LMFDB - Embedded newform 6384.2.a.cd.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6384,2,Mod(1,6384)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6384, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6384.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6384 = 2^{4} \cdot 3 \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6384.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:	$50.9764966504$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.401584.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 7x^{3} + 8x^{2} + 12x - 4 $ x^5 - 2x^4 - 7x^3 + 8x^2 + 12x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 3192)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.292040$ of defining polynomial
Character	$\chi$	$=$	6384.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +0.950852 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +0.950852 q^{5} -1.00000 q^{7} +1.00000 q^{9} -1.41592 q^{11} +1.22854 q^{13} -0.950852 q^{15} +6.23411 q^{17} -1.00000 q^{19} +1.00000 q^{21} +2.69918 q^{23} -4.09588 q^{25} -1.00000 q^{27} -5.55174 q^{29} -10.2808 q^{31} +1.41592 q^{33} -0.950852 q^{35} -5.74034 q^{37} -1.22854 q^{39} +11.3984 q^{41} -5.37235 q^{43} +0.950852 q^{45} -2.09030 q^{47} +1.00000 q^{49} -6.23411 q^{51} -5.55174 q^{53} -1.34633 q^{55} +1.00000 q^{57} +9.20177 q^{59} -2.00000 q^{61} -1.00000 q^{63} +1.16816 q^{65} +9.88415 q^{67} -2.69918 q^{69} +1.27527 q^{71} +0.0982955 q^{73} +4.09588 q^{75} +1.41592 q^{77} +2.69918 q^{79} +1.00000 q^{81} -18.1183 q^{83} +5.92772 q^{85} +5.55174 q^{87} -9.85544 q^{89} -1.22854 q^{91} +10.2808 q^{93} -0.950852 q^{95} +9.60768 q^{97} -1.41592 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 5 q^{3} + 2 q^{5} - 5 q^{7} + 5 q^{9}+O(q^{10}) $ 5 * q - 5 * q^3 + 2 * q^5 - 5 * q^7 + 5 * q^9	$ 5 q - 5 q^{3} + 2 q^{5} - 5 q^{7} + 5 q^{9} - 6 q^{11} - 2 q^{15} + 6 q^{17} - 5 q^{19} + 5 q^{21} - 10 q^{23} + 7 q^{25} - 5 q^{27} + 4 q^{29} + 4 q^{31} + 6 q^{33} - 2 q^{35} + 6 q^{37} + 10 q^{41} - 4 q^{43} + 2 q^{45} - 2 q^{47} + 5 q^{49} - 6 q^{51} + 4 q^{53} - 8 q^{55} + 5 q^{57} - 12 q^{59} - 10 q^{61} - 5 q^{63} + 8 q^{65} - 2 q^{67} + 10 q^{69} - 30 q^{71} + 6 q^{73} - 7 q^{75} + 6 q^{77} - 10 q^{79} + 5 q^{81} - 14 q^{83} - 4 q^{87} + 10 q^{89} - 4 q^{93} - 2 q^{95} - 8 q^{97} - 6 q^{99}+O(q^{100}) $ 5 * q - 5 * q^3 + 2 * q^5 - 5 * q^7 + 5 * q^9 - 6 * q^11 - 2 * q^15 + 6 * q^17 - 5 * q^19 + 5 * q^21 - 10 * q^23 + 7 * q^25 - 5 * q^27 + 4 * q^29 + 4 * q^31 + 6 * q^33 - 2 * q^35 + 6 * q^37 + 10 * q^41 - 4 * q^43 + 2 * q^45 - 2 * q^47 + 5 * q^49 - 6 * q^51 + 4 * q^53 - 8 * q^55 + 5 * q^57 - 12 * q^59 - 10 * q^61 - 5 * q^63 + 8 * q^65 - 2 * q^67 + 10 * q^69 - 30 * q^71 + 6 * q^73 - 7 * q^75 + 6 * q^77 - 10 * q^79 + 5 * q^81 - 14 * q^83 - 4 * q^87 + 10 * q^89 - 4 * q^93 - 2 * q^95 - 8 * q^97 - 6 * 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
$5$	0.950852	0.425234	0.212617	−	0.977136i	$-0.431801\pi$
$5$	0.950852	0.425234	0.212617	+	0.977136i	$0.431801\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.41592	−0.426916	−0.213458	−	0.976952i	$-0.568473\pi$
$11$	−1.41592	−0.426916	−0.213458	+	0.976952i	$0.568473\pi$
$12$	0	0
$13$	1.22854	0.340735	0.170368	−	0.985381i	$-0.445504\pi$
$13$	1.22854	0.340735	0.170368	+	0.985381i	$0.445504\pi$
$14$	0	0
$15$	−0.950852	−0.245509
$16$	0	0
$17$	6.23411	1.51199	0.755997	−	0.654575i	$-0.227152\pi$
$17$	6.23411	1.51199	0.755997	+	0.654575i	$0.227152\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	2.69918	0.562818	0.281409	−	0.959588i	$-0.409198\pi$
$23$	2.69918	0.562818	0.281409	+	0.959588i	$0.409198\pi$
$24$	0	0
$25$	−4.09588	−0.819176
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−5.55174	−1.03093	−0.515466	−	0.856910i	$-0.672381\pi$
$29$	−5.55174	−1.03093	−0.515466	+	0.856910i	$0.672381\pi$
$30$	0	0
$31$	−10.2808	−1.84649	−0.923247	−	0.384206i	$-0.874475\pi$
$31$	−10.2808	−1.84649	−0.923247	+	0.384206i	$0.874475\pi$
$32$	0	0
$33$	1.41592	0.246480
$34$	0	0
$35$	−0.950852	−0.160723
$36$	0	0
$37$	−5.74034	−0.943706	−0.471853	−	0.881677i	$-0.656415\pi$
$37$	−5.74034	−0.943706	−0.471853	+	0.881677i	$0.656415\pi$
$38$	0	0
$39$	−1.22854	−0.196724
$40$	0	0
$41$	11.3984	1.78013	0.890063	−	0.455838i	$-0.150660\pi$
$41$	11.3984	1.78013	0.890063	+	0.455838i	$0.150660\pi$
$42$	0	0
$43$	−5.37235	−0.819275	−0.409638	−	0.912248i	$-0.634345\pi$
$43$	−5.37235	−0.819275	−0.409638	+	0.912248i	$0.634345\pi$
$44$	0	0
$45$	0.950852	0.141745
$46$	0	0
$47$	−2.09030	−0.304902	−0.152451	−	0.988311i	$-0.548717\pi$
$47$	−2.09030	−0.304902	−0.152451	+	0.988311i	$0.548717\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−6.23411	−0.872951
$52$	0	0
$53$	−5.55174	−0.762590	−0.381295	−	0.924453i	$-0.624522\pi$
$53$	−5.55174	−0.762590	−0.381295	+	0.924453i	$0.624522\pi$
$54$	0	0
$55$	−1.34633	−0.181539
$56$	0	0
$57$	1.00000	0.132453
$58$	0	0
$59$	9.20177	1.19797	0.598984	−	0.800761i	$-0.295571\pi$
$59$	9.20177	1.19797	0.598984	+	0.800761i	$0.295571\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	1.16816	0.144892
$66$	0	0
$67$	9.88415	1.20754	0.603770	−	0.797158i	$-0.293664\pi$
$67$	9.88415	1.20754	0.603770	+	0.797158i	$0.293664\pi$
$68$	0	0
$69$	−2.69918	−0.324943
$70$	0	0
$71$	1.27527	0.151347	0.0756734	−	0.997133i	$-0.475889\pi$
$71$	1.27527	0.151347	0.0756734	+	0.997133i	$0.475889\pi$
$72$	0	0
$73$	0.0982955	0.0115046	0.00575231	−	0.999983i	$-0.498169\pi$
$73$	0.0982955	0.0115046	0.00575231	+	0.999983i	$0.498169\pi$
$74$	0	0
$75$	4.09588	0.472951
$76$	0	0
$77$	1.41592	0.161359
$78$	0	0
$79$	2.69918	0.303682	0.151841	−	0.988405i	$-0.451480\pi$
$79$	2.69918	0.303682	0.151841	+	0.988405i	$0.451480\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−18.1183	−1.98874	−0.994369	−	0.105975i	$-0.966204\pi$
$83$	−18.1183	−1.98874	−0.994369	+	0.105975i	$0.966204\pi$
$84$	0	0
$85$	5.92772	0.642952
$86$	0	0
$87$	5.55174	0.595209
$88$	0	0
$89$	−9.85544	−1.04467	−0.522337	−	0.852739i	$-0.674940\pi$
$89$	−9.85544	−1.04467	−0.522337	+	0.852739i	$0.674940\pi$
$90$	0	0
$91$	−1.22854	−0.128786
$92$	0	0
$93$	10.2808	1.06607
$94$	0	0
$95$	−0.950852	−0.0975554
$96$	0	0
$97$	9.60768	0.975512	0.487756	−	0.872980i	$-0.337816\pi$
$97$	9.60768	0.975512	0.487756	+	0.872980i	$0.337816\pi$
$98$	0	0
$99$	−1.41592	−0.142305
$100$	0	0
$101$	11.5014	1.14443	0.572216	−	0.820103i	$-0.306084\pi$
$101$	11.5014	1.14443	0.572216	+	0.820103i	$0.306084\pi$
$102$	0	0
$103$	−1.28326	−0.126444	−0.0632218	−	0.998000i	$-0.520138\pi$
$103$	−1.28326	−0.126444	−0.0632218	+	0.998000i	$0.520138\pi$
$104$	0	0
$105$	0.950852	0.0927937
$106$	0	0
$107$	−12.4819	−1.20667	−0.603334	−	0.797488i	$-0.706161\pi$
$107$	−12.4819	−1.20667	−0.603334	+	0.797488i	$0.706161\pi$
$108$	0	0
$109$	13.0404	1.24904	0.624522	−	0.781007i	$-0.285294\pi$
$109$	13.0404	1.24904	0.624522	+	0.781007i	$0.285294\pi$
$110$	0	0
$111$	5.74034	0.544849
$112$	0	0
$113$	3.84663	0.361860	0.180930	−	0.983496i	$-0.442089\pi$
$113$	3.84663	0.361860	0.180930	+	0.983496i	$0.442089\pi$
$114$	0	0
$115$	2.56652	0.239330
$116$	0	0
$117$	1.22854	0.113578
$118$	0	0
$119$	−6.23411	−0.571480
$120$	0	0
$121$	−8.99517	−0.817743
$122$	0	0
$123$	−11.3984	−1.02776
$124$	0	0
$125$	−8.64884	−0.773576
$126$	0	0
$127$	−3.25456	−0.288795	−0.144398	−	0.989520i	$-0.546124\pi$
$127$	−3.25456	−0.288795	−0.144398	+	0.989520i	$0.546124\pi$
$128$	0	0
$129$	5.37235	0.473009
$130$	0	0
$131$	−9.65003	−0.843127	−0.421564	−	0.906799i	$-0.638519\pi$
$131$	−9.65003	−0.843127	−0.421564	+	0.906799i	$0.638519\pi$
$132$	0	0
$133$	1.00000	0.0867110
$134$	0	0
$135$	−0.950852	−0.0818363
$136$	0	0
$137$	−7.20177	−0.615289	−0.307645	−	0.951501i	$-0.599541\pi$
$137$	−7.20177	−0.615289	−0.307645	+	0.951501i	$0.599541\pi$
$138$	0	0
$139$	19.0348	1.61451	0.807253	−	0.590205i	$-0.200953\pi$
$139$	19.0348	1.61451	0.807253	+	0.590205i	$0.200953\pi$
$140$	0	0
$141$	2.09030	0.176035
$142$	0	0
$143$	−1.73951	−0.145465
$144$	0	0
$145$	−5.27888	−0.438387
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	4.01681	0.329070	0.164535	−	0.986371i	$-0.447388\pi$
$149$	4.01681	0.329070	0.164535	+	0.986371i	$0.447388\pi$
$150$	0	0
$151$	−4.40430	−0.358416	−0.179208	−	0.983811i	$-0.557354\pi$
$151$	−4.40430	−0.358416	−0.179208	+	0.983811i	$0.557354\pi$
$152$	0	0
$153$	6.23411	0.503998
$154$	0	0
$155$	−9.77557	−0.785192
$156$	0	0
$157$	−9.91285	−0.791132	−0.395566	−	0.918438i	$-0.629452\pi$
$157$	−9.91285	−0.791132	−0.395566	+	0.918438i	$0.629452\pi$
$158$	0	0
$159$	5.55174	0.440282
$160$	0	0
$161$	−2.69918	−0.212725
$162$	0	0
$163$	−6.54051	−0.512292	−0.256146	−	0.966638i	$-0.582453\pi$
$163$	−6.54051	−0.512292	−0.256146	+	0.966638i	$0.582453\pi$
$164$	0	0
$165$	1.34633	0.104812
$166$	0	0
$167$	8.27164	0.640078	0.320039	−	0.947404i	$-0.396304\pi$
$167$	8.27164	0.640078	0.320039	+	0.947404i	$0.396304\pi$
$168$	0	0
$169$	−11.4907	−0.883899
$170$	0	0
$171$	−1.00000	−0.0764719
$172$	0	0
$173$	4.36993	0.332240	0.166120	−	0.986106i	$-0.446876\pi$
$173$	4.36993	0.332240	0.166120	+	0.986106i	$0.446876\pi$
$174$	0	0
$175$	4.09588	0.309619
$176$	0	0
$177$	−9.20177	−0.691648
$178$	0	0
$179$	−8.54173	−0.638439	−0.319219	−	0.947681i	$-0.603421\pi$
$179$	−8.54173	−0.638439	−0.319219	+	0.947681i	$0.603421\pi$
$180$	0	0
$181$	−16.2373	−1.20691	−0.603454	−	0.797398i	$-0.706209\pi$
$181$	−16.2373	−1.20691	−0.603454	+	0.797398i	$0.706209\pi$
$182$	0	0
$183$	2.00000	0.147844
$184$	0	0
$185$	−5.45821	−0.401296
$186$	0	0
$187$	−8.82701	−0.645495
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	−20.9597	−1.51659	−0.758294	−	0.651912i	$-0.773967\pi$
$191$	−20.9597	−1.51659	−0.758294	+	0.651912i	$0.773967\pi$
$192$	0	0
$193$	2.57135	0.185090	0.0925451	−	0.995708i	$-0.470500\pi$
$193$	2.57135	0.185090	0.0925451	+	0.995708i	$0.470500\pi$
$194$	0	0
$195$	−1.16816	−0.0836536
$196$	0	0
$197$	−17.2784	−1.23104	−0.615519	−	0.788122i	$-0.711053\pi$
$197$	−17.2784	−1.23104	−0.615519	+	0.788122i	$0.711053\pi$
$198$	0	0
$199$	−13.0396	−0.924351	−0.462176	−	0.886788i	$-0.652931\pi$
$199$	−13.0396	−0.924351	−0.462176	+	0.886788i	$0.652931\pi$
$200$	0	0
$201$	−9.88415	−0.697174
$202$	0	0
$203$	5.55174	0.389656
$204$	0	0
$205$	10.8382	0.756970
$206$	0	0
$207$	2.69918	0.187606
$208$	0	0
$209$	1.41592	0.0979413
$210$	0	0
$211$	9.86817	0.679353	0.339676	−	0.940542i	$-0.389682\pi$
$211$	9.86817	0.679353	0.339676	+	0.940542i	$0.389682\pi$
$212$	0	0
$213$	−1.27527	−0.0873801
$214$	0	0
$215$	−5.10831	−0.348384
$216$	0	0
$217$	10.2808	0.697909
$218$	0	0
$219$	−0.0982955	−0.00664220
$220$	0	0
$221$	7.65885	0.515190
$222$	0	0
$223$	−6.17140	−0.413268	−0.206634	−	0.978418i	$-0.566251\pi$
$223$	−6.17140	−0.413268	−0.206634	+	0.978418i	$0.566251\pi$
$224$	0	0
$225$	−4.09588	−0.273059
$226$	0	0
$227$	−2.27647	−0.151094	−0.0755472	−	0.997142i	$-0.524070\pi$
$227$	−2.27647	−0.151094	−0.0755472	+	0.997142i	$0.524070\pi$
$228$	0	0
$229$	−21.1219	−1.39577	−0.697887	−	0.716208i	$-0.745876\pi$
$229$	−21.1219	−1.39577	−0.697887	+	0.716208i	$0.745876\pi$
$230$	0	0
$231$	−1.41592	−0.0931607
$232$	0	0
$233$	11.5790	0.758564	0.379282	−	0.925281i	$-0.376171\pi$
$233$	11.5790	0.758564	0.379282	+	0.925281i	$0.376171\pi$
$234$	0	0
$235$	−1.98757	−0.129655
$236$	0	0
$237$	−2.69918	−0.175331
$238$	0	0
$239$	−13.5037	−0.873484	−0.436742	−	0.899587i	$-0.643868\pi$
$239$	−13.5037	−0.873484	−0.436742	+	0.899587i	$0.643868\pi$
$240$	0	0
$241$	−18.2126	−1.17318	−0.586590	−	0.809884i	$-0.699530\pi$
$241$	−18.2126	−1.17318	−0.586590	+	0.809884i	$0.699530\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	0.950852	0.0607477
$246$	0	0
$247$	−1.22854	−0.0781701
$248$	0	0
$249$	18.1183	1.14820
$250$	0	0
$251$	−16.3451	−1.03170	−0.515848	−	0.856680i	$-0.672523\pi$
$251$	−16.3451	−1.03170	−0.515848	+	0.856680i	$0.672523\pi$
$252$	0	0
$253$	−3.82183	−0.240276
$254$	0	0
$255$	−5.92772	−0.371208
$256$	0	0
$257$	13.2129	0.824200	0.412100	−	0.911139i	$-0.364796\pi$
$257$	13.2129	0.824200	0.412100	+	0.911139i	$0.364796\pi$
$258$	0	0
$259$	5.74034	0.356687
$260$	0	0
$261$	−5.55174	−0.343644
$262$	0	0
$263$	19.7068	1.21517	0.607587	−	0.794253i	$-0.292138\pi$
$263$	19.7068	1.21517	0.607587	+	0.794253i	$0.292138\pi$
$264$	0	0
$265$	−5.27888	−0.324279
$266$	0	0
$267$	9.85544	0.603143
$268$	0	0
$269$	−14.4108	−0.878643	−0.439321	−	0.898330i	$-0.644781\pi$
$269$	−14.4108	−0.878643	−0.439321	+	0.898330i	$0.644781\pi$
$270$	0	0
$271$	19.4623	1.18225	0.591124	−	0.806581i	$-0.298684\pi$
$271$	19.4623	1.18225	0.591124	+	0.806581i	$0.298684\pi$
$272$	0	0
$273$	1.22854	0.0743546
$274$	0	0
$275$	5.79944	0.349719
$276$	0	0
$277$	17.0087	1.02196	0.510978	−	0.859594i	$-0.329283\pi$
$277$	17.0087	1.02196	0.510978	+	0.859594i	$0.329283\pi$
$278$	0	0
$279$	−10.2808	−0.615498
$280$	0	0
$281$	12.0609	0.719490	0.359745	−	0.933051i	$-0.382864\pi$
$281$	12.0609	0.719490	0.359745	+	0.933051i	$0.382864\pi$
$282$	0	0
$283$	−24.6984	−1.46817	−0.734085	−	0.679058i	$-0.762389\pi$
$283$	−24.6984	−1.46817	−0.734085	+	0.679058i	$0.762389\pi$
$284$	0	0
$285$	0.950852	0.0563236
$286$	0	0
$287$	−11.3984	−0.672824
$288$	0	0
$289$	21.8642	1.28613
$290$	0	0
$291$	−9.60768	−0.563212
$292$	0	0
$293$	−16.9933	−0.992760	−0.496380	−	0.868105i	$-0.665338\pi$
$293$	−16.9933	−0.992760	−0.496380	+	0.868105i	$0.665338\pi$
$294$	0	0
$295$	8.74953	0.509417
$296$	0	0
$297$	1.41592	0.0821600
$298$	0	0
$299$	3.31605	0.191772
$300$	0	0
$301$	5.37235	0.309657
$302$	0	0
$303$	−11.5014	−0.660738
$304$	0	0
$305$	−1.90170	−0.108891
$306$	0	0
$307$	−17.8246	−1.01730	−0.508652	−	0.860972i	$-0.669856\pi$
$307$	−17.8246	−1.01730	−0.508652	+	0.860972i	$0.669856\pi$
$308$	0	0
$309$	1.28326	0.0730022
$310$	0	0
$311$	−10.5065	−0.595769	−0.297884	−	0.954602i	$-0.596281\pi$
$311$	−10.5065	−0.595769	−0.297884	+	0.954602i	$0.596281\pi$
$312$	0	0
$313$	−16.9852	−0.960058	−0.480029	−	0.877253i	$-0.659374\pi$
$313$	−16.9852	−0.960058	−0.480029	+	0.877253i	$0.659374\pi$
$314$	0	0
$315$	−0.950852	−0.0535745
$316$	0	0
$317$	−19.9401	−1.11995	−0.559974	−	0.828511i	$-0.689189\pi$
$317$	−19.9401	−1.11995	−0.559974	+	0.828511i	$0.689189\pi$
$318$	0	0
$319$	7.86082	0.440121
$320$	0	0
$321$	12.4819	0.696670
$322$	0	0
$323$	−6.23411	−0.346875
$324$	0	0
$325$	−5.03195	−0.279122
$326$	0	0
$327$	−13.0404	−0.721136
$328$	0	0
$329$	2.09030	0.115242
$330$	0	0
$331$	30.7632	1.69090	0.845449	−	0.534056i	$-0.179333\pi$
$331$	30.7632	1.69090	0.845449	+	0.534056i	$0.179333\pi$
$332$	0	0
$333$	−5.74034	−0.314569
$334$	0	0
$335$	9.39836	0.513488
$336$	0	0
$337$	−8.32850	−0.453682	−0.226841	−	0.973932i	$-0.572840\pi$
$337$	−8.32850	−0.453682	−0.226841	+	0.973932i	$0.572840\pi$
$338$	0	0
$339$	−3.84663	−0.208920
$340$	0	0
$341$	14.5569	0.788298
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−2.56652	−0.138177
$346$	0	0
$347$	18.5146	0.993914	0.496957	−	0.867775i	$-0.334451\pi$
$347$	18.5146	0.993914	0.496957	+	0.867775i	$0.334451\pi$
$348$	0	0
$349$	25.4271	1.36108	0.680542	−	0.732709i	$-0.261744\pi$
$349$	25.4271	1.36108	0.680542	+	0.732709i	$0.261744\pi$
$350$	0	0
$351$	−1.22854	−0.0655746
$352$	0	0
$353$	10.4147	0.554320	0.277160	−	0.960824i	$-0.410607\pi$
$353$	10.4147	0.554320	0.277160	+	0.960824i	$0.410607\pi$
$354$	0	0
$355$	1.21259	0.0643578
$356$	0	0
$357$	6.23411	0.329944
$358$	0	0
$359$	−1.19137	−0.0628782	−0.0314391	−	0.999506i	$-0.510009\pi$
$359$	−1.19137	−0.0628782	−0.0314391	+	0.999506i	$0.510009\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	8.99517	0.472124
$364$	0	0
$365$	0.0934645	0.00489216
$366$	0	0
$367$	3.24804	0.169546	0.0847730	−	0.996400i	$-0.472983\pi$
$367$	3.24804	0.169546	0.0847730	+	0.996400i	$0.472983\pi$
$368$	0	0
$369$	11.3984	0.593375
$370$	0	0
$371$	5.55174	0.288232
$372$	0	0
$373$	−15.4614	−0.800563	−0.400281	−	0.916392i	$-0.631088\pi$
$373$	−15.4614	−0.800563	−0.400281	+	0.916392i	$0.631088\pi$
$374$	0	0
$375$	8.64884	0.446624
$376$	0	0
$377$	−6.82053	−0.351275
$378$	0	0
$379$	6.64393	0.341276	0.170638	−	0.985334i	$-0.445417\pi$
$379$	6.64393	0.341276	0.170638	+	0.985334i	$0.445417\pi$
$380$	0	0
$381$	3.25456	0.166736
$382$	0	0
$383$	21.2952	1.08814	0.544068	−	0.839041i	$-0.316883\pi$
$383$	21.2952	1.08814	0.544068	+	0.839041i	$0.316883\pi$
$384$	0	0
$385$	1.34633	0.0686154
$386$	0	0
$387$	−5.37235	−0.273092
$388$	0	0
$389$	−7.04524	−0.357208	−0.178604	−	0.983921i	$-0.557158\pi$
$389$	−7.04524	−0.357208	−0.178604	+	0.983921i	$0.557158\pi$
$390$	0	0
$391$	16.8270	0.850978
$392$	0	0
$393$	9.65003	0.486780
$394$	0	0
$395$	2.56652	0.129136
$396$	0	0
$397$	−30.6488	−1.53822	−0.769111	−	0.639116i	$-0.779300\pi$
$397$	−30.6488	−1.53822	−0.769111	+	0.639116i	$0.779300\pi$
$398$	0	0
$399$	−1.00000	−0.0500626
$400$	0	0
$401$	−19.4782	−0.972694	−0.486347	−	0.873766i	$-0.661671\pi$
$401$	−19.4782	−0.972694	−0.486347	+	0.873766i	$0.661671\pi$
$402$	0	0
$403$	−12.6304	−0.629166
$404$	0	0
$405$	0.950852	0.0472482
$406$	0	0
$407$	8.12786	0.402883
$408$	0	0
$409$	20.1404	0.995877	0.497939	−	0.867212i	$-0.334090\pi$
$409$	20.1404	0.995877	0.497939	+	0.867212i	$0.334090\pi$
$410$	0	0
$411$	7.20177	0.355237
$412$	0	0
$413$	−9.20177	−0.452790
$414$	0	0
$415$	−17.2278	−0.845679
$416$	0	0
$417$	−19.0348	−0.932136
$418$	0	0
$419$	−29.1010	−1.42168	−0.710838	−	0.703356i	$-0.751684\pi$
$419$	−29.1010	−1.42168	−0.710838	+	0.703356i	$0.751684\pi$
$420$	0	0
$421$	3.78660	0.184548	0.0922738	−	0.995734i	$-0.470586\pi$
$421$	3.78660	0.184548	0.0922738	+	0.995734i	$0.470586\pi$
$422$	0	0
$423$	−2.09030	−0.101634
$424$	0	0
$425$	−25.5342	−1.23859
$426$	0	0
$427$	2.00000	0.0967868
$428$	0	0
$429$	1.73951	0.0839845
$430$	0	0
$431$	−20.4131	−0.983267	−0.491633	−	0.870802i	$-0.663600\pi$
$431$	−20.4131	−0.983267	−0.491633	+	0.870802i	$0.663600\pi$
$432$	0	0
$433$	−31.3488	−1.50653	−0.753265	−	0.657717i	$-0.771522\pi$
$433$	−31.3488	−1.50653	−0.753265	+	0.657717i	$0.771522\pi$
$434$	0	0
$435$	5.27888	0.253103
$436$	0	0
$437$	−2.69918	−0.129119
$438$	0	0
$439$	15.7143	0.750004	0.375002	−	0.927024i	$-0.377642\pi$
$439$	15.7143	0.750004	0.375002	+	0.927024i	$0.377642\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−27.2641	−1.29536	−0.647678	−	0.761914i	$-0.724260\pi$
$443$	−27.2641	−1.29536	−0.647678	+	0.761914i	$0.724260\pi$
$444$	0	0
$445$	−9.37107	−0.444231
$446$	0	0
$447$	−4.01681	−0.189988
$448$	0	0
$449$	−11.0372	−0.520880	−0.260440	−	0.965490i	$-0.583868\pi$
$449$	−11.0372	−0.520880	−0.260440	+	0.965490i	$0.583868\pi$
$450$	0	0
$451$	−16.1392	−0.759964
$452$	0	0
$453$	4.40430	0.206932
$454$	0	0
$455$	−1.16816	−0.0547641
$456$	0	0
$457$	−29.4171	−1.37607	−0.688037	−	0.725675i	$-0.741527\pi$
$457$	−29.4171	−1.37607	−0.688037	+	0.725675i	$0.741527\pi$
$458$	0	0
$459$	−6.23411	−0.290984
$460$	0	0
$461$	8.67772	0.404162	0.202081	−	0.979369i	$-0.435230\pi$
$461$	8.67772	0.404162	0.202081	+	0.979369i	$0.435230\pi$
$462$	0	0
$463$	33.6540	1.56403	0.782017	−	0.623256i	$-0.214191\pi$
$463$	33.6540	1.56403	0.782017	+	0.623256i	$0.214191\pi$
$464$	0	0
$465$	9.77557	0.453331
$466$	0	0
$467$	−37.0251	−1.71332	−0.856660	−	0.515882i	$-0.827464\pi$
$467$	−37.0251	−1.71332	−0.856660	+	0.515882i	$0.827464\pi$
$468$	0	0
$469$	−9.88415	−0.456408
$470$	0	0
$471$	9.91285	0.456760
$472$	0	0
$473$	7.60682	0.349762
$474$	0	0
$475$	4.09588	0.187932
$476$	0	0
$477$	−5.55174	−0.254197
$478$	0	0
$479$	−31.8177	−1.45379	−0.726894	−	0.686749i	$-0.759037\pi$
$479$	−31.8177	−1.45379	−0.726894	+	0.686749i	$0.759037\pi$
$480$	0	0
$481$	−7.05223	−0.321554
$482$	0	0
$483$	2.69918	0.122817
$484$	0	0
$485$	9.13548	0.414821
$486$	0	0
$487$	18.4452	0.835831	0.417915	−	0.908486i	$-0.362761\pi$
$487$	18.4452	0.835831	0.417915	+	0.908486i	$0.362761\pi$
$488$	0	0
$489$	6.54051	0.295772
$490$	0	0
$491$	17.0908	0.771295	0.385647	−	0.922646i	$-0.373978\pi$
$491$	17.0908	0.771295	0.385647	+	0.922646i	$0.373978\pi$
$492$	0	0
$493$	−34.6102	−1.55876
$494$	0	0
$495$	−1.34633	−0.0605131
$496$	0	0
$497$	−1.27527	−0.0572037
$498$	0	0
$499$	−31.1282	−1.39349	−0.696745	−	0.717319i	$-0.745369\pi$
$499$	−31.1282	−1.39349	−0.696745	+	0.717319i	$0.745369\pi$
$500$	0	0
$501$	−8.27164	−0.369549
$502$	0	0
$503$	−34.6106	−1.54321	−0.771604	−	0.636103i	$-0.780545\pi$
$503$	−34.6106	−1.54321	−0.771604	+	0.636103i	$0.780545\pi$
$504$	0	0
$505$	10.9361	0.486651
$506$	0	0
$507$	11.4907	0.510320
$508$	0	0
$509$	−37.3001	−1.65330	−0.826648	−	0.562719i	$-0.809755\pi$
$509$	−37.3001	−1.65330	−0.826648	+	0.562719i	$0.809755\pi$
$510$	0	0
$511$	−0.0982955	−0.00434834
$512$	0	0
$513$	1.00000	0.0441511
$514$	0	0
$515$	−1.22019	−0.0537681
$516$	0	0
$517$	2.95971	0.130168
$518$	0	0
$519$	−4.36993	−0.191819
$520$	0	0
$521$	27.8482	1.22005	0.610025	−	0.792382i	$-0.291159\pi$
$521$	27.8482	1.22005	0.610025	+	0.792382i	$0.291159\pi$
$522$	0	0
$523$	19.6328	0.858484	0.429242	−	0.903190i	$-0.358781\pi$
$523$	19.6328	0.858484	0.429242	+	0.903190i	$0.358781\pi$
$524$	0	0
$525$	−4.09588	−0.178759
$526$	0	0
$527$	−64.0920	−2.79189
$528$	0	0
$529$	−15.7144	−0.683235
$530$	0	0
$531$	9.20177	0.399323
$532$	0	0
$533$	14.0033	0.606552
$534$	0	0
$535$	−11.8684	−0.513117
$536$	0	0
$537$	8.54173	0.368603
$538$	0	0
$539$	−1.41592	−0.0609880
$540$	0	0
$541$	−8.69266	−0.373727	−0.186863	−	0.982386i	$-0.559832\pi$
$541$	−8.69266	−0.373727	−0.186863	+	0.982386i	$0.559832\pi$
$542$	0	0
$543$	16.2373	0.696808
$544$	0	0
$545$	12.3995	0.531136
$546$	0	0
$547$	5.30647	0.226888	0.113444	−	0.993544i	$-0.463812\pi$
$547$	5.30647	0.226888	0.113444	+	0.993544i	$0.463812\pi$
$548$	0	0
$549$	−2.00000	−0.0853579
$550$	0	0
$551$	5.55174	0.236512
$552$	0	0
$553$	−2.69918	−0.114781
$554$	0	0
$555$	5.45821	0.231688
$556$	0	0
$557$	5.29441	0.224331	0.112166	−	0.993690i	$-0.464221\pi$
$557$	5.29441	0.224331	0.112166	+	0.993690i	$0.464221\pi$
$558$	0	0
$559$	−6.60014	−0.279156
$560$	0	0
$561$	8.82701	0.372677
$562$	0	0
$563$	17.9240	0.755407	0.377703	−	0.925927i	$-0.376714\pi$
$563$	17.9240	0.755407	0.377703	+	0.925927i	$0.376714\pi$
$564$	0	0
$565$	3.65757	0.153875
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	8.11826	0.340335	0.170168	−	0.985415i	$-0.445569\pi$
$569$	8.11826	0.340335	0.170168	+	0.985415i	$0.445569\pi$
$570$	0	0
$571$	−26.7096	−1.11776	−0.558881	−	0.829248i	$-0.688769\pi$
$571$	−26.7096	−1.11776	−0.558881	+	0.829248i	$0.688769\pi$
$572$	0	0
$573$	20.9597	0.875603
$574$	0	0
$575$	−11.0555	−0.461047
$576$	0	0
$577$	12.7447	0.530569	0.265284	−	0.964170i	$-0.414534\pi$
$577$	12.7447	0.530569	0.265284	+	0.964170i	$0.414534\pi$
$578$	0	0
$579$	−2.57135	−0.106862
$580$	0	0
$581$	18.1183	0.751672
$582$	0	0
$583$	7.86082	0.325562
$584$	0	0
$585$	1.16816	0.0482974
$586$	0	0
$587$	45.7834	1.88968	0.944842	−	0.327526i	$-0.106215\pi$
$587$	45.7834	1.88968	0.944842	+	0.327526i	$0.106215\pi$
$588$	0	0
$589$	10.2808	0.423615
$590$	0	0
$591$	17.2784	0.710740
$592$	0	0
$593$	−24.8183	−1.01916	−0.509582	−	0.860422i	$-0.670200\pi$
$593$	−24.8183	−1.01916	−0.509582	+	0.860422i	$0.670200\pi$
$594$	0	0
$595$	−5.92772	−0.243013
$596$	0	0
$597$	13.0396	0.533675
$598$	0	0
$599$	3.08351	0.125989	0.0629944	−	0.998014i	$-0.479935\pi$
$599$	3.08351	0.125989	0.0629944	+	0.998014i	$0.479935\pi$
$600$	0	0
$601$	−21.8730	−0.892218	−0.446109	−	0.894979i	$-0.647191\pi$
$601$	−21.8730	−0.892218	−0.446109	+	0.894979i	$0.647191\pi$
$602$	0	0
$603$	9.88415	0.402514
$604$	0	0
$605$	−8.55308	−0.347732
$606$	0	0
$607$	38.2884	1.55408	0.777040	−	0.629451i	$-0.216720\pi$
$607$	38.2884	1.55408	0.777040	+	0.629451i	$0.216720\pi$
$608$	0	0
$609$	−5.55174	−0.224968
$610$	0	0
$611$	−2.56802	−0.103891
$612$	0	0
$613$	−41.8119	−1.68877	−0.844384	−	0.535738i	$-0.820034\pi$
$613$	−41.8119	−1.68877	−0.844384	+	0.535738i	$0.820034\pi$
$614$	0	0
$615$	−10.8382	−0.437037
$616$	0	0
$617$	−11.9378	−0.480596	−0.240298	−	0.970699i	$-0.577245\pi$
$617$	−11.9378	−0.480596	−0.240298	+	0.970699i	$0.577245\pi$
$618$	0	0
$619$	29.9576	1.20410	0.602049	−	0.798459i	$-0.294351\pi$
$619$	29.9576	1.20410	0.602049	+	0.798459i	$0.294351\pi$
$620$	0	0
$621$	−2.69918	−0.108314
$622$	0	0
$623$	9.85544	0.394850
$624$	0	0
$625$	12.2556	0.490225
$626$	0	0
$627$	−1.41592	−0.0565464
$628$	0	0
$629$	−35.7859	−1.42688
$630$	0	0
$631$	39.9038	1.58854	0.794272	−	0.607563i	$-0.207853\pi$
$631$	39.9038	1.58854	0.794272	+	0.607563i	$0.207853\pi$
$632$	0	0
$633$	−9.86817	−0.392224
$634$	0	0
$635$	−3.09460	−0.122806
$636$	0	0
$637$	1.22854	0.0486765
$638$	0	0
$639$	1.27527	0.0504489
$640$	0	0
$641$	−7.92411	−0.312983	−0.156492	−	0.987679i	$-0.550018\pi$
$641$	−7.92411	−0.312983	−0.156492	+	0.987679i	$0.550018\pi$
$642$	0	0
$643$	7.51508	0.296366	0.148183	−	0.988960i	$-0.452658\pi$
$643$	7.51508	0.296366	0.148183	+	0.988960i	$0.452658\pi$
$644$	0	0
$645$	5.10831	0.201139
$646$	0	0
$647$	−36.0096	−1.41569	−0.707843	−	0.706370i	$-0.750331\pi$
$647$	−36.0096	−1.41569	−0.707843	+	0.706370i	$0.750331\pi$
$648$	0	0
$649$	−13.0290	−0.511432
$650$	0	0
$651$	−10.2808	−0.402938
$652$	0	0
$653$	−14.2597	−0.558024	−0.279012	−	0.960288i	$-0.590007\pi$
$653$	−14.2597	−0.558024	−0.279012	+	0.960288i	$0.590007\pi$
$654$	0	0
$655$	−9.17576	−0.358526
$656$	0	0
$657$	0.0982955	0.00383487
$658$	0	0
$659$	44.5792	1.73656	0.868280	−	0.496074i	$-0.165226\pi$
$659$	44.5792	1.73656	0.868280	+	0.496074i	$0.165226\pi$
$660$	0	0
$661$	2.25214	0.0875981	0.0437990	−	0.999040i	$-0.486054\pi$
$661$	2.25214	0.0875981	0.0437990	+	0.999040i	$0.486054\pi$
$662$	0	0
$663$	−7.65885	−0.297445
$664$	0	0
$665$	0.950852	0.0368725
$666$	0	0
$667$	−14.9852	−0.580228
$668$	0	0
$669$	6.17140	0.238600
$670$	0	0
$671$	2.83184	0.109322
$672$	0	0
$673$	42.8606	1.65216	0.826078	−	0.563556i	$-0.190567\pi$
$673$	42.8606	1.65216	0.826078	+	0.563556i	$0.190567\pi$
$674$	0	0
$675$	4.09588	0.157650
$676$	0	0
$677$	−34.6031	−1.32991	−0.664953	−	0.746885i	$-0.731548\pi$
$677$	−34.6031	−1.32991	−0.664953	+	0.746885i	$0.731548\pi$
$678$	0	0
$679$	−9.60768	−0.368709
$680$	0	0
$681$	2.27647	0.0872344
$682$	0	0
$683$	−18.3300	−0.701380	−0.350690	−	0.936492i	$-0.614053\pi$
$683$	−18.3300	−0.701380	−0.350690	+	0.936492i	$0.614053\pi$
$684$	0	0
$685$	−6.84782	−0.261642
$686$	0	0
$687$	21.1219	0.805851
$688$	0	0
$689$	−6.82053	−0.259841
$690$	0	0
$691$	−2.94129	−0.111892	−0.0559459	−	0.998434i	$-0.517817\pi$
$691$	−2.94129	−0.111892	−0.0559459	+	0.998434i	$0.517817\pi$
$692$	0	0
$693$	1.41592	0.0537864
$694$	0	0
$695$	18.0992	0.686543
$696$	0	0
$697$	71.0587	2.69154
$698$	0	0
$699$	−11.5790	−0.437957
$700$	0	0
$701$	−36.2884	−1.37060	−0.685298	−	0.728263i	$-0.740328\pi$
$701$	−36.2884	−1.37060	−0.685298	+	0.728263i	$0.740328\pi$
$702$	0	0
$703$	5.74034	0.216501
$704$	0	0
$705$	1.98757	0.0748563
$706$	0	0
$707$	−11.5014	−0.432554
$708$	0	0
$709$	−39.9613	−1.50078	−0.750390	−	0.660996i	$-0.770134\pi$
$709$	−39.9613	−1.50078	−0.750390	+	0.660996i	$0.770134\pi$
$710$	0	0
$711$	2.69918	0.101227
$712$	0	0
$713$	−27.7499	−1.03924
$714$	0	0
$715$	−1.65402	−0.0618568
$716$	0	0
$717$	13.5037	0.504306
$718$	0	0
$719$	32.4004	1.20833	0.604165	−	0.796859i	$-0.293507\pi$
$719$	32.4004	1.20833	0.604165	+	0.796859i	$0.293507\pi$
$720$	0	0
$721$	1.28326	0.0477912
$722$	0	0
$723$	18.2126	0.677336
$724$	0	0
$725$	22.7393	0.844515
$726$	0	0
$727$	43.5030	1.61344	0.806718	−	0.590936i	$-0.201242\pi$
$727$	43.5030	1.61344	0.806718	+	0.590936i	$0.201242\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−33.4918	−1.23874
$732$	0	0
$733$	−24.2988	−0.897495	−0.448748	−	0.893658i	$-0.648130\pi$
$733$	−24.2988	−0.897495	−0.448748	+	0.893658i	$0.648130\pi$
$734$	0	0
$735$	−0.950852	−0.0350727
$736$	0	0
$737$	−13.9952	−0.515519
$738$	0	0
$739$	26.3608	0.969699	0.484850	−	0.874598i	$-0.338874\pi$
$739$	26.3608	0.969699	0.484850	+	0.874598i	$0.338874\pi$
$740$	0	0
$741$	1.22854	0.0451315
$742$	0	0
$743$	7.51812	0.275813	0.137907	−	0.990445i	$-0.455963\pi$
$743$	7.51812	0.275813	0.137907	+	0.990445i	$0.455963\pi$
$744$	0	0
$745$	3.81939	0.139932
$746$	0	0
$747$	−18.1183	−0.662913
$748$	0	0
$749$	12.4819	0.456078
$750$	0	0
$751$	−22.2295	−0.811164	−0.405582	−	0.914059i	$-0.632931\pi$
$751$	−22.2295	−0.811164	−0.405582	+	0.914059i	$0.632931\pi$
$752$	0	0
$753$	16.3451	0.595650
$754$	0	0
$755$	−4.18783	−0.152411
$756$	0	0
$757$	54.4323	1.97838	0.989188	−	0.146651i	$-0.0468493\pi$
$757$	54.4323	1.97838	0.989188	+	0.146651i	$0.0468493\pi$
$758$	0	0
$759$	3.82183	0.138724
$760$	0	0
$761$	−24.5881	−0.891317	−0.445658	−	0.895203i	$-0.647030\pi$
$761$	−24.5881	−0.891317	−0.445658	+	0.895203i	$0.647030\pi$
$762$	0	0
$763$	−13.0404	−0.472094
$764$	0	0
$765$	5.92772	0.214317
$766$	0	0
$767$	11.3047	0.408190
$768$	0	0
$769$	−31.4656	−1.13468	−0.567339	−	0.823484i	$-0.692027\pi$
$769$	−31.4656	−1.13468	−0.567339	+	0.823484i	$0.692027\pi$
$770$	0	0
$771$	−13.2129	−0.475852
$772$	0	0
$773$	50.4507	1.81459	0.907294	−	0.420498i	$-0.138144\pi$
$773$	50.4507	1.81459	0.907294	+	0.420498i	$0.138144\pi$
$774$	0	0
$775$	42.1091	1.51260
$776$	0	0
$777$	−5.74034	−0.205934
$778$	0	0
$779$	−11.3984	−0.408389
$780$	0	0
$781$	−1.80568	−0.0646124
$782$	0	0
$783$	5.55174	0.198403
$784$	0	0
$785$	−9.42566	−0.336416
$786$	0	0
$787$	−15.3421	−0.546886	−0.273443	−	0.961888i	$-0.588162\pi$
$787$	−15.3421	−0.546886	−0.273443	+	0.961888i	$0.588162\pi$
$788$	0	0
$789$	−19.7068	−0.701581
$790$	0	0
$791$	−3.84663	−0.136770
$792$	0	0
$793$	−2.45708	−0.0872534
$794$	0	0
$795$	5.27888	0.187223
$796$	0	0
$797$	−43.8322	−1.55262	−0.776308	−	0.630354i	$-0.782910\pi$
$797$	−43.8322	−1.55262	−0.776308	+	0.630354i	$0.782910\pi$
$798$	0	0
$799$	−13.0312	−0.461011
$800$	0	0
$801$	−9.85544	−0.348225
$802$	0	0
$803$	−0.139179	−0.00491151
$804$	0	0
$805$	−2.56652	−0.0904581
$806$	0	0
$807$	14.4108	0.507285
$808$	0	0
$809$	43.4758	1.52853	0.764265	−	0.644903i	$-0.223102\pi$
$809$	43.4758	1.52853	0.764265	+	0.644903i	$0.223102\pi$
$810$	0	0
$811$	−35.2713	−1.23854	−0.619271	−	0.785177i	$-0.712572\pi$
$811$	−35.2713	−1.23854	−0.619271	+	0.785177i	$0.712572\pi$
$812$	0	0
$813$	−19.4623	−0.682571
$814$	0	0
$815$	−6.21906	−0.217844
$816$	0	0
$817$	5.37235	0.187955
$818$	0	0
$819$	−1.22854	−0.0429286
$820$	0	0
$821$	−24.2037	−0.844716	−0.422358	−	0.906429i	$-0.638798\pi$
$821$	−24.2037	−0.844716	−0.422358	+	0.906429i	$0.638798\pi$
$822$	0	0
$823$	21.3723	0.744993	0.372497	−	0.928034i	$-0.378502\pi$
$823$	21.3723	0.744993	0.372497	+	0.928034i	$0.378502\pi$
$824$	0	0
$825$	−5.79944	−0.201911
$826$	0	0
$827$	21.4311	0.745234	0.372617	−	0.927985i	$-0.378461\pi$
$827$	21.4311	0.745234	0.372617	+	0.927985i	$0.378461\pi$
$828$	0	0
$829$	−36.5210	−1.26843	−0.634214	−	0.773158i	$-0.718676\pi$
$829$	−36.5210	−1.26843	−0.634214	+	0.773158i	$0.718676\pi$
$830$	0	0
$831$	−17.0087	−0.590026
$832$	0	0
$833$	6.23411	0.215999
$834$	0	0
$835$	7.86510	0.272183
$836$	0	0
$837$	10.2808	0.355358
$838$	0	0
$839$	−21.4320	−0.739914	−0.369957	−	0.929049i	$-0.620628\pi$
$839$	−21.4320	−0.739914	−0.369957	+	0.929049i	$0.620628\pi$
$840$	0	0
$841$	1.82181	0.0628209
$842$	0	0
$843$	−12.0609	−0.415398
$844$	0	0
$845$	−10.9260	−0.375864
$846$	0	0
$847$	8.99517	0.309078
$848$	0	0
$849$	24.6984	0.847648
$850$	0	0
$851$	−15.4942	−0.531135
$852$	0	0
$853$	29.8777	1.02299	0.511497	−	0.859285i	$-0.329091\pi$
$853$	29.8777	1.02299	0.511497	+	0.859285i	$0.329091\pi$
$854$	0	0
$855$	−0.950852	−0.0325185
$856$	0	0
$857$	16.6001	0.567050	0.283525	−	0.958965i	$-0.408496\pi$
$857$	16.6001	0.567050	0.283525	+	0.958965i	$0.408496\pi$
$858$	0	0
$859$	31.9649	1.09063	0.545314	−	0.838232i	$-0.316410\pi$
$859$	31.9649	1.09063	0.545314	+	0.838232i	$0.316410\pi$
$860$	0	0
$861$	11.3984	0.388455
$862$	0	0
$863$	−17.4613	−0.594388	−0.297194	−	0.954817i	$-0.596051\pi$
$863$	−17.4613	−0.594388	−0.297194	+	0.954817i	$0.596051\pi$
$864$	0	0
$865$	4.15516	0.141280
$866$	0	0
$867$	−21.8642	−0.742546
$868$	0	0
$869$	−3.82183	−0.129647
$870$	0	0
$871$	12.1431	0.411452
$872$	0	0
$873$	9.60768	0.325171
$874$	0	0
$875$	8.64884	0.292384
$876$	0	0
$877$	−38.8637	−1.31233	−0.656167	−	0.754615i	$-0.727823\pi$
$877$	−38.8637	−1.31233	−0.656167	+	0.754615i	$0.727823\pi$
$878$	0	0
$879$	16.9933	0.573170
$880$	0	0
$881$	30.0438	1.01220	0.506101	−	0.862474i	$-0.331086\pi$
$881$	30.0438	1.01220	0.506101	+	0.862474i	$0.331086\pi$
$882$	0	0
$883$	27.2880	0.918313	0.459157	−	0.888355i	$-0.348152\pi$
$883$	27.2880	0.918313	0.459157	+	0.888355i	$0.348152\pi$
$884$	0	0
$885$	−8.74953	−0.294112
$886$	0	0
$887$	1.60277	0.0538158	0.0269079	−	0.999638i	$-0.491434\pi$
$887$	1.60277	0.0538158	0.0269079	+	0.999638i	$0.491434\pi$
$888$	0	0
$889$	3.25456	0.109154
$890$	0	0
$891$	−1.41592	−0.0474351
$892$	0	0
$893$	2.09030	0.0699494
$894$	0	0
$895$	−8.12192	−0.271486
$896$	0	0
$897$	−3.31605	−0.110720
$898$	0	0
$899$	57.0766	1.90361
$900$	0	0
$901$	−34.6102	−1.15303
$902$	0	0
$903$	−5.37235	−0.178781
$904$	0	0
$905$	−15.4392	−0.513218
$906$	0	0
$907$	6.38211	0.211914	0.105957	−	0.994371i	$-0.466209\pi$
$907$	6.38211	0.211914	0.105957	+	0.994371i	$0.466209\pi$
$908$	0	0
$909$	11.5014	0.381477
$910$	0	0
$911$	49.3472	1.63494	0.817472	−	0.575968i	$-0.195375\pi$
$911$	49.3472	1.63494	0.817472	+	0.575968i	$0.195375\pi$
$912$	0	0
$913$	25.6540	0.849024
$914$	0	0
$915$	1.90170	0.0628684
$916$	0	0
$917$	9.65003	0.318672
$918$	0	0
$919$	41.6540	1.37404	0.687020	−	0.726639i	$-0.258919\pi$
$919$	41.6540	1.37404	0.687020	+	0.726639i	$0.258919\pi$
$920$	0	0
$921$	17.8246	0.587340
$922$	0	0
$923$	1.56672	0.0515692
$924$	0	0
$925$	23.5117	0.773061
$926$	0	0
$927$	−1.28326	−0.0421478
$928$	0	0
$929$	−6.79487	−0.222932	−0.111466	−	0.993768i	$-0.535555\pi$
$929$	−6.79487	−0.222932	−0.111466	+	0.993768i	$0.535555\pi$
$930$	0	0
$931$	−1.00000	−0.0327737
$932$	0	0
$933$	10.5065	0.343967
$934$	0	0
$935$	−8.39318	−0.274486
$936$	0	0
$937$	−15.9616	−0.521441	−0.260721	−	0.965414i	$-0.583960\pi$
$937$	−15.9616	−0.521441	−0.260721	+	0.965414i	$0.583960\pi$
$938$	0	0
$939$	16.9852	0.554290
$940$	0	0
$941$	−56.4732	−1.84097	−0.920487	−	0.390773i	$-0.872208\pi$
$941$	−56.4732	−1.84097	−0.920487	+	0.390773i	$0.872208\pi$
$942$	0	0
$943$	30.7663	1.00189
$944$	0	0
$945$	0.950852	0.0309312
$946$	0	0
$947$	−6.23162	−0.202500	−0.101250	−	0.994861i	$-0.532284\pi$
$947$	−6.23162	−0.202500	−0.101250	+	0.994861i	$0.532284\pi$
$948$	0	0
$949$	0.120760	0.00392003
$950$	0	0
$951$	19.9401	0.646602
$952$	0	0
$953$	−11.1467	−0.361077	−0.180538	−	0.983568i	$-0.557784\pi$
$953$	−11.1467	−0.361077	−0.180538	+	0.983568i	$0.557784\pi$
$954$	0	0
$955$	−19.9295	−0.644905
$956$	0	0
$957$	−7.86082	−0.254104
$958$	0	0
$959$	7.20177	0.232557
$960$	0	0
$961$	74.6958	2.40954
$962$	0	0
$963$	−12.4819	−0.402223
$964$	0	0
$965$	2.44498	0.0787066
$966$	0	0
$967$	−25.8469	−0.831180	−0.415590	−	0.909552i	$-0.636425\pi$
$967$	−25.8469	−0.831180	−0.415590	+	0.909552i	$0.636425\pi$
$968$	0	0
$969$	6.23411	0.200269
$970$	0	0
$971$	−25.4320	−0.816151	−0.408076	−	0.912948i	$-0.633800\pi$
$971$	−25.4320	−0.816151	−0.408076	+	0.912948i	$0.633800\pi$
$972$	0	0
$973$	−19.0348	−0.610226
$974$	0	0
$975$	5.03195	0.161151
$976$	0	0
$977$	47.5055	1.51983	0.759917	−	0.650020i	$-0.225239\pi$
$977$	47.5055	1.51983	0.759917	+	0.650020i	$0.225239\pi$
$978$	0	0
$979$	13.9545	0.445988
$980$	0	0
$981$	13.0404	0.416348
$982$	0	0
$983$	−41.5333	−1.32471	−0.662353	−	0.749192i	$-0.730442\pi$
$983$	−41.5333	−1.32471	−0.662353	+	0.749192i	$0.730442\pi$
$984$	0	0
$985$	−16.4292	−0.523479
$986$	0	0
$987$	−2.09030	−0.0665351
$988$	0	0
$989$	−14.5009	−0.461103
$990$	0	0
$991$	32.7800	1.04129	0.520645	−	0.853773i	$-0.325691\pi$
$991$	32.7800	1.04129	0.520645	+	0.853773i	$0.325691\pi$
$992$	0	0
$993$	−30.7632	−0.976240
$994$	0	0
$995$	−12.3987	−0.393066
$996$	0	0
$997$	21.9129	0.693987	0.346994	−	0.937868i	$-0.387203\pi$
$997$	21.9129	0.693987	0.346994	+	0.937868i	$0.387203\pi$
$998$	0	0
$999$	5.74034	0.181616

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6384.2.a.cd.1.3		5
4.3	odd	2		3192.2.a.bb.1.3	✓	5
12.11	even	2		9576.2.a.cn.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3192.2.a.bb.1.3	✓	5	4.3	odd	2
6384.2.a.cd.1.3		5	1.1	even	1	trivial
9576.2.a.cn.1.3		5	12.11	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:	\( N \)	\(=\)	\( 6384 = 2^{4} \cdot 3 \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6384.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +0.950852 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +0.950852 q^{5} -1.00000 q^{7} +1.00000 q^{9} -1.41592 q^{11} +1.22854 q^{13} -0.950852 q^{15} +6.23411 q^{17} -1.00000 q^{19} +1.00000 q^{21} +2.69918 q^{23} -4.09588 q^{25} -1.00000 q^{27} -5.55174 q^{29} -10.2808 q^{31} +1.41592 q^{33} -0.950852 q^{35} -5.74034 q^{37} -1.22854 q^{39} +11.3984 q^{41} -5.37235 q^{43} +0.950852 q^{45} -2.09030 q^{47} +1.00000 q^{49} -6.23411 q^{51} -5.55174 q^{53} -1.34633 q^{55} +1.00000 q^{57} +9.20177 q^{59} -2.00000 q^{61} -1.00000 q^{63} +1.16816 q^{65} +9.88415 q^{67} -2.69918 q^{69} +1.27527 q^{71} +0.0982955 q^{73} +4.09588 q^{75} +1.41592 q^{77} +2.69918 q^{79} +1.00000 q^{81} -18.1183 q^{83} +5.92772 q^{85} +5.55174 q^{87} -9.85544 q^{89} -1.22854 q^{91} +10.2808 q^{93} -0.950852 q^{95} +9.60768 q^{97} -1.41592 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 5 q^{3} + 2 q^{5} - 5 q^{7} + 5 q^{9}+O(q^{10}) \) 5 * q - 5 * q^3 + 2 * q^5 - 5 * q^7 + 5 * q^9	\( 5 q - 5 q^{3} + 2 q^{5} - 5 q^{7} + 5 q^{9} - 6 q^{11} - 2 q^{15} + 6 q^{17} - 5 q^{19} + 5 q^{21} - 10 q^{23} + 7 q^{25} - 5 q^{27} + 4 q^{29} + 4 q^{31} + 6 q^{33} - 2 q^{35} + 6 q^{37} + 10 q^{41} - 4 q^{43} + 2 q^{45} - 2 q^{47} + 5 q^{49} - 6 q^{51} + 4 q^{53} - 8 q^{55} + 5 q^{57} - 12 q^{59} - 10 q^{61} - 5 q^{63} + 8 q^{65} - 2 q^{67} + 10 q^{69} - 30 q^{71} + 6 q^{73} - 7 q^{75} + 6 q^{77} - 10 q^{79} + 5 q^{81} - 14 q^{83} - 4 q^{87} + 10 q^{89} - 4 q^{93} - 2 q^{95} - 8 q^{97} - 6 q^{99}+O(q^{100}) \) 5 * q - 5 * q^3 + 2 * q^5 - 5 * q^7 + 5 * q^9 - 6 * q^11 - 2 * q^15 + 6 * q^17 - 5 * q^19 + 5 * q^21 - 10 * q^23 + 7 * q^25 - 5 * q^27 + 4 * q^29 + 4 * q^31 + 6 * q^33 - 2 * q^35 + 6 * q^37 + 10 * q^41 - 4 * q^43 + 2 * q^45 - 2 * q^47 + 5 * q^49 - 6 * q^51 + 4 * q^53 - 8 * q^55 + 5 * q^57 - 12 * q^59 - 10 * q^61 - 5 * q^63 + 8 * q^65 - 2 * q^67 + 10 * q^69 - 30 * q^71 + 6 * q^73 - 7 * q^75 + 6 * q^77 - 10 * q^79 + 5 * q^81 - 14 * q^83 - 4 * q^87 + 10 * q^89 - 4 * q^93 - 2 * q^95 - 8 * q^97 - 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more