LMFDB - Embedded newform 1480.2.a.f.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1480,2,Mod(1,1480)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1480, 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("1480.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1480 = 2^{3} \cdot 5 \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1480.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:	$11.8178594991$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.568.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 6x - 2 $ x^3 - x^2 - 6*x - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.363328$ of defining polynomial
Character	$\chi$	$=$	1480.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.363328 q^{3} +1.00000 q^{5} +0.363328 q^{7} -2.86799 q^{9} +O(q^{10})$	$q-0.363328 q^{3} +1.00000 q^{5} +0.363328 q^{7} -2.86799 q^{9} -1.14134 q^{11} -0.363328 q^{15} +4.77801 q^{19} -0.132007 q^{21} +4.00000 q^{23} +1.00000 q^{25} +2.13201 q^{27} +4.28267 q^{29} +5.50466 q^{31} +0.414680 q^{33} +0.363328 q^{35} +1.00000 q^{37} +3.86799 q^{41} -2.28267 q^{43} -2.86799 q^{45} +12.9287 q^{47} -6.86799 q^{49} -6.15066 q^{53} -1.14134 q^{55} -1.73599 q^{57} +7.78734 q^{59} -0.546687 q^{61} -1.04202 q^{63} +0.212663 q^{67} -1.45331 q^{69} +3.58532 q^{71} +8.69735 q^{73} -0.363328 q^{75} -0.414680 q^{77} -7.06068 q^{79} +7.82936 q^{81} -0.466031 q^{83} -1.55602 q^{87} -16.0187 q^{89} -2.00000 q^{93} +4.77801 q^{95} +6.00000 q^{97} +3.27334 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{3} + 3 q^{5} - q^{7} + 4 q^{9}+O(q^{10}) $ 3 * q + q^3 + 3 * q^5 - q^7 + 4 * q^9	$ 3 q + q^{3} + 3 q^{5} - q^{7} + 4 q^{9} + 5 q^{11} + q^{15} + 8 q^{19} - 13 q^{21} + 12 q^{23} + 3 q^{25} + 19 q^{27} - 4 q^{29} + 6 q^{31} - 3 q^{33} - q^{35} + 3 q^{37} - q^{41} + 10 q^{43} + 4 q^{45} + 3 q^{47} - 8 q^{49} + 11 q^{53} + 5 q^{55} + 20 q^{57} - 4 q^{59} - 10 q^{61} - 22 q^{63} + 28 q^{67} + 4 q^{69} + 15 q^{71} + 5 q^{73} + q^{75} + 3 q^{77} + 2 q^{79} + 15 q^{81} + 5 q^{83} + 8 q^{87} - 6 q^{89} - 6 q^{93} + 8 q^{95} + 18 q^{97} + 14 q^{99}+O(q^{100}) $ 3 * q + q^3 + 3 * q^5 - q^7 + 4 * q^9 + 5 * q^11 + q^15 + 8 * q^19 - 13 * q^21 + 12 * q^23 + 3 * q^25 + 19 * q^27 - 4 * q^29 + 6 * q^31 - 3 * q^33 - q^35 + 3 * q^37 - q^41 + 10 * q^43 + 4 * q^45 + 3 * q^47 - 8 * q^49 + 11 * q^53 + 5 * q^55 + 20 * q^57 - 4 * q^59 - 10 * q^61 - 22 * q^63 + 28 * q^67 + 4 * q^69 + 15 * q^71 + 5 * q^73 + q^75 + 3 * q^77 + 2 * q^79 + 15 * q^81 + 5 * q^83 + 8 * q^87 - 6 * q^89 - 6 * q^93 + 8 * q^95 + 18 * q^97 + 14 * 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
$3$	−0.363328	−0.209768	−0.104884	−	0.994484i	$-0.533447\pi$
$3$	−0.363328	−0.209768	−0.104884	+	0.994484i	$0.533447\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	0.363328	0.137325	0.0686626	−	0.997640i	$-0.478127\pi$
$7$	0.363328	0.137325	0.0686626	+	0.997640i	$0.478127\pi$
$8$	0	0
$9$	−2.86799	−0.955998
$10$	0	0
$11$	−1.14134	−0.344126	−0.172063	−	0.985086i	$-0.555043\pi$
$11$	−1.14134	−0.344126	−0.172063	+	0.985086i	$0.555043\pi$
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	−0.363328	−0.0938109
$16$	0	0
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	4.77801	1.09615	0.548075	−	0.836429i	$-0.315361\pi$
$19$	4.77801	1.09615	0.548075	+	0.836429i	$0.315361\pi$
$20$	0	0
$21$	−0.132007	−0.0288064
$22$	0	0
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	2.13201	0.410305
$28$	0	0
$29$	4.28267	0.795272	0.397636	−	0.917543i	$-0.369831\pi$
$29$	4.28267	0.795272	0.397636	+	0.917543i	$0.369831\pi$
$30$	0	0
$31$	5.50466	0.988667	0.494333	−	0.869272i	$-0.335412\pi$
$31$	5.50466	0.988667	0.494333	+	0.869272i	$0.335412\pi$
$32$	0	0
$33$	0.414680	0.0721865
$34$	0	0
$35$	0.363328	0.0614137
$36$	0	0
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	0	0
$39$	0	0
$40$	0	0
$41$	3.86799	0.604079	0.302039	−	0.953295i	$-0.402333\pi$
$41$	3.86799	0.604079	0.302039	+	0.953295i	$0.402333\pi$
$42$	0	0
$43$	−2.28267	−0.348104	−0.174052	−	0.984736i	$-0.555686\pi$
$43$	−2.28267	−0.348104	−0.174052	+	0.984736i	$0.555686\pi$
$44$	0	0
$45$	−2.86799	−0.427535
$46$	0	0
$47$	12.9287	1.88584	0.942920	−	0.333018i	$-0.108067\pi$
$47$	12.9287	1.88584	0.942920	+	0.333018i	$0.108067\pi$
$48$	0	0
$49$	−6.86799	−0.981142
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−6.15066	−0.844859	−0.422429	−	0.906396i	$-0.638823\pi$
$53$	−6.15066	−0.844859	−0.422429	+	0.906396i	$0.638823\pi$
$54$	0	0
$55$	−1.14134	−0.153898
$56$	0	0
$57$	−1.73599	−0.229937
$58$	0	0
$59$	7.78734	1.01382	0.506912	−	0.861998i	$-0.330787\pi$
$59$	7.78734	1.01382	0.506912	+	0.861998i	$0.330787\pi$
$60$	0	0
$61$	−0.546687	−0.0699961	−0.0349981	−	0.999387i	$-0.511143\pi$
$61$	−0.546687	−0.0699961	−0.0349981	+	0.999387i	$0.511143\pi$
$62$	0	0
$63$	−1.04202	−0.131283
$64$	0	0
$65$	0	0
$66$	0	0
$67$	0.212663	0.0259810	0.0129905	−	0.999916i	$-0.495865\pi$
$67$	0.212663	0.0259810	0.0129905	+	0.999916i	$0.495865\pi$
$68$	0	0
$69$	−1.45331	−0.174958
$70$	0	0
$71$	3.58532	0.425499	0.212750	−	0.977107i	$-0.431758\pi$
$71$	3.58532	0.425499	0.212750	+	0.977107i	$0.431758\pi$
$72$	0	0
$73$	8.69735	1.01795	0.508974	−	0.860782i	$-0.330025\pi$
$73$	8.69735	1.01795	0.508974	+	0.860782i	$0.330025\pi$
$74$	0	0
$75$	−0.363328	−0.0419535
$76$	0	0
$77$	−0.414680	−0.0472571
$78$	0	0
$79$	−7.06068	−0.794389	−0.397194	−	0.917735i	$-0.630016\pi$
$79$	−7.06068	−0.794389	−0.397194	+	0.917735i	$0.630016\pi$
$80$	0	0
$81$	7.82936	0.869929
$82$	0	0
$83$	−0.466031	−0.0511536	−0.0255768	−	0.999673i	$-0.508142\pi$
$83$	−0.466031	−0.0511536	−0.0255768	+	0.999673i	$0.508142\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−1.55602	−0.166822
$88$	0	0
$89$	−16.0187	−1.69797	−0.848987	−	0.528414i	$-0.822787\pi$
$89$	−16.0187	−1.69797	−0.848987	+	0.528414i	$0.822787\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−2.00000	−0.207390
$94$	0	0
$95$	4.77801	0.490213
$96$	0	0
$97$	6.00000	0.609208	0.304604	−	0.952479i	$-0.401476\pi$
$97$	6.00000	0.609208	0.304604	+	0.952479i	$0.401476\pi$
$98$	0	0
$99$	3.27334	0.328983
$100$	0	0
$101$	13.6040	1.35365	0.676823	−	0.736146i	$-0.263356\pi$
$101$	13.6040	1.35365	0.676823	+	0.736146i	$0.263356\pi$
$102$	0	0
$103$	6.17997	0.608930	0.304465	−	0.952523i	$-0.401522\pi$
$103$	6.17997	0.608930	0.304465	+	0.952523i	$0.401522\pi$
$104$	0	0
$105$	−0.132007	−0.0128826
$106$	0	0
$107$	18.7967	1.81714	0.908571	−	0.417730i	$-0.137174\pi$
$107$	18.7967	1.81714	0.908571	+	0.417730i	$0.137174\pi$
$108$	0	0
$109$	−4.54669	−0.435494	−0.217747	−	0.976005i	$-0.569871\pi$
$109$	−4.54669	−0.435494	−0.217747	+	0.976005i	$0.569871\pi$
$110$	0	0
$111$	−0.363328	−0.0344856
$112$	0	0
$113$	−8.82936	−0.830596	−0.415298	−	0.909685i	$-0.636323\pi$
$113$	−8.82936	−0.830596	−0.415298	+	0.909685i	$0.636323\pi$
$114$	0	0
$115$	4.00000	0.373002
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−9.69735	−0.881577
$122$	0	0
$123$	−1.40535	−0.126716
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	10.7487	0.953793	0.476897	−	0.878959i	$-0.341762\pi$
$127$	10.7487	0.953793	0.476897	+	0.878959i	$0.341762\pi$
$128$	0	0
$129$	0.829359	0.0730210
$130$	0	0
$131$	−5.50466	−0.480945	−0.240472	−	0.970656i	$-0.577302\pi$
$131$	−5.50466	−0.480945	−0.240472	+	0.970656i	$0.577302\pi$
$132$	0	0
$133$	1.73599	0.150529
$134$	0	0
$135$	2.13201	0.183494
$136$	0	0
$137$	3.45331	0.295036	0.147518	−	0.989059i	$-0.452871\pi$
$137$	3.45331	0.295036	0.147518	+	0.989059i	$0.452871\pi$
$138$	0	0
$139$	10.0187	0.849771	0.424886	−	0.905247i	$-0.360314\pi$
$139$	10.0187	0.849771	0.424886	+	0.905247i	$0.360314\pi$
$140$	0	0
$141$	−4.69735	−0.395588
$142$	0	0
$143$	0	0
$144$	0	0
$145$	4.28267	0.355657
$146$	0	0
$147$	2.49534	0.205812
$148$	0	0
$149$	−1.86799	−0.153032	−0.0765160	−	0.997068i	$-0.524380\pi$
$149$	−1.86799	−0.153032	−0.0765160	+	0.997068i	$0.524380\pi$
$150$	0	0
$151$	−4.46264	−0.363165	−0.181582	−	0.983376i	$-0.558122\pi$
$151$	−4.46264	−0.363165	−0.181582	+	0.983376i	$0.558122\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	5.50466	0.442145
$156$	0	0
$157$	20.1693	1.60969	0.804844	−	0.593487i	$-0.202249\pi$
$157$	20.1693	1.60969	0.804844	+	0.593487i	$0.202249\pi$
$158$	0	0
$159$	2.23471	0.177224
$160$	0	0
$161$	1.45331	0.114537
$162$	0	0
$163$	9.55602	0.748485	0.374242	−	0.927331i	$-0.377903\pi$
$163$	9.55602	0.748485	0.374242	+	0.927331i	$0.377903\pi$
$164$	0	0
$165$	0.414680	0.0322828
$166$	0	0
$167$	−15.8387	−1.22563	−0.612817	−	0.790225i	$-0.709964\pi$
$167$	−15.8387	−1.22563	−0.612817	+	0.790225i	$0.709964\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	0	0
$171$	−13.7033	−1.04792
$172$	0	0
$173$	15.8680	1.20642	0.603211	−	0.797582i	$-0.293888\pi$
$173$	15.8680	1.20642	0.603211	+	0.797582i	$0.293888\pi$
$174$	0	0
$175$	0.363328	0.0274650
$176$	0	0
$177$	−2.82936	−0.212668
$178$	0	0
$179$	−9.34335	−0.698355	−0.349177	−	0.937057i	$-0.613539\pi$
$179$	−9.34335	−0.698355	−0.349177	+	0.937057i	$0.613539\pi$
$180$	0	0
$181$	−19.2627	−1.43178	−0.715892	−	0.698211i	$-0.753980\pi$
$181$	−19.2627	−1.43178	−0.715892	+	0.698211i	$0.753980\pi$
$182$	0	0
$183$	0.198627	0.0146829
$184$	0	0
$185$	1.00000	0.0735215
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0.774618	0.0563452
$190$	0	0
$191$	3.32469	0.240566	0.120283	−	0.992740i	$-0.461620\pi$
$191$	3.32469	0.240566	0.120283	+	0.992740i	$0.461620\pi$
$192$	0	0
$193$	−1.73599	−0.124959	−0.0624795	−	0.998046i	$-0.519901\pi$
$193$	−1.73599	−0.124959	−0.0624795	+	0.998046i	$0.519901\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−2.41468	−0.172039	−0.0860194	−	0.996293i	$-0.527415\pi$
$197$	−2.41468	−0.172039	−0.0860194	+	0.996293i	$0.527415\pi$
$198$	0	0
$199$	4.05135	0.287193	0.143596	−	0.989636i	$-0.454133\pi$
$199$	4.05135	0.287193	0.143596	+	0.989636i	$0.454133\pi$
$200$	0	0
$201$	−0.0772666	−0.00544997
$202$	0	0
$203$	1.55602	0.109211
$204$	0	0
$205$	3.86799	0.270152
$206$	0	0
$207$	−11.4720	−0.797357
$208$	0	0
$209$	−5.45331	−0.377214
$210$	0	0
$211$	−1.56666	−0.107854	−0.0539268	−	0.998545i	$-0.517174\pi$
$211$	−1.56666	−0.107854	−0.0539268	+	0.998545i	$0.517174\pi$
$212$	0	0
$213$	−1.30265	−0.0892560
$214$	0	0
$215$	−2.28267	−0.155677
$216$	0	0
$217$	2.00000	0.135769
$218$	0	0
$219$	−3.15999	−0.213533
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−17.0900	−1.14443	−0.572215	−	0.820104i	$-0.693916\pi$
$223$	−17.0900	−1.14443	−0.572215	+	0.820104i	$0.693916\pi$
$224$	0	0
$225$	−2.86799	−0.191200
$226$	0	0
$227$	4.99067	0.331242	0.165621	−	0.986189i	$-0.447037\pi$
$227$	4.99067	0.331242	0.165621	+	0.986189i	$0.447037\pi$
$228$	0	0
$229$	−28.1693	−1.86148	−0.930741	−	0.365680i	$-0.880836\pi$
$229$	−28.1693	−1.86148	−0.930741	+	0.365680i	$0.880836\pi$
$230$	0	0
$231$	0.150665	0.00991302
$232$	0	0
$233$	−6.00000	−0.393073	−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000	−0.393073	−0.196537	+	0.980497i	$0.562969\pi$
$234$	0	0
$235$	12.9287	0.843374
$236$	0	0
$237$	2.56534	0.166637
$238$	0	0
$239$	−20.9766	−1.35687	−0.678433	−	0.734662i	$-0.737341\pi$
$239$	−20.9766	−1.35687	−0.678433	+	0.734662i	$0.737341\pi$
$240$	0	0
$241$	2.30133	0.148242	0.0741208	−	0.997249i	$-0.476385\pi$
$241$	2.30133	0.148242	0.0741208	+	0.997249i	$0.476385\pi$
$242$	0	0
$243$	−9.24065	−0.592788
$244$	0	0
$245$	−6.86799	−0.438780
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0.169322	0.0107304
$250$	0	0
$251$	26.7967	1.69139	0.845695	−	0.533666i	$-0.179186\pi$
$251$	26.7967	1.69139	0.845695	+	0.533666i	$0.179186\pi$
$252$	0	0
$253$	−4.56534	−0.287021
$254$	0	0
$255$	0	0
$256$	0	0
$257$	16.8294	1.04979	0.524893	−	0.851168i	$-0.324105\pi$
$257$	16.8294	1.04979	0.524893	+	0.851168i	$0.324105\pi$
$258$	0	0
$259$	0.363328	0.0225761
$260$	0	0
$261$	−12.2827	−0.760278
$262$	0	0
$263$	1.91934	0.118352	0.0591759	−	0.998248i	$-0.481153\pi$
$263$	1.91934	0.118352	0.0591759	+	0.998248i	$0.481153\pi$
$264$	0	0
$265$	−6.15066	−0.377832
$266$	0	0
$267$	5.82003	0.356180
$268$	0	0
$269$	12.3013	0.750025	0.375013	−	0.927020i	$-0.377638\pi$
$269$	12.3013	0.750025	0.375013	+	0.927020i	$0.377638\pi$
$270$	0	0
$271$	−27.8867	−1.69399	−0.846997	−	0.531598i	$-0.821592\pi$
$271$	−27.8867	−1.69399	−0.846997	+	0.531598i	$0.821592\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1.14134	−0.0688252
$276$	0	0
$277$	−16.3013	−0.979452	−0.489726	−	0.871876i	$-0.662903\pi$
$277$	−16.3013	−0.979452	−0.489726	+	0.871876i	$0.662903\pi$
$278$	0	0
$279$	−15.7873	−0.945163
$280$	0	0
$281$	−8.90663	−0.531325	−0.265662	−	0.964066i	$-0.585591\pi$
$281$	−8.90663	−0.531325	−0.265662	+	0.964066i	$0.585591\pi$
$282$	0	0
$283$	−15.5747	−0.925818	−0.462909	−	0.886406i	$-0.653194\pi$
$283$	−15.5747	−0.925818	−0.462909	+	0.886406i	$0.653194\pi$
$284$	0	0
$285$	−1.73599	−0.102831
$286$	0	0
$287$	1.40535	0.0829552
$288$	0	0
$289$	−17.0000	−1.00000
$290$	0	0
$291$	−2.17997	−0.127792
$292$	0	0
$293$	−10.0000	−0.584206	−0.292103	−	0.956387i	$-0.594355\pi$
$293$	−10.0000	−0.584206	−0.292103	+	0.956387i	$0.594355\pi$
$294$	0	0
$295$	7.78734	0.453396
$296$	0	0
$297$	−2.43334	−0.141197
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−0.829359	−0.0478035
$302$	0	0
$303$	−4.94271	−0.283951
$304$	0	0
$305$	−0.546687	−0.0313032
$306$	0	0
$307$	11.3727	0.649072	0.324536	−	0.945873i	$-0.394792\pi$
$307$	11.3727	0.649072	0.324536	+	0.945873i	$0.394792\pi$
$308$	0	0
$309$	−2.24536	−0.127734
$310$	0	0
$311$	1.24065	0.0703508	0.0351754	−	0.999381i	$-0.488801\pi$
$311$	1.24065	0.0703508	0.0351754	+	0.999381i	$0.488801\pi$
$312$	0	0
$313$	−12.9066	−0.729526	−0.364763	−	0.931100i	$-0.618850\pi$
$313$	−12.9066	−0.729526	−0.364763	+	0.931100i	$0.618850\pi$
$314$	0	0
$315$	−1.04202	−0.0587113
$316$	0	0
$317$	−0.0186574	−0.00104790	−0.000523952	−	1.00000i	$-0.500167\pi$
$317$	−0.0186574	−0.00104790	−0.000523952		1.00000i	$0.500167\pi$
$318$	0	0
$319$	−4.88797	−0.273674
$320$	0	0
$321$	−6.82936	−0.381178
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0	0
$326$	0	0
$327$	1.65194	0.0913525
$328$	0	0
$329$	4.69735	0.258973
$330$	0	0
$331$	9.24065	0.507912	0.253956	−	0.967216i	$-0.418268\pi$
$331$	9.24065	0.507912	0.253956	+	0.967216i	$0.418268\pi$
$332$	0	0
$333$	−2.86799	−0.157165
$334$	0	0
$335$	0.212663	0.0116190
$336$	0	0
$337$	25.6226	1.39575	0.697877	−	0.716218i	$-0.254128\pi$
$337$	25.6226	1.39575	0.697877	+	0.716218i	$0.254128\pi$
$338$	0	0
$339$	3.20796	0.174232
$340$	0	0
$341$	−6.28267	−0.340226
$342$	0	0
$343$	−5.03863	−0.272061
$344$	0	0
$345$	−1.45331	−0.0782437
$346$	0	0
$347$	9.45331	0.507480	0.253740	−	0.967272i	$-0.418339\pi$
$347$	9.45331	0.507480	0.253740	+	0.967272i	$0.418339\pi$
$348$	0	0
$349$	−16.2640	−0.870593	−0.435296	−	0.900287i	$-0.643356\pi$
$349$	−16.2640	−0.870593	−0.435296	+	0.900287i	$0.643356\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	5.43466	0.289258	0.144629	−	0.989486i	$-0.453801\pi$
$353$	5.43466	0.289258	0.144629	+	0.989486i	$0.453801\pi$
$354$	0	0
$355$	3.58532	0.190289
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−2.85866	−0.150875	−0.0754373	−	0.997151i	$-0.524035\pi$
$359$	−2.85866	−0.150875	−0.0754373	+	0.997151i	$0.524035\pi$
$360$	0	0
$361$	3.82936	0.201545
$362$	0	0
$363$	3.52332	0.184926
$364$	0	0
$365$	8.69735	0.455240
$366$	0	0
$367$	−4.77801	−0.249410	−0.124705	−	0.992194i	$-0.539798\pi$
$367$	−4.77801	−0.249410	−0.124705	+	0.992194i	$0.539798\pi$
$368$	0	0
$369$	−11.0934	−0.577498
$370$	0	0
$371$	−2.23471	−0.116020
$372$	0	0
$373$	0.433337	0.0224373	0.0112187	−	0.999937i	$-0.496429\pi$
$373$	0.433337	0.0224373	0.0112187	+	0.999937i	$0.496429\pi$
$374$	0	0
$375$	−0.363328	−0.0187622
$376$	0	0
$377$	0	0
$378$	0	0
$379$	14.8587	0.763238	0.381619	−	0.924320i	$-0.375367\pi$
$379$	14.8587	0.763238	0.381619	+	0.924320i	$0.375367\pi$
$380$	0	0
$381$	−3.90531	−0.200075
$382$	0	0
$383$	−7.73599	−0.395290	−0.197645	−	0.980274i	$-0.563329\pi$
$383$	−7.73599	−0.395290	−0.197645	+	0.980274i	$0.563329\pi$
$384$	0	0
$385$	−0.414680	−0.0211340
$386$	0	0
$387$	6.54669	0.332787
$388$	0	0
$389$	−15.7546	−0.798792	−0.399396	−	0.916778i	$-0.630780\pi$
$389$	−15.7546	−0.798792	−0.399396	+	0.916778i	$0.630780\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	2.00000	0.100887
$394$	0	0
$395$	−7.06068	−0.355261
$396$	0	0
$397$	5.58532	0.280319	0.140160	−	0.990129i	$-0.455238\pi$
$397$	5.58532	0.280319	0.140160	+	0.990129i	$0.455238\pi$
$398$	0	0
$399$	−0.630732	−0.0315761
$400$	0	0
$401$	−1.11203	−0.0555322	−0.0277661	−	0.999614i	$-0.508839\pi$
$401$	−1.11203	−0.0555322	−0.0277661	+	0.999614i	$0.508839\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	7.82936	0.389044
$406$	0	0
$407$	−1.14134	−0.0565739
$408$	0	0
$409$	−16.8480	−0.833081	−0.416540	−	0.909117i	$-0.636758\pi$
$409$	−16.8480	−0.833081	−0.416540	+	0.909117i	$0.636758\pi$
$410$	0	0
$411$	−1.25469	−0.0618891
$412$	0	0
$413$	2.82936	0.139224
$414$	0	0
$415$	−0.466031	−0.0228766
$416$	0	0
$417$	−3.64006	−0.178255
$418$	0	0
$419$	26.8960	1.31395	0.656977	−	0.753910i	$-0.271835\pi$
$419$	26.8960	1.31395	0.656977	+	0.753910i	$0.271835\pi$
$420$	0	0
$421$	16.8480	0.821122	0.410561	−	0.911833i	$-0.365333\pi$
$421$	16.8480	0.821122	0.410561	+	0.911833i	$0.365333\pi$
$422$	0	0
$423$	−37.0793	−1.80286
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−0.198627	−0.00961223
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−3.11929	−0.150251	−0.0751254	−	0.997174i	$-0.523936\pi$
$431$	−3.11929	−0.150251	−0.0751254	+	0.997174i	$0.523936\pi$
$432$	0	0
$433$	8.43334	0.405280	0.202640	−	0.979253i	$-0.435048\pi$
$433$	8.43334	0.405280	0.202640	+	0.979253i	$0.435048\pi$
$434$	0	0
$435$	−1.55602	−0.0746052
$436$	0	0
$437$	19.1120	0.914252
$438$	0	0
$439$	−25.3434	−1.20957	−0.604786	−	0.796388i	$-0.706741\pi$
$439$	−25.3434	−1.20957	−0.604786	+	0.796388i	$0.706741\pi$
$440$	0	0
$441$	19.6974	0.937969
$442$	0	0
$443$	−26.8446	−1.27543	−0.637713	−	0.770274i	$-0.720120\pi$
$443$	−26.8446	−1.27543	−0.637713	+	0.770274i	$0.720120\pi$
$444$	0	0
$445$	−16.0187	−0.759357
$446$	0	0
$447$	0.678694	0.0321011
$448$	0	0
$449$	14.0000	0.660701	0.330350	−	0.943858i	$-0.392833\pi$
$449$	14.0000	0.660701	0.330350	+	0.943858i	$0.392833\pi$
$450$	0	0
$451$	−4.41468	−0.207879
$452$	0	0
$453$	1.62140	0.0761802
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−19.6587	−0.919596	−0.459798	−	0.888024i	$-0.652078\pi$
$457$	−19.6587	−0.919596	−0.459798	+	0.888024i	$0.652078\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	38.6027	1.79791	0.898953	−	0.438046i	$-0.144329\pi$
$461$	38.6027	1.79791	0.898953	+	0.438046i	$0.144329\pi$
$462$	0	0
$463$	−37.7546	−1.75461	−0.877304	−	0.479936i	$-0.840660\pi$
$463$	−37.7546	−1.75461	−0.877304	+	0.479936i	$0.840660\pi$
$464$	0	0
$465$	−2.00000	−0.0927478
$466$	0	0
$467$	14.9507	0.691837	0.345918	−	0.938265i	$-0.387567\pi$
$467$	14.9507	0.691837	0.345918	+	0.938265i	$0.387567\pi$
$468$	0	0
$469$	0.0772666	0.00356784
$470$	0	0
$471$	−7.32808	−0.337660
$472$	0	0
$473$	2.60530	0.119792
$474$	0	0
$475$	4.77801	0.219230
$476$	0	0
$477$	17.6401	0.807683
$478$	0	0
$479$	8.21266	0.375246	0.187623	−	0.982241i	$-0.439922\pi$
$479$	8.21266	0.375246	0.187623	+	0.982241i	$0.439922\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	−0.528030	−0.0240262
$484$	0	0
$485$	6.00000	0.272446
$486$	0	0
$487$	−27.5747	−1.24953	−0.624764	−	0.780814i	$-0.714805\pi$
$487$	−27.5747	−1.24953	−0.624764	+	0.780814i	$0.714805\pi$
$488$	0	0
$489$	−3.47197	−0.157008
$490$	0	0
$491$	−25.4906	−1.15038	−0.575188	−	0.818021i	$-0.695071\pi$
$491$	−25.4906	−1.15038	−0.575188	+	0.818021i	$0.695071\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	3.27334	0.147126
$496$	0	0
$497$	1.30265	0.0584318
$498$	0	0
$499$	21.4020	0.958083	0.479042	−	0.877792i	$-0.340984\pi$
$499$	21.4020	0.958083	0.479042	+	0.877792i	$0.340984\pi$
$500$	0	0
$501$	5.75464	0.257098
$502$	0	0
$503$	−5.45331	−0.243151	−0.121576	−	0.992582i	$-0.538795\pi$
$503$	−5.45331	−0.243151	−0.121576	+	0.992582i	$0.538795\pi$
$504$	0	0
$505$	13.6040	0.605369
$506$	0	0
$507$	4.72327	0.209768
$508$	0	0
$509$	−36.4333	−1.61488	−0.807440	−	0.589950i	$-0.799147\pi$
$509$	−36.4333	−1.61488	−0.807440	+	0.589950i	$0.799147\pi$
$510$	0	0
$511$	3.15999	0.139790
$512$	0	0
$513$	10.1867	0.449756
$514$	0	0
$515$	6.17997	0.272322
$516$	0	0
$517$	−14.7560	−0.648966
$518$	0	0
$519$	−5.76529	−0.253068
$520$	0	0
$521$	17.6040	0.771244	0.385622	−	0.922657i	$-0.373987\pi$
$521$	17.6040	0.771244	0.385622	+	0.922657i	$0.373987\pi$
$522$	0	0
$523$	−5.71733	−0.250001	−0.125001	−	0.992157i	$-0.539893\pi$
$523$	−5.71733	−0.250001	−0.125001	+	0.992157i	$0.539893\pi$
$524$	0	0
$525$	−0.132007	−0.00576128
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	−22.3340	−0.969214
$532$	0	0
$533$	0	0
$534$	0	0
$535$	18.7967	0.812651
$536$	0	0
$537$	3.39470	0.146492
$538$	0	0
$539$	7.83869	0.337636
$540$	0	0
$541$	−8.90663	−0.382926	−0.191463	−	0.981500i	$-0.561323\pi$
$541$	−8.90663	−0.382926	−0.191463	+	0.981500i	$0.561323\pi$
$542$	0	0
$543$	6.99868	0.300342
$544$	0	0
$545$	−4.54669	−0.194759
$546$	0	0
$547$	21.8133	0.932667	0.466334	−	0.884609i	$-0.345575\pi$
$547$	21.8133	0.932667	0.466334	+	0.884609i	$0.345575\pi$
$548$	0	0
$549$	1.56789	0.0669161
$550$	0	0
$551$	20.4626	0.871738
$552$	0	0
$553$	−2.56534	−0.109090
$554$	0	0
$555$	−0.363328	−0.0154224
$556$	0	0
$557$	41.1307	1.74276	0.871382	−	0.490606i	$-0.163224\pi$
$557$	41.1307	1.74276	0.871382	+	0.490606i	$0.163224\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	14.5467	0.613070	0.306535	−	0.951859i	$-0.400830\pi$
$563$	14.5467	0.613070	0.306535	+	0.951859i	$0.400830\pi$
$564$	0	0
$565$	−8.82936	−0.371454
$566$	0	0
$567$	2.84463	0.119463
$568$	0	0
$569$	24.5467	1.02905	0.514525	−	0.857475i	$-0.327968\pi$
$569$	24.5467	1.02905	0.514525	+	0.857475i	$0.327968\pi$
$570$	0	0
$571$	−18.1693	−0.760362	−0.380181	−	0.924912i	$-0.624138\pi$
$571$	−18.1693	−0.760362	−0.380181	+	0.924912i	$0.624138\pi$
$572$	0	0
$573$	−1.20796	−0.0504631
$574$	0	0
$575$	4.00000	0.166812
$576$	0	0
$577$	9.65872	0.402098	0.201049	−	0.979581i	$-0.435565\pi$
$577$	9.65872	0.402098	0.201049	+	0.979581i	$0.435565\pi$
$578$	0	0
$579$	0.630732	0.0262123
$580$	0	0
$581$	−0.169322	−0.00702467
$582$	0	0
$583$	7.01998	0.290738
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−30.3200	−1.25144	−0.625720	−	0.780048i	$-0.715195\pi$
$587$	−30.3200	−1.25144	−0.625720	+	0.780048i	$0.715195\pi$
$588$	0	0
$589$	26.3013	1.08373
$590$	0	0
$591$	0.877321	0.0360882
$592$	0	0
$593$	29.8280	1.22489	0.612445	−	0.790513i	$-0.290186\pi$
$593$	29.8280	1.22489	0.612445	+	0.790513i	$0.290186\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−1.47197	−0.0602437
$598$	0	0
$599$	−7.95204	−0.324911	−0.162456	−	0.986716i	$-0.551941\pi$
$599$	−7.95204	−0.324911	−0.162456	+	0.986716i	$0.551941\pi$
$600$	0	0
$601$	−25.3947	−1.03587	−0.517936	−	0.855420i	$-0.673299\pi$
$601$	−25.3947	−1.03587	−0.517936	+	0.855420i	$0.673299\pi$
$602$	0	0
$603$	−0.609917	−0.0248377
$604$	0	0
$605$	−9.69735	−0.394253
$606$	0	0
$607$	−29.4533	−1.19547	−0.597737	−	0.801693i	$-0.703933\pi$
$607$	−29.4533	−1.19547	−0.597737	+	0.801693i	$0.703933\pi$
$608$	0	0
$609$	−0.565344	−0.0229089
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−1.32131	−0.0533670	−0.0266835	−	0.999644i	$-0.508495\pi$
$613$	−1.32131	−0.0533670	−0.0266835	+	0.999644i	$0.508495\pi$
$614$	0	0
$615$	−1.40535	−0.0566692
$616$	0	0
$617$	43.6999	1.75929	0.879646	−	0.475629i	$-0.157780\pi$
$617$	43.6999	1.75929	0.879646	+	0.475629i	$0.157780\pi$
$618$	0	0
$619$	0.209274	0.00841143	0.00420572	−	0.999991i	$-0.498661\pi$
$619$	0.209274	0.00841143	0.00420572	+	0.999991i	$0.498661\pi$
$620$	0	0
$621$	8.52803	0.342218
$622$	0	0
$623$	−5.82003	−0.233175
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	1.98134	0.0791272
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−26.0700	−1.03783	−0.518915	−	0.854826i	$-0.673664\pi$
$631$	−26.0700	−1.03783	−0.518915	+	0.854826i	$0.673664\pi$
$632$	0	0
$633$	0.569213	0.0226242
$634$	0	0
$635$	10.7487	0.426549
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−10.2827	−0.406776
$640$	0	0
$641$	−24.0921	−0.951579	−0.475789	−	0.879559i	$-0.657838\pi$
$641$	−24.0921	−0.951579	−0.475789	+	0.879559i	$0.657838\pi$
$642$	0	0
$643$	29.5933	1.16705	0.583524	−	0.812096i	$-0.301674\pi$
$643$	29.5933	1.16705	0.583524	+	0.812096i	$0.301674\pi$
$644$	0	0
$645$	0.829359	0.0326560
$646$	0	0
$647$	35.7733	1.40639	0.703197	−	0.710995i	$-0.251755\pi$
$647$	35.7733	1.40639	0.703197	+	0.710995i	$0.251755\pi$
$648$	0	0
$649$	−8.88797	−0.348883
$650$	0	0
$651$	−0.726656	−0.0284799
$652$	0	0
$653$	24.3786	0.954008	0.477004	−	0.878901i	$-0.341723\pi$
$653$	24.3786	0.954008	0.477004	+	0.878901i	$0.341723\pi$
$654$	0	0
$655$	−5.50466	−0.215085
$656$	0	0
$657$	−24.9439	−0.973156
$658$	0	0
$659$	−17.5454	−0.683471	−0.341735	−	0.939796i	$-0.611015\pi$
$659$	−17.5454	−0.683471	−0.341735	+	0.939796i	$0.611015\pi$
$660$	0	0
$661$	32.0560	1.24683	0.623416	−	0.781890i	$-0.285744\pi$
$661$	32.0560	1.24683	0.623416	+	0.781890i	$0.285744\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1.73599	0.0673186
$666$	0	0
$667$	17.1307	0.663303
$668$	0	0
$669$	6.20927	0.240064
$670$	0	0
$671$	0.623954	0.0240875
$672$	0	0
$673$	23.2440	0.895992	0.447996	−	0.894036i	$-0.352138\pi$
$673$	23.2440	0.895992	0.447996	+	0.894036i	$0.352138\pi$
$674$	0	0
$675$	2.13201	0.0820610
$676$	0	0
$677$	20.0734	0.771483	0.385742	−	0.922607i	$-0.373946\pi$
$677$	20.0734	0.771483	0.385742	+	0.922607i	$0.373946\pi$
$678$	0	0
$679$	2.17997	0.0836595
$680$	0	0
$681$	−1.81325	−0.0694840
$682$	0	0
$683$	−25.4533	−0.973944	−0.486972	−	0.873418i	$-0.661899\pi$
$683$	−25.4533	−0.973944	−0.486972	+	0.873418i	$0.661899\pi$
$684$	0	0
$685$	3.45331	0.131944
$686$	0	0
$687$	10.2347	0.390479
$688$	0	0
$689$	0	0
$690$	0	0
$691$	1.19608	0.0455009	0.0227505	−	0.999741i	$-0.492758\pi$
$691$	1.19608	0.0455009	0.0227505	+	0.999741i	$0.492758\pi$
$692$	0	0
$693$	1.18930	0.0451777
$694$	0	0
$695$	10.0187	0.380029
$696$	0	0
$697$	0	0
$698$	0	0
$699$	2.17997	0.0824540
$700$	0	0
$701$	29.9414	1.13087	0.565435	−	0.824793i	$-0.308708\pi$
$701$	29.9414	1.13087	0.565435	+	0.824793i	$0.308708\pi$
$702$	0	0
$703$	4.77801	0.180206
$704$	0	0
$705$	−4.69735	−0.176913
$706$	0	0
$707$	4.94271	0.185890
$708$	0	0
$709$	−26.0959	−0.980053	−0.490026	−	0.871708i	$-0.663013\pi$
$709$	−26.0959	−0.980053	−0.490026	+	0.871708i	$0.663013\pi$
$710$	0	0
$711$	20.2500	0.759434
$712$	0	0
$713$	22.0187	0.824605
$714$	0	0
$715$	0	0
$716$	0	0
$717$	7.62140	0.284627
$718$	0	0
$719$	18.5360	0.691278	0.345639	−	0.938368i	$-0.387662\pi$
$719$	18.5360	0.691278	0.345639	+	0.938368i	$0.387662\pi$
$720$	0	0
$721$	2.24536	0.0836215
$722$	0	0
$723$	−0.836138	−0.0310963
$724$	0	0
$725$	4.28267	0.159054
$726$	0	0
$727$	−36.0959	−1.33872	−0.669362	−	0.742937i	$-0.733432\pi$
$727$	−36.0959	−1.33872	−0.669362	+	0.742937i	$0.733432\pi$
$728$	0	0
$729$	−20.1307	−0.745581
$730$	0	0
$731$	0	0
$732$	0	0
$733$	8.13201	0.300363	0.150181	−	0.988658i	$-0.452014\pi$
$733$	8.13201	0.300363	0.150181	+	0.988658i	$0.452014\pi$
$734$	0	0
$735$	2.49534	0.0920418
$736$	0	0
$737$	−0.242720	−0.00894072
$738$	0	0
$739$	0.942709	0.0346781	0.0173391	−	0.999850i	$-0.494481\pi$
$739$	0.942709	0.0346781	0.0173391	+	0.999850i	$0.494481\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−34.8446	−1.27833	−0.639163	−	0.769072i	$-0.720719\pi$
$743$	−34.8446	−1.27833	−0.639163	+	0.769072i	$0.720719\pi$
$744$	0	0
$745$	−1.86799	−0.0684380
$746$	0	0
$747$	1.33657	0.0489027
$748$	0	0
$749$	6.82936	0.249539
$750$	0	0
$751$	−10.8587	−0.396238	−0.198119	−	0.980178i	$-0.563483\pi$
$751$	−10.8587	−0.396238	−0.198119	+	0.980178i	$0.563483\pi$
$752$	0	0
$753$	−9.73599	−0.354799
$754$	0	0
$755$	−4.46264	−0.162412
$756$	0	0
$757$	−38.3386	−1.39344	−0.696721	−	0.717342i	$-0.745358\pi$
$757$	−38.3386	−1.39344	−0.696721	+	0.717342i	$0.745358\pi$
$758$	0	0
$759$	1.65872	0.0602077
$760$	0	0
$761$	−24.4333	−0.885708	−0.442854	−	0.896594i	$-0.646034\pi$
$761$	−24.4333	−0.885708	−0.442854	+	0.896594i	$0.646034\pi$
$762$	0	0
$763$	−1.65194	−0.0598042
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	8.84802	0.319068	0.159534	−	0.987192i	$-0.449001\pi$
$769$	8.84802	0.319068	0.159534	+	0.987192i	$0.449001\pi$
$770$	0	0
$771$	−6.11458	−0.220211
$772$	0	0
$773$	5.58532	0.200890	0.100445	−	0.994943i	$-0.467973\pi$
$773$	5.58532	0.200890	0.100445	+	0.994943i	$0.467973\pi$
$774$	0	0
$775$	5.50466	0.197733
$776$	0	0
$777$	−0.132007	−0.00473574
$778$	0	0
$779$	18.4813	0.662161
$780$	0	0
$781$	−4.09206	−0.146425
$782$	0	0
$783$	9.13069	0.326304
$784$	0	0
$785$	20.1693	0.719874
$786$	0	0
$787$	27.4754	0.979391	0.489695	−	0.871894i	$-0.337108\pi$
$787$	27.4754	0.979391	0.489695	+	0.871894i	$0.337108\pi$
$788$	0	0
$789$	−0.697352	−0.0248264
$790$	0	0
$791$	−3.20796	−0.114062
$792$	0	0
$793$	0	0
$794$	0	0
$795$	2.23471	0.0792570
$796$	0	0
$797$	−2.90663	−0.102958	−0.0514790	−	0.998674i	$-0.516394\pi$
$797$	−2.90663	−0.102958	−0.0514790	+	0.998674i	$0.516394\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	45.9414	1.62326
$802$	0	0
$803$	−9.92660	−0.350302
$804$	0	0
$805$	1.45331	0.0512226
$806$	0	0
$807$	−4.46942	−0.157331
$808$	0	0
$809$	2.82936	0.0994750	0.0497375	−	0.998762i	$-0.484162\pi$
$809$	2.82936	0.0994750	0.0497375	+	0.998762i	$0.484162\pi$
$810$	0	0
$811$	37.9053	1.33104	0.665518	−	0.746382i	$-0.268211\pi$
$811$	37.9053	1.33104	0.665518	+	0.746382i	$0.268211\pi$
$812$	0	0
$813$	10.1320	0.355345
$814$	0	0
$815$	9.55602	0.334733
$816$	0	0
$817$	−10.9066	−0.381575
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−14.2093	−0.495907	−0.247954	−	0.968772i	$-0.579758\pi$
$821$	−14.2093	−0.495907	−0.247954	+	0.968772i	$0.579758\pi$
$822$	0	0
$823$	26.3713	0.919247	0.459624	−	0.888114i	$-0.347984\pi$
$823$	26.3713	0.919247	0.459624	+	0.888114i	$0.347984\pi$
$824$	0	0
$825$	0.414680	0.0144373
$826$	0	0
$827$	8.10270	0.281759	0.140879	−	0.990027i	$-0.455007\pi$
$827$	8.10270	0.281759	0.140879	+	0.990027i	$0.455007\pi$
$828$	0	0
$829$	−5.20796	−0.180880	−0.0904399	−	0.995902i	$-0.528827\pi$
$829$	−5.20796	−0.180880	−0.0904399	+	0.995902i	$0.528827\pi$
$830$	0	0
$831$	5.92273	0.205457
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−15.8387	−0.548120
$836$	0	0
$837$	11.7360	0.405655
$838$	0	0
$839$	9.98134	0.344594	0.172297	−	0.985045i	$-0.444881\pi$
$839$	9.98134	0.344594	0.172297	+	0.985045i	$0.444881\pi$
$840$	0	0
$841$	−10.6587	−0.367542
$842$	0	0
$843$	3.23603	0.111455
$844$	0	0
$845$	−13.0000	−0.447214
$846$	0	0
$847$	−3.52332	−0.121063
$848$	0	0
$849$	5.65872	0.194207
$850$	0	0
$851$	4.00000	0.137118
$852$	0	0
$853$	19.4320	0.665340	0.332670	−	0.943043i	$-0.392051\pi$
$853$	19.4320	0.665340	0.332670	+	0.943043i	$0.392051\pi$
$854$	0	0
$855$	−13.7033	−0.468643
$856$	0	0
$857$	18.5653	0.634180	0.317090	−	0.948395i	$-0.397294\pi$
$857$	18.5653	0.634180	0.317090	+	0.948395i	$0.397294\pi$
$858$	0	0
$859$	17.7687	0.606260	0.303130	−	0.952949i	$-0.401968\pi$
$859$	17.7687	0.606260	0.303130	+	0.952949i	$0.401968\pi$
$860$	0	0
$861$	−0.510604	−0.0174013
$862$	0	0
$863$	−39.4647	−1.34339	−0.671697	−	0.740826i	$-0.734434\pi$
$863$	−39.4647	−1.34339	−0.671697	+	0.740826i	$0.734434\pi$
$864$	0	0
$865$	15.8680	0.539528
$866$	0	0
$867$	6.17658	0.209768
$868$	0	0
$869$	8.05861	0.273370
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−17.2080	−0.582401
$874$	0	0
$875$	0.363328	0.0122827
$876$	0	0
$877$	10.0373	0.338936	0.169468	−	0.985536i	$-0.445795\pi$
$877$	10.0373	0.338936	0.169468	+	0.985536i	$0.445795\pi$
$878$	0	0
$879$	3.63328	0.122548
$880$	0	0
$881$	52.8667	1.78112	0.890562	−	0.454862i	$-0.150312\pi$
$881$	52.8667	1.78112	0.890562	+	0.454862i	$0.150312\pi$
$882$	0	0
$883$	33.5560	1.12925	0.564625	−	0.825348i	$-0.309021\pi$
$883$	33.5560	1.12925	0.564625	+	0.825348i	$0.309021\pi$
$884$	0	0
$885$	−2.82936	−0.0951079
$886$	0	0
$887$	−53.3259	−1.79051	−0.895255	−	0.445555i	$-0.853006\pi$
$887$	−53.3259	−1.79051	−0.895255	+	0.445555i	$0.853006\pi$
$888$	0	0
$889$	3.90531	0.130980
$890$	0	0
$891$	−8.93593	−0.299365
$892$	0	0
$893$	61.7733	2.06716
$894$	0	0
$895$	−9.34335	−0.312314
$896$	0	0
$897$	0	0
$898$	0	0
$899$	23.5747	0.786259
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0.301330	0.0100276
$904$	0	0
$905$	−19.2627	−0.640314
$906$	0	0
$907$	−14.2173	−0.472077	−0.236039	−	0.971744i	$-0.575849\pi$
$907$	−14.2173	−0.472077	−0.236039	+	0.971744i	$0.575849\pi$
$908$	0	0
$909$	−39.0161	−1.29408
$910$	0	0
$911$	−8.08867	−0.267989	−0.133995	−	0.990982i	$-0.542781\pi$
$911$	−8.08867	−0.267989	−0.133995	+	0.990982i	$0.542781\pi$
$912$	0	0
$913$	0.531898	0.0176033
$914$	0	0
$915$	0.198627	0.00656640
$916$	0	0
$917$	−2.00000	−0.0660458
$918$	0	0
$919$	39.3620	1.29843	0.649216	−	0.760604i	$-0.275097\pi$
$919$	39.3620	1.29843	0.649216	+	0.760604i	$0.275097\pi$
$920$	0	0
$921$	−4.13201	−0.136154
$922$	0	0
$923$	0	0
$924$	0	0
$925$	1.00000	0.0328798
$926$	0	0
$927$	−17.7241	−0.582136
$928$	0	0
$929$	30.0000	0.984268	0.492134	−	0.870519i	$-0.336217\pi$
$929$	30.0000	0.984268	0.492134	+	0.870519i	$0.336217\pi$
$930$	0	0
$931$	−32.8153	−1.07548
$932$	0	0
$933$	−0.450763	−0.0147573
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−0.0946926	−0.00309347	−0.00154674	−	0.999999i	$-0.500492\pi$
$937$	−0.0946926	−0.00309347	−0.00154674	+	0.999999i	$0.500492\pi$
$938$	0	0
$939$	4.68934	0.153031
$940$	0	0
$941$	−20.8667	−0.680234	−0.340117	−	0.940383i	$-0.610467\pi$
$941$	−20.8667	−0.680234	−0.340117	+	0.940383i	$0.610467\pi$
$942$	0	0
$943$	15.4720	0.503837
$944$	0	0
$945$	0.774618	0.0251983
$946$	0	0
$947$	18.4813	0.600562	0.300281	−	0.953851i	$-0.402920\pi$
$947$	18.4813	0.600562	0.300281	+	0.953851i	$0.402920\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0.00677877	0.000219817 0
$952$	0	0
$953$	−0.660037	−0.0213807	−0.0106903	−	0.999943i	$-0.503403\pi$
$953$	−0.660037	−0.0213807	−0.0106903	+	0.999943i	$0.503403\pi$
$954$	0	0
$955$	3.32469	0.107585
$956$	0	0
$957$	1.77594	0.0574079
$958$	0	0
$959$	1.25469	0.0405159
$960$	0	0
$961$	−0.698670	−0.0225378
$962$	0	0
$963$	−53.9087	−1.73718
$964$	0	0
$965$	−1.73599	−0.0558833
$966$	0	0
$967$	48.4413	1.55777	0.778884	−	0.627168i	$-0.215786\pi$
$967$	48.4413	1.55777	0.778884	+	0.627168i	$0.215786\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−15.1120	−0.484968	−0.242484	−	0.970155i	$-0.577962\pi$
$971$	−15.1120	−0.484968	−0.242484	+	0.970155i	$0.577962\pi$
$972$	0	0
$973$	3.64006	0.116695
$974$	0	0
$975$	0	0
$976$	0	0
$977$	10.2640	0.328375	0.164187	−	0.986429i	$-0.447500\pi$
$977$	10.2640	0.328375	0.164187	+	0.986429i	$0.447500\pi$
$978$	0	0
$979$	18.2827	0.584317
$980$	0	0
$981$	13.0399	0.416331
$982$	0	0
$983$	7.10864	0.226730	0.113365	−	0.993553i	$-0.463837\pi$
$983$	7.10864	0.226730	0.113365	+	0.993553i	$0.463837\pi$
$984$	0	0
$985$	−2.41468	−0.0769381
$986$	0	0
$987$	−1.70668	−0.0543242
$988$	0	0
$989$	−9.13069	−0.290339
$990$	0	0
$991$	3.94865	0.125433	0.0627165	−	0.998031i	$-0.480024\pi$
$991$	3.94865	0.125433	0.0627165	+	0.998031i	$0.480024\pi$
$992$	0	0
$993$	−3.35739	−0.106544
$994$	0	0
$995$	4.05135	0.128436
$996$	0	0
$997$	−49.7733	−1.57634	−0.788168	−	0.615460i	$-0.788970\pi$
$997$	−49.7733	−1.57634	−0.788168	+	0.615460i	$0.788970\pi$
$998$	0	0
$999$	2.13201	0.0674537

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1480.2.a.f.1.2	✓	3
4.3	odd	2		2960.2.a.s.1.2		3
5.4	even	2		7400.2.a.m.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1480.2.a.f.1.2	✓	3	1.1	even	1	trivial
2960.2.a.s.1.2		3	4.3	odd	2
7400.2.a.m.1.2		3	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:	\( N \)	\(=\)	\( 1480 = 2^{3} \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1480.a (trivial)

\(f(q)\)	\(=\)	\(q-0.363328 q^{3} +1.00000 q^{5} +0.363328 q^{7} -2.86799 q^{9} +O(q^{10})\)	\(q-0.363328 q^{3} +1.00000 q^{5} +0.363328 q^{7} -2.86799 q^{9} -1.14134 q^{11} -0.363328 q^{15} +4.77801 q^{19} -0.132007 q^{21} +4.00000 q^{23} +1.00000 q^{25} +2.13201 q^{27} +4.28267 q^{29} +5.50466 q^{31} +0.414680 q^{33} +0.363328 q^{35} +1.00000 q^{37} +3.86799 q^{41} -2.28267 q^{43} -2.86799 q^{45} +12.9287 q^{47} -6.86799 q^{49} -6.15066 q^{53} -1.14134 q^{55} -1.73599 q^{57} +7.78734 q^{59} -0.546687 q^{61} -1.04202 q^{63} +0.212663 q^{67} -1.45331 q^{69} +3.58532 q^{71} +8.69735 q^{73} -0.363328 q^{75} -0.414680 q^{77} -7.06068 q^{79} +7.82936 q^{81} -0.466031 q^{83} -1.55602 q^{87} -16.0187 q^{89} -2.00000 q^{93} +4.77801 q^{95} +6.00000 q^{97} +3.27334 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{3} + 3 q^{5} - q^{7} + 4 q^{9}+O(q^{10}) \) 3 * q + q^3 + 3 * q^5 - q^7 + 4 * q^9	\( 3 q + q^{3} + 3 q^{5} - q^{7} + 4 q^{9} + 5 q^{11} + q^{15} + 8 q^{19} - 13 q^{21} + 12 q^{23} + 3 q^{25} + 19 q^{27} - 4 q^{29} + 6 q^{31} - 3 q^{33} - q^{35} + 3 q^{37} - q^{41} + 10 q^{43} + 4 q^{45} + 3 q^{47} - 8 q^{49} + 11 q^{53} + 5 q^{55} + 20 q^{57} - 4 q^{59} - 10 q^{61} - 22 q^{63} + 28 q^{67} + 4 q^{69} + 15 q^{71} + 5 q^{73} + q^{75} + 3 q^{77} + 2 q^{79} + 15 q^{81} + 5 q^{83} + 8 q^{87} - 6 q^{89} - 6 q^{93} + 8 q^{95} + 18 q^{97} + 14 q^{99}+O(q^{100}) \) 3 * q + q^3 + 3 * q^5 - q^7 + 4 * q^9 + 5 * q^11 + q^15 + 8 * q^19 - 13 * q^21 + 12 * q^23 + 3 * q^25 + 19 * q^27 - 4 * q^29 + 6 * q^31 - 3 * q^33 - q^35 + 3 * q^37 - q^41 + 10 * q^43 + 4 * q^45 + 3 * q^47 - 8 * q^49 + 11 * q^53 + 5 * q^55 + 20 * q^57 - 4 * q^59 - 10 * q^61 - 22 * q^63 + 28 * q^67 + 4 * q^69 + 15 * q^71 + 5 * q^73 + q^75 + 3 * q^77 + 2 * q^79 + 15 * q^81 + 5 * q^83 + 8 * q^87 - 6 * q^89 - 6 * q^93 + 8 * q^95 + 18 * q^97 + 14 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more