LMFDB - Embedded newform 3015.2.a.l.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3015,2,Mod(1,3015)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3015, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3015.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3015 = 3^{2} \cdot 5 \cdot 67 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3015.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:	$24.0748962094$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 3x^{6} - 6x^{5} + 15x^{4} + 14x^{3} - 15x^{2} - 6x + 4 $ x^7 - 3x^6 - 6x^5 + 15x^4 + 14x^3 - 15x^2 - 6x + 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1005)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.67802$ of defining polynomial
Character	$\chi$	$=$	3015.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.67802 q^{2} +5.17182 q^{4} -1.00000 q^{5} -2.93755 q^{7} -8.49420 q^{8} +O(q^{10})$	\(q-2.67802 q^{2} +5.17182 q^{4} -1.00000 q^{5} -2.93755 q^{7} -8.49420 q^{8} +2.67802 q^{10} +1.71548 q^{11} -3.32239 q^{13} +7.86684 q^{14} +12.4041 q^{16} -4.08812 q^{17} +3.94442 q^{19} -5.17182 q^{20} -4.59411 q^{22} +1.28493 q^{23} +1.00000 q^{25} +8.89743 q^{26} -15.1925 q^{28} +6.22248 q^{29} -1.65693 q^{31} -16.2300 q^{32} +10.9481 q^{34} +2.93755 q^{35} +0.816404 q^{37} -10.5633 q^{38} +8.49420 q^{40} +4.01555 q^{41} +4.24766 q^{43} +8.87216 q^{44} -3.44107 q^{46} +7.68857 q^{47} +1.62923 q^{49} -2.67802 q^{50} -17.1828 q^{52} +12.5579 q^{53} -1.71548 q^{55} +24.9522 q^{56} -16.6640 q^{58} -2.41729 q^{59} -13.8271 q^{61} +4.43730 q^{62} +18.6561 q^{64} +3.32239 q^{65} +1.00000 q^{67} -21.1430 q^{68} -7.86684 q^{70} -7.62533 q^{71} +15.6054 q^{73} -2.18635 q^{74} +20.3998 q^{76} -5.03933 q^{77} -13.5024 q^{79} -12.4041 q^{80} -10.7537 q^{82} -9.15183 q^{83} +4.08812 q^{85} -11.3753 q^{86} -14.5717 q^{88} +4.23914 q^{89} +9.75969 q^{91} +6.64541 q^{92} -20.5902 q^{94} -3.94442 q^{95} -7.19034 q^{97} -4.36311 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 4 q^{2} + 8 q^{4} - 7 q^{5} + 3 q^{7} - 9 q^{8}+O(q^{10}) $ 7 * q - 4 * q^2 + 8 * q^4 - 7 * q^5 + 3 * q^7 - 9 * q^8	$ 7 q - 4 q^{2} + 8 q^{4} - 7 q^{5} + 3 q^{7} - 9 q^{8} + 4 q^{10} - 5 q^{11} - q^{13} + 4 q^{14} + 6 q^{16} - 11 q^{17} + 8 q^{19} - 8 q^{20} - 3 q^{22} - 11 q^{23} + 7 q^{25} + 5 q^{26} - 17 q^{28} - 3 q^{31} - 22 q^{32} + 4 q^{34} - 3 q^{35} - 5 q^{37} + 9 q^{40} - q^{41} - 3 q^{43} + 9 q^{44} - 12 q^{46} - 10 q^{47} - 4 q^{49} - 4 q^{50} - 6 q^{52} - 3 q^{53} + 5 q^{55} + 12 q^{56} - 24 q^{58} + 4 q^{59} - 9 q^{61} - 20 q^{62} - 3 q^{64} + q^{65} + 7 q^{67} + 3 q^{68} - 4 q^{70} - 6 q^{71} - 9 q^{73} - 2 q^{74} + 2 q^{76} - 17 q^{77} - 11 q^{79} - 6 q^{80} - 16 q^{82} - 30 q^{83} + 11 q^{85} + 11 q^{86} - 25 q^{88} - 13 q^{89} - 5 q^{91} - 10 q^{92} - 25 q^{94} - 8 q^{95} - 7 q^{97} + 10 q^{98}+O(q^{100}) $ 7 * q - 4 * q^2 + 8 * q^4 - 7 * q^5 + 3 * q^7 - 9 * q^8 + 4 * q^10 - 5 * q^11 - q^13 + 4 * q^14 + 6 * q^16 - 11 * q^17 + 8 * q^19 - 8 * q^20 - 3 * q^22 - 11 * q^23 + 7 * q^25 + 5 * q^26 - 17 * q^28 - 3 * q^31 - 22 * q^32 + 4 * q^34 - 3 * q^35 - 5 * q^37 + 9 * q^40 - q^41 - 3 * q^43 + 9 * q^44 - 12 * q^46 - 10 * q^47 - 4 * q^49 - 4 * q^50 - 6 * q^52 - 3 * q^53 + 5 * q^55 + 12 * q^56 - 24 * q^58 + 4 * q^59 - 9 * q^61 - 20 * q^62 - 3 * q^64 + q^65 + 7 * q^67 + 3 * q^68 - 4 * q^70 - 6 * q^71 - 9 * q^73 - 2 * q^74 + 2 * q^76 - 17 * q^77 - 11 * q^79 - 6 * q^80 - 16 * q^82 - 30 * q^83 + 11 * q^85 + 11 * q^86 - 25 * q^88 - 13 * q^89 - 5 * q^91 - 10 * q^92 - 25 * q^94 - 8 * q^95 - 7 * q^97 + 10 * q^98

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 $

$2$	−2.67802	−1.89365	−0.946825	−	0.321750i	$-0.895729\pi$
$2$	−2.67802	−1.89365	−0.946825	+	0.321750i	$0.895729\pi$
$3$	0	0
$3$	0	0
$4$	5.17182	2.58591
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−2.93755	−1.11029	−0.555146	−	0.831753i	$-0.687337\pi$
$7$	−2.93755	−1.11029	−0.555146	+	0.831753i	$0.687337\pi$
$8$	−8.49420	−3.00315
$9$	0	0
$10$	2.67802	0.846866
$11$	1.71548	0.517238	0.258619	−	0.965979i	$-0.416733\pi$
$11$	1.71548	0.517238	0.258619	+	0.965979i	$0.416733\pi$
$12$	0	0
$13$	−3.32239	−0.921464	−0.460732	−	0.887539i	$-0.652413\pi$
$13$	−3.32239	−0.921464	−0.460732	+	0.887539i	$0.652413\pi$
$14$	7.86684	2.10250
$15$	0	0
$16$	12.4041	3.10101
$17$	−4.08812	−0.991516	−0.495758	−	0.868461i	$-0.665110\pi$
$17$	−4.08812	−0.991516	−0.495758	+	0.868461i	$0.665110\pi$
$18$	0	0
$19$	3.94442	0.904913	0.452457	−	0.891786i	$-0.350548\pi$
$19$	3.94442	0.904913	0.452457	+	0.891786i	$0.350548\pi$
$20$	−5.17182	−1.15645
$21$	0	0
$22$	−4.59411	−0.979467
$23$	1.28493	0.267926	0.133963	−	0.990986i	$-0.457230\pi$
$23$	1.28493	0.267926	0.133963	+	0.990986i	$0.457230\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	8.89743	1.74493
$27$	0	0
$28$	−15.1925	−2.87111
$29$	6.22248	1.15549	0.577743	−	0.816219i	$-0.303934\pi$
$29$	6.22248	1.15549	0.577743	+	0.816219i	$0.303934\pi$
$30$	0	0
$31$	−1.65693	−0.297594	−0.148797	−	0.988868i	$-0.547540\pi$
$31$	−1.65693	−0.297594	−0.148797	+	0.988868i	$0.547540\pi$
$32$	−16.2300	−2.86908
$33$	0	0
$34$	10.9481	1.87758
$35$	2.93755	0.496537
$36$	0	0
$37$	0.816404	0.134216	0.0671080	−	0.997746i	$-0.478623\pi$
$37$	0.816404	0.134216	0.0671080	+	0.997746i	$0.478623\pi$
$38$	−10.5633	−1.71359
$39$	0	0
$40$	8.49420	1.34305
$41$	4.01555	0.627123	0.313561	−	0.949568i	$-0.398478\pi$
$41$	4.01555	0.627123	0.313561	+	0.949568i	$0.398478\pi$
$42$	0	0
$43$	4.24766	0.647762	0.323881	−	0.946098i	$-0.395012\pi$
$43$	4.24766	0.647762	0.323881	+	0.946098i	$0.395012\pi$
$44$	8.87216	1.33753
$45$	0	0
$46$	−3.44107	−0.507358
$47$	7.68857	1.12149	0.560746	−	0.827988i	$-0.310514\pi$
$47$	7.68857	1.12149	0.560746	+	0.827988i	$0.310514\pi$
$48$	0	0
$49$	1.62923	0.232747
$50$	−2.67802	−0.378730
$51$	0	0
$52$	−17.1828	−2.38282
$53$	12.5579	1.72496	0.862479	−	0.506093i	$-0.168911\pi$
$53$	12.5579	1.72496	0.862479	+	0.506093i	$0.168911\pi$
$54$	0	0
$55$	−1.71548	−0.231316
$56$	24.9522	3.33438
$57$	0	0
$58$	−16.6640	−2.18809
$59$	−2.41729	−0.314704	−0.157352	−	0.987543i	$-0.550296\pi$
$59$	−2.41729	−0.314704	−0.157352	+	0.987543i	$0.550296\pi$
$60$	0	0
$61$	−13.8271	−1.77038	−0.885189	−	0.465231i	$-0.845971\pi$
$61$	−13.8271	−1.77038	−0.885189	+	0.465231i	$0.845971\pi$
$62$	4.43730	0.563538
$63$	0	0
$64$	18.6561	2.33201
$65$	3.32239	0.412091
$66$	0	0
$67$	1.00000	0.122169
$67$	1.00000	0.122169
$68$	−21.1430	−2.56397
$69$	0	0
$70$	−7.86684	−0.940268
$71$	−7.62533	−0.904960	−0.452480	−	0.891774i	$-0.649461\pi$
$71$	−7.62533	−0.904960	−0.452480	+	0.891774i	$0.649461\pi$
$72$	0	0
$73$	15.6054	1.82648	0.913239	−	0.407424i	$-0.133573\pi$
$73$	15.6054	1.82648	0.913239	+	0.407424i	$0.133573\pi$
$74$	−2.18635	−0.254158
$75$	0	0
$76$	20.3998	2.34002
$77$	−5.03933	−0.574284
$78$	0	0
$79$	−13.5024	−1.51914	−0.759570	−	0.650426i	$-0.774590\pi$
$79$	−13.5024	−1.51914	−0.759570	+	0.650426i	$0.774590\pi$
$80$	−12.4041	−1.38682
$81$	0	0
$82$	−10.7537	−1.18755
$83$	−9.15183	−1.00454	−0.502272	−	0.864710i	$-0.667502\pi$
$83$	−9.15183	−1.00454	−0.502272	+	0.864710i	$0.667502\pi$
$84$	0	0
$85$	4.08812	0.443419
$86$	−11.3753	−1.22663
$87$	0	0
$88$	−14.5717	−1.55334
$89$	4.23914	0.449348	0.224674	−	0.974434i	$-0.427868\pi$
$89$	4.23914	0.449348	0.224674	+	0.974434i	$0.427868\pi$
$90$	0	0
$91$	9.75969	1.02309
$92$	6.64541	0.692832
$93$	0	0
$94$	−20.5902	−2.12371
$95$	−3.94442	−0.404689
$96$	0	0
$97$	−7.19034	−0.730069	−0.365034	−	0.930994i	$-0.618943\pi$
$97$	−7.19034	−0.730069	−0.365034	+	0.930994i	$0.618943\pi$
$98$	−4.36311	−0.440740
$99$	0	0
$100$	5.17182	0.517182
$101$	−6.22093	−0.619006	−0.309503	−	0.950898i	$-0.600163\pi$
$101$	−6.22093	−0.619006	−0.309503	+	0.950898i	$0.600163\pi$
$102$	0	0
$103$	12.3039	1.21234	0.606169	−	0.795336i	$-0.292705\pi$
$103$	12.3039	1.21234	0.606169	+	0.795336i	$0.292705\pi$
$104$	28.2210	2.76730
$105$	0	0
$106$	−33.6303	−3.26647
$107$	−18.6337	−1.80139	−0.900694	−	0.434453i	$-0.856942\pi$
$107$	−18.6337	−1.80139	−0.900694	+	0.434453i	$0.856942\pi$
$108$	0	0
$109$	−3.12201	−0.299035	−0.149517	−	0.988759i	$-0.547772\pi$
$109$	−3.12201	−0.299035	−0.149517	+	0.988759i	$0.547772\pi$
$110$	4.59411	0.438031
$111$	0	0
$112$	−36.4376	−3.44303
$113$	−14.9635	−1.40765	−0.703825	−	0.710374i	$-0.748526\pi$
$113$	−14.9635	−1.40765	−0.703825	+	0.710374i	$0.748526\pi$
$114$	0	0
$115$	−1.28493	−0.119820
$116$	32.1815	2.98798
$117$	0	0
$118$	6.47355	0.595939
$119$	12.0091	1.10087
$120$	0	0
$121$	−8.05712	−0.732465
$122$	37.0293	3.35248
$123$	0	0
$124$	−8.56935	−0.769550
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	8.49151	0.753499	0.376750	−	0.926315i	$-0.377042\pi$
$127$	8.49151	0.753499	0.376750	+	0.926315i	$0.377042\pi$
$128$	−17.5016	−1.54694
$129$	0	0
$130$	−8.89743	−0.780356
$131$	−17.3874	−1.51914	−0.759572	−	0.650424i	$-0.774591\pi$
$131$	−17.3874	−1.51914	−0.759572	+	0.650424i	$0.774591\pi$
$132$	0	0
$133$	−11.5870	−1.00472
$134$	−2.67802	−0.231346
$135$	0	0
$136$	34.7254	2.97767
$137$	10.4138	0.889711	0.444856	−	0.895602i	$-0.353255\pi$
$137$	10.4138	0.889711	0.444856	+	0.895602i	$0.353255\pi$
$138$	0	0
$139$	−1.76685	−0.149862	−0.0749310	−	0.997189i	$-0.523874\pi$
$139$	−1.76685	−0.149862	−0.0749310	+	0.997189i	$0.523874\pi$
$140$	15.1925	1.28400
$141$	0	0
$142$	20.4208	1.71368
$143$	−5.69950	−0.476616
$144$	0	0
$145$	−6.22248	−0.516749
$146$	−41.7918	−3.45871
$147$	0	0
$148$	4.22229	0.347070
$149$	8.58033	0.702928	0.351464	−	0.936201i	$-0.385684\pi$
$149$	8.58033	0.702928	0.351464	+	0.936201i	$0.385684\pi$
$150$	0	0
$151$	16.7257	1.36112	0.680561	−	0.732692i	$-0.261736\pi$
$151$	16.7257	1.36112	0.680561	+	0.732692i	$0.261736\pi$
$152$	−33.5047	−2.71759
$153$	0	0
$154$	13.4954	1.08749
$155$	1.65693	0.133088
$156$	0	0
$157$	23.5429	1.87893	0.939465	−	0.342644i	$-0.111323\pi$
$157$	23.5429	1.87893	0.939465	+	0.342644i	$0.111323\pi$
$158$	36.1598	2.87672
$159$	0	0
$160$	16.2300	1.28309
$161$	−3.77454	−0.297476
$162$	0	0
$163$	−2.44795	−0.191738	−0.0958691	−	0.995394i	$-0.530563\pi$
$163$	−2.44795	−0.191738	−0.0958691	+	0.995394i	$0.530563\pi$
$164$	20.7677	1.62168
$165$	0	0
$166$	24.5088	1.90225
$167$	3.27391	0.253342	0.126671	−	0.991945i	$-0.459571\pi$
$167$	3.27391	0.253342	0.126671	+	0.991945i	$0.459571\pi$
$168$	0	0
$169$	−1.96175	−0.150904
$170$	−10.9481	−0.839681
$171$	0	0
$172$	21.9681	1.67505
$173$	−8.10304	−0.616063	−0.308032	−	0.951376i	$-0.599670\pi$
$173$	−8.10304	−0.616063	−0.308032	+	0.951376i	$0.599670\pi$
$174$	0	0
$175$	−2.93755	−0.222058
$176$	21.2789	1.60396
$177$	0	0
$178$	−11.3525	−0.850908
$179$	6.23102	0.465729	0.232864	−	0.972509i	$-0.425190\pi$
$179$	6.23102	0.465729	0.232864	+	0.972509i	$0.425190\pi$
$180$	0	0
$181$	17.9465	1.33396	0.666978	−	0.745077i	$-0.267587\pi$
$181$	17.9465	1.33396	0.666978	+	0.745077i	$0.267587\pi$
$182$	−26.1367	−1.93738
$183$	0	0
$184$	−10.9144	−0.804623
$185$	−0.816404	−0.0600232
$186$	0	0
$187$	−7.01311	−0.512849
$188$	39.7639	2.90008
$189$	0	0
$190$	10.5633	0.766340
$191$	−25.9356	−1.87664	−0.938318	−	0.345775i	$-0.887616\pi$
$191$	−25.9356	−1.87664	−0.938318	+	0.345775i	$0.887616\pi$
$192$	0	0
$193$	−21.1578	−1.52297	−0.761485	−	0.648182i	$-0.775530\pi$
$193$	−21.1578	−1.52297	−0.761485	+	0.648182i	$0.775530\pi$
$194$	19.2559	1.38249
$195$	0	0
$196$	8.42606	0.601861
$197$	16.2849	1.16025	0.580125	−	0.814527i	$-0.303004\pi$
$197$	16.2849	1.16025	0.580125	+	0.814527i	$0.303004\pi$
$198$	0	0
$199$	−10.8832	−0.771489	−0.385744	−	0.922606i	$-0.626055\pi$
$199$	−10.8832	−0.771489	−0.385744	+	0.922606i	$0.626055\pi$
$200$	−8.49420	−0.600631
$201$	0	0
$202$	16.6598	1.17218
$203$	−18.2789	−1.28293
$204$	0	0
$205$	−4.01555	−0.280458
$206$	−32.9501	−2.29574
$207$	0	0
$208$	−41.2110	−2.85747
$209$	6.76659	0.468055
$210$	0	0
$211$	−14.1129	−0.971571	−0.485785	−	0.874078i	$-0.661466\pi$
$211$	−14.1129	−0.971571	−0.485785	+	0.874078i	$0.661466\pi$
$212$	64.9471	4.46058
$213$	0	0
$214$	49.9015	3.41120
$215$	−4.24766	−0.289688
$216$	0	0
$217$	4.86733	0.330416
$218$	8.36083	0.566267
$219$	0	0
$220$	−8.87216	−0.598161
$221$	13.5823	0.913646
$222$	0	0
$223$	−8.30108	−0.555881	−0.277941	−	0.960598i	$-0.589652\pi$
$223$	−8.30108	−0.555881	−0.277941	+	0.960598i	$0.589652\pi$
$224$	47.6764	3.18551
$225$	0	0
$226$	40.0727	2.66559
$227$	−4.80453	−0.318888	−0.159444	−	0.987207i	$-0.550970\pi$
$227$	−4.80453	−0.318888	−0.159444	+	0.987207i	$0.550970\pi$
$228$	0	0
$229$	15.0700	0.995852	0.497926	−	0.867220i	$-0.334095\pi$
$229$	15.0700	0.995852	0.497926	+	0.867220i	$0.334095\pi$
$230$	3.44107	0.226897
$231$	0	0
$232$	−52.8550	−3.47010
$233$	−22.5918	−1.48004	−0.740019	−	0.672586i	$-0.765184\pi$
$233$	−22.5918	−1.48004	−0.740019	+	0.672586i	$0.765184\pi$
$234$	0	0
$235$	−7.68857	−0.501547
$236$	−12.5018	−0.813795
$237$	0	0
$238$	−32.1606	−2.08466
$239$	5.34326	0.345626	0.172813	−	0.984955i	$-0.444714\pi$
$239$	5.34326	0.345626	0.172813	+	0.984955i	$0.444714\pi$
$240$	0	0
$241$	−3.95333	−0.254656	−0.127328	−	0.991861i	$-0.540640\pi$
$241$	−3.95333	−0.254656	−0.127328	+	0.991861i	$0.540640\pi$
$242$	21.5772	1.38703
$243$	0	0
$244$	−71.5112	−4.57804
$245$	−1.62923	−0.104087
$246$	0	0
$247$	−13.1049	−0.833845
$248$	14.0743	0.893720
$249$	0	0
$250$	2.67802	0.169373
$251$	−10.0497	−0.634333	−0.317166	−	0.948370i	$-0.602731\pi$
$251$	−10.0497	−0.634333	−0.317166	+	0.948370i	$0.602731\pi$
$252$	0	0
$253$	2.20427	0.138581
$254$	−22.7405	−1.42686
$255$	0	0
$256$	9.55753	0.597346
$257$	12.8152	0.799392	0.399696	−	0.916648i	$-0.369116\pi$
$257$	12.8152	0.799392	0.399696	+	0.916648i	$0.369116\pi$
$258$	0	0
$259$	−2.39823	−0.149019
$260$	17.1828	1.06563
$261$	0	0
$262$	46.5639	2.87673
$263$	−8.19776	−0.505495	−0.252748	−	0.967532i	$-0.581334\pi$
$263$	−8.19776	−0.505495	−0.252748	+	0.967532i	$0.581334\pi$
$264$	0	0
$265$	−12.5579	−0.771425
$266$	31.0302	1.90258
$267$	0	0
$268$	5.17182	0.315919
$269$	4.69407	0.286203	0.143101	−	0.989708i	$-0.454293\pi$
$269$	4.69407	0.286203	0.143101	+	0.989708i	$0.454293\pi$
$270$	0	0
$271$	30.8826	1.87598	0.937991	−	0.346659i	$-0.112684\pi$
$271$	30.8826	1.87598	0.937991	+	0.346659i	$0.112684\pi$
$272$	−50.7093	−3.07470
$273$	0	0
$274$	−27.8884	−1.68480
$275$	1.71548	0.103448
$276$	0	0
$277$	−28.9141	−1.73728	−0.868639	−	0.495446i	$-0.835005\pi$
$277$	−28.9141	−1.73728	−0.868639	+	0.495446i	$0.835005\pi$
$278$	4.73166	0.283786
$279$	0	0
$280$	−24.9522	−1.49118
$281$	−8.20399	−0.489409	−0.244704	−	0.969598i	$-0.578691\pi$
$281$	−8.20399	−0.489409	−0.244704	+	0.969598i	$0.578691\pi$
$282$	0	0
$283$	5.78294	0.343760	0.171880	−	0.985118i	$-0.445016\pi$
$283$	5.78294	0.343760	0.171880	+	0.985118i	$0.445016\pi$
$284$	−39.4368	−2.34014
$285$	0	0
$286$	15.2634	0.902543
$287$	−11.7959	−0.696289
$288$	0	0
$289$	−0.287241	−0.0168965
$290$	16.6640	0.978541
$291$	0	0
$292$	80.7085	4.72311
$293$	−26.8384	−1.56791	−0.783957	−	0.620815i	$-0.786802\pi$
$293$	−26.8384	−1.56791	−0.783957	+	0.620815i	$0.786802\pi$
$294$	0	0
$295$	2.41729	0.140740
$296$	−6.93470	−0.403071
$297$	0	0
$298$	−22.9783	−1.33110
$299$	−4.26903	−0.246884
$300$	0	0
$301$	−12.4777	−0.719204
$302$	−44.7919	−2.57749
$303$	0	0
$304$	48.9268	2.80615
$305$	13.8271	0.791737
$306$	0	0
$307$	1.37245	0.0783301	0.0391650	−	0.999233i	$-0.487530\pi$
$307$	1.37245	0.0783301	0.0391650	+	0.999233i	$0.487530\pi$
$308$	−26.0625	−1.48505
$309$	0	0
$310$	−4.43730	−0.252022
$311$	26.5753	1.50695	0.753474	−	0.657478i	$-0.228377\pi$
$311$	26.5753	1.50695	0.753474	+	0.657478i	$0.228377\pi$
$312$	0	0
$313$	−23.9168	−1.35186	−0.675929	−	0.736967i	$-0.736257\pi$
$313$	−23.9168	−1.35186	−0.675929	+	0.736967i	$0.736257\pi$
$314$	−63.0486	−3.55804
$315$	0	0
$316$	−69.8320	−3.92835
$317$	−12.0685	−0.677836	−0.338918	−	0.940816i	$-0.610061\pi$
$317$	−12.0685	−0.677836	−0.338918	+	0.940816i	$0.610061\pi$
$318$	0	0
$319$	10.6746	0.597661
$320$	−18.6561	−1.04291
$321$	0	0
$322$	10.1083	0.563315
$323$	−16.1253	−0.897235
$324$	0	0
$325$	−3.32239	−0.184293
$326$	6.55567	0.363085
$327$	0	0
$328$	−34.1089	−1.88335
$329$	−22.5856	−1.24518
$330$	0	0
$331$	−20.4573	−1.12444	−0.562218	−	0.826989i	$-0.690052\pi$
$331$	−20.4573	−1.12444	−0.562218	+	0.826989i	$0.690052\pi$
$332$	−47.3316	−2.59766
$333$	0	0
$334$	−8.76760	−0.479742
$335$	−1.00000	−0.0546358
$336$	0	0
$337$	−16.9672	−0.924261	−0.462131	−	0.886812i	$-0.652915\pi$
$337$	−16.9672	−0.924261	−0.462131	+	0.886812i	$0.652915\pi$
$338$	5.25362	0.285759
$339$	0	0
$340$	21.1430	1.14664
$341$	−2.84244	−0.153927
$342$	0	0
$343$	15.7769	0.851875
$344$	−36.0805	−1.94533
$345$	0	0
$346$	21.7002	1.16661
$347$	1.93074	0.103648	0.0518238	−	0.998656i	$-0.483497\pi$
$347$	1.93074	0.103648	0.0518238	+	0.998656i	$0.483497\pi$
$348$	0	0
$349$	−5.47085	−0.292848	−0.146424	−	0.989222i	$-0.546776\pi$
$349$	−5.47085	−0.292848	−0.146424	+	0.989222i	$0.546776\pi$
$350$	7.86684	0.420500
$351$	0	0
$352$	−27.8422	−1.48399
$353$	−14.9217	−0.794200	−0.397100	−	0.917775i	$-0.629983\pi$
$353$	−14.9217	−0.794200	−0.397100	+	0.917775i	$0.629983\pi$
$354$	0	0
$355$	7.62533	0.404711
$356$	21.9241	1.16197
$357$	0	0
$358$	−16.6868	−0.881927
$359$	27.5188	1.45239	0.726193	−	0.687491i	$-0.241288\pi$
$359$	27.5188	1.45239	0.726193	+	0.687491i	$0.241288\pi$
$360$	0	0
$361$	−3.44152	−0.181132
$362$	−48.0613	−2.52605
$363$	0	0
$364$	50.4753	2.64563
$365$	−15.6054	−0.816826
$366$	0	0
$367$	−24.7563	−1.29227	−0.646135	−	0.763223i	$-0.723616\pi$
$367$	−24.7563	−1.29227	−0.646135	+	0.763223i	$0.723616\pi$
$368$	15.9383	0.830842
$369$	0	0
$370$	2.18635	0.113663
$371$	−36.8895	−1.91521
$372$	0	0
$373$	−23.3059	−1.20673	−0.603366	−	0.797465i	$-0.706174\pi$
$373$	−23.3059	−1.20673	−0.603366	+	0.797465i	$0.706174\pi$
$374$	18.7813	0.971157
$375$	0	0
$376$	−65.3082	−3.36802
$377$	−20.6735	−1.06474
$378$	0	0
$379$	−29.5945	−1.52017	−0.760085	−	0.649824i	$-0.774843\pi$
$379$	−29.5945	−1.52017	−0.760085	+	0.649824i	$0.774843\pi$
$380$	−20.3998	−1.04649
$381$	0	0
$382$	69.4562	3.55369
$383$	−20.1023	−1.02718	−0.513590	−	0.858036i	$-0.671685\pi$
$383$	−20.1023	−1.02718	−0.513590	+	0.858036i	$0.671685\pi$
$384$	0	0
$385$	5.03933	0.256828
$386$	56.6611	2.88397
$387$	0	0
$388$	−37.1871	−1.88789
$389$	−38.9487	−1.97478	−0.987388	−	0.158321i	$-0.949392\pi$
$389$	−38.9487	−1.97478	−0.987388	+	0.158321i	$0.949392\pi$
$390$	0	0
$391$	−5.25294	−0.265653
$392$	−13.8390	−0.698974
$393$	0	0
$394$	−43.6113	−2.19711
$395$	13.5024	0.679380
$396$	0	0
$397$	−7.23673	−0.363201	−0.181601	−	0.983372i	$-0.558128\pi$
$397$	−7.23673	−0.363201	−0.181601	+	0.983372i	$0.558128\pi$
$398$	29.1454	1.46093
$399$	0	0
$400$	12.4041	0.620203
$401$	−16.1117	−0.804578	−0.402289	−	0.915513i	$-0.631785\pi$
$401$	−16.1117	−0.804578	−0.402289	+	0.915513i	$0.631785\pi$
$402$	0	0
$403$	5.50497	0.274222
$404$	−32.1735	−1.60069
$405$	0	0
$406$	48.9513	2.42941
$407$	1.40053	0.0694215
$408$	0	0
$409$	−6.12555	−0.302889	−0.151445	−	0.988466i	$-0.548393\pi$
$409$	−6.12555	−0.302889	−0.151445	+	0.988466i	$0.548393\pi$
$410$	10.7537	0.531089
$411$	0	0
$412$	63.6335	3.13500
$413$	7.10091	0.349413
$414$	0	0
$415$	9.15183	0.449246
$416$	53.9222	2.64375
$417$	0	0
$418$	−18.1211	−0.886332
$419$	−19.7767	−0.966156	−0.483078	−	0.875577i	$-0.660481\pi$
$419$	−19.7767	−0.966156	−0.483078	+	0.875577i	$0.660481\pi$
$420$	0	0
$421$	3.09670	0.150924	0.0754619	−	0.997149i	$-0.475957\pi$
$421$	3.09670	0.150924	0.0754619	+	0.997149i	$0.475957\pi$
$422$	37.7946	1.83981
$423$	0	0
$424$	−106.669	−5.18031
$425$	−4.08812	−0.198303
$426$	0	0
$427$	40.6179	1.96564
$428$	−96.3701	−4.65823
$429$	0	0
$430$	11.3753	0.548567
$431$	14.2396	0.685896	0.342948	−	0.939354i	$-0.388575\pi$
$431$	14.2396	0.685896	0.342948	+	0.939354i	$0.388575\pi$
$432$	0	0
$433$	33.0204	1.58686	0.793428	−	0.608664i	$-0.208294\pi$
$433$	33.0204	1.58686	0.793428	+	0.608664i	$0.208294\pi$
$434$	−13.0348	−0.625691
$435$	0	0
$436$	−16.1465	−0.773276
$437$	5.06830	0.242450
$438$	0	0
$439$	26.7281	1.27566	0.637832	−	0.770175i	$-0.279831\pi$
$439$	26.7281	1.27566	0.637832	+	0.770175i	$0.279831\pi$
$440$	14.5717	0.694677
$441$	0	0
$442$	−36.3738	−1.73013
$443$	3.44565	0.163708	0.0818539	−	0.996644i	$-0.473916\pi$
$443$	3.44565	0.163708	0.0818539	+	0.996644i	$0.473916\pi$
$444$	0	0
$445$	−4.23914	−0.200955
$446$	22.2305	1.05264
$447$	0	0
$448$	−54.8033	−2.58921
$449$	−6.75694	−0.318880	−0.159440	−	0.987208i	$-0.550969\pi$
$449$	−6.75694	−0.318880	−0.159440	+	0.987208i	$0.550969\pi$
$450$	0	0
$451$	6.88860	0.324372
$452$	−77.3886	−3.64005
$453$	0	0
$454$	12.8667	0.603862
$455$	−9.75969	−0.457541
$456$	0	0
$457$	27.6885	1.29522	0.647608	−	0.761974i	$-0.275770\pi$
$457$	27.6885	1.29522	0.647608	+	0.761974i	$0.275770\pi$
$458$	−40.3578	−1.88579
$459$	0	0
$460$	−6.64541	−0.309844
$461$	−15.0380	−0.700389	−0.350194	−	0.936677i	$-0.613885\pi$
$461$	−15.0380	−0.700389	−0.350194	+	0.936677i	$0.613885\pi$
$462$	0	0
$463$	−13.2650	−0.616476	−0.308238	−	0.951309i	$-0.599739\pi$
$463$	−13.2650	−0.616476	−0.308238	+	0.951309i	$0.599739\pi$
$464$	77.1840	3.58318
$465$	0	0
$466$	60.5014	2.80267
$467$	17.4982	0.809718	0.404859	−	0.914379i	$-0.367321\pi$
$467$	17.4982	0.809718	0.404859	+	0.914379i	$0.367321\pi$
$468$	0	0
$469$	−2.93755	−0.135644
$470$	20.5902	0.949754
$471$	0	0
$472$	20.5329	0.945104
$473$	7.28678	0.335047
$474$	0	0
$475$	3.94442	0.180983
$476$	62.1088	2.84675
$477$	0	0
$478$	−14.3094	−0.654495
$479$	−0.729119	−0.0333143	−0.0166572	−	0.999861i	$-0.505302\pi$
$479$	−0.729119	−0.0333143	−0.0166572	+	0.999861i	$0.505302\pi$
$480$	0	0
$481$	−2.71241	−0.123675
$482$	10.5871	0.482229
$483$	0	0
$484$	−41.6699	−1.89409
$485$	7.19034	0.326497
$486$	0	0
$487$	22.0260	0.998096	0.499048	−	0.866574i	$-0.333683\pi$
$487$	22.0260	0.998096	0.499048	+	0.866574i	$0.333683\pi$
$488$	117.450	5.31672
$489$	0	0
$490$	4.36311	0.197105
$491$	4.03954	0.182302	0.0911509	−	0.995837i	$-0.470945\pi$
$491$	4.03954	0.182302	0.0911509	+	0.995837i	$0.470945\pi$
$492$	0	0
$493$	−25.4383	−1.14568
$494$	35.0952	1.57901
$495$	0	0
$496$	−20.5527	−0.922842
$497$	22.3998	1.00477
$498$	0	0
$499$	−39.9303	−1.78752	−0.893762	−	0.448542i	$-0.851944\pi$
$499$	−39.9303	−1.78752	−0.893762	+	0.448542i	$0.851944\pi$
$500$	−5.17182	−0.231291
$501$	0	0
$502$	26.9134	1.20120
$503$	12.7228	0.567281	0.283640	−	0.958931i	$-0.408458\pi$
$503$	12.7228	0.567281	0.283640	+	0.958931i	$0.408458\pi$
$504$	0	0
$505$	6.22093	0.276828
$506$	−5.90309	−0.262425
$507$	0	0
$508$	43.9165	1.94848
$509$	33.4517	1.48272	0.741360	−	0.671107i	$-0.234181\pi$
$509$	33.4517	1.48272	0.741360	+	0.671107i	$0.234181\pi$
$510$	0	0
$511$	−45.8418	−2.02792
$512$	9.40791	0.415775
$513$	0	0
$514$	−34.3195	−1.51377
$515$	−12.3039	−0.542174
$516$	0	0
$517$	13.1896	0.580078
$518$	6.42252	0.282189
$519$	0	0
$520$	−28.2210	−1.23757
$521$	10.5574	0.462527	0.231264	−	0.972891i	$-0.425714\pi$
$521$	10.5574	0.462527	0.231264	+	0.972891i	$0.425714\pi$
$522$	0	0
$523$	−1.26165	−0.0551680	−0.0275840	−	0.999619i	$-0.508781\pi$
$523$	−1.26165	−0.0551680	−0.0275840	+	0.999619i	$0.508781\pi$
$524$	−89.9244	−3.92837
$525$	0	0
$526$	21.9538	0.957231
$527$	6.77374	0.295069
$528$	0	0
$529$	−21.3490	−0.928216
$530$	33.6303	1.46081
$531$	0	0
$532$	−59.9256	−2.59811
$533$	−13.3412	−0.577871
$534$	0	0
$535$	18.6337	0.805606
$536$	−8.49420	−0.366894
$537$	0	0
$538$	−12.5708	−0.541968
$539$	2.79491	0.120385
$540$	0	0
$541$	−29.1980	−1.25532	−0.627660	−	0.778488i	$-0.715987\pi$
$541$	−29.1980	−1.25532	−0.627660	+	0.778488i	$0.715987\pi$
$542$	−82.7043	−3.55245
$543$	0	0
$544$	66.3501	2.84473
$545$	3.12201	0.133732
$546$	0	0
$547$	32.1623	1.37516	0.687580	−	0.726109i	$-0.258673\pi$
$547$	32.1623	1.37516	0.687580	+	0.726109i	$0.258673\pi$
$548$	53.8583	2.30071
$549$	0	0
$550$	−4.59411	−0.195893
$551$	24.5441	1.04561
$552$	0	0
$553$	39.6641	1.68669
$554$	77.4326	3.28979
$555$	0	0
$556$	−9.13780	−0.387529
$557$	−34.1182	−1.44564	−0.722818	−	0.691039i	$-0.757153\pi$
$557$	−34.1182	−1.44564	−0.722818	+	0.691039i	$0.757153\pi$
$558$	0	0
$559$	−14.1124	−0.596889
$560$	36.4376	1.53977
$561$	0	0
$562$	21.9705	0.926769
$563$	−39.6874	−1.67262	−0.836312	−	0.548253i	$-0.815293\pi$
$563$	−39.6874	−1.67262	−0.836312	+	0.548253i	$0.815293\pi$
$564$	0	0
$565$	14.9635	0.629520
$566$	−15.4868	−0.650961
$567$	0	0
$568$	64.7711	2.71774
$569$	3.65190	0.153096	0.0765479	−	0.997066i	$-0.475610\pi$
$569$	3.65190	0.153096	0.0765479	+	0.997066i	$0.475610\pi$
$570$	0	0
$571$	−8.83791	−0.369855	−0.184927	−	0.982752i	$-0.559205\pi$
$571$	−8.83791	−0.369855	−0.184927	+	0.982752i	$0.559205\pi$
$572$	−29.4768	−1.23248
$573$	0	0
$574$	31.5897	1.31853
$575$	1.28493	0.0535852
$576$	0	0
$577$	−4.88707	−0.203451	−0.101726	−	0.994812i	$-0.532436\pi$
$577$	−4.88707	−0.203451	−0.101726	+	0.994812i	$0.532436\pi$
$578$	0.769239	0.0319961
$579$	0	0
$580$	−32.1815	−1.33627
$581$	26.8840	1.11534
$582$	0	0
$583$	21.5428	0.892213
$584$	−132.556	−5.48520
$585$	0	0
$586$	71.8738	2.96908
$587$	40.3399	1.66500	0.832502	−	0.554022i	$-0.186908\pi$
$587$	40.3399	1.66500	0.832502	+	0.554022i	$0.186908\pi$
$588$	0	0
$589$	−6.53564	−0.269296
$590$	−6.47355	−0.266512
$591$	0	0
$592$	10.1267	0.416205
$593$	−0.00336085	−0.000138014 0	−6.90068e−5	−	1.00000i	$-0.500022\pi$
$593$	−0.00336085	−0.000138014 0	−6.90068e−5		1.00000i	$0.500022\pi$
$594$	0	0
$595$	−12.0091	−0.492325
$596$	44.3759	1.81771
$597$	0	0
$598$	11.4326	0.467512
$599$	14.4315	0.589654	0.294827	−	0.955551i	$-0.404738\pi$
$599$	14.4315	0.589654	0.294827	+	0.955551i	$0.404738\pi$
$600$	0	0
$601$	−13.0069	−0.530564	−0.265282	−	0.964171i	$-0.585465\pi$
$601$	−13.0069	−0.530564	−0.265282	+	0.964171i	$0.585465\pi$
$602$	33.4156	1.36192
$603$	0	0
$604$	86.5025	3.51973
$605$	8.05712	0.327568
$606$	0	0
$607$	5.78556	0.234829	0.117414	−	0.993083i	$-0.462539\pi$
$607$	5.78556	0.234829	0.117414	+	0.993083i	$0.462539\pi$
$608$	−64.0178	−2.59627
$609$	0	0
$610$	−37.0293	−1.49927
$611$	−25.5444	−1.03342
$612$	0	0
$613$	−38.7419	−1.56477	−0.782386	−	0.622794i	$-0.785998\pi$
$613$	−38.7419	−1.56477	−0.782386	+	0.622794i	$0.785998\pi$
$614$	−3.67546	−0.148330
$615$	0	0
$616$	42.8050	1.72466
$617$	−28.7393	−1.15700	−0.578501	−	0.815682i	$-0.696362\pi$
$617$	−28.7393	−1.15700	−0.578501	+	0.815682i	$0.696362\pi$
$618$	0	0
$619$	−39.0204	−1.56836	−0.784181	−	0.620533i	$-0.786916\pi$
$619$	−39.0204	−1.56836	−0.784181	+	0.620533i	$0.786916\pi$
$620$	8.56935	0.344153
$621$	0	0
$622$	−71.1694	−2.85363
$623$	−12.4527	−0.498907
$624$	0	0
$625$	1.00000	0.0400000
$626$	64.0498	2.55994
$627$	0	0
$628$	121.760	4.85874
$629$	−3.33756	−0.133077
$630$	0	0
$631$	21.3961	0.851764	0.425882	−	0.904779i	$-0.359964\pi$
$631$	21.3961	0.851764	0.425882	+	0.904779i	$0.359964\pi$
$632$	114.692	4.56221
$633$	0	0
$634$	32.3198	1.28358
$635$	−8.49151	−0.336975
$636$	0	0
$637$	−5.41292	−0.214468
$638$	−28.5867	−1.13176
$639$	0	0
$640$	17.5016	0.691812
$641$	39.7885	1.57155	0.785775	−	0.618513i	$-0.212264\pi$
$641$	39.7885	1.57155	0.785775	+	0.618513i	$0.212264\pi$
$642$	0	0
$643$	36.8386	1.45277	0.726386	−	0.687287i	$-0.241198\pi$
$643$	36.8386	1.45277	0.726386	+	0.687287i	$0.241198\pi$
$644$	−19.5213	−0.769245
$645$	0	0
$646$	43.1839	1.69905
$647$	−14.6683	−0.576669	−0.288334	−	0.957530i	$-0.593101\pi$
$647$	−14.6683	−0.576669	−0.288334	+	0.957530i	$0.593101\pi$
$648$	0	0
$649$	−4.14682	−0.162777
$650$	8.89743	0.348986
$651$	0	0
$652$	−12.6603	−0.495817
$653$	25.0074	0.978616	0.489308	−	0.872111i	$-0.337249\pi$
$653$	25.0074	0.978616	0.489308	+	0.872111i	$0.337249\pi$
$654$	0	0
$655$	17.3874	0.679382
$656$	49.8090	1.94472
$657$	0	0
$658$	60.4848	2.35794
$659$	20.9641	0.816645	0.408322	−	0.912838i	$-0.366114\pi$
$659$	20.9641	0.816645	0.408322	+	0.912838i	$0.366114\pi$
$660$	0	0
$661$	−1.24040	−0.0482459	−0.0241229	−	0.999709i	$-0.507679\pi$
$661$	−1.24040	−0.0482459	−0.0241229	+	0.999709i	$0.507679\pi$
$662$	54.7852	2.12929
$663$	0	0
$664$	77.7375	3.01680
$665$	11.5870	0.449323
$666$	0	0
$667$	7.99544	0.309585
$668$	16.9320	0.655120
$669$	0	0
$670$	2.67802	0.103461
$671$	−23.7202	−0.915707
$672$	0	0
$673$	−22.8502	−0.880809	−0.440405	−	0.897799i	$-0.645165\pi$
$673$	−22.8502	−0.880809	−0.440405	+	0.897799i	$0.645165\pi$
$674$	45.4385	1.75023
$675$	0	0
$676$	−10.1458	−0.390224
$677$	−35.4330	−1.36180	−0.680901	−	0.732376i	$-0.738411\pi$
$677$	−35.4330	−1.36180	−0.680901	+	0.732376i	$0.738411\pi$
$678$	0	0
$679$	21.1220	0.810589
$680$	−34.7254	−1.33166
$681$	0	0
$682$	7.61212	0.291483
$683$	−20.3470	−0.778555	−0.389278	−	0.921121i	$-0.627275\pi$
$683$	−20.3470	−0.778555	−0.389278	+	0.921121i	$0.627275\pi$
$684$	0	0
$685$	−10.4138	−0.397891
$686$	−42.2510	−1.61315
$687$	0	0
$688$	52.6881	2.00872
$689$	−41.7221	−1.58949
$690$	0	0
$691$	36.2888	1.38049	0.690246	−	0.723575i	$-0.257502\pi$
$691$	36.2888	1.38049	0.690246	+	0.723575i	$0.257502\pi$
$692$	−41.9075	−1.59308
$693$	0	0
$694$	−5.17057	−0.196272
$695$	1.76685	0.0670203
$696$	0	0
$697$	−16.4161	−0.621802
$698$	14.6511	0.554551
$699$	0	0
$700$	−15.1925	−0.574222
$701$	−34.7845	−1.31379	−0.656895	−	0.753982i	$-0.728131\pi$
$701$	−34.7845	−1.31379	−0.656895	+	0.753982i	$0.728131\pi$
$702$	0	0
$703$	3.22024	0.121454
$704$	32.0042	1.20620
$705$	0	0
$706$	39.9606	1.50394
$707$	18.2743	0.687277
$708$	0	0
$709$	36.0123	1.35247	0.676236	−	0.736685i	$-0.263610\pi$
$709$	36.0123	1.35247	0.676236	+	0.736685i	$0.263610\pi$
$710$	−20.4208	−0.766380
$711$	0	0
$712$	−36.0081	−1.34946
$713$	−2.12904	−0.0797331
$714$	0	0
$715$	5.69950	0.213149
$716$	32.2257	1.20433
$717$	0	0
$718$	−73.6960	−2.75031
$719$	24.9530	0.930590	0.465295	−	0.885156i	$-0.345948\pi$
$719$	24.9530	0.930590	0.465295	+	0.885156i	$0.345948\pi$
$720$	0	0
$721$	−36.1433	−1.34605
$722$	9.21647	0.343001
$723$	0	0
$724$	92.8162	3.44949
$725$	6.22248	0.231097
$726$	0	0
$727$	21.6836	0.804201	0.402100	−	0.915596i	$-0.368280\pi$
$727$	21.6836	0.804201	0.402100	+	0.915596i	$0.368280\pi$
$728$	−82.9008	−3.07251
$729$	0	0
$730$	41.7918	1.54678
$731$	−17.3649	−0.642266
$732$	0	0
$733$	24.4842	0.904345	0.452172	−	0.891931i	$-0.350649\pi$
$733$	24.4842	0.904345	0.452172	+	0.891931i	$0.350649\pi$
$734$	66.2980	2.44711
$735$	0	0
$736$	−20.8543	−0.768700
$737$	1.71548	0.0631906
$738$	0	0
$739$	16.1989	0.595884	0.297942	−	0.954584i	$-0.403700\pi$
$739$	16.1989	0.595884	0.297942	+	0.954584i	$0.403700\pi$
$740$	−4.22229	−0.155214
$741$	0	0
$742$	98.7909	3.62673
$743$	−41.1179	−1.50847	−0.754234	−	0.656605i	$-0.771992\pi$
$743$	−41.1179	−1.50847	−0.754234	+	0.656605i	$0.771992\pi$
$744$	0	0
$745$	−8.58033	−0.314359
$746$	62.4137	2.28513
$747$	0	0
$748$	−36.2705	−1.32618
$749$	54.7375	2.00007
$750$	0	0
$751$	−13.9897	−0.510492	−0.255246	−	0.966876i	$-0.582157\pi$
$751$	−13.9897	−0.510492	−0.255246	+	0.966876i	$0.582157\pi$
$752$	95.3694	3.47776
$753$	0	0
$754$	55.3641	2.01624
$755$	−16.7257	−0.608712
$756$	0	0
$757$	−31.9542	−1.16139	−0.580697	−	0.814120i	$-0.697220\pi$
$757$	−31.9542	−1.16139	−0.580697	+	0.814120i	$0.697220\pi$
$758$	79.2549	2.87867
$759$	0	0
$760$	33.5047	1.21534
$761$	41.5290	1.50542	0.752712	−	0.658349i	$-0.228745\pi$
$761$	41.5290	1.50542	0.752712	+	0.658349i	$0.228745\pi$
$762$	0	0
$763$	9.17109	0.332016
$764$	−134.134	−4.85281
$765$	0	0
$766$	53.8345	1.94512
$767$	8.03116	0.289988
$768$	0	0
$769$	−16.3690	−0.590280	−0.295140	−	0.955454i	$-0.595366\pi$
$769$	−16.3690	−0.590280	−0.295140	+	0.955454i	$0.595366\pi$
$770$	−13.4954	−0.486342
$771$	0	0
$772$	−109.424	−3.93826
$773$	31.3105	1.12616	0.563080	−	0.826402i	$-0.309616\pi$
$773$	31.3105	1.12616	0.563080	+	0.826402i	$0.309616\pi$
$774$	0	0
$775$	−1.65693	−0.0595187
$776$	61.0762	2.19251
$777$	0	0
$778$	104.305	3.73953
$779$	15.8390	0.567492
$780$	0	0
$781$	−13.0811	−0.468080
$782$	14.0675	0.503053
$783$	0	0
$784$	20.2090	0.721750
$785$	−23.5429	−0.840283
$786$	0	0
$787$	−9.52277	−0.339450	−0.169725	−	0.985491i	$-0.554288\pi$
$787$	−9.52277	−0.339450	−0.169725	+	0.985491i	$0.554288\pi$
$788$	84.2224	3.00030
$789$	0	0
$790$	−36.1598	−1.28651
$791$	43.9561	1.56290
$792$	0	0
$793$	45.9390	1.63134
$794$	19.3802	0.687776
$795$	0	0
$796$	−56.2858	−1.99500
$797$	−13.3603	−0.473245	−0.236622	−	0.971602i	$-0.576040\pi$
$797$	−13.3603	−0.473245	−0.236622	+	0.971602i	$0.576040\pi$
$798$	0	0
$799$	−31.4318	−1.11198
$800$	−16.2300	−0.573815
$801$	0	0
$802$	43.1475	1.52359
$803$	26.7709	0.944723
$804$	0	0
$805$	3.77454	0.133035
$806$	−14.7424	−0.519280
$807$	0	0
$808$	52.8419	1.85897
$809$	−50.0231	−1.75872	−0.879359	−	0.476159i	$-0.842029\pi$
$809$	−50.0231	−1.75872	−0.879359	+	0.476159i	$0.842029\pi$
$810$	0	0
$811$	−2.88647	−0.101358	−0.0506788	−	0.998715i	$-0.516138\pi$
$811$	−2.88647	−0.101358	−0.0506788	+	0.998715i	$0.516138\pi$
$812$	−94.5350	−3.31753
$813$	0	0
$814$	−3.75065	−0.131460
$815$	2.44795	0.0857479
$816$	0	0
$817$	16.7546	0.586168
$818$	16.4044	0.573566
$819$	0	0
$820$	−20.7677	−0.725238
$821$	−30.6578	−1.06996	−0.534982	−	0.844863i	$-0.679682\pi$
$821$	−30.6578	−1.06996	−0.534982	+	0.844863i	$0.679682\pi$
$822$	0	0
$823$	−31.9639	−1.11419	−0.557095	−	0.830449i	$-0.688084\pi$
$823$	−31.9639	−1.11419	−0.557095	+	0.830449i	$0.688084\pi$
$824$	−104.512	−3.64084
$825$	0	0
$826$	−19.0164	−0.661666
$827$	38.5687	1.34117	0.670583	−	0.741835i	$-0.266044\pi$
$827$	38.5687	1.34117	0.670583	+	0.741835i	$0.266044\pi$
$828$	0	0
$829$	−3.09182	−0.107383	−0.0536917	−	0.998558i	$-0.517099\pi$
$829$	−3.09182	−0.107383	−0.0536917	+	0.998558i	$0.517099\pi$
$830$	−24.5088	−0.850714
$831$	0	0
$832$	−61.9828	−2.14887
$833$	−6.66048	−0.230772
$834$	0	0
$835$	−3.27391	−0.113298
$836$	34.9956	1.21035
$837$	0	0
$838$	52.9626	1.82956
$839$	18.0696	0.623833	0.311917	−	0.950109i	$-0.399029\pi$
$839$	18.0696	0.623833	0.311917	+	0.950109i	$0.399029\pi$
$840$	0	0
$841$	9.71928	0.335148
$842$	−8.29303	−0.285797
$843$	0	0
$844$	−72.9892	−2.51239
$845$	1.96175	0.0674863
$846$	0	0
$847$	23.6682	0.813250
$848$	155.769	5.34912
$849$	0	0
$850$	10.9481	0.375517
$851$	1.04902	0.0359599
$852$	0	0
$853$	−41.1426	−1.40870	−0.704348	−	0.709855i	$-0.748760\pi$
$853$	−41.1426	−1.40870	−0.704348	+	0.709855i	$0.748760\pi$
$854$	−108.776	−3.72223
$855$	0	0
$856$	158.279	5.40985
$857$	−34.8300	−1.18977	−0.594885	−	0.803811i	$-0.702803\pi$
$857$	−34.8300	−1.18977	−0.594885	+	0.803811i	$0.702803\pi$
$858$	0	0
$859$	−9.22679	−0.314814	−0.157407	−	0.987534i	$-0.550313\pi$
$859$	−9.22679	−0.314814	−0.157407	+	0.987534i	$0.550313\pi$
$860$	−21.9681	−0.749106
$861$	0	0
$862$	−38.1339	−1.29885
$863$	−36.3399	−1.23703	−0.618513	−	0.785774i	$-0.712265\pi$
$863$	−36.3399	−1.23703	−0.618513	+	0.785774i	$0.712265\pi$
$864$	0	0
$865$	8.10304	0.275512
$866$	−88.4293	−3.00495
$867$	0	0
$868$	25.1729	0.854425
$869$	−23.1632	−0.785756
$870$	0	0
$871$	−3.32239	−0.112575
$872$	26.5190	0.898047
$873$	0	0
$874$	−13.5730	−0.459115
$875$	2.93755	0.0993075
$876$	0	0
$877$	−33.7559	−1.13986	−0.569928	−	0.821695i	$-0.693029\pi$
$877$	−33.7559	−1.13986	−0.569928	+	0.821695i	$0.693029\pi$
$878$	−71.5786	−2.41566
$879$	0	0
$880$	−21.2789	−0.717313
$881$	−2.19718	−0.0740250	−0.0370125	−	0.999315i	$-0.511784\pi$
$881$	−2.19718	−0.0740250	−0.0370125	+	0.999315i	$0.511784\pi$
$882$	0	0
$883$	0.891116	0.0299884	0.0149942	−	0.999888i	$-0.495227\pi$
$883$	0.891116	0.0299884	0.0149942	+	0.999888i	$0.495227\pi$
$884$	70.2453	2.36260
$885$	0	0
$886$	−9.22754	−0.310005
$887$	−6.83153	−0.229380	−0.114690	−	0.993401i	$-0.536588\pi$
$887$	−6.83153	−0.229380	−0.114690	+	0.993401i	$0.536588\pi$
$888$	0	0
$889$	−24.9443	−0.836604
$890$	11.3525	0.380538
$891$	0	0
$892$	−42.9316	−1.43746
$893$	30.3270	1.01485
$894$	0	0
$895$	−6.23102	−0.208280
$896$	51.4119	1.71755
$897$	0	0
$898$	18.0953	0.603847
$899$	−10.3102	−0.343865
$900$	0	0
$901$	−51.3382	−1.71032
$902$	−18.4478	−0.614246
$903$	0	0
$904$	127.103	4.22739
$905$	−17.9465	−0.596563
$906$	0	0
$907$	−31.4180	−1.04322	−0.521609	−	0.853185i	$-0.674668\pi$
$907$	−31.4180	−1.04322	−0.521609	+	0.853185i	$0.674668\pi$
$908$	−24.8482	−0.824616
$909$	0	0
$910$	26.1367	0.866423
$911$	−17.5160	−0.580332	−0.290166	−	0.956976i	$-0.593711\pi$
$911$	−17.5160	−0.580332	−0.290166	+	0.956976i	$0.593711\pi$
$912$	0	0
$913$	−15.6998	−0.519588
$914$	−74.1506	−2.45268
$915$	0	0
$916$	77.9391	2.57518
$917$	51.0764	1.68669
$918$	0	0
$919$	5.51226	0.181833	0.0909164	−	0.995859i	$-0.471020\pi$
$919$	5.51226	0.181833	0.0909164	+	0.995859i	$0.471020\pi$
$920$	10.9144	0.359838
$921$	0	0
$922$	40.2721	1.32629
$923$	25.3343	0.833889
$924$	0	0
$925$	0.816404	0.0268432
$926$	35.5239	1.16739
$927$	0	0
$928$	−100.991	−3.31518
$929$	52.5559	1.72430	0.862151	−	0.506651i	$-0.169117\pi$
$929$	52.5559	1.72430	0.862151	+	0.506651i	$0.169117\pi$
$930$	0	0
$931$	6.42636	0.210615
$932$	−116.841	−3.82724
$933$	0	0
$934$	−46.8605	−1.53332
$935$	7.01311	0.229353
$936$	0	0
$937$	−40.5829	−1.32578	−0.662892	−	0.748715i	$-0.730671\pi$
$937$	−40.5829	−1.32578	−0.662892	+	0.748715i	$0.730671\pi$
$938$	7.86684	0.256862
$939$	0	0
$940$	−39.7639	−1.29695
$941$	−10.3401	−0.337076	−0.168538	−	0.985695i	$-0.553905\pi$
$941$	−10.3401	−0.337076	−0.168538	+	0.985695i	$0.553905\pi$
$942$	0	0
$943$	5.15969	0.168022
$944$	−29.9841	−0.975901
$945$	0	0
$946$	−19.5142	−0.634461
$947$	8.99498	0.292298	0.146149	−	0.989263i	$-0.453312\pi$
$947$	8.99498	0.292298	0.146149	+	0.989263i	$0.453312\pi$
$948$	0	0
$949$	−51.8473	−1.68303
$950$	−10.5633	−0.342718
$951$	0	0
$952$	−102.008	−3.30609
$953$	−9.15335	−0.296506	−0.148253	−	0.988949i	$-0.547365\pi$
$953$	−9.15335	−0.296506	−0.148253	+	0.988949i	$0.547365\pi$
$954$	0	0
$955$	25.9356	0.839257
$956$	27.6343	0.893758
$957$	0	0
$958$	1.95260	0.0630856
$959$	−30.5911	−0.987839
$960$	0	0
$961$	−28.2546	−0.911438
$962$	7.26390	0.234197
$963$	0	0
$964$	−20.4459	−0.658517
$965$	21.1578	0.681093
$966$	0	0
$967$	28.3567	0.911889	0.455944	−	0.890008i	$-0.349302\pi$
$967$	28.3567	0.911889	0.455944	+	0.890008i	$0.349302\pi$
$968$	68.4388	2.19971
$969$	0	0
$970$	−19.2559	−0.618270
$971$	45.6039	1.46350	0.731749	−	0.681575i	$-0.238704\pi$
$971$	45.6039	1.46350	0.731749	+	0.681575i	$0.238704\pi$
$972$	0	0
$973$	5.19021	0.166390
$974$	−58.9863	−1.89004
$975$	0	0
$976$	−171.512	−5.48997
$977$	11.7573	0.376149	0.188074	−	0.982155i	$-0.439775\pi$
$977$	11.7573	0.376149	0.188074	+	0.982155i	$0.439775\pi$
$978$	0	0
$979$	7.27218	0.232420
$980$	−8.42606	−0.269160
$981$	0	0
$982$	−10.8180	−0.345216
$983$	−47.6788	−1.52072	−0.760359	−	0.649503i	$-0.774977\pi$
$983$	−47.6788	−1.52072	−0.760359	+	0.649503i	$0.774977\pi$
$984$	0	0
$985$	−16.2849	−0.518880
$986$	68.1243	2.16952
$987$	0	0
$988$	−67.7761	−2.15625
$989$	5.45793	0.173552
$990$	0	0
$991$	15.9231	0.505812	0.252906	−	0.967491i	$-0.418614\pi$
$991$	15.9231	0.505812	0.252906	+	0.967491i	$0.418614\pi$
$992$	26.8919	0.853819
$993$	0	0
$994$	−59.9873	−1.90268
$995$	10.8832	0.345020
$996$	0	0
$997$	16.9132	0.535646	0.267823	−	0.963468i	$-0.413696\pi$
$997$	16.9132	0.535646	0.267823	+	0.963468i	$0.413696\pi$
$998$	106.934	3.38494
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3015.2.a.l.1.1		7
3.2	odd	2		1005.2.a.i.1.7	✓	7
15.14	odd	2		5025.2.a.bb.1.1		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1005.2.a.i.1.7	✓	7	3.2	odd	2
3015.2.a.l.1.1		7	1.1	even	1	trivial
5025.2.a.bb.1.1		7	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3015 = 3^{2} \cdot 5 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3015.a (trivial)

\(f(q)\)	\(=\)	\(q-2.67802 q^{2} +5.17182 q^{4} -1.00000 q^{5} -2.93755 q^{7} -8.49420 q^{8} +O(q^{10})\)	\(q-2.67802 q^{2} +5.17182 q^{4} -1.00000 q^{5} -2.93755 q^{7} -8.49420 q^{8} +2.67802 q^{10} +1.71548 q^{11} -3.32239 q^{13} +7.86684 q^{14} +12.4041 q^{16} -4.08812 q^{17} +3.94442 q^{19} -5.17182 q^{20} -4.59411 q^{22} +1.28493 q^{23} +1.00000 q^{25} +8.89743 q^{26} -15.1925 q^{28} +6.22248 q^{29} -1.65693 q^{31} -16.2300 q^{32} +10.9481 q^{34} +2.93755 q^{35} +0.816404 q^{37} -10.5633 q^{38} +8.49420 q^{40} +4.01555 q^{41} +4.24766 q^{43} +8.87216 q^{44} -3.44107 q^{46} +7.68857 q^{47} +1.62923 q^{49} -2.67802 q^{50} -17.1828 q^{52} +12.5579 q^{53} -1.71548 q^{55} +24.9522 q^{56} -16.6640 q^{58} -2.41729 q^{59} -13.8271 q^{61} +4.43730 q^{62} +18.6561 q^{64} +3.32239 q^{65} +1.00000 q^{67} -21.1430 q^{68} -7.86684 q^{70} -7.62533 q^{71} +15.6054 q^{73} -2.18635 q^{74} +20.3998 q^{76} -5.03933 q^{77} -13.5024 q^{79} -12.4041 q^{80} -10.7537 q^{82} -9.15183 q^{83} +4.08812 q^{85} -11.3753 q^{86} -14.5717 q^{88} +4.23914 q^{89} +9.75969 q^{91} +6.64541 q^{92} -20.5902 q^{94} -3.94442 q^{95} -7.19034 q^{97} -4.36311 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 4 q^{2} + 8 q^{4} - 7 q^{5} + 3 q^{7} - 9 q^{8}+O(q^{10}) \) 7 * q - 4 * q^2 + 8 * q^4 - 7 * q^5 + 3 * q^7 - 9 * q^8	\( 7 q - 4 q^{2} + 8 q^{4} - 7 q^{5} + 3 q^{7} - 9 q^{8} + 4 q^{10} - 5 q^{11} - q^{13} + 4 q^{14} + 6 q^{16} - 11 q^{17} + 8 q^{19} - 8 q^{20} - 3 q^{22} - 11 q^{23} + 7 q^{25} + 5 q^{26} - 17 q^{28} - 3 q^{31} - 22 q^{32} + 4 q^{34} - 3 q^{35} - 5 q^{37} + 9 q^{40} - q^{41} - 3 q^{43} + 9 q^{44} - 12 q^{46} - 10 q^{47} - 4 q^{49} - 4 q^{50} - 6 q^{52} - 3 q^{53} + 5 q^{55} + 12 q^{56} - 24 q^{58} + 4 q^{59} - 9 q^{61} - 20 q^{62} - 3 q^{64} + q^{65} + 7 q^{67} + 3 q^{68} - 4 q^{70} - 6 q^{71} - 9 q^{73} - 2 q^{74} + 2 q^{76} - 17 q^{77} - 11 q^{79} - 6 q^{80} - 16 q^{82} - 30 q^{83} + 11 q^{85} + 11 q^{86} - 25 q^{88} - 13 q^{89} - 5 q^{91} - 10 q^{92} - 25 q^{94} - 8 q^{95} - 7 q^{97} + 10 q^{98}+O(q^{100}) \) 7 * q - 4 * q^2 + 8 * q^4 - 7 * q^5 + 3 * q^7 - 9 * q^8 + 4 * q^10 - 5 * q^11 - q^13 + 4 * q^14 + 6 * q^16 - 11 * q^17 + 8 * q^19 - 8 * q^20 - 3 * q^22 - 11 * q^23 + 7 * q^25 + 5 * q^26 - 17 * q^28 - 3 * q^31 - 22 * q^32 + 4 * q^34 - 3 * q^35 - 5 * q^37 + 9 * q^40 - q^41 - 3 * q^43 + 9 * q^44 - 12 * q^46 - 10 * q^47 - 4 * q^49 - 4 * q^50 - 6 * q^52 - 3 * q^53 + 5 * q^55 + 12 * q^56 - 24 * q^58 + 4 * q^59 - 9 * q^61 - 20 * q^62 - 3 * q^64 + q^65 + 7 * q^67 + 3 * q^68 - 4 * q^70 - 6 * q^71 - 9 * q^73 - 2 * q^74 + 2 * q^76 - 17 * q^77 - 11 * q^79 - 6 * q^80 - 16 * q^82 - 30 * q^83 + 11 * q^85 + 11 * q^86 - 25 * q^88 - 13 * q^89 - 5 * q^91 - 10 * q^92 - 25 * q^94 - 8 * q^95 - 7 * q^97 + 10 * q^98

Introduction

L-functions

Modular forms

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