LMFDB - Embedded newform 6000.2.a.bb.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6000,2,Mod(1,6000)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6000, 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("6000.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6000 = 2^{4} \cdot 3 \cdot 5^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6000.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:	$47.9102412128$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 750)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	6000.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +4.61803 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +4.61803 q^{7} +1.00000 q^{9} -0.854102 q^{11} -5.61803 q^{13} +5.70820 q^{17} +7.09017 q^{19} +4.61803 q^{21} +8.09017 q^{23} +1.00000 q^{27} -7.70820 q^{29} -2.47214 q^{31} -0.854102 q^{33} -0.618034 q^{37} -5.61803 q^{39} +2.61803 q^{41} -5.70820 q^{43} +6.38197 q^{47} +14.3262 q^{49} +5.70820 q^{51} -2.61803 q^{53} +7.09017 q^{57} +4.14590 q^{59} -1.70820 q^{61} +4.61803 q^{63} -1.23607 q^{67} +8.09017 q^{69} -4.47214 q^{71} +8.00000 q^{73} -3.94427 q^{77} -3.52786 q^{79} +1.00000 q^{81} +2.00000 q^{83} -7.70820 q^{87} -4.56231 q^{89} -25.9443 q^{91} -2.47214 q^{93} -8.18034 q^{97} -0.854102 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 7 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 + 7 * q^7 + 2 * q^9	$ 2 q + 2 q^{3} + 7 q^{7} + 2 q^{9} + 5 q^{11} - 9 q^{13} - 2 q^{17} + 3 q^{19} + 7 q^{21} + 5 q^{23} + 2 q^{27} - 2 q^{29} + 4 q^{31} + 5 q^{33} + q^{37} - 9 q^{39} + 3 q^{41} + 2 q^{43} + 15 q^{47} + 13 q^{49} - 2 q^{51} - 3 q^{53} + 3 q^{57} + 15 q^{59} + 10 q^{61} + 7 q^{63} + 2 q^{67} + 5 q^{69} + 16 q^{73} + 10 q^{77} - 16 q^{79} + 2 q^{81} + 4 q^{83} - 2 q^{87} + 11 q^{89} - 34 q^{91} + 4 q^{93} + 6 q^{97} + 5 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 + 7 * q^7 + 2 * q^9 + 5 * q^11 - 9 * q^13 - 2 * q^17 + 3 * q^19 + 7 * q^21 + 5 * q^23 + 2 * q^27 - 2 * q^29 + 4 * q^31 + 5 * q^33 + q^37 - 9 * q^39 + 3 * q^41 + 2 * q^43 + 15 * q^47 + 13 * q^49 - 2 * q^51 - 3 * q^53 + 3 * q^57 + 15 * q^59 + 10 * q^61 + 7 * q^63 + 2 * q^67 + 5 * q^69 + 16 * q^73 + 10 * q^77 - 16 * q^79 + 2 * q^81 + 4 * q^83 - 2 * q^87 + 11 * q^89 - 34 * q^91 + 4 * q^93 + 6 * q^97 + 5 * 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.61803	1.74545	0.872726	−	0.488210i	$-0.162350\pi$
$7$	4.61803	1.74545	0.872726	+	0.488210i	$0.162350\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−0.854102	−0.257521	−0.128761	−	0.991676i	$-0.541100\pi$
$11$	−0.854102	−0.257521	−0.128761	+	0.991676i	$0.541100\pi$
$12$	0	0
$13$	−5.61803	−1.55816	−0.779081	−	0.626923i	$-0.784314\pi$
$13$	−5.61803	−1.55816	−0.779081	+	0.626923i	$0.784314\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.70820	1.38444	0.692221	−	0.721685i	$-0.256632\pi$
$17$	5.70820	1.38444	0.692221	+	0.721685i	$0.256632\pi$
$18$	0	0
$19$	7.09017	1.62660	0.813298	−	0.581847i	$-0.197670\pi$
$19$	7.09017	1.62660	0.813298	+	0.581847i	$0.197670\pi$
$20$	0	0
$21$	4.61803	1.00774
$22$	0	0
$23$	8.09017	1.68692	0.843459	−	0.537194i	$-0.180516\pi$
$23$	8.09017	1.68692	0.843459	+	0.537194i	$0.180516\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−7.70820	−1.43138	−0.715689	−	0.698419i	$-0.753887\pi$
$29$	−7.70820	−1.43138	−0.715689	+	0.698419i	$0.753887\pi$
$30$	0	0
$31$	−2.47214	−0.444009	−0.222004	−	0.975046i	$-0.571260\pi$
$31$	−2.47214	−0.444009	−0.222004	+	0.975046i	$0.571260\pi$
$32$	0	0
$33$	−0.854102	−0.148680
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−0.618034	−0.101604	−0.0508021	−	0.998709i	$-0.516178\pi$
$37$	−0.618034	−0.101604	−0.0508021	+	0.998709i	$0.516178\pi$
$38$	0	0
$39$	−5.61803	−0.899605
$40$	0	0
$41$	2.61803	0.408868	0.204434	−	0.978880i	$-0.434465\pi$
$41$	2.61803	0.408868	0.204434	+	0.978880i	$0.434465\pi$
$42$	0	0
$43$	−5.70820	−0.870493	−0.435246	−	0.900311i	$-0.643339\pi$
$43$	−5.70820	−0.870493	−0.435246	+	0.900311i	$0.643339\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	6.38197	0.930905	0.465453	−	0.885073i	$-0.345892\pi$
$47$	6.38197	0.930905	0.465453	+	0.885073i	$0.345892\pi$
$48$	0	0
$49$	14.3262	2.04661
$50$	0	0
$51$	5.70820	0.799308
$52$	0	0
$53$	−2.61803	−0.359615	−0.179807	−	0.983702i	$-0.557547\pi$
$53$	−2.61803	−0.359615	−0.179807	+	0.983702i	$0.557547\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	7.09017	0.939116
$58$	0	0
$59$	4.14590	0.539750	0.269875	−	0.962895i	$-0.413018\pi$
$59$	4.14590	0.539750	0.269875	+	0.962895i	$0.413018\pi$
$60$	0	0
$61$	−1.70820	−0.218713	−0.109357	−	0.994003i	$-0.534879\pi$
$61$	−1.70820	−0.218713	−0.109357	+	0.994003i	$0.534879\pi$
$62$	0	0
$63$	4.61803	0.581818
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−1.23607	−0.151010	−0.0755049	−	0.997145i	$-0.524057\pi$
$67$	−1.23607	−0.151010	−0.0755049	+	0.997145i	$0.524057\pi$
$68$	0	0
$69$	8.09017	0.973942
$70$	0	0
$71$	−4.47214	−0.530745	−0.265372	−	0.964146i	$-0.585495\pi$
$71$	−4.47214	−0.530745	−0.265372	+	0.964146i	$0.585495\pi$
$72$	0	0
$73$	8.00000	0.936329	0.468165	−	0.883641i	$-0.344915\pi$
$73$	8.00000	0.936329	0.468165	+	0.883641i	$0.344915\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−3.94427	−0.449492
$78$	0	0
$79$	−3.52786	−0.396916	−0.198458	−	0.980109i	$-0.563593\pi$
$79$	−3.52786	−0.396916	−0.198458	+	0.980109i	$0.563593\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	2.00000	0.219529	0.109764	−	0.993958i	$-0.464990\pi$
$83$	2.00000	0.219529	0.109764	+	0.993958i	$0.464990\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−7.70820	−0.826406
$88$	0	0
$89$	−4.56231	−0.483603	−0.241802	−	0.970326i	$-0.577738\pi$
$89$	−4.56231	−0.483603	−0.241802	+	0.970326i	$0.577738\pi$
$90$	0	0
$91$	−25.9443	−2.71970
$92$	0	0
$93$	−2.47214	−0.256349
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−8.18034	−0.830588	−0.415294	−	0.909687i	$-0.636321\pi$
$97$	−8.18034	−0.830588	−0.415294	+	0.909687i	$0.636321\pi$
$98$	0	0
$99$	−0.854102	−0.0858405
$100$	0	0
$101$	−0.291796	−0.0290348	−0.0145174	−	0.999895i	$-0.504621\pi$
$101$	−0.291796	−0.0290348	−0.0145174	+	0.999895i	$0.504621\pi$
$102$	0	0
$103$	14.0902	1.38835	0.694173	−	0.719808i	$-0.255770\pi$
$103$	14.0902	1.38835	0.694173	+	0.719808i	$0.255770\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−2.00000	−0.193347	−0.0966736	−	0.995316i	$-0.530820\pi$
$107$	−2.00000	−0.193347	−0.0966736	+	0.995316i	$0.530820\pi$
$108$	0	0
$109$	11.7082	1.12144	0.560721	−	0.828005i	$-0.310524\pi$
$109$	11.7082	1.12144	0.560721	+	0.828005i	$0.310524\pi$
$110$	0	0
$111$	−0.618034	−0.0586612
$112$	0	0
$113$	9.70820	0.913271	0.456636	−	0.889654i	$-0.349054\pi$
$113$	9.70820	0.913271	0.456636	+	0.889654i	$0.349054\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−5.61803	−0.519387
$118$	0	0
$119$	26.3607	2.41648
$120$	0	0
$121$	−10.2705	−0.933683
$122$	0	0
$123$	2.61803	0.236060
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−14.4721	−1.28419	−0.642097	−	0.766623i	$-0.721935\pi$
$127$	−14.4721	−1.28419	−0.642097	+	0.766623i	$0.721935\pi$
$128$	0	0
$129$	−5.70820	−0.502579
$130$	0	0
$131$	4.67376	0.408348	0.204174	−	0.978935i	$-0.434549\pi$
$131$	4.67376	0.408348	0.204174	+	0.978935i	$0.434549\pi$
$132$	0	0
$133$	32.7426	2.83915
$134$	0	0
$135$	0	0
$136$	0	0
$137$	12.9443	1.10590	0.552952	−	0.833213i	$-0.313501\pi$
$137$	12.9443	1.10590	0.552952	+	0.833213i	$0.313501\pi$
$138$	0	0
$139$	0.381966	0.0323979	0.0161990	−	0.999869i	$-0.494843\pi$
$139$	0.381966	0.0323979	0.0161990	+	0.999869i	$0.494843\pi$
$140$	0	0
$141$	6.38197	0.537458
$142$	0	0
$143$	4.79837	0.401260
$144$	0	0
$145$	0	0
$146$	0	0
$147$	14.3262	1.18161
$148$	0	0
$149$	17.4164	1.42681	0.713404	−	0.700753i	$-0.247153\pi$
$149$	17.4164	1.42681	0.713404	+	0.700753i	$0.247153\pi$
$150$	0	0
$151$	13.7082	1.11556	0.557779	−	0.829990i	$-0.311654\pi$
$151$	13.7082	1.11556	0.557779	+	0.829990i	$0.311654\pi$
$152$	0	0
$153$	5.70820	0.461481
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−10.9443	−0.873448	−0.436724	−	0.899596i	$-0.643861\pi$
$157$	−10.9443	−0.873448	−0.436724	+	0.899596i	$0.643861\pi$
$158$	0	0
$159$	−2.61803	−0.207624
$160$	0	0
$161$	37.3607	2.94443
$162$	0	0
$163$	−24.0000	−1.87983	−0.939913	−	0.341415i	$-0.889094\pi$
$163$	−24.0000	−1.87983	−0.939913	+	0.341415i	$0.889094\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−13.0902	−1.01295	−0.506474	−	0.862255i	$-0.669051\pi$
$167$	−13.0902	−1.01295	−0.506474	+	0.862255i	$0.669051\pi$
$168$	0	0
$169$	18.5623	1.42787
$170$	0	0
$171$	7.09017	0.542199
$172$	0	0
$173$	−6.61803	−0.503160	−0.251580	−	0.967837i	$-0.580950\pi$
$173$	−6.61803	−0.503160	−0.251580	+	0.967837i	$0.580950\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	4.14590	0.311625
$178$	0	0
$179$	20.7984	1.55454	0.777272	−	0.629165i	$-0.216603\pi$
$179$	20.7984	1.55454	0.777272	+	0.629165i	$0.216603\pi$
$180$	0	0
$181$	−11.5279	−0.856859	−0.428430	−	0.903575i	$-0.640933\pi$
$181$	−11.5279	−0.856859	−0.428430	+	0.903575i	$0.640933\pi$
$182$	0	0
$183$	−1.70820	−0.126274
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−4.87539	−0.356524
$188$	0	0
$189$	4.61803	0.335913
$190$	0	0
$191$	4.00000	0.289430	0.144715	−	0.989473i	$-0.453773\pi$
$191$	4.00000	0.289430	0.144715	+	0.989473i	$0.453773\pi$
$192$	0	0
$193$	3.70820	0.266922	0.133461	−	0.991054i	$-0.457391\pi$
$193$	3.70820	0.266922	0.133461	+	0.991054i	$0.457391\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−2.94427	−0.209771	−0.104885	−	0.994484i	$-0.533448\pi$
$197$	−2.94427	−0.209771	−0.104885	+	0.994484i	$0.533448\pi$
$198$	0	0
$199$	9.23607	0.654727	0.327364	−	0.944898i	$-0.393840\pi$
$199$	9.23607	0.654727	0.327364	+	0.944898i	$0.393840\pi$
$200$	0	0
$201$	−1.23607	−0.0871855
$202$	0	0
$203$	−35.5967	−2.49840
$204$	0	0
$205$	0	0
$206$	0	0
$207$	8.09017	0.562306
$208$	0	0
$209$	−6.05573	−0.418883
$210$	0	0
$211$	2.43769	0.167818	0.0839089	−	0.996473i	$-0.473260\pi$
$211$	2.43769	0.167818	0.0839089	+	0.996473i	$0.473260\pi$
$212$	0	0
$213$	−4.47214	−0.306426
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−11.4164	−0.774996
$218$	0	0
$219$	8.00000	0.540590
$220$	0	0
$221$	−32.0689	−2.15719
$222$	0	0
$223$	−11.0557	−0.740346	−0.370173	−	0.928963i	$-0.620702\pi$
$223$	−11.0557	−0.740346	−0.370173	+	0.928963i	$0.620702\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	15.2361	1.01125	0.505627	−	0.862752i	$-0.331261\pi$
$227$	15.2361	1.01125	0.505627	+	0.862752i	$0.331261\pi$
$228$	0	0
$229$	−27.2361	−1.79981	−0.899905	−	0.436086i	$-0.856364\pi$
$229$	−27.2361	−1.79981	−0.899905	+	0.436086i	$0.856364\pi$
$230$	0	0
$231$	−3.94427	−0.259514
$232$	0	0
$233$	−6.47214	−0.424004	−0.212002	−	0.977269i	$-0.567998\pi$
$233$	−6.47214	−0.424004	−0.212002	+	0.977269i	$0.567998\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−3.52786	−0.229159
$238$	0	0
$239$	24.9443	1.61351	0.806755	−	0.590886i	$-0.201222\pi$
$239$	24.9443	1.61351	0.806755	+	0.590886i	$0.201222\pi$
$240$	0	0
$241$	3.90983	0.251854	0.125927	−	0.992039i	$-0.459809\pi$
$241$	3.90983	0.251854	0.125927	+	0.992039i	$0.459809\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−39.8328	−2.53450
$248$	0	0
$249$	2.00000	0.126745
$250$	0	0
$251$	24.0000	1.51487	0.757433	−	0.652913i	$-0.226453\pi$
$251$	24.0000	1.51487	0.757433	+	0.652913i	$0.226453\pi$
$252$	0	0
$253$	−6.90983	−0.434417
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−26.4721	−1.65129	−0.825643	−	0.564193i	$-0.809188\pi$
$257$	−26.4721	−1.65129	−0.825643	+	0.564193i	$0.809188\pi$
$258$	0	0
$259$	−2.85410	−0.177345
$260$	0	0
$261$	−7.70820	−0.477126
$262$	0	0
$263$	9.67376	0.596510	0.298255	−	0.954486i	$-0.403595\pi$
$263$	9.67376	0.596510	0.298255	+	0.954486i	$0.403595\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−4.56231	−0.279209
$268$	0	0
$269$	17.4164	1.06190	0.530949	−	0.847404i	$-0.321836\pi$
$269$	17.4164	1.06190	0.530949	+	0.847404i	$0.321836\pi$
$270$	0	0
$271$	−17.4164	−1.05797	−0.528986	−	0.848631i	$-0.677427\pi$
$271$	−17.4164	−1.05797	−0.528986	+	0.848631i	$0.677427\pi$
$272$	0	0
$273$	−25.9443	−1.57022
$274$	0	0
$275$	0	0
$276$	0	0
$277$	28.4508	1.70945	0.854723	−	0.519084i	$-0.173727\pi$
$277$	28.4508	1.70945	0.854723	+	0.519084i	$0.173727\pi$
$278$	0	0
$279$	−2.47214	−0.148003
$280$	0	0
$281$	−13.5066	−0.805735	−0.402867	−	0.915258i	$-0.631986\pi$
$281$	−13.5066	−0.805735	−0.402867	+	0.915258i	$0.631986\pi$
$282$	0	0
$283$	−14.1803	−0.842934	−0.421467	−	0.906844i	$-0.638485\pi$
$283$	−14.1803	−0.842934	−0.421467	+	0.906844i	$0.638485\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	12.0902	0.713660
$288$	0	0
$289$	15.5836	0.916682
$290$	0	0
$291$	−8.18034	−0.479540
$292$	0	0
$293$	−3.79837	−0.221903	−0.110952	−	0.993826i	$-0.535390\pi$
$293$	−3.79837	−0.221903	−0.110952	+	0.993826i	$0.535390\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−0.854102	−0.0495600
$298$	0	0
$299$	−45.4508	−2.62849
$300$	0	0
$301$	−26.3607	−1.51940
$302$	0	0
$303$	−0.291796	−0.0167632
$304$	0	0
$305$	0	0
$306$	0	0
$307$	16.1803	0.923461	0.461730	−	0.887020i	$-0.347229\pi$
$307$	16.1803	0.923461	0.461730	+	0.887020i	$0.347229\pi$
$308$	0	0
$309$	14.0902	0.801562
$310$	0	0
$311$	8.94427	0.507183	0.253592	−	0.967311i	$-0.418388\pi$
$311$	8.94427	0.507183	0.253592	+	0.967311i	$0.418388\pi$
$312$	0	0
$313$	−13.7082	−0.774833	−0.387417	−	0.921905i	$-0.626633\pi$
$313$	−13.7082	−0.774833	−0.387417	+	0.921905i	$0.626633\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−17.1459	−0.963010	−0.481505	−	0.876443i	$-0.659910\pi$
$317$	−17.1459	−0.963010	−0.481505	+	0.876443i	$0.659910\pi$
$318$	0	0
$319$	6.58359	0.368610
$320$	0	0
$321$	−2.00000	−0.111629
$322$	0	0
$323$	40.4721	2.25193
$324$	0	0
$325$	0	0
$326$	0	0
$327$	11.7082	0.647465
$328$	0	0
$329$	29.4721	1.62485
$330$	0	0
$331$	2.47214	0.135881	0.0679404	−	0.997689i	$-0.478357\pi$
$331$	2.47214	0.135881	0.0679404	+	0.997689i	$0.478357\pi$
$332$	0	0
$333$	−0.618034	−0.0338681
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−32.8328	−1.78852	−0.894259	−	0.447550i	$-0.852297\pi$
$337$	−32.8328	−1.78852	−0.894259	+	0.447550i	$0.852297\pi$
$338$	0	0
$339$	9.70820	0.527277
$340$	0	0
$341$	2.11146	0.114342
$342$	0	0
$343$	33.8328	1.82680
$344$	0	0
$345$	0	0
$346$	0	0
$347$	8.18034	0.439144	0.219572	−	0.975596i	$-0.429534\pi$
$347$	8.18034	0.439144	0.219572	+	0.975596i	$0.429534\pi$
$348$	0	0
$349$	−20.0000	−1.07058	−0.535288	−	0.844670i	$-0.679797\pi$
$349$	−20.0000	−1.07058	−0.535288	+	0.844670i	$0.679797\pi$
$350$	0	0
$351$	−5.61803	−0.299868
$352$	0	0
$353$	−31.1246	−1.65660	−0.828298	−	0.560288i	$-0.810691\pi$
$353$	−31.1246	−1.65660	−0.828298	+	0.560288i	$0.810691\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	26.3607	1.39516
$358$	0	0
$359$	−9.41641	−0.496979	−0.248489	−	0.968635i	$-0.579934\pi$
$359$	−9.41641	−0.496979	−0.248489	+	0.968635i	$0.579934\pi$
$360$	0	0
$361$	31.2705	1.64582
$362$	0	0
$363$	−10.2705	−0.539062
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−11.4164	−0.595932	−0.297966	−	0.954577i	$-0.596308\pi$
$367$	−11.4164	−0.595932	−0.297966	+	0.954577i	$0.596308\pi$
$368$	0	0
$369$	2.61803	0.136289
$370$	0	0
$371$	−12.0902	−0.627690
$372$	0	0
$373$	3.67376	0.190220	0.0951101	−	0.995467i	$-0.469680\pi$
$373$	3.67376	0.190220	0.0951101	+	0.995467i	$0.469680\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	43.3050	2.23032
$378$	0	0
$379$	28.9230	1.48567	0.742837	−	0.669472i	$-0.233480\pi$
$379$	28.9230	1.48567	0.742837	+	0.669472i	$0.233480\pi$
$380$	0	0
$381$	−14.4721	−0.741430
$382$	0	0
$383$	−2.03444	−0.103955	−0.0519776	−	0.998648i	$-0.516552\pi$
$383$	−2.03444	−0.103955	−0.0519776	+	0.998648i	$0.516552\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−5.70820	−0.290164
$388$	0	0
$389$	32.5410	1.64990	0.824948	−	0.565209i	$-0.191205\pi$
$389$	32.5410	1.64990	0.824948	+	0.565209i	$0.191205\pi$
$390$	0	0
$391$	46.1803	2.33544
$392$	0	0
$393$	4.67376	0.235760
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−7.67376	−0.385135	−0.192568	−	0.981284i	$-0.561681\pi$
$397$	−7.67376	−0.385135	−0.192568	+	0.981284i	$0.561681\pi$
$398$	0	0
$399$	32.7426	1.63918
$400$	0	0
$401$	14.3262	0.715418	0.357709	−	0.933833i	$-0.383558\pi$
$401$	14.3262	0.715418	0.357709	+	0.933833i	$0.383558\pi$
$402$	0	0
$403$	13.8885	0.691838
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0.527864	0.0261652
$408$	0	0
$409$	1.85410	0.0916794	0.0458397	−	0.998949i	$-0.485404\pi$
$409$	1.85410	0.0916794	0.0458397	+	0.998949i	$0.485404\pi$
$410$	0	0
$411$	12.9443	0.638494
$412$	0	0
$413$	19.1459	0.942108
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0.381966	0.0187050
$418$	0	0
$419$	17.8885	0.873913	0.436956	−	0.899483i	$-0.356056\pi$
$419$	17.8885	0.873913	0.436956	+	0.899483i	$0.356056\pi$
$420$	0	0
$421$	15.1246	0.737128	0.368564	−	0.929602i	$-0.379849\pi$
$421$	15.1246	0.737128	0.368564	+	0.929602i	$0.379849\pi$
$422$	0	0
$423$	6.38197	0.310302
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−7.88854	−0.381753
$428$	0	0
$429$	4.79837	0.231668
$430$	0	0
$431$	−12.7639	−0.614817	−0.307408	−	0.951578i	$-0.599462\pi$
$431$	−12.7639	−0.614817	−0.307408	+	0.951578i	$0.599462\pi$
$432$	0	0
$433$	6.36068	0.305675	0.152837	−	0.988251i	$-0.451159\pi$
$433$	6.36068	0.305675	0.152837	+	0.988251i	$0.451159\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	57.3607	2.74393
$438$	0	0
$439$	1.34752	0.0643138	0.0321569	−	0.999483i	$-0.489762\pi$
$439$	1.34752	0.0643138	0.0321569	+	0.999483i	$0.489762\pi$
$440$	0	0
$441$	14.3262	0.682202
$442$	0	0
$443$	27.1246	1.28873	0.644365	−	0.764718i	$-0.277122\pi$
$443$	27.1246	1.28873	0.644365	+	0.764718i	$0.277122\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	17.4164	0.823768
$448$	0	0
$449$	−3.56231	−0.168116	−0.0840578	−	0.996461i	$-0.526788\pi$
$449$	−3.56231	−0.168116	−0.0840578	+	0.996461i	$0.526788\pi$
$450$	0	0
$451$	−2.23607	−0.105292
$452$	0	0
$453$	13.7082	0.644068
$454$	0	0
$455$	0	0
$456$	0	0
$457$	1.88854	0.0883424	0.0441712	−	0.999024i	$-0.485935\pi$
$457$	1.88854	0.0883424	0.0441712	+	0.999024i	$0.485935\pi$
$458$	0	0
$459$	5.70820	0.266436
$460$	0	0
$461$	−29.5967	−1.37846	−0.689229	−	0.724544i	$-0.742051\pi$
$461$	−29.5967	−1.37846	−0.689229	+	0.724544i	$0.742051\pi$
$462$	0	0
$463$	−32.9443	−1.53105	−0.765525	−	0.643406i	$-0.777521\pi$
$463$	−32.9443	−1.53105	−0.765525	+	0.643406i	$0.777521\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	16.7639	0.775742	0.387871	−	0.921714i	$-0.373211\pi$
$467$	16.7639	0.775742	0.387871	+	0.921714i	$0.373211\pi$
$468$	0	0
$469$	−5.70820	−0.263580
$470$	0	0
$471$	−10.9443	−0.504285
$472$	0	0
$473$	4.87539	0.224171
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−2.61803	−0.119872
$478$	0	0
$479$	−31.1246	−1.42212	−0.711060	−	0.703131i	$-0.751785\pi$
$479$	−31.1246	−1.42212	−0.711060	+	0.703131i	$0.751785\pi$
$480$	0	0
$481$	3.47214	0.158316
$482$	0	0
$483$	37.3607	1.69997
$484$	0	0
$485$	0	0
$486$	0	0
$487$	11.7984	0.534635	0.267318	−	0.963608i	$-0.413863\pi$
$487$	11.7984	0.534635	0.267318	+	0.963608i	$0.413863\pi$
$488$	0	0
$489$	−24.0000	−1.08532
$490$	0	0
$491$	18.0344	0.813883	0.406941	−	0.913454i	$-0.366595\pi$
$491$	18.0344	0.813883	0.406941	+	0.913454i	$0.366595\pi$
$492$	0	0
$493$	−44.0000	−1.98166
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−20.6525	−0.926390
$498$	0	0
$499$	7.61803	0.341030	0.170515	−	0.985355i	$-0.445457\pi$
$499$	7.61803	0.341030	0.170515	+	0.985355i	$0.445457\pi$
$500$	0	0
$501$	−13.0902	−0.584826
$502$	0	0
$503$	−36.8541	−1.64324	−0.821622	−	0.570033i	$-0.806930\pi$
$503$	−36.8541	−1.64324	−0.821622	+	0.570033i	$0.806930\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	18.5623	0.824381
$508$	0	0
$509$	25.7082	1.13950	0.569748	−	0.821819i	$-0.307041\pi$
$509$	25.7082	1.13950	0.569748	+	0.821819i	$0.307041\pi$
$510$	0	0
$511$	36.9443	1.63432
$512$	0	0
$513$	7.09017	0.313039
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−5.45085	−0.239728
$518$	0	0
$519$	−6.61803	−0.290499
$520$	0	0
$521$	−10.0902	−0.442058	−0.221029	−	0.975267i	$-0.570942\pi$
$521$	−10.0902	−0.442058	−0.221029	+	0.975267i	$0.570942\pi$
$522$	0	0
$523$	−12.3607	−0.540495	−0.270247	−	0.962791i	$-0.587105\pi$
$523$	−12.3607	−0.540495	−0.270247	+	0.962791i	$0.587105\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−14.1115	−0.614705
$528$	0	0
$529$	42.4508	1.84569
$530$	0	0
$531$	4.14590	0.179917
$532$	0	0
$533$	−14.7082	−0.637083
$534$	0	0
$535$	0	0
$536$	0	0
$537$	20.7984	0.897516
$538$	0	0
$539$	−12.2361	−0.527045
$540$	0	0
$541$	13.4164	0.576816	0.288408	−	0.957508i	$-0.406874\pi$
$541$	13.4164	0.576816	0.288408	+	0.957508i	$0.406874\pi$
$542$	0	0
$543$	−11.5279	−0.494708
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−5.23607	−0.223878	−0.111939	−	0.993715i	$-0.535706\pi$
$547$	−5.23607	−0.223878	−0.111939	+	0.993715i	$0.535706\pi$
$548$	0	0
$549$	−1.70820	−0.0729044
$550$	0	0
$551$	−54.6525	−2.32827
$552$	0	0
$553$	−16.2918	−0.692798
$554$	0	0
$555$	0	0
$556$	0	0
$557$	42.4508	1.79870	0.899350	−	0.437229i	$-0.144040\pi$
$557$	42.4508	1.79870	0.899350	+	0.437229i	$0.144040\pi$
$558$	0	0
$559$	32.0689	1.35637
$560$	0	0
$561$	−4.87539	−0.205839
$562$	0	0
$563$	−10.9443	−0.461246	−0.230623	−	0.973043i	$-0.574076\pi$
$563$	−10.9443	−0.461246	−0.230623	+	0.973043i	$0.574076\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	4.61803	0.193939
$568$	0	0
$569$	−36.9230	−1.54789	−0.773946	−	0.633252i	$-0.781720\pi$
$569$	−36.9230	−1.54789	−0.773946	+	0.633252i	$0.781720\pi$
$570$	0	0
$571$	29.2705	1.22493	0.612466	−	0.790497i	$-0.290177\pi$
$571$	29.2705	1.22493	0.612466	+	0.790497i	$0.290177\pi$
$572$	0	0
$573$	4.00000	0.167102
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−4.76393	−0.198325	−0.0991625	−	0.995071i	$-0.531616\pi$
$577$	−4.76393	−0.198325	−0.0991625	+	0.995071i	$0.531616\pi$
$578$	0	0
$579$	3.70820	0.154108
$580$	0	0
$581$	9.23607	0.383177
$582$	0	0
$583$	2.23607	0.0926085
$584$	0	0
$585$	0	0
$586$	0	0
$587$	20.7639	0.857019	0.428510	−	0.903537i	$-0.359039\pi$
$587$	20.7639	0.857019	0.428510	+	0.903537i	$0.359039\pi$
$588$	0	0
$589$	−17.5279	−0.722223
$590$	0	0
$591$	−2.94427	−0.121111
$592$	0	0
$593$	31.3050	1.28554	0.642770	−	0.766059i	$-0.277785\pi$
$593$	31.3050	1.28554	0.642770	+	0.766059i	$0.277785\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	9.23607	0.378007
$598$	0	0
$599$	11.2361	0.459093	0.229547	−	0.973298i	$-0.426276\pi$
$599$	11.2361	0.459093	0.229547	+	0.973298i	$0.426276\pi$
$600$	0	0
$601$	30.5623	1.24666	0.623331	−	0.781958i	$-0.285779\pi$
$601$	30.5623	1.24666	0.623331	+	0.781958i	$0.285779\pi$
$602$	0	0
$603$	−1.23607	−0.0503366
$604$	0	0
$605$	0	0
$606$	0	0
$607$	36.9787	1.50092	0.750460	−	0.660916i	$-0.229832\pi$
$607$	36.9787	1.50092	0.750460	+	0.660916i	$0.229832\pi$
$608$	0	0
$609$	−35.5967	−1.44245
$610$	0	0
$611$	−35.8541	−1.45050
$612$	0	0
$613$	−10.7426	−0.433891	−0.216946	−	0.976184i	$-0.569609\pi$
$613$	−10.7426	−0.433891	−0.216946	+	0.976184i	$0.569609\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−14.4721	−0.582626	−0.291313	−	0.956628i	$-0.594092\pi$
$617$	−14.4721	−0.582626	−0.291313	+	0.956628i	$0.594092\pi$
$618$	0	0
$619$	5.14590	0.206831	0.103416	−	0.994638i	$-0.467023\pi$
$619$	5.14590	0.206831	0.103416	+	0.994638i	$0.467023\pi$
$620$	0	0
$621$	8.09017	0.324647
$622$	0	0
$623$	−21.0689	−0.844107
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−6.05573	−0.241842
$628$	0	0
$629$	−3.52786	−0.140665
$630$	0	0
$631$	−20.2918	−0.807804	−0.403902	−	0.914802i	$-0.632346\pi$
$631$	−20.2918	−0.807804	−0.403902	+	0.914802i	$0.632346\pi$
$632$	0	0
$633$	2.43769	0.0968896
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−80.4853	−3.18894
$638$	0	0
$639$	−4.47214	−0.176915
$640$	0	0
$641$	−35.8541	−1.41615	−0.708076	−	0.706136i	$-0.750437\pi$
$641$	−35.8541	−1.41615	−0.708076	+	0.706136i	$0.750437\pi$
$642$	0	0
$643$	−32.9443	−1.29920	−0.649598	−	0.760278i	$-0.725063\pi$
$643$	−32.9443	−1.29920	−0.649598	+	0.760278i	$0.725063\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	8.14590	0.320248	0.160124	−	0.987097i	$-0.448811\pi$
$647$	8.14590	0.320248	0.160124	+	0.987097i	$0.448811\pi$
$648$	0	0
$649$	−3.54102	−0.138997
$650$	0	0
$651$	−11.4164	−0.447444
$652$	0	0
$653$	−18.8541	−0.737818	−0.368909	−	0.929466i	$-0.620269\pi$
$653$	−18.8541	−0.737818	−0.368909	+	0.929466i	$0.620269\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	8.00000	0.312110
$658$	0	0
$659$	23.5066	0.915686	0.457843	−	0.889033i	$-0.348622\pi$
$659$	23.5066	0.915686	0.457843	+	0.889033i	$0.348622\pi$
$660$	0	0
$661$	−41.1246	−1.59956	−0.799781	−	0.600292i	$-0.795051\pi$
$661$	−41.1246	−1.59956	−0.799781	+	0.600292i	$0.795051\pi$
$662$	0	0
$663$	−32.0689	−1.24545
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−62.3607	−2.41462
$668$	0	0
$669$	−11.0557	−0.427439
$670$	0	0
$671$	1.45898	0.0563233
$672$	0	0
$673$	18.4721	0.712049	0.356024	−	0.934477i	$-0.384132\pi$
$673$	18.4721	0.712049	0.356024	+	0.934477i	$0.384132\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−17.7426	−0.681905	−0.340953	−	0.940080i	$-0.610750\pi$
$677$	−17.7426	−0.681905	−0.340953	+	0.940080i	$0.610750\pi$
$678$	0	0
$679$	−37.7771	−1.44975
$680$	0	0
$681$	15.2361	0.583847
$682$	0	0
$683$	5.88854	0.225319	0.112659	−	0.993634i	$-0.464063\pi$
$683$	5.88854	0.225319	0.112659	+	0.993634i	$0.464063\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−27.2361	−1.03912
$688$	0	0
$689$	14.7082	0.560338
$690$	0	0
$691$	−1.88854	−0.0718436	−0.0359218	−	0.999355i	$-0.511437\pi$
$691$	−1.88854	−0.0718436	−0.0359218	+	0.999355i	$0.511437\pi$
$692$	0	0
$693$	−3.94427	−0.149831
$694$	0	0
$695$	0	0
$696$	0	0
$697$	14.9443	0.566055
$698$	0	0
$699$	−6.47214	−0.244799
$700$	0	0
$701$	−4.29180	−0.162099	−0.0810495	−	0.996710i	$-0.525827\pi$
$701$	−4.29180	−0.162099	−0.0810495	+	0.996710i	$0.525827\pi$
$702$	0	0
$703$	−4.38197	−0.165269
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−1.34752	−0.0506789
$708$	0	0
$709$	−12.6525	−0.475174	−0.237587	−	0.971366i	$-0.576356\pi$
$709$	−12.6525	−0.475174	−0.237587	+	0.971366i	$0.576356\pi$
$710$	0	0
$711$	−3.52786	−0.132305
$712$	0	0
$713$	−20.0000	−0.749006
$714$	0	0
$715$	0	0
$716$	0	0
$717$	24.9443	0.931561
$718$	0	0
$719$	−47.8885	−1.78594	−0.892971	−	0.450115i	$-0.851383\pi$
$719$	−47.8885	−1.78594	−0.892971	+	0.450115i	$0.851383\pi$
$720$	0	0
$721$	65.0689	2.42329
$722$	0	0
$723$	3.90983	0.145408
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−28.3262	−1.05056	−0.525281	−	0.850929i	$-0.676040\pi$
$727$	−28.3262	−1.05056	−0.525281	+	0.850929i	$0.676040\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−32.5836	−1.20515
$732$	0	0
$733$	46.9230	1.73314	0.866570	−	0.499056i	$-0.166320\pi$
$733$	46.9230	1.73314	0.866570	+	0.499056i	$0.166320\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.05573	0.0388882
$738$	0	0
$739$	−7.67376	−0.282284	−0.141142	−	0.989989i	$-0.545077\pi$
$739$	−7.67376	−0.282284	−0.141142	+	0.989989i	$0.545077\pi$
$740$	0	0
$741$	−39.8328	−1.46330
$742$	0	0
$743$	−44.8541	−1.64554	−0.822769	−	0.568376i	$-0.807572\pi$
$743$	−44.8541	−1.64554	−0.822769	+	0.568376i	$0.807572\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	2.00000	0.0731762
$748$	0	0
$749$	−9.23607	−0.337479
$750$	0	0
$751$	−54.0689	−1.97300	−0.986501	−	0.163756i	$-0.947639\pi$
$751$	−54.0689	−1.97300	−0.986501	+	0.163756i	$0.947639\pi$
$752$	0	0
$753$	24.0000	0.874609
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−27.3262	−0.993189	−0.496595	−	0.867983i	$-0.665416\pi$
$757$	−27.3262	−0.993189	−0.496595	+	0.867983i	$0.665416\pi$
$758$	0	0
$759$	−6.90983	−0.250811
$760$	0	0
$761$	25.9656	0.941251	0.470625	−	0.882333i	$-0.344028\pi$
$761$	25.9656	0.941251	0.470625	+	0.882333i	$0.344028\pi$
$762$	0	0
$763$	54.0689	1.95743
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−23.2918	−0.841018
$768$	0	0
$769$	27.3262	0.985409	0.492705	−	0.870197i	$-0.336008\pi$
$769$	27.3262	0.985409	0.492705	+	0.870197i	$0.336008\pi$
$770$	0	0
$771$	−26.4721	−0.953371
$772$	0	0
$773$	9.05573	0.325712	0.162856	−	0.986650i	$-0.447929\pi$
$773$	9.05573	0.325712	0.162856	+	0.986650i	$0.447929\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−2.85410	−0.102390
$778$	0	0
$779$	18.5623	0.665064
$780$	0	0
$781$	3.81966	0.136678
$782$	0	0
$783$	−7.70820	−0.275469
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−37.4164	−1.33375	−0.666875	−	0.745169i	$-0.732369\pi$
$787$	−37.4164	−1.33375	−0.666875	+	0.745169i	$0.732369\pi$
$788$	0	0
$789$	9.67376	0.344395
$790$	0	0
$791$	44.8328	1.59407
$792$	0	0
$793$	9.59675	0.340791
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−48.8673	−1.73097	−0.865484	−	0.500937i	$-0.832989\pi$
$797$	−48.8673	−1.73097	−0.865484	+	0.500937i	$0.832989\pi$
$798$	0	0
$799$	36.4296	1.28879
$800$	0	0
$801$	−4.56231	−0.161201
$802$	0	0
$803$	−6.83282	−0.241125
$804$	0	0
$805$	0	0
$806$	0	0
$807$	17.4164	0.613087
$808$	0	0
$809$	30.6738	1.07843	0.539216	−	0.842167i	$-0.318721\pi$
$809$	30.6738	1.07843	0.539216	+	0.842167i	$0.318721\pi$
$810$	0	0
$811$	−30.2705	−1.06294	−0.531471	−	0.847077i	$-0.678360\pi$
$811$	−30.2705	−1.06294	−0.531471	+	0.847077i	$0.678360\pi$
$812$	0	0
$813$	−17.4164	−0.610820
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−40.4721	−1.41594
$818$	0	0
$819$	−25.9443	−0.906566
$820$	0	0
$821$	5.88854	0.205512	0.102756	−	0.994707i	$-0.467234\pi$
$821$	5.88854	0.205512	0.102756	+	0.994707i	$0.467234\pi$
$822$	0	0
$823$	29.9230	1.04305	0.521525	−	0.853236i	$-0.325363\pi$
$823$	29.9230	1.04305	0.521525	+	0.853236i	$0.325363\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	28.1803	0.979926	0.489963	−	0.871743i	$-0.337010\pi$
$827$	28.1803	0.979926	0.489963	+	0.871743i	$0.337010\pi$
$828$	0	0
$829$	50.3607	1.74910	0.874549	−	0.484937i	$-0.161157\pi$
$829$	50.3607	1.74910	0.874549	+	0.484937i	$0.161157\pi$
$830$	0	0
$831$	28.4508	0.986949
$832$	0	0
$833$	81.7771	2.83341
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−2.47214	−0.0854495
$838$	0	0
$839$	−54.6525	−1.88681	−0.943407	−	0.331639i	$-0.892399\pi$
$839$	−54.6525	−1.88681	−0.943407	+	0.331639i	$0.892399\pi$
$840$	0	0
$841$	30.4164	1.04884
$842$	0	0
$843$	−13.5066	−0.465191
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−47.4296	−1.62970
$848$	0	0
$849$	−14.1803	−0.486668
$850$	0	0
$851$	−5.00000	−0.171398
$852$	0	0
$853$	−14.7295	−0.504328	−0.252164	−	0.967684i	$-0.581142\pi$
$853$	−14.7295	−0.504328	−0.252164	+	0.967684i	$0.581142\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	9.70820	0.331626	0.165813	−	0.986157i	$-0.446975\pi$
$857$	9.70820	0.331626	0.165813	+	0.986157i	$0.446975\pi$
$858$	0	0
$859$	−28.0344	−0.956523	−0.478261	−	0.878218i	$-0.658733\pi$
$859$	−28.0344	−0.956523	−0.478261	+	0.878218i	$0.658733\pi$
$860$	0	0
$861$	12.0902	0.412032
$862$	0	0
$863$	−18.0344	−0.613900	−0.306950	−	0.951726i	$-0.599308\pi$
$863$	−18.0344	−0.613900	−0.306950	+	0.951726i	$0.599308\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	15.5836	0.529247
$868$	0	0
$869$	3.01316	0.102214
$870$	0	0
$871$	6.94427	0.235298
$872$	0	0
$873$	−8.18034	−0.276863
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−37.9787	−1.28245	−0.641225	−	0.767353i	$-0.721573\pi$
$877$	−37.9787	−1.28245	−0.641225	+	0.767353i	$0.721573\pi$
$878$	0	0
$879$	−3.79837	−0.128116
$880$	0	0
$881$	53.1591	1.79097	0.895487	−	0.445088i	$-0.146827\pi$
$881$	53.1591	1.79097	0.895487	+	0.445088i	$0.146827\pi$
$882$	0	0
$883$	19.4164	0.653414	0.326707	−	0.945126i	$-0.394061\pi$
$883$	19.4164	0.653414	0.326707	+	0.945126i	$0.394061\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	2.03444	0.0683099	0.0341549	−	0.999417i	$-0.489126\pi$
$887$	2.03444	0.0683099	0.0341549	+	0.999417i	$0.489126\pi$
$888$	0	0
$889$	−66.8328	−2.24150
$890$	0	0
$891$	−0.854102	−0.0286135
$892$	0	0
$893$	45.2492	1.51421
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−45.4508	−1.51756
$898$	0	0
$899$	19.0557	0.635544
$900$	0	0
$901$	−14.9443	−0.497866
$902$	0	0
$903$	−26.3607	−0.877228
$904$	0	0
$905$	0	0
$906$	0	0
$907$	14.6525	0.486527	0.243264	−	0.969960i	$-0.421782\pi$
$907$	14.6525	0.486527	0.243264	+	0.969960i	$0.421782\pi$
$908$	0	0
$909$	−0.291796	−0.00967826
$910$	0	0
$911$	46.9443	1.55533	0.777667	−	0.628677i	$-0.216403\pi$
$911$	46.9443	1.55533	0.777667	+	0.628677i	$0.216403\pi$
$912$	0	0
$913$	−1.70820	−0.0565333
$914$	0	0
$915$	0	0
$916$	0	0
$917$	21.5836	0.712753
$918$	0	0
$919$	−41.2361	−1.36025	−0.680126	−	0.733095i	$-0.738075\pi$
$919$	−41.2361	−1.36025	−0.680126	+	0.733095i	$0.738075\pi$
$920$	0	0
$921$	16.1803	0.533160
$922$	0	0
$923$	25.1246	0.826987
$924$	0	0
$925$	0	0
$926$	0	0
$927$	14.0902	0.462782
$928$	0	0
$929$	−58.8115	−1.92954	−0.964772	−	0.263088i	$-0.915259\pi$
$929$	−58.8115	−1.92954	−0.964772	+	0.263088i	$0.915259\pi$
$930$	0	0
$931$	101.575	3.32900
$932$	0	0
$933$	8.94427	0.292822
$934$	0	0
$935$	0	0
$936$	0	0
$937$	24.6525	0.805361	0.402681	−	0.915341i	$-0.368079\pi$
$937$	24.6525	0.805361	0.402681	+	0.915341i	$0.368079\pi$
$938$	0	0
$939$	−13.7082	−0.447350
$940$	0	0
$941$	4.18034	0.136275	0.0681376	−	0.997676i	$-0.478294\pi$
$941$	4.18034	0.136275	0.0681376	+	0.997676i	$0.478294\pi$
$942$	0	0
$943$	21.1803	0.689727
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−21.8885	−0.711282	−0.355641	−	0.934623i	$-0.615737\pi$
$947$	−21.8885	−0.711282	−0.355641	+	0.934623i	$0.615737\pi$
$948$	0	0
$949$	−44.9443	−1.45895
$950$	0	0
$951$	−17.1459	−0.555994
$952$	0	0
$953$	−44.5410	−1.44283	−0.721413	−	0.692506i	$-0.756507\pi$
$953$	−44.5410	−1.44283	−0.721413	+	0.692506i	$0.756507\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	6.58359	0.212817
$958$	0	0
$959$	59.7771	1.93030
$960$	0	0
$961$	−24.8885	−0.802856
$962$	0	0
$963$	−2.00000	−0.0644491
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−1.68692	−0.0542476	−0.0271238	−	0.999632i	$-0.508635\pi$
$967$	−1.68692	−0.0542476	−0.0271238	+	0.999632i	$0.508635\pi$
$968$	0	0
$969$	40.4721	1.30015
$970$	0	0
$971$	40.2705	1.29234	0.646171	−	0.763193i	$-0.276369\pi$
$971$	40.2705	1.29234	0.646171	+	0.763193i	$0.276369\pi$
$972$	0	0
$973$	1.76393	0.0565491
$974$	0	0
$975$	0	0
$976$	0	0
$977$	53.3050	1.70538	0.852688	−	0.522420i	$-0.174971\pi$
$977$	53.3050	1.70538	0.852688	+	0.522420i	$0.174971\pi$
$978$	0	0
$979$	3.89667	0.124538
$980$	0	0
$981$	11.7082	0.373814
$982$	0	0
$983$	29.2148	0.931807	0.465903	−	0.884836i	$-0.345729\pi$
$983$	29.2148	0.931807	0.465903	+	0.884836i	$0.345729\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	29.4721	0.938108
$988$	0	0
$989$	−46.1803	−1.46845
$990$	0	0
$991$	−51.8885	−1.64829	−0.824147	−	0.566376i	$-0.808345\pi$
$991$	−51.8885	−1.64829	−0.824147	+	0.566376i	$0.808345\pi$
$992$	0	0
$993$	2.47214	0.0784509
$994$	0	0
$995$	0	0
$996$	0	0
$997$	7.09017	0.224548	0.112274	−	0.993677i	$-0.464187\pi$
$997$	7.09017	0.224548	0.112274	+	0.993677i	$0.464187\pi$
$998$	0	0
$999$	−0.618034	−0.0195537

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6000.2.a.bb.1.2		2
4.3	odd	2		750.2.a.e.1.1	yes	2
5.2	odd	4		6000.2.f.k.1249.2		4
5.3	odd	4		6000.2.f.k.1249.3		4
5.4	even	2		6000.2.a.a.1.1		2
12.11	even	2		2250.2.a.a.1.1		2
20.3	even	4		750.2.c.a.499.2		4
20.7	even	4		750.2.c.a.499.3		4
20.19	odd	2		750.2.a.d.1.2	✓	2
60.23	odd	4		2250.2.c.g.1999.4		4
60.47	odd	4		2250.2.c.g.1999.1		4
60.59	even	2		2250.2.a.p.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
750.2.a.d.1.2	✓	2	20.19	odd	2
750.2.a.e.1.1	yes	2	4.3	odd	2
750.2.c.a.499.2		4	20.3	even	4
750.2.c.a.499.3		4	20.7	even	4
2250.2.a.a.1.1		2	12.11	even	2
2250.2.a.p.1.2		2	60.59	even	2
2250.2.c.g.1999.1		4	60.47	odd	4
2250.2.c.g.1999.4		4	60.23	odd	4
6000.2.a.a.1.1		2	5.4	even	2
6000.2.a.bb.1.2		2	1.1	even	1	trivial
6000.2.f.k.1249.2		4	5.2	odd	4
6000.2.f.k.1249.3		4	5.3	odd	4

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:	\( N \)	\(=\)	\( 6000 = 2^{4} \cdot 3 \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6000.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +4.61803 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +4.61803 q^{7} +1.00000 q^{9} -0.854102 q^{11} -5.61803 q^{13} +5.70820 q^{17} +7.09017 q^{19} +4.61803 q^{21} +8.09017 q^{23} +1.00000 q^{27} -7.70820 q^{29} -2.47214 q^{31} -0.854102 q^{33} -0.618034 q^{37} -5.61803 q^{39} +2.61803 q^{41} -5.70820 q^{43} +6.38197 q^{47} +14.3262 q^{49} +5.70820 q^{51} -2.61803 q^{53} +7.09017 q^{57} +4.14590 q^{59} -1.70820 q^{61} +4.61803 q^{63} -1.23607 q^{67} +8.09017 q^{69} -4.47214 q^{71} +8.00000 q^{73} -3.94427 q^{77} -3.52786 q^{79} +1.00000 q^{81} +2.00000 q^{83} -7.70820 q^{87} -4.56231 q^{89} -25.9443 q^{91} -2.47214 q^{93} -8.18034 q^{97} -0.854102 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 7 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 + 7 * q^7 + 2 * q^9	\( 2 q + 2 q^{3} + 7 q^{7} + 2 q^{9} + 5 q^{11} - 9 q^{13} - 2 q^{17} + 3 q^{19} + 7 q^{21} + 5 q^{23} + 2 q^{27} - 2 q^{29} + 4 q^{31} + 5 q^{33} + q^{37} - 9 q^{39} + 3 q^{41} + 2 q^{43} + 15 q^{47} + 13 q^{49} - 2 q^{51} - 3 q^{53} + 3 q^{57} + 15 q^{59} + 10 q^{61} + 7 q^{63} + 2 q^{67} + 5 q^{69} + 16 q^{73} + 10 q^{77} - 16 q^{79} + 2 q^{81} + 4 q^{83} - 2 q^{87} + 11 q^{89} - 34 q^{91} + 4 q^{93} + 6 q^{97} + 5 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 + 7 * q^7 + 2 * q^9 + 5 * q^11 - 9 * q^13 - 2 * q^17 + 3 * q^19 + 7 * q^21 + 5 * q^23 + 2 * q^27 - 2 * q^29 + 4 * q^31 + 5 * q^33 + q^37 - 9 * q^39 + 3 * q^41 + 2 * q^43 + 15 * q^47 + 13 * q^49 - 2 * q^51 - 3 * q^53 + 3 * q^57 + 15 * q^59 + 10 * q^61 + 7 * q^63 + 2 * q^67 + 5 * q^69 + 16 * q^73 + 10 * q^77 - 16 * q^79 + 2 * q^81 + 4 * q^83 - 2 * q^87 + 11 * q^89 - 34 * q^91 + 4 * q^93 + 6 * q^97 + 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more