6243	2.58447
$880$	0	0
$881$	−40.4549	−1.36296	−0.681479	−	0.731837i	$-0.738663\pi$
$881$	−40.4549	−1.36296	−0.681479	+	0.731837i	$0.738663\pi$
$882$	0	0
$883$	45.2872	1.52403	0.762017	−	0.647557i	$-0.224209\pi$
$883$	45.2872	1.52403	0.762017	+	0.647557i	$0.224209\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	0.589628	0.0197978	0.00989888	−	0.999951i	$-0.496849\pi$
$887$	0.589628	0.0197978	0.00989888	+	0.999951i	$0.496849\pi$
$888$	0	0
$889$	−1.09010	−0.0365609
$890$	0	0
$891$	−0.587479	−0.0196813
$892$	0	0
$893$	11.9078	0.398478
$894$	0	0
$895$	0	0
$896$	0	0
$897$	9.39142	0.313570
$898$	0	0
$899$	−16.6415	−0.555023
$900$	0	0
$901$	8.08463	0.269338
$902$	0	0
$903$	−0.738990	−0.0245921
$904$	0	0
$905$	0	0
$906$	0	0
$907$	38.4290	1.27601	0.638007	−	0.770031i	$-0.279759\pi$
$907$	38.4290	1.27601	0.638007	+	0.770031i	$0.279759\pi$
$908$	0	0
$909$	−53.0034	−1.75801
$910$	0	0
$911$	−16.9694	−0.562220	−0.281110	−	0.959676i	$-0.590703\pi$
$911$	−16.9694	−0.562220	−0.281110	+	0.959676i	$0.590703\pi$
$912$	0	0
$913$	2.58813	0.0856546
$914$	0	0
$915$	0	0
$916$	0	0
$917$	2.17326	0.0717672
$918$	0	0
$919$	20.1099	0.663364	0.331682	−	0.943391i	$-0.392384\pi$
$919$	20.1099	0.663364	0.331682	+	0.943391i	$0.392384\pi$
$920$	0	0
$921$	−47.0988	−1.55196
$922$	0	0
$923$	−2.46218	−0.0810435
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−53.4968	−1.75707
$928$	0	0
$929$	−6.46013	−0.211950	−0.105975	−	0.994369i	$-0.533796\pi$
$929$	−6.46013	−0.211950	−0.105975	+	0.994369i	$0.533796\pi$
$930$	0	0
$931$	−6.98373	−0.228883
$932$	0	0
$933$	62.5801	2.04878
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−40.2458	−1.31477	−0.657387	−	0.753553i	$-0.728338\pi$
$937$	−40.2458	−1.31477	−0.657387	+	0.753553i	$0.728338\pi$
$938$	0	0
$939$	−17.0388	−0.556041
$940$	0	0
$941$	40.2467	1.31201	0.656003	−	0.754759i	$-0.272246\pi$
$941$	40.2467	1.31201	0.656003	+	0.754759i	$0.272246\pi$
$942$	0	0
$943$	−74.4616	−2.42480
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−35.9987	−1.16980	−0.584900	−	0.811106i	$-0.698866\pi$
$947$	−35.9987	−1.16980	−0.584900	+	0.811106i	$0.698866\pi$
$948$	0	0
$949$	−5.14693	−0.167076
$950$	0	0
$951$	72.3738	2.34688
$952$	0	0
$953$	61.0774	1.97849	0.989246	−	0.146264i	$-0.0467247\pi$
$953$	61.0774	1.97849	0.989246	+	0.146264i	$0.0467247\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−44.6262	−1.44256
$958$	0	0
$959$	0.801275	0.0258745
$960$	0	0
$961$	−1.37993	−0.0445139
$962$	0	0
$963$	−62.9544	−2.02868
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−23.6689	−0.761142	−0.380571	−	0.924752i	$-0.624272\pi$
$967$	−23.6689	−0.761142	−0.380571	+	0.924752i	$0.624272\pi$
$968$	0	0
$969$	10.0225	0.321970
$970$	0	0
$971$	−52.4851	−1.68433	−0.842163	−	0.539223i	$-0.818718\pi$
$971$	−52.4851	−1.68433	−0.842163	+	0.539223i	$0.818718\pi$
$972$	0	0
$973$	0.239616	0.00768173
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−24.8813	−0.796025	−0.398012	−	0.917380i	$-0.630300\pi$
$977$	−24.8813	−0.796025	−0.398012	+	0.917380i	$0.630300\pi$
$978$	0	0
$979$	−65.0196	−2.07804
$980$	0	0
$981$	37.0133	1.18174
$982$	0	0
$983$	33.1585	1.05759	0.528795	−	0.848749i	$-0.322644\pi$
$983$	33.1585	1.05759	0.528795	+	0.848749i	$0.322644\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	4.25200	0.135343
$988$	0	0
$989$	13.4711	0.428357
$990$	0	0
$991$	32.2866	1.02562	0.512808	−	0.858503i	$-0.328605\pi$
$991$	32.2866	1.02562	0.512808	+	0.858503i	$0.328605\pi$
$992$	0	0
$993$	−64.9438	−2.06093
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−3.00638	−0.0952131	−0.0476066	−	0.998866i	$-0.515159\pi$
$997$	−3.00638	−0.0952131	−0.0476066	+	0.998866i	$0.515159\pi$
$998$	0	0
$999$	−25.7015	−0.813161

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3800.2.a.bb.1.6	✓	6
4.3	odd	2		7600.2.a.cm.1.1		6
5.2	odd	4		3800.2.d.p.3649.2		12
5.3	odd	4		3800.2.d.p.3649.11		12
5.4	even	2		3800.2.a.bd.1.1	yes	6
20.19	odd	2		7600.2.a.ci.1.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3800.2.a.bb.1.6	✓	6	1.1	even	1	trivial
3800.2.a.bd.1.1	yes	6	5.4	even	2
3800.2.d.p.3649.2		12	5.2	odd	4
3800.2.d.p.3649.11		12	5.3	odd	4
7600.2.a.ci.1.6		6	20.19	odd	2
7600.2.a.cm.1.1		6	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:	\( N \)	\(=\)	\( 3800 = 2^{3} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3800.a (trivial)

\(f(q)\)	\(=\)	\(q+2.79951 q^{3} +0.127550 q^{7} +4.83726 q^{9} +O(q^{10})\)	\(q+2.79951 q^{3} +0.127550 q^{7} +4.83726 q^{9} +5.21326 q^{11} -0.515371 q^{13} +3.58010 q^{17} +1.00000 q^{19} +0.357078 q^{21} -6.50922 q^{23} +5.14343 q^{27} -3.05772 q^{29} +5.44243 q^{31} +14.5946 q^{33} -4.99696 q^{37} -1.44279 q^{39} +11.4394 q^{41} -2.06955 q^{43} +11.9078 q^{47} -6.98373 q^{49} +10.0225 q^{51} +2.25821 q^{53} +2.79951 q^{57} +1.89137 q^{59} -5.83111 q^{61} +0.616994 q^{63} +0.432668 q^{67} -18.2226 q^{69} +4.77748 q^{71} +9.98684 q^{73} +0.664953 q^{77} -3.28280 q^{79} -0.112689 q^{81} +0.496452 q^{83} -8.56013 q^{87} -12.4720 q^{89} -0.0657358 q^{91} +15.2361 q^{93} +7.00361 q^{97} +25.2179 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{3} - 2 q^{7} + 10 q^{9}+O(q^{10}) \) 6 * q - 2 * q^3 - 2 * q^7 + 10 * q^9	\( 6 q - 2 q^{3} - 2 q^{7} + 10 q^{9} + 3 q^{11} + 3 q^{13} + 2 q^{17} + 6 q^{19} + 11 q^{21} - 4 q^{23} - 20 q^{27} + 7 q^{29} + 5 q^{31} + 16 q^{33} + 8 q^{39} + 11 q^{41} + 7 q^{43} - 20 q^{47} - 2 q^{49} + 13 q^{51} + 7 q^{53} - 2 q^{57} - 4 q^{59} + 13 q^{61} + q^{63} - 25 q^{67} + 7 q^{69} + 29 q^{71} + 19 q^{73} - 24 q^{77} + 28 q^{79} + 38 q^{81} + 15 q^{83} - 57 q^{87} - 12 q^{89} + 27 q^{93} + 13 q^{97} + 10 q^{99}+O(q^{100}) \) 6 * q - 2 * q^3 - 2 * q^7 + 10 * q^9 + 3 * q^11 + 3 * q^13 + 2 * q^17 + 6 * q^19 + 11 * q^21 - 4 * q^23 - 20 * q^27 + 7 * q^29 + 5 * q^31 + 16 * q^33 + 8 * q^39 + 11 * q^41 + 7 * q^43 - 20 * q^47 - 2 * q^49 + 13 * q^51 + 7 * q^53 - 2 * q^57 - 4 * q^59 + 13 * q^61 + q^63 - 25 * q^67 + 7 * q^69 + 29 * q^71 + 19 * q^73 - 24 * q^77 + 28 * q^79 + 38 * q^81 + 15 * q^83 - 57 * q^87 - 12 * q^89 + 27 * q^93 + 13 * q^97 + 10 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more