LMFDB - Embedded newform 6040.2.a.o.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6040,2,Mod(1,6040)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6040, 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("6040.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6040 = 2^{3} \cdot 5 \cdot 151 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6040.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:	$48.2296428209$
Analytic rank:	$0$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 5 x^{14} - 19 x^{13} + 119 x^{12} + 106 x^{11} - 1063 x^{10} - 48 x^{9} + 4510 x^{8} + \cdots + 272 $ x^15 - 5x^14 - 19x^13 + 119x^12 + 106x^11 - 1063x^10 - 48x^9 + 4510x^8 - 1130x^7 - 9512x^6 + 2839x^5 + 9383x^4 - 1781x^3 - 3692x^2 - 92x + 272
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$1.14431$ of defining polynomial
Character	$\chi$	$=$	6040.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.14431 q^{3} +1.00000 q^{5} +0.357411 q^{7} -1.69055 q^{9} +O(q^{10})$	$q+1.14431 q^{3} +1.00000 q^{5} +0.357411 q^{7} -1.69055 q^{9} +3.73642 q^{11} -2.83332 q^{13} +1.14431 q^{15} +6.01349 q^{17} -6.73253 q^{19} +0.408991 q^{21} +3.51951 q^{23} +1.00000 q^{25} -5.36746 q^{27} -0.466531 q^{29} +7.02851 q^{31} +4.27564 q^{33} +0.357411 q^{35} +8.99325 q^{37} -3.24221 q^{39} -0.120392 q^{41} +8.01683 q^{43} -1.69055 q^{45} -6.49046 q^{47} -6.87226 q^{49} +6.88132 q^{51} -2.45510 q^{53} +3.73642 q^{55} -7.70413 q^{57} +3.14997 q^{59} -0.565494 q^{61} -0.604220 q^{63} -2.83332 q^{65} +3.57444 q^{67} +4.02743 q^{69} -0.0633229 q^{71} +11.5054 q^{73} +1.14431 q^{75} +1.33544 q^{77} -6.68216 q^{79} -1.07041 q^{81} +11.0664 q^{83} +6.01349 q^{85} -0.533858 q^{87} +4.85315 q^{89} -1.01266 q^{91} +8.04282 q^{93} -6.73253 q^{95} +11.1480 q^{97} -6.31660 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q + 5 q^{3} + 15 q^{5} + 7 q^{7} + 18 q^{9}+O(q^{10}) $ 15 * q + 5 * q^3 + 15 * q^5 + 7 * q^7 + 18 * q^9	$ 15 q + 5 q^{3} + 15 q^{5} + 7 q^{7} + 18 q^{9} + 7 q^{11} + 2 q^{13} + 5 q^{15} - 3 q^{17} + 8 q^{19} + 7 q^{21} + 15 q^{23} + 15 q^{25} + 23 q^{27} + 5 q^{29} + 27 q^{31} - 5 q^{33} + 7 q^{35} - 4 q^{37} + 11 q^{39} + 20 q^{41} + 25 q^{43} + 18 q^{45} + 35 q^{47} - 14 q^{49} + 25 q^{51} - 2 q^{53} + 7 q^{55} - 24 q^{57} + 39 q^{59} + 23 q^{61} + 39 q^{63} + 2 q^{65} + 32 q^{67} + 13 q^{69} + 30 q^{71} + 7 q^{73} + 5 q^{75} - 4 q^{77} + 38 q^{79} + 11 q^{81} + 29 q^{83} - 3 q^{85} + 4 q^{87} + 19 q^{89} + 16 q^{91} + 8 q^{93} + 8 q^{95} - 8 q^{97} + 16 q^{99}+O(q^{100}) $ 15 * q + 5 * q^3 + 15 * q^5 + 7 * q^7 + 18 * q^9 + 7 * q^11 + 2 * q^13 + 5 * q^15 - 3 * q^17 + 8 * q^19 + 7 * q^21 + 15 * q^23 + 15 * q^25 + 23 * q^27 + 5 * q^29 + 27 * q^31 - 5 * q^33 + 7 * q^35 - 4 * q^37 + 11 * q^39 + 20 * q^41 + 25 * q^43 + 18 * q^45 + 35 * q^47 - 14 * q^49 + 25 * q^51 - 2 * q^53 + 7 * q^55 - 24 * q^57 + 39 * q^59 + 23 * q^61 + 39 * q^63 + 2 * q^65 + 32 * q^67 + 13 * q^69 + 30 * q^71 + 7 * q^73 + 5 * q^75 - 4 * q^77 + 38 * q^79 + 11 * q^81 + 29 * q^83 - 3 * q^85 + 4 * q^87 + 19 * q^89 + 16 * q^91 + 8 * q^93 + 8 * q^95 - 8 * q^97 + 16 * 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
$3$	1.14431	0.660670	0.330335	−	0.943864i	$-0.392838\pi$
$3$	1.14431	0.660670	0.330335	+	0.943864i	$0.392838\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	0.357411	0.135089	0.0675444	−	0.997716i	$-0.478484\pi$
$7$	0.357411	0.135089	0.0675444	+	0.997716i	$0.478484\pi$
$8$	0	0
$9$	−1.69055	−0.563516
$10$	0	0
$11$	3.73642	1.12657	0.563287	−	0.826261i	$-0.309536\pi$
$11$	3.73642	1.12657	0.563287	+	0.826261i	$0.309536\pi$
$12$	0	0
$13$	−2.83332	−0.785823	−0.392911	−	0.919576i	$-0.628532\pi$
$13$	−2.83332	−0.785823	−0.392911	+	0.919576i	$0.628532\pi$
$14$	0	0
$15$	1.14431	0.295460
$16$	0	0
$17$	6.01349	1.45849	0.729243	−	0.684255i	$-0.239872\pi$
$17$	6.01349	1.45849	0.729243	+	0.684255i	$0.239872\pi$
$18$	0	0
$19$	−6.73253	−1.54455	−0.772275	−	0.635289i	$-0.780881\pi$
$19$	−6.73253	−1.54455	−0.772275	+	0.635289i	$0.780881\pi$
$20$	0	0
$21$	0.408991	0.0892491
$22$	0	0
$23$	3.51951	0.733869	0.366935	−	0.930247i	$-0.380407\pi$
$23$	3.51951	0.733869	0.366935	+	0.930247i	$0.380407\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−5.36746	−1.03297
$28$	0	0
$29$	−0.466531	−0.0866327	−0.0433163	−	0.999061i	$-0.513792\pi$
$29$	−0.466531	−0.0866327	−0.0433163	+	0.999061i	$0.513792\pi$
$30$	0	0
$31$	7.02851	1.26236	0.631179	−	0.775637i	$-0.282571\pi$
$31$	7.02851	1.26236	0.631179	+	0.775637i	$0.282571\pi$
$32$	0	0
$33$	4.27564	0.744294
$34$	0	0
$35$	0.357411	0.0604135
$36$	0	0
$37$	8.99325	1.47848	0.739241	−	0.673441i	$-0.235185\pi$
$37$	8.99325	1.47848	0.739241	+	0.673441i	$0.235185\pi$
$38$	0	0
$39$	−3.24221	−0.519169
$40$	0	0
$41$	−0.120392	−0.0188021	−0.00940105	−	0.999956i	$-0.502992\pi$
$41$	−0.120392	−0.0188021	−0.00940105	+	0.999956i	$0.502992\pi$
$42$	0	0
$43$	8.01683	1.22255	0.611277	−	0.791416i	$-0.290656\pi$
$43$	8.01683	1.22255	0.611277	+	0.791416i	$0.290656\pi$
$44$	0	0
$45$	−1.69055	−0.252012
$46$	0	0
$47$	−6.49046	−0.946731	−0.473365	−	0.880866i	$-0.656961\pi$
$47$	−6.49046	−0.946731	−0.473365	+	0.880866i	$0.656961\pi$
$48$	0	0
$49$	−6.87226	−0.981751
$50$	0	0
$51$	6.88132	0.963578
$52$	0	0
$53$	−2.45510	−0.337234	−0.168617	−	0.985682i	$-0.553930\pi$
$53$	−2.45510	−0.337234	−0.168617	+	0.985682i	$0.553930\pi$
$54$	0	0
$55$	3.73642	0.503819
$56$	0	0
$57$	−7.70413	−1.02044
$58$	0	0
$59$	3.14997	0.410091	0.205046	−	0.978752i	$-0.434266\pi$
$59$	3.14997	0.410091	0.205046	+	0.978752i	$0.434266\pi$
$60$	0	0
$61$	−0.565494	−0.0724041	−0.0362021	−	0.999344i	$-0.511526\pi$
$61$	−0.565494	−0.0724041	−0.0362021	+	0.999344i	$0.511526\pi$
$62$	0	0
$63$	−0.604220	−0.0761246
$64$	0	0
$65$	−2.83332	−0.351431
$66$	0	0
$67$	3.57444	0.436687	0.218343	−	0.975872i	$-0.429935\pi$
$67$	3.57444	0.436687	0.218343	+	0.975872i	$0.429935\pi$
$68$	0	0
$69$	4.02743	0.484845
$70$	0	0
$71$	−0.0633229	−0.00751504	−0.00375752	−	0.999993i	$-0.501196\pi$
$71$	−0.0633229	−0.00751504	−0.00375752	+	0.999993i	$0.501196\pi$
$72$	0	0
$73$	11.5054	1.34660	0.673302	−	0.739367i	$-0.264875\pi$
$73$	11.5054	1.34660	0.673302	+	0.739367i	$0.264875\pi$
$74$	0	0
$75$	1.14431	0.132134
$76$	0	0
$77$	1.33544	0.152188
$78$	0	0
$79$	−6.68216	−0.751802	−0.375901	−	0.926660i	$-0.622667\pi$
$79$	−6.68216	−0.751802	−0.375901	+	0.926660i	$0.622667\pi$
$80$	0	0
$81$	−1.07041	−0.118935
$82$	0	0
$83$	11.0664	1.21470	0.607349	−	0.794435i	$-0.292233\pi$
$83$	11.0664	1.21470	0.607349	+	0.794435i	$0.292233\pi$
$84$	0	0
$85$	6.01349	0.652255
$86$	0	0
$87$	−0.533858	−0.0572356
$88$	0	0
$89$	4.85315	0.514433	0.257217	−	0.966354i	$-0.417195\pi$
$89$	4.85315	0.514433	0.257217	+	0.966354i	$0.417195\pi$
$90$	0	0
$91$	−1.01266	−0.106156
$92$	0	0
$93$	8.04282	0.834002
$94$	0	0
$95$	−6.73253	−0.690743
$96$	0	0
$97$	11.1480	1.13191	0.565956	−	0.824435i	$-0.308507\pi$
$97$	11.1480	1.13191	0.565956	+	0.824435i	$0.308507\pi$
$98$	0	0
$99$	−6.31660	−0.634842
$100$	0	0
$101$	13.6849	1.36170	0.680850	−	0.732422i	$-0.261610\pi$
$101$	13.6849	1.36170	0.680850	+	0.732422i	$0.261610\pi$
$102$	0	0
$103$	11.9551	1.17797	0.588983	−	0.808145i	$-0.299528\pi$
$103$	11.9551	1.17797	0.588983	+	0.808145i	$0.299528\pi$
$104$	0	0
$105$	0.408991	0.0399134
$106$	0	0
$107$	−15.8133	−1.52873	−0.764367	−	0.644781i	$-0.776948\pi$
$107$	−15.8133	−1.52873	−0.764367	+	0.644781i	$0.776948\pi$
$108$	0	0
$109$	7.93525	0.760059	0.380029	−	0.924974i	$-0.375914\pi$
$109$	7.93525	0.760059	0.380029	+	0.924974i	$0.375914\pi$
$110$	0	0
$111$	10.2911	0.976788
$112$	0	0
$113$	−0.0548040	−0.00515552	−0.00257776	−	0.999997i	$-0.500821\pi$
$113$	−0.0548040	−0.00515552	−0.00257776	+	0.999997i	$0.500821\pi$
$114$	0	0
$115$	3.51951	0.328196
$116$	0	0
$117$	4.78987	0.442823
$118$	0	0
$119$	2.14929	0.197025
$120$	0	0
$121$	2.96087	0.269170
$122$	0	0
$123$	−0.137766	−0.0124220
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−10.5565	−0.936742	−0.468371	−	0.883532i	$-0.655159\pi$
$127$	−10.5565	−0.936742	−0.468371	+	0.883532i	$0.655159\pi$
$128$	0	0
$129$	9.17376	0.807705
$130$	0	0
$131$	6.94450	0.606744	0.303372	−	0.952872i	$-0.401888\pi$
$131$	6.94450	0.606744	0.303372	+	0.952872i	$0.401888\pi$
$132$	0	0
$133$	−2.40628	−0.208651
$134$	0	0
$135$	−5.36746	−0.461957
$136$	0	0
$137$	−19.5262	−1.66823	−0.834116	−	0.551588i	$-0.814022\pi$
$137$	−19.5262	−1.66823	−0.834116	+	0.551588i	$0.814022\pi$
$138$	0	0
$139$	6.59806	0.559640	0.279820	−	0.960052i	$-0.409725\pi$
$139$	6.59806	0.559640	0.279820	+	0.960052i	$0.409725\pi$
$140$	0	0
$141$	−7.42712	−0.625476
$142$	0	0
$143$	−10.5865	−0.885288
$144$	0	0
$145$	−0.466531	−0.0387433
$146$	0	0
$147$	−7.86402	−0.648613
$148$	0	0
$149$	−3.54496	−0.290414	−0.145207	−	0.989401i	$-0.546385\pi$
$149$	−3.54496	−0.290414	−0.145207	+	0.989401i	$0.546385\pi$
$150$	0	0
$151$	−1.00000	−0.0813788
$151$	−1.00000	−0.0813788
$152$	0	0
$153$	−10.1661	−0.821879
$154$	0	0
$155$	7.02851	0.564544
$156$	0	0
$157$	20.1476	1.60796	0.803978	−	0.594658i	$-0.202713\pi$
$157$	20.1476	1.60796	0.803978	+	0.594658i	$0.202713\pi$
$158$	0	0
$159$	−2.80940	−0.222800
$160$	0	0
$161$	1.25791	0.0991375
$162$	0	0
$163$	9.81711	0.768935	0.384468	−	0.923138i	$-0.374385\pi$
$163$	9.81711	0.768935	0.384468	+	0.923138i	$0.374385\pi$
$164$	0	0
$165$	4.27564	0.332858
$166$	0	0
$167$	15.0051	1.16113	0.580566	−	0.814213i	$-0.302831\pi$
$167$	15.0051	1.16113	0.580566	+	0.814213i	$0.302831\pi$
$168$	0	0
$169$	−4.97228	−0.382483
$170$	0	0
$171$	11.3817	0.870377
$172$	0	0
$173$	6.26810	0.476555	0.238277	−	0.971197i	$-0.423417\pi$
$173$	6.26810	0.476555	0.238277	+	0.971197i	$0.423417\pi$
$174$	0	0
$175$	0.357411	0.0270178
$176$	0	0
$177$	3.60456	0.270935
$178$	0	0
$179$	−11.2839	−0.843397	−0.421699	−	0.906736i	$-0.638566\pi$
$179$	−11.2839	−0.843397	−0.421699	+	0.906736i	$0.638566\pi$
$180$	0	0
$181$	−18.1749	−1.35093	−0.675464	−	0.737393i	$-0.736057\pi$
$181$	−18.1749	−1.35093	−0.675464	+	0.737393i	$0.736057\pi$
$182$	0	0
$183$	−0.647103	−0.0478352
$184$	0	0
$185$	8.99325	0.661197
$186$	0	0
$187$	22.4690	1.64309
$188$	0	0
$189$	−1.91839	−0.139542
$190$	0	0
$191$	−1.99915	−0.144653	−0.0723266	−	0.997381i	$-0.523042\pi$
$191$	−1.99915	−0.144653	−0.0723266	+	0.997381i	$0.523042\pi$
$192$	0	0
$193$	−18.8908	−1.35979	−0.679894	−	0.733311i	$-0.737974\pi$
$193$	−18.8908	−1.35979	−0.679894	+	0.733311i	$0.737974\pi$
$194$	0	0
$195$	−3.24221	−0.232180
$196$	0	0
$197$	−5.65204	−0.402691	−0.201346	−	0.979520i	$-0.564531\pi$
$197$	−5.65204	−0.402691	−0.201346	+	0.979520i	$0.564531\pi$
$198$	0	0
$199$	12.6919	0.899707	0.449854	−	0.893102i	$-0.351476\pi$
$199$	12.6919	0.899707	0.449854	+	0.893102i	$0.351476\pi$
$200$	0	0
$201$	4.09028	0.288506
$202$	0	0
$203$	−0.166744	−0.0117031
$204$	0	0
$205$	−0.120392	−0.00840855
$206$	0	0
$207$	−5.94990	−0.413547
$208$	0	0
$209$	−25.1556	−1.74005
$210$	0	0
$211$	25.8944	1.78264	0.891320	−	0.453374i	$-0.149780\pi$
$211$	25.8944	1.78264	0.891320	+	0.453374i	$0.149780\pi$
$212$	0	0
$213$	−0.0724612	−0.00496496
$214$	0	0
$215$	8.01683	0.546743
$216$	0	0
$217$	2.51207	0.170530
$218$	0	0
$219$	13.1658	0.889661
$220$	0	0
$221$	−17.0382	−1.14611
$222$	0	0
$223$	12.8694	0.861799	0.430900	−	0.902400i	$-0.358196\pi$
$223$	12.8694	0.861799	0.430900	+	0.902400i	$0.358196\pi$
$224$	0	0
$225$	−1.69055	−0.112703
$226$	0	0
$227$	−15.2262	−1.01060	−0.505299	−	0.862944i	$-0.668618\pi$
$227$	−15.2262	−1.01060	−0.505299	+	0.862944i	$0.668618\pi$
$228$	0	0
$229$	6.50525	0.429879	0.214940	−	0.976627i	$-0.431045\pi$
$229$	6.50525	0.429879	0.214940	+	0.976627i	$0.431045\pi$
$230$	0	0
$231$	1.52816	0.100546
$232$	0	0
$233$	3.41303	0.223595	0.111797	−	0.993731i	$-0.464339\pi$
$233$	3.41303	0.223595	0.111797	+	0.993731i	$0.464339\pi$
$234$	0	0
$235$	−6.49046	−0.423391
$236$	0	0
$237$	−7.64649	−0.496693
$238$	0	0
$239$	1.97655	0.127852	0.0639262	−	0.997955i	$-0.479638\pi$
$239$	1.97655	0.127852	0.0639262	+	0.997955i	$0.479638\pi$
$240$	0	0
$241$	13.9571	0.899053	0.449526	−	0.893267i	$-0.351593\pi$
$241$	13.9571	0.899053	0.449526	+	0.893267i	$0.351593\pi$
$242$	0	0
$243$	14.8775	0.954391
$244$	0	0
$245$	−6.87226	−0.439052
$246$	0	0
$247$	19.0754	1.21374
$248$	0	0
$249$	12.6635	0.802514
$250$	0	0
$251$	5.51803	0.348295	0.174147	−	0.984720i	$-0.444283\pi$
$251$	5.51803	0.348295	0.174147	+	0.984720i	$0.444283\pi$
$252$	0	0
$253$	13.1504	0.826759
$254$	0	0
$255$	6.88132	0.430925
$256$	0	0
$257$	−10.4334	−0.650818	−0.325409	−	0.945573i	$-0.605502\pi$
$257$	−10.4334	−0.650818	−0.325409	+	0.945573i	$0.605502\pi$
$258$	0	0
$259$	3.21429	0.199726
$260$	0	0
$261$	0.788693	0.0488189
$262$	0	0
$263$	17.8149	1.09851	0.549257	−	0.835654i	$-0.314911\pi$
$263$	17.8149	1.09851	0.549257	+	0.835654i	$0.314911\pi$
$264$	0	0
$265$	−2.45510	−0.150816
$266$	0	0
$267$	5.55353	0.339870
$268$	0	0
$269$	12.4445	0.758757	0.379378	−	0.925242i	$-0.376138\pi$
$269$	12.4445	0.758757	0.379378	+	0.925242i	$0.376138\pi$
$270$	0	0
$271$	−18.0106	−1.09406	−0.547032	−	0.837112i	$-0.684242\pi$
$271$	−18.0106	−1.09406	−0.547032	+	0.837112i	$0.684242\pi$
$272$	0	0
$273$	−1.15880	−0.0701339
$274$	0	0
$275$	3.73642	0.225315
$276$	0	0
$277$	17.9484	1.07842	0.539209	−	0.842172i	$-0.318723\pi$
$277$	17.9484	1.07842	0.539209	+	0.842172i	$0.318723\pi$
$278$	0	0
$279$	−11.8820	−0.711359
$280$	0	0
$281$	7.18289	0.428496	0.214248	−	0.976779i	$-0.431270\pi$
$281$	7.18289	0.428496	0.214248	+	0.976779i	$0.431270\pi$
$282$	0	0
$283$	−2.57454	−0.153041	−0.0765204	−	0.997068i	$-0.524381\pi$
$283$	−2.57454	−0.153041	−0.0765204	+	0.997068i	$0.524381\pi$
$284$	0	0
$285$	−7.70413	−0.456353
$286$	0	0
$287$	−0.0430295	−0.00253995
$288$	0	0
$289$	19.1621	1.12718
$290$	0	0
$291$	12.7569	0.747820
$292$	0	0
$293$	−18.8870	−1.10339	−0.551696	−	0.834046i	$-0.686019\pi$
$293$	−18.8870	−1.10339	−0.551696	+	0.834046i	$0.686019\pi$
$294$	0	0
$295$	3.14997	0.183398
$296$	0	0
$297$	−20.0551	−1.16371
$298$	0	0
$299$	−9.97192	−0.576691
$300$	0	0
$301$	2.86530	0.165153
$302$	0	0
$303$	15.6598	0.899635
$304$	0	0
$305$	−0.565494	−0.0323801
$306$	0	0
$307$	−5.91704	−0.337703	−0.168852	−	0.985641i	$-0.554006\pi$
$307$	−5.91704	−0.337703	−0.168852	+	0.985641i	$0.554006\pi$
$308$	0	0
$309$	13.6803	0.778247
$310$	0	0
$311$	13.2578	0.751782	0.375891	−	0.926664i	$-0.377337\pi$
$311$	13.2578	0.751782	0.375891	+	0.926664i	$0.377337\pi$
$312$	0	0
$313$	−10.2101	−0.577110	−0.288555	−	0.957463i	$-0.593175\pi$
$313$	−10.2101	−0.577110	−0.288555	+	0.957463i	$0.593175\pi$
$314$	0	0
$315$	−0.604220	−0.0340440
$316$	0	0
$317$	2.77522	0.155872	0.0779360	−	0.996958i	$-0.475167\pi$
$317$	2.77522	0.155872	0.0779360	+	0.996958i	$0.475167\pi$
$318$	0	0
$319$	−1.74316	−0.0975982
$320$	0	0
$321$	−18.0954	−1.00999
$322$	0	0
$323$	−40.4860	−2.25270
$324$	0	0
$325$	−2.83332	−0.157165
$326$	0	0
$327$	9.08041	0.502148
$328$	0	0
$329$	−2.31976	−0.127893
$330$	0	0
$331$	−22.6014	−1.24228	−0.621142	−	0.783698i	$-0.713331\pi$
$331$	−22.6014	−1.24228	−0.621142	+	0.783698i	$0.713331\pi$
$332$	0	0
$333$	−15.2035	−0.833147
$334$	0	0
$335$	3.57444	0.195292
$336$	0	0
$337$	−23.7430	−1.29337	−0.646683	−	0.762759i	$-0.723844\pi$
$337$	−23.7430	−1.29337	−0.646683	+	0.762759i	$0.723844\pi$
$338$	0	0
$339$	−0.0627129	−0.00340610
$340$	0	0
$341$	26.2615	1.42214
$342$	0	0
$343$	−4.95810	−0.267712
$344$	0	0
$345$	4.02743	0.216829
$346$	0	0
$347$	24.3756	1.30855	0.654274	−	0.756257i	$-0.272974\pi$
$347$	24.3756	1.30855	0.654274	+	0.756257i	$0.272974\pi$
$348$	0	0
$349$	−23.6707	−1.26707	−0.633533	−	0.773716i	$-0.718396\pi$
$349$	−23.6707	−1.26707	−0.633533	+	0.773716i	$0.718396\pi$
$350$	0	0
$351$	15.2077	0.811729
$352$	0	0
$353$	−35.5003	−1.88949	−0.944745	−	0.327807i	$-0.893691\pi$
$353$	−35.5003	−1.88949	−0.944745	+	0.327807i	$0.893691\pi$
$354$	0	0
$355$	−0.0633229	−0.00336083
$356$	0	0
$357$	2.45946	0.130169
$358$	0	0
$359$	2.30006	0.121392	0.0606962	−	0.998156i	$-0.480668\pi$
$359$	2.30006	0.121392	0.0606962	+	0.998156i	$0.480668\pi$
$360$	0	0
$361$	26.3270	1.38563
$362$	0	0
$363$	3.38816	0.177832
$364$	0	0
$365$	11.5054	0.602220
$366$	0	0
$367$	3.27097	0.170743	0.0853715	−	0.996349i	$-0.472792\pi$
$367$	3.27097	0.170743	0.0853715	+	0.996349i	$0.472792\pi$
$368$	0	0
$369$	0.203529	0.0105953
$370$	0	0
$371$	−0.877480	−0.0455565
$372$	0	0
$373$	−6.13428	−0.317621	−0.158811	−	0.987309i	$-0.550766\pi$
$373$	−6.13428	−0.317621	−0.158811	+	0.987309i	$0.550766\pi$
$374$	0	0
$375$	1.14431	0.0590921
$376$	0	0
$377$	1.32183	0.0680779
$378$	0	0
$379$	15.7016	0.806537	0.403268	−	0.915082i	$-0.367874\pi$
$379$	15.7016	0.806537	0.403268	+	0.915082i	$0.367874\pi$
$380$	0	0
$381$	−12.0800	−0.618877
$382$	0	0
$383$	−11.7440	−0.600092	−0.300046	−	0.953925i	$-0.597002\pi$
$383$	−11.7440	−0.600092	−0.300046	+	0.953925i	$0.597002\pi$
$384$	0	0
$385$	1.33544	0.0680603
$386$	0	0
$387$	−13.5528	−0.688929
$388$	0	0
$389$	−12.8303	−0.650522	−0.325261	−	0.945624i	$-0.605452\pi$
$389$	−12.8303	−0.650522	−0.325261	+	0.945624i	$0.605452\pi$
$390$	0	0
$391$	21.1646	1.07034
$392$	0	0
$393$	7.94668	0.400857
$394$	0	0
$395$	−6.68216	−0.336216
$396$	0	0
$397$	−13.1141	−0.658176	−0.329088	−	0.944299i	$-0.606741\pi$
$397$	−13.1141	−0.658176	−0.329088	+	0.944299i	$0.606741\pi$
$398$	0	0
$399$	−2.75354	−0.137850
$400$	0	0
$401$	17.1154	0.854700	0.427350	−	0.904086i	$-0.359447\pi$
$401$	17.1154	0.854700	0.427350	+	0.904086i	$0.359447\pi$
$402$	0	0
$403$	−19.9141	−0.991990
$404$	0	0
$405$	−1.07041	−0.0531892
$406$	0	0
$407$	33.6026	1.66562
$408$	0	0
$409$	−2.93576	−0.145164	−0.0725819	−	0.997362i	$-0.523124\pi$
$409$	−2.93576	−0.145164	−0.0725819	+	0.997362i	$0.523124\pi$
$410$	0	0
$411$	−22.3441	−1.10215
$412$	0	0
$413$	1.12584	0.0553987
$414$	0	0
$415$	11.0664	0.543229
$416$	0	0
$417$	7.55025	0.369737
$418$	0	0
$419$	−27.1927	−1.32845	−0.664226	−	0.747532i	$-0.731239\pi$
$419$	−27.1927	−1.32845	−0.664226	+	0.747532i	$0.731239\pi$
$420$	0	0
$421$	−6.10763	−0.297668	−0.148834	−	0.988862i	$-0.547552\pi$
$421$	−6.10763	−0.297668	−0.148834	+	0.988862i	$0.547552\pi$
$422$	0	0
$423$	10.9724	0.533497
$424$	0	0
$425$	6.01349	0.291697
$426$	0	0
$427$	−0.202114	−0.00978099
$428$	0	0
$429$	−12.1143	−0.584883
$430$	0	0
$431$	25.6139	1.23378	0.616889	−	0.787050i	$-0.288393\pi$
$431$	25.6139	1.23378	0.616889	+	0.787050i	$0.288393\pi$
$432$	0	0
$433$	−6.75023	−0.324395	−0.162198	−	0.986758i	$-0.551858\pi$
$433$	−6.75023	−0.324395	−0.162198	+	0.986758i	$0.551858\pi$
$434$	0	0
$435$	−0.533858	−0.0255965
$436$	0	0
$437$	−23.6952	−1.13350
$438$	0	0
$439$	15.7357	0.751025	0.375513	−	0.926817i	$-0.377467\pi$
$439$	15.7357	0.751025	0.375513	+	0.926817i	$0.377467\pi$
$440$	0	0
$441$	11.6179	0.553232
$442$	0	0
$443$	−0.587239	−0.0279006	−0.0139503	−	0.999903i	$-0.504441\pi$
$443$	−0.587239	−0.0279006	−0.0139503	+	0.999903i	$0.504441\pi$
$444$	0	0
$445$	4.85315	0.230061
$446$	0	0
$447$	−4.05654	−0.191868
$448$	0	0
$449$	−16.6245	−0.784560	−0.392280	−	0.919846i	$-0.628314\pi$
$449$	−16.6245	−0.784560	−0.392280	+	0.919846i	$0.628314\pi$
$450$	0	0
$451$	−0.449836	−0.0211820
$452$	0	0
$453$	−1.14431	−0.0537645
$454$	0	0
$455$	−1.01266	−0.0474743
$456$	0	0
$457$	−2.15521	−0.100816	−0.0504082	−	0.998729i	$-0.516052\pi$
$457$	−2.15521	−0.100816	−0.0504082	+	0.998729i	$0.516052\pi$
$458$	0	0
$459$	−32.2772	−1.50657
$460$	0	0
$461$	−5.03259	−0.234391	−0.117196	−	0.993109i	$-0.537390\pi$
$461$	−5.03259	−0.234391	−0.117196	+	0.993109i	$0.537390\pi$
$462$	0	0
$463$	−19.9057	−0.925097	−0.462549	−	0.886594i	$-0.653065\pi$
$463$	−19.9057	−0.925097	−0.462549	+	0.886594i	$0.653065\pi$
$464$	0	0
$465$	8.04282	0.372977
$466$	0	0
$467$	8.61087	0.398464	0.199232	−	0.979952i	$-0.436155\pi$
$467$	8.61087	0.398464	0.199232	+	0.979952i	$0.436155\pi$
$468$	0	0
$469$	1.27754	0.0589915
$470$	0	0
$471$	23.0552	1.06233
$472$	0	0
$473$	29.9543	1.37730
$474$	0	0
$475$	−6.73253	−0.308910
$476$	0	0
$477$	4.15046	0.190036
$478$	0	0
$479$	20.1391	0.920177	0.460089	−	0.887873i	$-0.347818\pi$
$479$	20.1391	0.920177	0.460089	+	0.887873i	$0.347818\pi$
$480$	0	0
$481$	−25.4808	−1.16182
$482$	0	0
$483$	1.43945	0.0654972
$484$	0	0
$485$	11.1480	0.506207
$486$	0	0
$487$	35.9167	1.62754	0.813770	−	0.581188i	$-0.197412\pi$
$487$	35.9167	1.62754	0.813770	+	0.581188i	$0.197412\pi$
$488$	0	0
$489$	11.2339	0.508012
$490$	0	0
$491$	−34.8053	−1.57074	−0.785370	−	0.619027i	$-0.787527\pi$
$491$	−34.8053	−1.57074	−0.785370	+	0.619027i	$0.787527\pi$
$492$	0	0
$493$	−2.80548	−0.126353
$494$	0	0
$495$	−6.31660	−0.283910
$496$	0	0
$497$	−0.0226323	−0.00101520
$498$	0	0
$499$	13.9683	0.625306	0.312653	−	0.949867i	$-0.398782\pi$
$499$	13.9683	0.625306	0.312653	+	0.949867i	$0.398782\pi$
$500$	0	0
$501$	17.1706	0.767125
$502$	0	0
$503$	−4.06271	−0.181147	−0.0905736	−	0.995890i	$-0.528870\pi$
$503$	−4.06271	−0.181147	−0.0905736	+	0.995890i	$0.528870\pi$
$504$	0	0
$505$	13.6849	0.608971
$506$	0	0
$507$	−5.68985	−0.252695
$508$	0	0
$509$	−22.8458	−1.01262	−0.506310	−	0.862351i	$-0.668991\pi$
$509$	−22.8458	−1.01262	−0.506310	+	0.862351i	$0.668991\pi$
$510$	0	0
$511$	4.11216	0.181911
$512$	0	0
$513$	36.1366	1.59547
$514$	0	0
$515$	11.9551	0.526803
$516$	0	0
$517$	−24.2511	−1.06656
$518$	0	0
$519$	7.17267	0.314845
$520$	0	0
$521$	−15.2065	−0.666210	−0.333105	−	0.942890i	$-0.608096\pi$
$521$	−15.2065	−0.666210	−0.333105	+	0.942890i	$0.608096\pi$
$522$	0	0
$523$	17.0578	0.745885	0.372942	−	0.927855i	$-0.378349\pi$
$523$	17.0578	0.745885	0.372942	+	0.927855i	$0.378349\pi$
$524$	0	0
$525$	0.408991	0.0178498
$526$	0	0
$527$	42.2659	1.84113
$528$	0	0
$529$	−10.6130	−0.461436
$530$	0	0
$531$	−5.32517	−0.231093
$532$	0	0
$533$	0.341110	0.0147751
$534$	0	0
$535$	−15.8133	−0.683671
$536$	0	0
$537$	−12.9123	−0.557207
$538$	0	0
$539$	−25.6777	−1.10602
$540$	0	0
$541$	5.24030	0.225298	0.112649	−	0.993635i	$-0.464066\pi$
$541$	5.24030	0.225298	0.112649	+	0.993635i	$0.464066\pi$
$542$	0	0
$543$	−20.7978	−0.892518
$544$	0	0
$545$	7.93525	0.339909
$546$	0	0
$547$	41.5284	1.77562	0.887812	−	0.460206i	$-0.152224\pi$
$547$	41.5284	1.77562	0.887812	+	0.460206i	$0.152224\pi$
$548$	0	0
$549$	0.955995	0.0408009
$550$	0	0
$551$	3.14094	0.133808
$552$	0	0
$553$	−2.38828	−0.101560
$554$	0	0
$555$	10.2911	0.436833
$556$	0	0
$557$	42.1100	1.78426	0.892128	−	0.451783i	$-0.149212\pi$
$557$	42.1100	1.78426	0.892128	+	0.451783i	$0.149212\pi$
$558$	0	0
$559$	−22.7143	−0.960711
$560$	0	0
$561$	25.7115	1.08554
$562$	0	0
$563$	32.7862	1.38178	0.690888	−	0.722962i	$-0.257220\pi$
$563$	32.7862	1.38178	0.690888	+	0.722962i	$0.257220\pi$
$564$	0	0
$565$	−0.0548040	−0.00230562
$566$	0	0
$567$	−0.382578	−0.0160668
$568$	0	0
$569$	16.7394	0.701754	0.350877	−	0.936422i	$-0.385884\pi$
$569$	16.7394	0.701754	0.350877	+	0.936422i	$0.385884\pi$
$570$	0	0
$571$	−35.9566	−1.50474	−0.752369	−	0.658742i	$-0.771089\pi$
$571$	−35.9566	−1.50474	−0.752369	+	0.658742i	$0.771089\pi$
$572$	0	0
$573$	−2.28765	−0.0955679
$574$	0	0
$575$	3.51951	0.146774
$576$	0	0
$577$	−13.7877	−0.573989	−0.286995	−	0.957932i	$-0.592656\pi$
$577$	−13.7877	−0.573989	−0.286995	+	0.957932i	$0.592656\pi$
$578$	0	0
$579$	−21.6170	−0.898370
$580$	0	0
$581$	3.95526	0.164092
$582$	0	0
$583$	−9.17329	−0.379919
$584$	0	0
$585$	4.78987	0.198037
$586$	0	0
$587$	−30.1510	−1.24447	−0.622233	−	0.782832i	$-0.713774\pi$
$587$	−30.1510	−1.24447	−0.622233	+	0.782832i	$0.713774\pi$
$588$	0	0
$589$	−47.3197	−1.94977
$590$	0	0
$591$	−6.46771	−0.266046
$592$	0	0
$593$	−20.0139	−0.821873	−0.410937	−	0.911664i	$-0.634798\pi$
$593$	−20.0139	−0.821873	−0.410937	+	0.911664i	$0.634798\pi$
$594$	0	0
$595$	2.14929	0.0881123
$596$	0	0
$597$	14.5236	0.594410
$598$	0	0
$599$	−2.29374	−0.0937197	−0.0468598	−	0.998901i	$-0.514921\pi$
$599$	−2.29374	−0.0937197	−0.0468598	+	0.998901i	$0.514921\pi$
$600$	0	0
$601$	−22.7458	−0.927821	−0.463910	−	0.885882i	$-0.653554\pi$
$601$	−22.7458	−0.927821	−0.463910	+	0.885882i	$0.653554\pi$
$602$	0	0
$603$	−6.04275	−0.246080
$604$	0	0
$605$	2.96087	0.120376
$606$	0	0
$607$	7.22590	0.293290	0.146645	−	0.989189i	$-0.453152\pi$
$607$	7.22590	0.293290	0.146645	+	0.989189i	$0.453152\pi$
$608$	0	0
$609$	−0.190807	−0.00773189
$610$	0	0
$611$	18.3896	0.743962
$612$	0	0
$613$	−38.0983	−1.53878	−0.769389	−	0.638781i	$-0.779439\pi$
$613$	−38.0983	−1.53878	−0.769389	+	0.638781i	$0.779439\pi$
$614$	0	0
$615$	−0.137766	−0.00555528
$616$	0	0
$617$	2.02577	0.0815546	0.0407773	−	0.999168i	$-0.487017\pi$
$617$	2.02577	0.0815546	0.0407773	+	0.999168i	$0.487017\pi$
$618$	0	0
$619$	23.2764	0.935559	0.467780	−	0.883845i	$-0.345054\pi$
$619$	23.2764	0.935559	0.467780	+	0.883845i	$0.345054\pi$
$620$	0	0
$621$	−18.8908	−0.758063
$622$	0	0
$623$	1.73457	0.0694941
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−28.7859	−1.14960
$628$	0	0
$629$	54.0808	2.15634
$630$	0	0
$631$	−20.1944	−0.803926	−0.401963	−	0.915656i	$-0.631672\pi$
$631$	−20.1944	−0.803926	−0.401963	+	0.915656i	$0.631672\pi$
$632$	0	0
$633$	29.6313	1.17774
$634$	0	0
$635$	−10.5565	−0.418924
$636$	0	0
$637$	19.4713	0.771482
$638$	0	0
$639$	0.107050	0.00423484
$640$	0	0
$641$	42.6872	1.68604	0.843021	−	0.537881i	$-0.180775\pi$
$641$	42.6872	1.68604	0.843021	+	0.537881i	$0.180775\pi$
$642$	0	0
$643$	−42.5616	−1.67847	−0.839233	−	0.543773i	$-0.816995\pi$
$643$	−42.5616	−1.67847	−0.839233	+	0.543773i	$0.816995\pi$
$644$	0	0
$645$	9.17376	0.361217
$646$	0	0
$647$	32.2264	1.26695	0.633474	−	0.773764i	$-0.281628\pi$
$647$	32.2264	1.26695	0.633474	+	0.773764i	$0.281628\pi$
$648$	0	0
$649$	11.7696	0.461998
$650$	0	0
$651$	2.87460	0.112664
$652$	0	0
$653$	29.0511	1.13686	0.568429	−	0.822733i	$-0.307552\pi$
$653$	29.0511	1.13686	0.568429	+	0.822733i	$0.307552\pi$
$654$	0	0
$655$	6.94450	0.271344
$656$	0	0
$657$	−19.4504	−0.758832
$658$	0	0
$659$	23.7346	0.924568	0.462284	−	0.886732i	$-0.347030\pi$
$659$	23.7346	0.924568	0.462284	+	0.886732i	$0.347030\pi$
$660$	0	0
$661$	37.0445	1.44086	0.720431	−	0.693526i	$-0.243944\pi$
$661$	37.0445	1.44086	0.720431	+	0.693526i	$0.243944\pi$
$662$	0	0
$663$	−19.4970	−0.757201
$664$	0	0
$665$	−2.40628	−0.0933117
$666$	0	0
$667$	−1.64196	−0.0635771
$668$	0	0
$669$	14.7266	0.569365
$670$	0	0
$671$	−2.11293	−0.0815687
$672$	0	0
$673$	−27.5860	−1.06336	−0.531681	−	0.846945i	$-0.678439\pi$
$673$	−27.5860	−1.06336	−0.531681	+	0.846945i	$0.678439\pi$
$674$	0	0
$675$	−5.36746	−0.206593
$676$	0	0
$677$	−31.9389	−1.22751	−0.613756	−	0.789495i	$-0.710342\pi$
$677$	−31.9389	−1.22751	−0.613756	+	0.789495i	$0.710342\pi$
$678$	0	0
$679$	3.98444	0.152909
$680$	0	0
$681$	−17.4235	−0.667672
$682$	0	0
$683$	−9.42287	−0.360556	−0.180278	−	0.983616i	$-0.557700\pi$
$683$	−9.42287	−0.360556	−0.180278	+	0.983616i	$0.557700\pi$
$684$	0	0
$685$	−19.5262	−0.746056
$686$	0	0
$687$	7.44405	0.284008
$688$	0	0
$689$	6.95609	0.265006
$690$	0	0
$691$	32.5167	1.23699	0.618497	−	0.785787i	$-0.287742\pi$
$691$	32.5167	1.23699	0.618497	+	0.785787i	$0.287742\pi$
$692$	0	0
$693$	−2.25762	−0.0857601
$694$	0	0
$695$	6.59806	0.250279
$696$	0	0
$697$	−0.723977	−0.0274226
$698$	0	0
$699$	3.90557	0.147722
$700$	0	0
$701$	−8.12370	−0.306828	−0.153414	−	0.988162i	$-0.549027\pi$
$701$	−8.12370	−0.306828	−0.153414	+	0.988162i	$0.549027\pi$
$702$	0	0
$703$	−60.5474	−2.28359
$704$	0	0
$705$	−7.42712	−0.279722
$706$	0	0
$707$	4.89115	0.183951
$708$	0	0
$709$	−24.5182	−0.920801	−0.460400	−	0.887711i	$-0.652294\pi$
$709$	−24.5182	−0.920801	−0.460400	+	0.887711i	$0.652294\pi$
$710$	0	0
$711$	11.2965	0.423652
$712$	0	0
$713$	24.7370	0.926406
$714$	0	0
$715$	−10.5865	−0.395913
$716$	0	0
$717$	2.26179	0.0844682
$718$	0	0
$719$	−33.7583	−1.25897	−0.629487	−	0.777011i	$-0.716735\pi$
$719$	−33.7583	−1.25897	−0.629487	+	0.777011i	$0.716735\pi$
$720$	0	0
$721$	4.27287	0.159130
$722$	0	0
$723$	15.9712	0.593977
$724$	0	0
$725$	−0.466531	−0.0173265
$726$	0	0
$727$	−34.8495	−1.29250	−0.646249	−	0.763127i	$-0.723663\pi$
$727$	−34.8495	−1.29250	−0.646249	+	0.763127i	$0.723663\pi$
$728$	0	0
$729$	20.2357	0.749472
$730$	0	0
$731$	48.2091	1.78308
$732$	0	0
$733$	−37.0027	−1.36673	−0.683364	−	0.730078i	$-0.739484\pi$
$733$	−37.0027	−1.36673	−0.683364	+	0.730078i	$0.739484\pi$
$734$	0	0
$735$	−7.86402	−0.290069
$736$	0	0
$737$	13.3556	0.491960
$738$	0	0
$739$	−16.6855	−0.613786	−0.306893	−	0.951744i	$-0.599289\pi$
$739$	−16.6855	−0.613786	−0.306893	+	0.951744i	$0.599289\pi$
$740$	0	0
$741$	21.8283	0.801882
$742$	0	0
$743$	21.0351	0.771702	0.385851	−	0.922561i	$-0.373908\pi$
$743$	21.0351	0.771702	0.385851	+	0.922561i	$0.373908\pi$
$744$	0	0
$745$	−3.54496	−0.129877
$746$	0	0
$747$	−18.7083	−0.684501
$748$	0	0
$749$	−5.65187	−0.206515
$750$	0	0
$751$	2.38269	0.0869456	0.0434728	−	0.999055i	$-0.486158\pi$
$751$	2.38269	0.0869456	0.0434728	+	0.999055i	$0.486158\pi$
$752$	0	0
$753$	6.31436	0.230108
$754$	0	0
$755$	−1.00000	−0.0363937
$756$	0	0
$757$	−7.13496	−0.259325	−0.129662	−	0.991558i	$-0.541389\pi$
$757$	−7.13496	−0.259325	−0.129662	+	0.991558i	$0.541389\pi$
$758$	0	0
$759$	15.0482	0.546214
$760$	0	0
$761$	−41.3251	−1.49803	−0.749017	−	0.662551i	$-0.769474\pi$
$761$	−41.3251	−1.49803	−0.749017	+	0.662551i	$0.769474\pi$
$762$	0	0
$763$	2.83615	0.102675
$764$	0	0
$765$	−10.1661	−0.367556
$766$	0	0
$767$	−8.92489	−0.322259
$768$	0	0
$769$	−7.98034	−0.287778	−0.143889	−	0.989594i	$-0.545961\pi$
$769$	−7.98034	−0.287778	−0.143889	+	0.989594i	$0.545961\pi$
$770$	0	0
$771$	−11.9391	−0.429976
$772$	0	0
$773$	28.4742	1.02415	0.512074	−	0.858942i	$-0.328877\pi$
$773$	28.4742	1.02415	0.512074	+	0.858942i	$0.328877\pi$
$774$	0	0
$775$	7.02851	0.252472
$776$	0	0
$777$	3.67815	0.131953
$778$	0	0
$779$	0.810544	0.0290408
$780$	0	0
$781$	−0.236601	−0.00846625
$782$	0	0
$783$	2.50409	0.0894887
$784$	0	0
$785$	20.1476	0.719100
$786$	0	0
$787$	−2.02507	−0.0721859	−0.0360929	−	0.999348i	$-0.511491\pi$
$787$	−2.02507	−0.0721859	−0.0360929	+	0.999348i	$0.511491\pi$
$788$	0	0
$789$	20.3858	0.725755
$790$	0	0
$791$	−0.0195876	−0.000696453 0
$792$	0	0
$793$	1.60223	0.0568968
$794$	0	0
$795$	−2.80940	−0.0996392
$796$	0	0
$797$	−13.8061	−0.489036	−0.244518	−	0.969645i	$-0.578630\pi$
$797$	−13.8061	−0.489036	−0.244518	+	0.969645i	$0.578630\pi$
$798$	0	0
$799$	−39.0303	−1.38079
$800$	0	0
$801$	−8.20448	−0.289891
$802$	0	0
$803$	42.9890	1.51705
$804$	0	0
$805$	1.25791	0.0443356
$806$	0	0
$807$	14.2405	0.501288
$808$	0	0
$809$	33.6900	1.18448	0.592238	−	0.805763i	$-0.298244\pi$
$809$	33.6900	1.18448	0.592238	+	0.805763i	$0.298244\pi$
$810$	0	0
$811$	37.1624	1.30495	0.652475	−	0.757810i	$-0.273731\pi$
$811$	37.1624	1.30495	0.652475	+	0.757810i	$0.273731\pi$
$812$	0	0
$813$	−20.6097	−0.722815
$814$	0	0
$815$	9.81711	0.343878
$816$	0	0
$817$	−53.9736	−1.88830
$818$	0	0
$819$	1.71195	0.0598204
$820$	0	0
$821$	13.9805	0.487922	0.243961	−	0.969785i	$-0.421553\pi$
$821$	13.9805	0.487922	0.243961	+	0.969785i	$0.421553\pi$
$822$	0	0
$823$	−8.82111	−0.307485	−0.153742	−	0.988111i	$-0.549133\pi$
$823$	−8.82111	−0.307485	−0.153742	+	0.988111i	$0.549133\pi$
$824$	0	0
$825$	4.27564	0.148859
$826$	0	0
$827$	23.9490	0.832790	0.416395	−	0.909184i	$-0.363293\pi$
$827$	23.9490	0.832790	0.416395	+	0.909184i	$0.363293\pi$
$828$	0	0
$829$	−5.35186	−0.185878	−0.0929389	−	0.995672i	$-0.529626\pi$
$829$	−5.35186	−0.185878	−0.0929389	+	0.995672i	$0.529626\pi$
$830$	0	0
$831$	20.5386	0.712478
$832$	0	0
$833$	−41.3263	−1.43187
$834$	0	0
$835$	15.0051	0.519274
$836$	0	0
$837$	−37.7252	−1.30398
$838$	0	0
$839$	2.68043	0.0925386	0.0462693	−	0.998929i	$-0.485267\pi$
$839$	2.68043	0.0925386	0.0462693	+	0.998929i	$0.485267\pi$
$840$	0	0
$841$	−28.7823	−0.992495
$842$	0	0
$843$	8.21948	0.283094
$844$	0	0
$845$	−4.97228	−0.171052
$846$	0	0
$847$	1.05825	0.0363618
$848$	0	0
$849$	−2.94609	−0.101109
$850$	0	0
$851$	31.6519	1.08501
$852$	0	0
$853$	35.3484	1.21031	0.605153	−	0.796109i	$-0.293112\pi$
$853$	35.3484	1.21031	0.605153	+	0.796109i	$0.293112\pi$
$854$	0	0
$855$	11.3817	0.389245
$856$	0	0
$857$	18.5386	0.633268	0.316634	−	0.948548i	$-0.397447\pi$
$857$	18.5386	0.633268	0.316634	+	0.948548i	$0.397447\pi$
$858$	0	0
$859$	−4.39419	−0.149928	−0.0749640	−	0.997186i	$-0.523884\pi$
$859$	−4.39419	−0.149928	−0.0749640	+	0.997186i	$0.523884\pi$
$860$	0	0
$861$	−0.0492393	−0.00167807
$862$	0	0
$863$	3.32972	0.113345	0.0566725	−	0.998393i	$-0.481951\pi$
$863$	3.32972	0.113345	0.0566725	+	0.998393i	$0.481951\pi$
$864$	0	0
$865$	6.26810	0.213122
$866$	0	0
$867$	21.9274	0.744695
$868$	0	0
$869$	−24.9674	−0.846961
$870$	0	0
$871$	−10.1275	−0.343158
$872$	0	0
$873$	−18.8463	−0.637850
$874$	0	0
$875$	0.357411	0.0120827
$876$	0	0
$877$	−6.30793	−0.213004	−0.106502	−	0.994313i	$-0.533965\pi$
$877$	−6.30793	−0.213004	−0.106502	+	0.994313i	$0.533965\pi$
$878$	0	0
$879$	−21.6127	−0.728977
$880$	0	0
$881$	−41.1150	−1.38520	−0.692600	−	0.721322i	$-0.743535\pi$
$881$	−41.1150	−1.38520	−0.692600	+	0.721322i	$0.743535\pi$
$882$	0	0
$883$	10.9417	0.368219	0.184110	−	0.982906i	$-0.441060\pi$
$883$	10.9417	0.368219	0.184110	+	0.982906i	$0.441060\pi$
$884$	0	0
$885$	3.60456	0.121166
$886$	0	0
$887$	55.7722	1.87265	0.936323	−	0.351139i	$-0.114206\pi$
$887$	55.7722	1.87265	0.936323	+	0.351139i	$0.114206\pi$
$888$	0	0
$889$	−3.77303	−0.126543
$890$	0	0
$891$	−3.99952	−0.133989
$892$	0	0
$893$	43.6972	1.46227
$894$	0	0
$895$	−11.2839	−0.377179
$896$	0	0
$897$	−11.4110	−0.381002
$898$	0	0
$899$	−3.27902	−0.109361
$900$	0	0
$901$	−14.7637	−0.491851
$902$	0	0
$903$	3.27881	0.109112
$904$	0	0
$905$	−18.1749	−0.604154
$906$	0	0
$907$	−39.1885	−1.30123	−0.650616	−	0.759407i	$-0.725489\pi$
$907$	−39.1885	−1.30123	−0.650616	+	0.759407i	$0.725489\pi$
$908$	0	0
$909$	−23.1350	−0.767340
$910$	0	0
$911$	4.88529	0.161857	0.0809285	−	0.996720i	$-0.474211\pi$
$911$	4.88529	0.161857	0.0809285	+	0.996720i	$0.474211\pi$
$912$	0	0
$913$	41.3489	1.36845
$914$	0	0
$915$	−0.647103	−0.0213926
$916$	0	0
$917$	2.48204	0.0819643
$918$	0	0
$919$	46.1408	1.52204	0.761022	−	0.648726i	$-0.224698\pi$
$919$	46.1408	1.52204	0.761022	+	0.648726i	$0.224698\pi$
$920$	0	0
$921$	−6.77095	−0.223110
$922$	0	0
$923$	0.179414	0.00590549
$924$	0	0
$925$	8.99325	0.295696
$926$	0	0
$927$	−20.2106	−0.663803
$928$	0	0
$929$	−50.2824	−1.64971	−0.824856	−	0.565342i	$-0.808744\pi$
$929$	−50.2824	−1.64971	−0.824856	+	0.565342i	$0.808744\pi$
$930$	0	0
$931$	46.2677	1.51636
$932$	0	0
$933$	15.1711	0.496680
$934$	0	0
$935$	22.4690	0.734814
$936$	0	0
$937$	−31.5532	−1.03080	−0.515399	−	0.856951i	$-0.672356\pi$
$937$	−31.5532	−1.03080	−0.515399	+	0.856951i	$0.672356\pi$
$938$	0	0
$939$	−11.6836	−0.381279
$940$	0	0
$941$	6.54466	0.213350	0.106675	−	0.994294i	$-0.465980\pi$
$941$	6.54466	0.213350	0.106675	+	0.994294i	$0.465980\pi$
$942$	0	0
$943$	−0.423722	−0.0137983
$944$	0	0
$945$	−1.91839	−0.0624052
$946$	0	0
$947$	−22.1236	−0.718921	−0.359460	−	0.933160i	$-0.617039\pi$
$947$	−22.1236	−0.718921	−0.359460	+	0.933160i	$0.617039\pi$
$948$	0	0
$949$	−32.5985	−1.05819
$950$	0	0
$951$	3.17572	0.102980
$952$	0	0
$953$	−46.9000	−1.51924	−0.759619	−	0.650368i	$-0.774615\pi$
$953$	−46.9000	−1.51924	−0.759619	+	0.650368i	$0.774615\pi$
$954$	0	0
$955$	−1.99915	−0.0646908
$956$	0	0
$957$	−1.99472	−0.0644802
$958$	0	0
$959$	−6.97887	−0.225360
$960$	0	0
$961$	18.4000	0.593549
$962$	0	0
$963$	26.7332	0.861465
$964$	0	0
$965$	−18.8908	−0.608115
$966$	0	0
$967$	−1.73034	−0.0556441	−0.0278221	−	0.999613i	$-0.508857\pi$
$967$	−1.73034	−0.0556441	−0.0278221	+	0.999613i	$0.508857\pi$
$968$	0	0
$969$	−46.3287	−1.48829
$970$	0	0
$971$	−2.05863	−0.0660645	−0.0330322	−	0.999454i	$-0.510516\pi$
$971$	−2.05863	−0.0660645	−0.0330322	+	0.999454i	$0.510516\pi$
$972$	0	0
$973$	2.35822	0.0756011
$974$	0	0
$975$	−3.24221	−0.103834
$976$	0	0
$977$	−22.8000	−0.729436	−0.364718	−	0.931118i	$-0.618835\pi$
$977$	−22.8000	−0.729436	−0.364718	+	0.931118i	$0.618835\pi$
$978$	0	0
$979$	18.1334	0.579547
$980$	0	0
$981$	−13.4149	−0.428305
$982$	0	0
$983$	37.4814	1.19547	0.597736	−	0.801693i	$-0.296067\pi$
$983$	37.4814	1.19547	0.597736	+	0.801693i	$0.296067\pi$
$984$	0	0
$985$	−5.65204	−0.180089
$986$	0	0
$987$	−2.65454	−0.0844948
$988$	0	0
$989$	28.2153	0.897195
$990$	0	0
$991$	25.9993	0.825894	0.412947	−	0.910755i	$-0.364500\pi$
$991$	25.9993	0.825894	0.412947	+	0.910755i	$0.364500\pi$
$992$	0	0
$993$	−25.8631	−0.820739
$994$	0	0
$995$	12.6919	0.402361
$996$	0	0
$997$	−28.0016	−0.886820	−0.443410	−	0.896319i	$-0.646231\pi$
$997$	−28.0016	−0.886820	−0.443410	+	0.896319i	$0.646231\pi$
$998$	0	0
$999$	−48.2709	−1.52722

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6040.2.a.o.1.9	✓	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6040.2.a.o.1.9	✓	15	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 6040 = 2^{3} \cdot 5 \cdot 151 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6040.a (trivial)

\(f(q)\)	\(=\)	\(q+1.14431 q^{3} +1.00000 q^{5} +0.357411 q^{7} -1.69055 q^{9} +O(q^{10})\)	\(q+1.14431 q^{3} +1.00000 q^{5} +0.357411 q^{7} -1.69055 q^{9} +3.73642 q^{11} -2.83332 q^{13} +1.14431 q^{15} +6.01349 q^{17} -6.73253 q^{19} +0.408991 q^{21} +3.51951 q^{23} +1.00000 q^{25} -5.36746 q^{27} -0.466531 q^{29} +7.02851 q^{31} +4.27564 q^{33} +0.357411 q^{35} +8.99325 q^{37} -3.24221 q^{39} -0.120392 q^{41} +8.01683 q^{43} -1.69055 q^{45} -6.49046 q^{47} -6.87226 q^{49} +6.88132 q^{51} -2.45510 q^{53} +3.73642 q^{55} -7.70413 q^{57} +3.14997 q^{59} -0.565494 q^{61} -0.604220 q^{63} -2.83332 q^{65} +3.57444 q^{67} +4.02743 q^{69} -0.0633229 q^{71} +11.5054 q^{73} +1.14431 q^{75} +1.33544 q^{77} -6.68216 q^{79} -1.07041 q^{81} +11.0664 q^{83} +6.01349 q^{85} -0.533858 q^{87} +4.85315 q^{89} -1.01266 q^{91} +8.04282 q^{93} -6.73253 q^{95} +11.1480 q^{97} -6.31660 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + 5 q^{3} + 15 q^{5} + 7 q^{7} + 18 q^{9}+O(q^{10}) \) 15 * q + 5 * q^3 + 15 * q^5 + 7 * q^7 + 18 * q^9	\( 15 q + 5 q^{3} + 15 q^{5} + 7 q^{7} + 18 q^{9} + 7 q^{11} + 2 q^{13} + 5 q^{15} - 3 q^{17} + 8 q^{19} + 7 q^{21} + 15 q^{23} + 15 q^{25} + 23 q^{27} + 5 q^{29} + 27 q^{31} - 5 q^{33} + 7 q^{35} - 4 q^{37} + 11 q^{39} + 20 q^{41} + 25 q^{43} + 18 q^{45} + 35 q^{47} - 14 q^{49} + 25 q^{51} - 2 q^{53} + 7 q^{55} - 24 q^{57} + 39 q^{59} + 23 q^{61} + 39 q^{63} + 2 q^{65} + 32 q^{67} + 13 q^{69} + 30 q^{71} + 7 q^{73} + 5 q^{75} - 4 q^{77} + 38 q^{79} + 11 q^{81} + 29 q^{83} - 3 q^{85} + 4 q^{87} + 19 q^{89} + 16 q^{91} + 8 q^{93} + 8 q^{95} - 8 q^{97} + 16 q^{99}+O(q^{100}) \) 15 * q + 5 * q^3 + 15 * q^5 + 7 * q^7 + 18 * q^9 + 7 * q^11 + 2 * q^13 + 5 * q^15 - 3 * q^17 + 8 * q^19 + 7 * q^21 + 15 * q^23 + 15 * q^25 + 23 * q^27 + 5 * q^29 + 27 * q^31 - 5 * q^33 + 7 * q^35 - 4 * q^37 + 11 * q^39 + 20 * q^41 + 25 * q^43 + 18 * q^45 + 35 * q^47 - 14 * q^49 + 25 * q^51 - 2 * q^53 + 7 * q^55 - 24 * q^57 + 39 * q^59 + 23 * q^61 + 39 * q^63 + 2 * q^65 + 32 * q^67 + 13 * q^69 + 30 * q^71 + 7 * q^73 + 5 * q^75 - 4 * q^77 + 38 * q^79 + 11 * q^81 + 29 * q^83 - 3 * q^85 + 4 * q^87 + 19 * q^89 + 16 * q^91 + 8 * q^93 + 8 * q^95 - 8 * q^97 + 16 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more