LMFDB - Embedded newform 6160.2.a.bd.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6160,2,Mod(1,6160)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6160, 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("6160.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6160 = 2^{4} \cdot 5 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6160.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:	$49.1878476451$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.404.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x - 1 $ x^3 - x^2 - 5*x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3080)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.65544$ of defining polynomial
Character	$\chi$	$=$	6160.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.65544 q^{3} -1.00000 q^{5} +1.00000 q^{7} +4.05137 q^{9} +O(q^{10})$	$q-2.65544 q^{3} -1.00000 q^{5} +1.00000 q^{7} +4.05137 q^{9} +1.00000 q^{11} -2.65544 q^{13} +2.65544 q^{15} -5.44731 q^{17} +4.10275 q^{19} -2.65544 q^{21} +0.259511 q^{23} +1.00000 q^{25} -2.79186 q^{27} -2.00000 q^{29} +4.91495 q^{31} -2.65544 q^{33} -1.00000 q^{35} -2.25951 q^{37} +7.05137 q^{39} -1.08505 q^{41} +4.53235 q^{43} -4.05137 q^{45} -4.13642 q^{47} +1.00000 q^{49} +14.4650 q^{51} -4.36226 q^{53} -1.00000 q^{55} -10.8946 q^{57} -5.97966 q^{59} +3.80957 q^{61} +4.05137 q^{63} +2.65544 q^{65} -7.15412 q^{67} -0.689115 q^{69} +16.2055 q^{71} -2.55269 q^{73} -2.65544 q^{75} +1.00000 q^{77} +3.63774 q^{79} -4.74049 q^{81} -6.79186 q^{83} +5.44731 q^{85} +5.31088 q^{87} +5.31088 q^{89} -2.65544 q^{91} -13.0514 q^{93} -4.10275 q^{95} +17.5164 q^{97} +4.05137 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 2 q^{3} - 3 q^{5} + 3 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q - 2 * q^3 - 3 * q^5 + 3 * q^7 + 3 * q^9	$ 3 q - 2 q^{3} - 3 q^{5} + 3 q^{7} + 3 q^{9} + 3 q^{11} - 2 q^{13} + 2 q^{15} - 4 q^{17} - 6 q^{19} - 2 q^{21} - 2 q^{23} + 3 q^{25} - 2 q^{27} - 6 q^{29} + 6 q^{31} - 2 q^{33} - 3 q^{35} - 4 q^{37} + 12 q^{39} - 12 q^{41} + 10 q^{43} - 3 q^{45} - 12 q^{47} + 3 q^{49} + 4 q^{51} + 8 q^{53} - 3 q^{55} - 8 q^{57} - 2 q^{59} - 22 q^{61} + 3 q^{63} + 2 q^{65} + 6 q^{67} - 14 q^{69} + 12 q^{71} - 20 q^{73} - 2 q^{75} + 3 q^{77} + 32 q^{79} - 17 q^{81} - 14 q^{83} + 4 q^{85} + 4 q^{87} + 4 q^{89} - 2 q^{91} - 30 q^{93} + 6 q^{95} + 4 q^{97} + 3 q^{99}+O(q^{100}) $ 3 * q - 2 * q^3 - 3 * q^5 + 3 * q^7 + 3 * q^9 + 3 * q^11 - 2 * q^13 + 2 * q^15 - 4 * q^17 - 6 * q^19 - 2 * q^21 - 2 * q^23 + 3 * q^25 - 2 * q^27 - 6 * q^29 + 6 * q^31 - 2 * q^33 - 3 * q^35 - 4 * q^37 + 12 * q^39 - 12 * q^41 + 10 * q^43 - 3 * q^45 - 12 * q^47 + 3 * q^49 + 4 * q^51 + 8 * q^53 - 3 * q^55 - 8 * q^57 - 2 * q^59 - 22 * q^61 + 3 * q^63 + 2 * q^65 + 6 * q^67 - 14 * q^69 + 12 * q^71 - 20 * q^73 - 2 * q^75 + 3 * q^77 + 32 * q^79 - 17 * q^81 - 14 * q^83 + 4 * q^85 + 4 * q^87 + 4 * q^89 - 2 * q^91 - 30 * q^93 + 6 * q^95 + 4 * q^97 + 3 * 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
$3$	−2.65544	−1.53312	−0.766560	−	0.642172i	$-0.778033\pi$
$3$	−2.65544	−1.53312	−0.766560	+	0.642172i	$0.778033\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	4.05137	1.35046
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−2.65544	−0.736487	−0.368244	−	0.929729i	$-0.620041\pi$
$13$	−2.65544	−0.736487	−0.368244	+	0.929729i	$0.620041\pi$
$14$	0	0
$15$	2.65544	0.685632
$16$	0	0
$17$	−5.44731	−1.32117	−0.660583	−	0.750753i	$-0.729691\pi$
$17$	−5.44731	−1.32117	−0.660583	+	0.750753i	$0.729691\pi$
$18$	0	0
$19$	4.10275	0.941235	0.470618	−	0.882337i	$-0.344031\pi$
$19$	4.10275	0.941235	0.470618	+	0.882337i	$0.344031\pi$
$20$	0	0
$21$	−2.65544	−0.579465
$22$	0	0
$23$	0.259511	0.0541117	0.0270558	−	0.999634i	$-0.491387\pi$
$23$	0.259511	0.0541117	0.0270558	+	0.999634i	$0.491387\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−2.79186	−0.537294
$28$	0	0
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	0	0
$31$	4.91495	0.882752	0.441376	−	0.897322i	$-0.354491\pi$
$31$	4.91495	0.882752	0.441376	+	0.897322i	$0.354491\pi$
$32$	0	0
$33$	−2.65544	−0.462253
$34$	0	0
$35$	−1.00000	−0.169031
$36$	0	0
$37$	−2.25951	−0.371461	−0.185731	−	0.982601i	$-0.559465\pi$
$37$	−2.25951	−0.371461	−0.185731	+	0.982601i	$0.559465\pi$
$38$	0	0
$39$	7.05137	1.12912
$40$	0	0
$41$	−1.08505	−0.169456	−0.0847279	−	0.996404i	$-0.527002\pi$
$41$	−1.08505	−0.169456	−0.0847279	+	0.996404i	$0.527002\pi$
$42$	0	0
$43$	4.53235	0.691177	0.345589	−	0.938386i	$-0.387679\pi$
$43$	4.53235	0.691177	0.345589	+	0.938386i	$0.387679\pi$
$44$	0	0
$45$	−4.05137	−0.603943
$46$	0	0
$47$	−4.13642	−0.603359	−0.301680	−	0.953409i	$-0.597547\pi$
$47$	−4.13642	−0.603359	−0.301680	+	0.953409i	$0.597547\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	14.4650	2.02551
$52$	0	0
$53$	−4.36226	−0.599202	−0.299601	−	0.954065i	$-0.596854\pi$
$53$	−4.36226	−0.599202	−0.299601	+	0.954065i	$0.596854\pi$
$54$	0	0
$55$	−1.00000	−0.134840
$56$	0	0
$57$	−10.8946	−1.44303
$58$	0	0
$59$	−5.97966	−0.778485	−0.389243	−	0.921135i	$-0.627263\pi$
$59$	−5.97966	−0.778485	−0.389243	+	0.921135i	$0.627263\pi$
$60$	0	0
$61$	3.80957	0.487765	0.243882	−	0.969805i	$-0.421579\pi$
$61$	3.80957	0.487765	0.243882	+	0.969805i	$0.421579\pi$
$62$	0	0
$63$	4.05137	0.510425
$64$	0	0
$65$	2.65544	0.329367
$66$	0	0
$67$	−7.15412	−0.874015	−0.437008	−	0.899458i	$-0.643962\pi$
$67$	−7.15412	−0.874015	−0.437008	+	0.899458i	$0.643962\pi$
$68$	0	0
$69$	−0.689115	−0.0829597
$70$	0	0
$71$	16.2055	1.92324	0.961619	−	0.274387i	$-0.0884750\pi$
$71$	16.2055	1.92324	0.961619	+	0.274387i	$0.0884750\pi$
$72$	0	0
$73$	−2.55269	−0.298770	−0.149385	−	0.988779i	$-0.547729\pi$
$73$	−2.55269	−0.298770	−0.149385	+	0.988779i	$0.547729\pi$
$74$	0	0
$75$	−2.65544	−0.306624
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	3.63774	0.409278	0.204639	−	0.978838i	$-0.434398\pi$
$79$	3.63774	0.409278	0.204639	+	0.978838i	$0.434398\pi$
$80$	0	0
$81$	−4.74049	−0.526721
$82$	0	0
$83$	−6.79186	−0.745504	−0.372752	−	0.927931i	$-0.621586\pi$
$83$	−6.79186	−0.745504	−0.372752	+	0.927931i	$0.621586\pi$
$84$	0	0
$85$	5.44731	0.590843
$86$	0	0
$87$	5.31088	0.569387
$88$	0	0
$89$	5.31088	0.562953	0.281476	−	0.959568i	$-0.409176\pi$
$89$	5.31088	0.562953	0.281476	+	0.959568i	$0.409176\pi$
$90$	0	0
$91$	−2.65544	−0.278366
$92$	0	0
$93$	−13.0514	−1.35336
$94$	0	0
$95$	−4.10275	−0.420933
$96$	0	0
$97$	17.5164	1.77852	0.889260	−	0.457403i	$-0.151220\pi$
$97$	17.5164	1.77852	0.889260	+	0.457403i	$0.151220\pi$
$98$	0	0
$99$	4.05137	0.407178
$100$	0	0
$101$	−7.43397	−0.739708	−0.369854	−	0.929090i	$-0.620592\pi$
$101$	−7.43397	−0.739708	−0.369854	+	0.929090i	$0.620592\pi$
$102$	0	0
$103$	−4.23917	−0.417698	−0.208849	−	0.977948i	$-0.566972\pi$
$103$	−4.23917	−0.417698	−0.208849	+	0.977948i	$0.566972\pi$
$104$	0	0
$105$	2.65544	0.259145
$106$	0	0
$107$	0.519021	0.0501757	0.0250878	−	0.999685i	$-0.492013\pi$
$107$	0.519021	0.0501757	0.0250878	+	0.999685i	$0.492013\pi$
$108$	0	0
$109$	3.31088	0.317125	0.158563	−	0.987349i	$-0.449314\pi$
$109$	3.31088	0.317125	0.158563	+	0.987349i	$0.449314\pi$
$110$	0	0
$111$	6.00000	0.569495
$112$	0	0
$113$	11.5164	1.08337	0.541685	−	0.840582i	$-0.317787\pi$
$113$	11.5164	1.08337	0.541685	+	0.840582i	$0.317787\pi$
$114$	0	0
$115$	−0.259511	−0.0241995
$116$	0	0
$117$	−10.7582	−0.994595
$118$	0	0
$119$	−5.44731	−0.499354
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	2.88128	0.259796
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	20.9974	1.86321	0.931607	−	0.363467i	$-0.118407\pi$
$127$	20.9974	1.86321	0.931607	+	0.363467i	$0.118407\pi$
$128$	0	0
$129$	−12.0354	−1.05966
$130$	0	0
$131$	1.58373	0.138371	0.0691855	−	0.997604i	$-0.477960\pi$
$131$	1.58373	0.138371	0.0691855	+	0.997604i	$0.477960\pi$
$132$	0	0
$133$	4.10275	0.355753
$134$	0	0
$135$	2.79186	0.240285
$136$	0	0
$137$	8.15676	0.696879	0.348440	−	0.937331i	$-0.386712\pi$
$137$	8.15676	0.696879	0.348440	+	0.937331i	$0.386712\pi$
$138$	0	0
$139$	−11.9327	−1.01211	−0.506057	−	0.862500i	$-0.668898\pi$
$139$	−11.9327	−1.01211	−0.506057	+	0.862500i	$0.668898\pi$
$140$	0	0
$141$	10.9840	0.925022
$142$	0	0
$143$	−2.65544	−0.222059
$144$	0	0
$145$	2.00000	0.166091
$146$	0	0
$147$	−2.65544	−0.219017
$148$	0	0
$149$	−4.17009	−0.341627	−0.170814	−	0.985303i	$-0.554640\pi$
$149$	−4.17009	−0.341627	−0.170814	+	0.985303i	$0.554640\pi$
$150$	0	0
$151$	21.8786	1.78046	0.890229	−	0.455513i	$-0.150544\pi$
$151$	21.8786	1.78046	0.890229	+	0.455513i	$0.150544\pi$
$152$	0	0
$153$	−22.0691	−1.78418
$154$	0	0
$155$	−4.91495	−0.394779
$156$	0	0
$157$	−12.2055	−0.974105	−0.487052	−	0.873373i	$-0.661928\pi$
$157$	−12.2055	−0.974105	−0.487052	+	0.873373i	$0.661928\pi$
$158$	0	0
$159$	11.5837	0.918649
$160$	0	0
$161$	0.259511	0.0204523
$162$	0	0
$163$	19.1541	1.50027	0.750133	−	0.661287i	$-0.229989\pi$
$163$	19.1541	1.50027	0.750133	+	0.661287i	$0.229989\pi$
$164$	0	0
$165$	2.65544	0.206726
$166$	0	0
$167$	−3.03804	−0.235091	−0.117545	−	0.993068i	$-0.537503\pi$
$167$	−3.03804	−0.235091	−0.117545	+	0.993068i	$0.537503\pi$
$168$	0	0
$169$	−5.94863	−0.457587
$170$	0	0
$171$	16.6218	1.27110
$172$	0	0
$173$	−3.07171	−0.233538	−0.116769	−	0.993159i	$-0.537254\pi$
$173$	−3.07171	−0.233538	−0.116769	+	0.993159i	$0.537254\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	15.8786	1.19351
$178$	0	0
$179$	−21.7626	−1.62661	−0.813305	−	0.581838i	$-0.802334\pi$
$179$	−21.7626	−1.62661	−0.813305	+	0.581838i	$0.802334\pi$
$180$	0	0
$181$	−14.3082	−1.06352	−0.531762	−	0.846894i	$-0.678470\pi$
$181$	−14.3082	−1.06352	−0.531762	+	0.846894i	$0.678470\pi$
$182$	0	0
$183$	−10.1161	−0.747802
$184$	0	0
$185$	2.25951	0.166123
$186$	0	0
$187$	−5.44731	−0.398346
$188$	0	0
$189$	−2.79186	−0.203078
$190$	0	0
$191$	17.3109	1.25257	0.626286	−	0.779594i	$-0.284574\pi$
$191$	17.3109	1.25257	0.626286	+	0.779594i	$0.284574\pi$
$192$	0	0
$193$	−4.42960	−0.318850	−0.159425	−	0.987210i	$-0.550964\pi$
$193$	−4.42960	−0.318850	−0.159425	+	0.987210i	$0.550964\pi$
$194$	0	0
$195$	−7.05137	−0.504959
$196$	0	0
$197$	3.57040	0.254380	0.127190	−	0.991878i	$-0.459404\pi$
$197$	3.57040	0.254380	0.127190	+	0.991878i	$0.459404\pi$
$198$	0	0
$199$	10.0824	0.714723	0.357361	−	0.933966i	$-0.383676\pi$
$199$	10.0824	0.714723	0.357361	+	0.933966i	$0.383676\pi$
$200$	0	0
$201$	18.9974	1.33997
$202$	0	0
$203$	−2.00000	−0.140372
$204$	0	0
$205$	1.08505	0.0757830
$206$	0	0
$207$	1.05137	0.0730756
$208$	0	0
$209$	4.10275	0.283793
$210$	0	0
$211$	−15.9593	−1.09868	−0.549342	−	0.835597i	$-0.685122\pi$
$211$	−15.9593	−1.09868	−0.549342	+	0.835597i	$0.685122\pi$
$212$	0	0
$213$	−43.0328	−2.94856
$214$	0	0
$215$	−4.53235	−0.309104
$216$	0	0
$217$	4.91495	0.333649
$218$	0	0
$219$	6.77853	0.458051
$220$	0	0
$221$	14.4650	0.973022
$222$	0	0
$223$	−21.2772	−1.42483	−0.712414	−	0.701760i	$-0.752398\pi$
$223$	−21.2772	−1.42483	−0.712414	+	0.701760i	$0.752398\pi$
$224$	0	0
$225$	4.05137	0.270092
$226$	0	0
$227$	2.82727	0.187652	0.0938261	−	0.995589i	$-0.470090\pi$
$227$	2.82727	0.187652	0.0938261	+	0.995589i	$0.470090\pi$
$228$	0	0
$229$	2.24618	0.148432	0.0742158	−	0.997242i	$-0.476355\pi$
$229$	2.24618	0.148432	0.0742158	+	0.997242i	$0.476355\pi$
$230$	0	0
$231$	−2.65544	−0.174715
$232$	0	0
$233$	−26.9840	−1.76778	−0.883891	−	0.467692i	$-0.845085\pi$
$233$	−26.9840	−1.76778	−0.883891	+	0.467692i	$0.845085\pi$
$234$	0	0
$235$	4.13642	0.269830
$236$	0	0
$237$	−9.65981	−0.627472
$238$	0	0
$239$	−15.6191	−1.01032	−0.505159	−	0.863026i	$-0.668566\pi$
$239$	−15.6191	−1.01032	−0.505159	+	0.863026i	$0.668566\pi$
$240$	0	0
$241$	2.60143	0.167573	0.0837864	−	0.996484i	$-0.473299\pi$
$241$	2.60143	0.167573	0.0837864	+	0.996484i	$0.473299\pi$
$242$	0	0
$243$	20.9637	1.34482
$244$	0	0
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	−10.8946	−0.693208
$248$	0	0
$249$	18.0354	1.14295
$250$	0	0
$251$	−21.1878	−1.33736	−0.668681	−	0.743549i	$-0.733141\pi$
$251$	−21.1878	−1.33736	−0.668681	+	0.743549i	$0.733141\pi$
$252$	0	0
$253$	0.259511	0.0163153
$254$	0	0
$255$	−14.4650	−0.905834
$256$	0	0
$257$	−1.06471	−0.0664146	−0.0332073	−	0.999448i	$-0.510572\pi$
$257$	−1.06471	−0.0664146	−0.0332073	+	0.999448i	$0.510572\pi$
$258$	0	0
$259$	−2.25951	−0.140399
$260$	0	0
$261$	−8.10275	−0.501548
$262$	0	0
$263$	−24.6572	−1.52043	−0.760213	−	0.649674i	$-0.774906\pi$
$263$	−24.6572	−1.52043	−0.760213	+	0.649674i	$0.774906\pi$
$264$	0	0
$265$	4.36226	0.267971
$266$	0	0
$267$	−14.1027	−0.863074
$268$	0	0
$269$	16.4110	1.00060	0.500298	−	0.865853i	$-0.333224\pi$
$269$	16.4110	1.00060	0.500298	+	0.865853i	$0.333224\pi$
$270$	0	0
$271$	−16.4783	−1.00099	−0.500494	−	0.865740i	$-0.666848\pi$
$271$	−16.4783	−1.00099	−0.500494	+	0.865740i	$0.666848\pi$
$272$	0	0
$273$	7.05137	0.426769
$274$	0	0
$275$	1.00000	0.0603023
$276$	0	0
$277$	5.74049	0.344913	0.172456	−	0.985017i	$-0.444830\pi$
$277$	5.74049	0.344913	0.172456	+	0.985017i	$0.444830\pi$
$278$	0	0
$279$	19.9123	1.19212
$280$	0	0
$281$	−13.6865	−0.816467	−0.408233	−	0.912878i	$-0.633855\pi$
$281$	−13.6865	−0.816467	−0.408233	+	0.912878i	$0.633855\pi$
$282$	0	0
$283$	−11.1408	−0.662251	−0.331126	−	0.943587i	$-0.607428\pi$
$283$	−11.1408	−0.662251	−0.331126	+	0.943587i	$0.607428\pi$
$284$	0	0
$285$	10.8946	0.645341
$286$	0	0
$287$	−1.08505	−0.0640483
$288$	0	0
$289$	12.6731	0.745479
$290$	0	0
$291$	−46.5137	−2.72668
$292$	0	0
$293$	−25.9663	−1.51697	−0.758485	−	0.651691i	$-0.774060\pi$
$293$	−25.9663	−1.51697	−0.758485	+	0.651691i	$0.774060\pi$
$294$	0	0
$295$	5.97966	0.348149
$296$	0	0
$297$	−2.79186	−0.162000
$298$	0	0
$299$	−0.689115	−0.0398526
$300$	0	0
$301$	4.53235	0.261240
$302$	0	0
$303$	19.7405	1.13406
$304$	0	0
$305$	−3.80957	−0.218135
$306$	0	0
$307$	−15.5571	−0.887888	−0.443944	−	0.896054i	$-0.646421\pi$
$307$	−15.5571	−0.887888	−0.443944	+	0.896054i	$0.646421\pi$
$308$	0	0
$309$	11.2569	0.640381
$310$	0	0
$311$	−4.53936	−0.257404	−0.128702	−	0.991683i	$-0.541081\pi$
$311$	−4.53936	−0.257404	−0.128702	+	0.991683i	$0.541081\pi$
$312$	0	0
$313$	−23.7626	−1.34314	−0.671570	−	0.740941i	$-0.734380\pi$
$313$	−23.7626	−1.34314	−0.671570	+	0.740941i	$0.734380\pi$
$314$	0	0
$315$	−4.05137	−0.228269
$316$	0	0
$317$	11.5837	0.650607	0.325303	−	0.945610i	$-0.394534\pi$
$317$	11.5837	0.650607	0.325303	+	0.945610i	$0.394534\pi$
$318$	0	0
$319$	−2.00000	−0.111979
$320$	0	0
$321$	−1.37823	−0.0769253
$322$	0	0
$323$	−22.3489	−1.24353
$324$	0	0
$325$	−2.65544	−0.147297
$326$	0	0
$327$	−8.79186	−0.486191
$328$	0	0
$329$	−4.13642	−0.228048
$330$	0	0
$331$	26.0354	1.43104	0.715518	−	0.698595i	$-0.246191\pi$
$331$	26.0354	1.43104	0.715518	+	0.698595i	$0.246191\pi$
$332$	0	0
$333$	−9.15412	−0.501643
$334$	0	0
$335$	7.15412	0.390871
$336$	0	0
$337$	−3.36490	−0.183298	−0.0916488	−	0.995791i	$-0.529214\pi$
$337$	−3.36490	−0.183298	−0.0916488	+	0.995791i	$0.529214\pi$
$338$	0	0
$339$	−30.5811	−1.66094
$340$	0	0
$341$	4.91495	0.266160
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0.689115	0.0371007
$346$	0	0
$347$	−17.0514	−0.915366	−0.457683	−	0.889116i	$-0.651320\pi$
$347$	−17.0514	−0.915366	−0.457683	+	0.889116i	$0.651320\pi$
$348$	0	0
$349$	−14.4987	−0.776097	−0.388048	−	0.921639i	$-0.626851\pi$
$349$	−14.4987	−0.776097	−0.388048	+	0.921639i	$0.626851\pi$
$350$	0	0
$351$	7.41363	0.395710
$352$	0	0
$353$	−1.14079	−0.0607182	−0.0303591	−	0.999539i	$-0.509665\pi$
$353$	−1.14079	−0.0607182	−0.0303591	+	0.999539i	$0.509665\pi$
$354$	0	0
$355$	−16.2055	−0.860098
$356$	0	0
$357$	14.4650	0.765569
$358$	0	0
$359$	11.5297	0.608515	0.304258	−	0.952590i	$-0.401592\pi$
$359$	11.5297	0.608515	0.304258	+	0.952590i	$0.401592\pi$
$360$	0	0
$361$	−2.16745	−0.114077
$362$	0	0
$363$	−2.65544	−0.139375
$364$	0	0
$365$	2.55269	0.133614
$366$	0	0
$367$	−26.2392	−1.36967	−0.684837	−	0.728697i	$-0.740127\pi$
$367$	−26.2392	−1.36967	−0.684837	+	0.728697i	$0.740127\pi$
$368$	0	0
$369$	−4.39593	−0.228843
$370$	0	0
$371$	−4.36226	−0.226477
$372$	0	0
$373$	−30.1922	−1.56329	−0.781646	−	0.623723i	$-0.785619\pi$
$373$	−30.1922	−1.56329	−0.781646	+	0.623723i	$0.785619\pi$
$374$	0	0
$375$	2.65544	0.137126
$376$	0	0
$377$	5.31088	0.273524
$378$	0	0
$379$	6.62177	0.340137	0.170069	−	0.985432i	$-0.445601\pi$
$379$	6.62177	0.340137	0.170069	+	0.985432i	$0.445601\pi$
$380$	0	0
$381$	−55.7573	−2.85653
$382$	0	0
$383$	31.7910	1.62444	0.812221	−	0.583350i	$-0.198258\pi$
$383$	31.7910	1.62444	0.812221	+	0.583350i	$0.198258\pi$
$384$	0	0
$385$	−1.00000	−0.0509647
$386$	0	0
$387$	18.3623	0.933406
$388$	0	0
$389$	−22.9707	−1.16466	−0.582330	−	0.812952i	$-0.697859\pi$
$389$	−22.9707	−1.16466	−0.582330	+	0.812952i	$0.697859\pi$
$390$	0	0
$391$	−1.41363	−0.0714905
$392$	0	0
$393$	−4.20550	−0.212139
$394$	0	0
$395$	−3.63774	−0.183035
$396$	0	0
$397$	−20.7245	−1.04013	−0.520067	−	0.854126i	$-0.674093\pi$
$397$	−20.7245	−1.04013	−0.520067	+	0.854126i	$0.674093\pi$
$398$	0	0
$399$	−10.8946	−0.545413
$400$	0	0
$401$	3.24354	0.161975	0.0809873	−	0.996715i	$-0.474193\pi$
$401$	3.24354	0.161975	0.0809873	+	0.996715i	$0.474193\pi$
$402$	0	0
$403$	−13.0514	−0.650135
$404$	0	0
$405$	4.74049	0.235557
$406$	0	0
$407$	−2.25951	−0.112000
$408$	0	0
$409$	−2.77152	−0.137043	−0.0685215	−	0.997650i	$-0.521828\pi$
$409$	−2.77152	−0.137043	−0.0685215	+	0.997650i	$0.521828\pi$
$410$	0	0
$411$	−21.6598	−1.06840
$412$	0	0
$413$	−5.97966	−0.294240
$414$	0	0
$415$	6.79186	0.333399
$416$	0	0
$417$	31.6865	1.55169
$418$	0	0
$419$	−27.2586	−1.33167	−0.665835	−	0.746099i	$-0.731924\pi$
$419$	−27.2586	−1.33167	−0.665835	+	0.746099i	$0.731924\pi$
$420$	0	0
$421$	−5.63774	−0.274767	−0.137383	−	0.990518i	$-0.543869\pi$
$421$	−5.63774	−0.274767	−0.137383	+	0.990518i	$0.543869\pi$
$422$	0	0
$423$	−16.7582	−0.814811
$424$	0	0
$425$	−5.44731	−0.264233
$426$	0	0
$427$	3.80957	0.184358
$428$	0	0
$429$	7.05137	0.340444
$430$	0	0
$431$	−14.1515	−0.681653	−0.340826	−	0.940126i	$-0.610707\pi$
$431$	−14.1515	−0.681653	−0.340826	+	0.940126i	$0.610707\pi$
$432$	0	0
$433$	−13.7272	−0.659685	−0.329843	−	0.944036i	$-0.606996\pi$
$433$	−13.7272	−0.659685	−0.329843	+	0.944036i	$0.606996\pi$
$434$	0	0
$435$	−5.31088	−0.254637
$436$	0	0
$437$	1.06471	0.0509318
$438$	0	0
$439$	−19.8246	−0.946178	−0.473089	−	0.881015i	$-0.656861\pi$
$439$	−19.8246	−0.946178	−0.473089	+	0.881015i	$0.656861\pi$
$440$	0	0
$441$	4.05137	0.192923
$442$	0	0
$443$	−7.74049	−0.367762	−0.183881	−	0.982949i	$-0.558866\pi$
$443$	−7.74049	−0.367762	−0.183881	+	0.982949i	$0.558866\pi$
$444$	0	0
$445$	−5.31088	−0.251760
$446$	0	0
$447$	11.0734	0.523756
$448$	0	0
$449$	12.7785	0.603056	0.301528	−	0.953457i	$-0.402503\pi$
$449$	12.7785	0.603056	0.301528	+	0.953457i	$0.402503\pi$
$450$	0	0
$451$	−1.08505	−0.0510929
$452$	0	0
$453$	−58.0975	−2.72966
$454$	0	0
$455$	2.65544	0.124489
$456$	0	0
$457$	2.19216	0.102545	0.0512726	−	0.998685i	$-0.483672\pi$
$457$	2.19216	0.102545	0.0512726	+	0.998685i	$0.483672\pi$
$458$	0	0
$459$	15.2081	0.709855
$460$	0	0
$461$	14.0824	0.655883	0.327942	−	0.944698i	$-0.393645\pi$
$461$	14.0824	0.655883	0.327942	+	0.944698i	$0.393645\pi$
$462$	0	0
$463$	−28.6032	−1.32930	−0.664651	−	0.747154i	$-0.731420\pi$
$463$	−28.6032	−1.32930	−0.664651	+	0.747154i	$0.731420\pi$
$464$	0	0
$465$	13.0514	0.605243
$466$	0	0
$467$	−5.13906	−0.237807	−0.118904	−	0.992906i	$-0.537938\pi$
$467$	−5.13906	−0.237807	−0.118904	+	0.992906i	$0.537938\pi$
$468$	0	0
$469$	−7.15412	−0.330347
$470$	0	0
$471$	32.4110	1.49342
$472$	0	0
$473$	4.53235	0.208398
$474$	0	0
$475$	4.10275	0.188247
$476$	0	0
$477$	−17.6731	−0.809198
$478$	0	0
$479$	20.6218	0.942233	0.471116	−	0.882071i	$-0.343851\pi$
$479$	20.6218	0.942233	0.471116	+	0.882071i	$0.343851\pi$
$480$	0	0
$481$	6.00000	0.273576
$482$	0	0
$483$	−0.689115	−0.0313558
$484$	0	0
$485$	−17.5164	−0.795378
$486$	0	0
$487$	−10.0221	−0.454143	−0.227072	−	0.973878i	$-0.572915\pi$
$487$	−10.0221	−0.454143	−0.227072	+	0.973878i	$0.572915\pi$
$488$	0	0
$489$	−50.8627	−2.30009
$490$	0	0
$491$	24.2276	1.09337	0.546687	−	0.837337i	$-0.315889\pi$
$491$	24.2276	1.09337	0.546687	+	0.837337i	$0.315889\pi$
$492$	0	0
$493$	10.8946	0.490669
$494$	0	0
$495$	−4.05137	−0.182096
$496$	0	0
$497$	16.2055	0.726916
$498$	0	0
$499$	31.3463	1.40325	0.701626	−	0.712545i	$-0.252458\pi$
$499$	31.3463	1.40325	0.701626	+	0.712545i	$0.252458\pi$
$500$	0	0
$501$	8.06735	0.360422
$502$	0	0
$503$	−37.4136	−1.66819	−0.834096	−	0.551620i	$-0.814010\pi$
$503$	−37.4136	−1.66819	−0.834096	+	0.551620i	$0.814010\pi$
$504$	0	0
$505$	7.43397	0.330807
$506$	0	0
$507$	15.7962	0.701535
$508$	0	0
$509$	−23.7219	−1.05145	−0.525727	−	0.850653i	$-0.676207\pi$
$509$	−23.7219	−1.05145	−0.525727	+	0.850653i	$0.676207\pi$
$510$	0	0
$511$	−2.55269	−0.112925
$512$	0	0
$513$	−11.4543	−0.505720
$514$	0	0
$515$	4.23917	0.186800
$516$	0	0
$517$	−4.13642	−0.181920
$518$	0	0
$519$	8.15676	0.358042
$520$	0	0
$521$	−3.97334	−0.174075	−0.0870375	−	0.996205i	$-0.527740\pi$
$521$	−3.97334	−0.174075	−0.0870375	+	0.996205i	$0.527740\pi$
$522$	0	0
$523$	27.7626	1.21397	0.606986	−	0.794713i	$-0.292378\pi$
$523$	27.7626	1.21397	0.606986	+	0.794713i	$0.292378\pi$
$524$	0	0
$525$	−2.65544	−0.115893
$526$	0	0
$527$	−26.7733	−1.16626
$528$	0	0
$529$	−22.9327	−0.997072
$530$	0	0
$531$	−24.2258	−1.05131
$532$	0	0
$533$	2.88128	0.124802
$534$	0	0
$535$	−0.519021	−0.0224392
$536$	0	0
$537$	57.7892	2.49379
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	2.79186	0.120032	0.0600158	−	0.998197i	$-0.480885\pi$
$541$	2.79186	0.120032	0.0600158	+	0.998197i	$0.480885\pi$
$542$	0	0
$543$	37.9947	1.63051
$544$	0	0
$545$	−3.31088	−0.141823
$546$	0	0
$547$	−2.48362	−0.106192	−0.0530959	−	0.998589i	$-0.516909\pi$
$547$	−2.48362	−0.106192	−0.0530959	+	0.998589i	$0.516909\pi$
$548$	0	0
$549$	15.4340	0.658706
$550$	0	0
$551$	−8.20550	−0.349566
$552$	0	0
$553$	3.63774	0.154692
$554$	0	0
$555$	−6.00000	−0.254686
$556$	0	0
$557$	14.4650	0.612902	0.306451	−	0.951886i	$-0.400858\pi$
$557$	14.4650	0.612902	0.306451	+	0.951886i	$0.400858\pi$
$558$	0	0
$559$	−12.0354	−0.509043
$560$	0	0
$561$	14.4650	0.610713
$562$	0	0
$563$	16.6572	0.702016	0.351008	−	0.936372i	$-0.385839\pi$
$563$	16.6572	0.702016	0.351008	+	0.936372i	$0.385839\pi$
$564$	0	0
$565$	−11.5164	−0.484498
$566$	0	0
$567$	−4.74049	−0.199082
$568$	0	0
$569$	−5.72716	−0.240095	−0.120047	−	0.992768i	$-0.538305\pi$
$569$	−5.72716	−0.240095	−0.120047	+	0.992768i	$0.538305\pi$
$570$	0	0
$571$	28.1648	1.17866	0.589330	−	0.807892i	$-0.299392\pi$
$571$	28.1648	1.17866	0.589330	+	0.807892i	$0.299392\pi$
$572$	0	0
$573$	−45.9681	−1.92034
$574$	0	0
$575$	0.259511	0.0108223
$576$	0	0
$577$	14.1701	0.589909	0.294954	−	0.955511i	$-0.404696\pi$
$577$	14.1701	0.589909	0.294954	+	0.955511i	$0.404696\pi$
$578$	0	0
$579$	11.7626	0.488835
$580$	0	0
$581$	−6.79186	−0.281774
$582$	0	0
$583$	−4.36226	−0.180666
$584$	0	0
$585$	10.7582	0.444796
$586$	0	0
$587$	−15.1071	−0.623537	−0.311769	−	0.950158i	$-0.600921\pi$
$587$	−15.1071	−0.623537	−0.311769	+	0.950158i	$0.600921\pi$
$588$	0	0
$589$	20.1648	0.830877
$590$	0	0
$591$	−9.48098	−0.389995
$592$	0	0
$593$	3.96633	0.162878	0.0814388	−	0.996678i	$-0.474048\pi$
$593$	3.96633	0.162878	0.0814388	+	0.996678i	$0.474048\pi$
$594$	0	0
$595$	5.44731	0.223318
$596$	0	0
$597$	−26.7733	−1.09576
$598$	0	0
$599$	−14.1027	−0.576223	−0.288111	−	0.957597i	$-0.593027\pi$
$599$	−14.1027	−0.576223	−0.288111	+	0.957597i	$0.593027\pi$
$600$	0	0
$601$	−4.22584	−0.172376	−0.0861878	−	0.996279i	$-0.527468\pi$
$601$	−4.22584	−0.172376	−0.0861878	+	0.996279i	$0.527468\pi$
$602$	0	0
$603$	−28.9840	−1.18032
$604$	0	0
$605$	−1.00000	−0.0406558
$606$	0	0
$607$	38.4110	1.55905	0.779527	−	0.626369i	$-0.215459\pi$
$607$	38.4110	1.55905	0.779527	+	0.626369i	$0.215459\pi$
$608$	0	0
$609$	5.31088	0.215208
$610$	0	0
$611$	10.9840	0.444366
$612$	0	0
$613$	18.8052	0.759535	0.379767	−	0.925082i	$-0.376004\pi$
$613$	18.8052	0.759535	0.379767	+	0.925082i	$0.376004\pi$
$614$	0	0
$615$	−2.88128	−0.116184
$616$	0	0
$617$	6.77853	0.272893	0.136447	−	0.990647i	$-0.456432\pi$
$617$	6.77853	0.272893	0.136447	+	0.990647i	$0.456432\pi$
$618$	0	0
$619$	18.5341	0.744948	0.372474	−	0.928043i	$-0.378510\pi$
$619$	18.5341	0.744948	0.372474	+	0.928043i	$0.378510\pi$
$620$	0	0
$621$	−0.724518	−0.0290739
$622$	0	0
$623$	5.31088	0.212776
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−10.8946	−0.435089
$628$	0	0
$629$	12.3082	0.490762
$630$	0	0
$631$	−13.9947	−0.557121	−0.278561	−	0.960419i	$-0.589857\pi$
$631$	−13.9947	−0.557121	−0.278561	+	0.960419i	$0.589857\pi$
$632$	0	0
$633$	42.3791	1.68442
$634$	0	0
$635$	−20.9974	−0.833255
$636$	0	0
$637$	−2.65544	−0.105212
$638$	0	0
$639$	65.6545	2.59725
$640$	0	0
$641$	−28.9840	−1.14480	−0.572400	−	0.819974i	$-0.693988\pi$
$641$	−28.9840	−1.14480	−0.572400	+	0.819974i	$0.693988\pi$
$642$	0	0
$643$	17.2011	0.678346	0.339173	−	0.940724i	$-0.389853\pi$
$643$	17.2011	0.678346	0.339173	+	0.940724i	$0.389853\pi$
$644$	0	0
$645$	12.0354	0.473894
$646$	0	0
$647$	−8.38260	−0.329554	−0.164777	−	0.986331i	$-0.552690\pi$
$647$	−8.38260	−0.329554	−0.164777	+	0.986331i	$0.552690\pi$
$648$	0	0
$649$	−5.97966	−0.234722
$650$	0	0
$651$	−13.0514	−0.511524
$652$	0	0
$653$	0.348927	0.0136546	0.00682728	−	0.999977i	$-0.497827\pi$
$653$	0.348927	0.0136546	0.00682728	+	0.999977i	$0.497827\pi$
$654$	0	0
$655$	−1.58373	−0.0618814
$656$	0	0
$657$	−10.3419	−0.403477
$658$	0	0
$659$	−19.7492	−0.769321	−0.384660	−	0.923058i	$-0.625681\pi$
$659$	−19.7492	−0.769321	−0.384660	+	0.923058i	$0.625681\pi$
$660$	0	0
$661$	−1.33755	−0.0520246	−0.0260123	−	0.999662i	$-0.508281\pi$
$661$	−1.33755	−0.0520246	−0.0260123	+	0.999662i	$0.508281\pi$
$662$	0	0
$663$	−38.4110	−1.49176
$664$	0	0
$665$	−4.10275	−0.159098
$666$	0	0
$667$	−0.519021	−0.0200966
$668$	0	0
$669$	56.5004	2.18443
$670$	0	0
$671$	3.80957	0.147067
$672$	0	0
$673$	−51.1488	−1.97164	−0.985822	−	0.167797i	$-0.946335\pi$
$673$	−51.1488	−1.97164	−0.985822	+	0.167797i	$0.946335\pi$
$674$	0	0
$675$	−2.79186	−0.107459
$676$	0	0
$677$	22.9016	0.880181	0.440090	−	0.897953i	$-0.354946\pi$
$677$	22.9016	0.880181	0.440090	+	0.897953i	$0.354946\pi$
$678$	0	0
$679$	17.5164	0.672217
$680$	0	0
$681$	−7.50764	−0.287694
$682$	0	0
$683$	−43.8520	−1.67795	−0.838975	−	0.544171i	$-0.816844\pi$
$683$	−43.8520	−1.67795	−0.838975	+	0.544171i	$0.816844\pi$
$684$	0	0
$685$	−8.15676	−0.311654
$686$	0	0
$687$	−5.96460	−0.227564
$688$	0	0
$689$	11.5837	0.441305
$690$	0	0
$691$	−16.9504	−0.644822	−0.322411	−	0.946600i	$-0.604493\pi$
$691$	−16.9504	−0.644822	−0.322411	+	0.946600i	$0.604493\pi$
$692$	0	0
$693$	4.05137	0.153899
$694$	0	0
$695$	11.9327	0.452631
$696$	0	0
$697$	5.91058	0.223879
$698$	0	0
$699$	71.6545	2.71022
$700$	0	0
$701$	7.96460	0.300819	0.150409	−	0.988624i	$-0.451941\pi$
$701$	7.96460	0.300819	0.150409	+	0.988624i	$0.451941\pi$
$702$	0	0
$703$	−9.27020	−0.349632
$704$	0	0
$705$	−10.9840	−0.413682
$706$	0	0
$707$	−7.43397	−0.279583
$708$	0	0
$709$	32.3349	1.21436	0.607182	−	0.794563i	$-0.292300\pi$
$709$	32.3349	1.21436	0.607182	+	0.794563i	$0.292300\pi$
$710$	0	0
$711$	14.7379	0.552713
$712$	0	0
$713$	1.27548	0.0477672
$714$	0	0
$715$	2.65544	0.0993079
$716$	0	0
$717$	41.4757	1.54894
$718$	0	0
$719$	34.2206	1.27621	0.638106	−	0.769949i	$-0.279718\pi$
$719$	34.2206	1.27621	0.638106	+	0.769949i	$0.279718\pi$
$720$	0	0
$721$	−4.23917	−0.157875
$722$	0	0
$723$	−6.90794	−0.256909
$724$	0	0
$725$	−2.00000	−0.0742781
$726$	0	0
$727$	27.2772	1.01166	0.505828	−	0.862634i	$-0.331187\pi$
$727$	27.2772	1.01166	0.505828	+	0.862634i	$0.331187\pi$
$728$	0	0
$729$	−41.4464	−1.53505
$730$	0	0
$731$	−24.6891	−0.913160
$732$	0	0
$733$	25.9256	0.957586	0.478793	−	0.877928i	$-0.341075\pi$
$733$	25.9256	0.957586	0.478793	+	0.877928i	$0.341075\pi$
$734$	0	0
$735$	2.65544	0.0979475
$736$	0	0
$737$	−7.15412	−0.263525
$738$	0	0
$739$	20.5457	0.755785	0.377893	−	0.925849i	$-0.376649\pi$
$739$	20.5457	0.755785	0.377893	+	0.925849i	$0.376649\pi$
$740$	0	0
$741$	28.9300	1.06277
$742$	0	0
$743$	−42.3082	−1.55214	−0.776069	−	0.630647i	$-0.782789\pi$
$743$	−42.3082	−1.55214	−0.776069	+	0.630647i	$0.782789\pi$
$744$	0	0
$745$	4.17009	0.152780
$746$	0	0
$747$	−27.5164	−1.00677
$748$	0	0
$749$	0.519021	0.0189646
$750$	0	0
$751$	−37.9274	−1.38399	−0.691995	−	0.721902i	$-0.743268\pi$
$751$	−37.9274	−1.38399	−0.691995	+	0.721902i	$0.743268\pi$
$752$	0	0
$753$	56.2630	2.05034
$754$	0	0
$755$	−21.8786	−0.796245
$756$	0	0
$757$	36.3756	1.32209	0.661047	−	0.750345i	$-0.270113\pi$
$757$	36.3756	1.32209	0.661047	+	0.750345i	$0.270113\pi$
$758$	0	0
$759$	−0.689115	−0.0250133
$760$	0	0
$761$	29.6661	1.07540	0.537698	−	0.843137i	$-0.319294\pi$
$761$	29.6661	1.07540	0.537698	+	0.843137i	$0.319294\pi$
$762$	0	0
$763$	3.31088	0.119862
$764$	0	0
$765$	22.0691	0.797909
$766$	0	0
$767$	15.8786	0.573344
$768$	0	0
$769$	−35.9797	−1.29746	−0.648730	−	0.761019i	$-0.724699\pi$
$769$	−35.9797	−1.29746	−0.648730	+	0.761019i	$0.724699\pi$
$770$	0	0
$771$	2.82727	0.101822
$772$	0	0
$773$	36.3436	1.30719	0.653595	−	0.756844i	$-0.273260\pi$
$773$	36.3436	1.30719	0.653595	+	0.756844i	$0.273260\pi$
$774$	0	0
$775$	4.91495	0.176550
$776$	0	0
$777$	6.00000	0.215249
$778$	0	0
$779$	−4.45168	−0.159498
$780$	0	0
$781$	16.2055	0.579878
$782$	0	0
$783$	5.58373	0.199546
$784$	0	0
$785$	12.2055	0.435633
$786$	0	0
$787$	−24.1648	−0.861383	−0.430691	−	0.902499i	$-0.641730\pi$
$787$	−24.1648	−0.861383	−0.430691	+	0.902499i	$0.641730\pi$
$788$	0	0
$789$	65.4757	2.33100
$790$	0	0
$791$	11.5164	0.409475
$792$	0	0
$793$	−10.1161	−0.359233
$794$	0	0
$795$	−11.5837	−0.410832
$796$	0	0
$797$	−3.79450	−0.134408	−0.0672041	−	0.997739i	$-0.521408\pi$
$797$	−3.79450	−0.134408	−0.0672041	+	0.997739i	$0.521408\pi$
$798$	0	0
$799$	22.5324	0.797137
$800$	0	0
$801$	21.5164	0.760244
$802$	0	0
$803$	−2.55269	−0.0900826
$804$	0	0
$805$	−0.259511	−0.00914654
$806$	0	0
$807$	−43.5784	−1.53403
$808$	0	0
$809$	−10.9300	−0.384279	−0.192139	−	0.981368i	$-0.561543\pi$
$809$	−10.9300	−0.384279	−0.192139	+	0.981368i	$0.561543\pi$
$810$	0	0
$811$	−3.31088	−0.116261	−0.0581304	−	0.998309i	$-0.518514\pi$
$811$	−3.31088	−0.116261	−0.0581304	+	0.998309i	$0.518514\pi$
$812$	0	0
$813$	43.7573	1.53463
$814$	0	0
$815$	−19.1541	−0.670940
$816$	0	0
$817$	18.5951	0.650560
$818$	0	0
$819$	−10.7582	−0.375922
$820$	0	0
$821$	−39.8566	−1.39100	−0.695502	−	0.718524i	$-0.744818\pi$
$821$	−39.8566	−1.39100	−0.695502	+	0.718524i	$0.744818\pi$
$822$	0	0
$823$	−3.18079	−0.110875	−0.0554376	−	0.998462i	$-0.517655\pi$
$823$	−3.18079	−0.110875	−0.0554376	+	0.998462i	$0.517655\pi$
$824$	0	0
$825$	−2.65544	−0.0924506
$826$	0	0
$827$	46.7059	1.62412	0.812062	−	0.583571i	$-0.198345\pi$
$827$	46.7059	1.62412	0.812062	+	0.583571i	$0.198345\pi$
$828$	0	0
$829$	−33.0328	−1.14728	−0.573638	−	0.819109i	$-0.694468\pi$
$829$	−33.0328	−1.14728	−0.573638	+	0.819109i	$0.694468\pi$
$830$	0	0
$831$	−15.2435	−0.528793
$832$	0	0
$833$	−5.44731	−0.188738
$834$	0	0
$835$	3.03804	0.105136
$836$	0	0
$837$	−13.7219	−0.474298
$838$	0	0
$839$	−40.0505	−1.38270	−0.691348	−	0.722522i	$-0.742983\pi$
$839$	−40.0505	−1.38270	−0.691348	+	0.722522i	$0.742983\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	36.3436	1.25174
$844$	0	0
$845$	5.94863	0.204639
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	29.5837	1.01531
$850$	0	0
$851$	−0.586367	−0.0201004
$852$	0	0
$853$	21.2099	0.726212	0.363106	−	0.931748i	$-0.381716\pi$
$853$	21.2099	0.726212	0.363106	+	0.931748i	$0.381716\pi$
$854$	0	0
$855$	−16.6218	−0.568453
$856$	0	0
$857$	−20.3686	−0.695778	−0.347889	−	0.937536i	$-0.613101\pi$
$857$	−20.3686	−0.695778	−0.347889	+	0.937536i	$0.613101\pi$
$858$	0	0
$859$	14.9910	0.511488	0.255744	−	0.966745i	$-0.417680\pi$
$859$	14.9910	0.511488	0.255744	+	0.966745i	$0.417680\pi$
$860$	0	0
$861$	2.88128	0.0981938
$862$	0	0
$863$	−12.9840	−0.441981	−0.220991	−	0.975276i	$-0.570929\pi$
$863$	−12.9840	−0.441981	−0.220991	+	0.975276i	$0.570929\pi$
$864$	0	0
$865$	3.07171	0.104441
$866$	0	0
$867$	−33.6528	−1.14291
$868$	0	0
$869$	3.63774	0.123402
$870$	0	0
$871$	18.9974	0.643701
$872$	0	0
$873$	70.9654	2.40182
$874$	0	0
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	−52.1462	−1.76085	−0.880426	−	0.474183i	$-0.842743\pi$
$877$	−52.1462	−1.76085	−0.880426	+	0.474183i	$0.842743\pi$
$878$	0	0
$879$	68.9521	2.32570
$880$	0	0
$881$	−23.6865	−0.798018	−0.399009	−	0.916947i	$-0.630646\pi$
$881$	−23.6865	−0.798018	−0.399009	+	0.916947i	$0.630646\pi$
$882$	0	0
$883$	−22.5004	−0.757199	−0.378600	−	0.925561i	$-0.623594\pi$
$883$	−22.5004	−0.757199	−0.378600	+	0.925561i	$0.623594\pi$
$884$	0	0
$885$	−15.8786	−0.533755
$886$	0	0
$887$	12.1382	0.407559	0.203780	−	0.979017i	$-0.434677\pi$
$887$	12.1382	0.407559	0.203780	+	0.979017i	$0.434677\pi$
$888$	0	0
$889$	20.9974	0.704229
$890$	0	0
$891$	−4.74049	−0.158812
$892$	0	0
$893$	−16.9707	−0.567903
$894$	0	0
$895$	21.7626	0.727442
$896$	0	0
$897$	1.82991	0.0610988
$898$	0	0
$899$	−9.82991	−0.327846
$900$	0	0
$901$	23.7626	0.791646
$902$	0	0
$903$	−12.0354	−0.400513
$904$	0	0
$905$	14.3082	0.475622
$906$	0	0
$907$	−19.6058	−0.651000	−0.325500	−	0.945542i	$-0.605533\pi$
$907$	−19.6058	−0.651000	−0.325500	+	0.945542i	$0.605533\pi$
$908$	0	0
$909$	−30.1178	−0.998945
$910$	0	0
$911$	47.5251	1.57458	0.787289	−	0.616584i	$-0.211484\pi$
$911$	47.5251	1.57458	0.787289	+	0.616584i	$0.211484\pi$
$912$	0	0
$913$	−6.79186	−0.224778
$914$	0	0
$915$	10.1161	0.334427
$916$	0	0
$917$	1.58373	0.0522993
$918$	0	0
$919$	8.38087	0.276459	0.138230	−	0.990400i	$-0.455859\pi$
$919$	8.38087	0.276459	0.138230	+	0.990400i	$0.455859\pi$
$920$	0	0
$921$	41.3109	1.36124
$922$	0	0
$923$	−43.0328	−1.41644
$924$	0	0
$925$	−2.25951	−0.0742922
$926$	0	0
$927$	−17.1745	−0.564083
$928$	0	0
$929$	5.61039	0.184071	0.0920355	−	0.995756i	$-0.470663\pi$
$929$	5.61039	0.184071	0.0920355	+	0.995756i	$0.470663\pi$
$930$	0	0
$931$	4.10275	0.134462
$932$	0	0
$933$	12.0540	0.394631
$934$	0	0
$935$	5.44731	0.178146
$936$	0	0
$937$	43.5448	1.42255	0.711273	−	0.702916i	$-0.248119\pi$
$937$	43.5448	1.42255	0.711273	+	0.702916i	$0.248119\pi$
$938$	0	0
$939$	63.1001	2.05919
$940$	0	0
$941$	−12.3959	−0.404096	−0.202048	−	0.979376i	$-0.564760\pi$
$941$	−12.3959	−0.404096	−0.202048	+	0.979376i	$0.564760\pi$
$942$	0	0
$943$	−0.281581	−0.00916954
$944$	0	0
$945$	2.79186	0.0908193
$946$	0	0
$947$	16.4917	0.535907	0.267954	−	0.963432i	$-0.413653\pi$
$947$	16.4917	0.535907	0.267954	+	0.963432i	$0.413653\pi$
$948$	0	0
$949$	6.77853	0.220040
$950$	0	0
$951$	−30.7599	−0.997459
$952$	0	0
$953$	−12.5944	−0.407973	−0.203987	−	0.978974i	$-0.565390\pi$
$953$	−12.5944	−0.407973	−0.203987	+	0.978974i	$0.565390\pi$
$954$	0	0
$955$	−17.3109	−0.560167
$956$	0	0
$957$	5.31088	0.171677
$958$	0	0
$959$	8.15676	0.263396
$960$	0	0
$961$	−6.84324	−0.220750
$962$	0	0
$963$	2.10275	0.0677601
$964$	0	0
$965$	4.42960	0.142594
$966$	0	0
$967$	−6.42960	−0.206762	−0.103381	−	0.994642i	$-0.532966\pi$
$967$	−6.42960	−0.206762	−0.103381	+	0.994642i	$0.532966\pi$
$968$	0	0
$969$	59.3463	1.90648
$970$	0	0
$971$	−30.3959	−0.975452	−0.487726	−	0.872997i	$-0.662173\pi$
$971$	−30.3959	−0.975452	−0.487726	+	0.872997i	$0.662173\pi$
$972$	0	0
$973$	−11.9327	−0.382543
$974$	0	0
$975$	7.05137	0.225825
$976$	0	0
$977$	10.5597	0.337835	0.168917	−	0.985630i	$-0.445973\pi$
$977$	10.5597	0.337835	0.168917	+	0.985630i	$0.445973\pi$
$978$	0	0
$979$	5.31088	0.169737
$980$	0	0
$981$	13.4136	0.428264
$982$	0	0
$983$	−55.8670	−1.78188	−0.890941	−	0.454119i	$-0.849954\pi$
$983$	−55.8670	−1.78188	−0.890941	+	0.454119i	$0.849954\pi$
$984$	0	0
$985$	−3.57040	−0.113762
$986$	0	0
$987$	10.9840	0.349625
$988$	0	0
$989$	1.17619	0.0374008
$990$	0	0
$991$	50.0354	1.58943	0.794713	−	0.606985i	$-0.207621\pi$
$991$	50.0354	1.58943	0.794713	+	0.606985i	$0.207621\pi$
$992$	0	0
$993$	−69.1355	−2.19395
$994$	0	0
$995$	−10.0824	−0.319634
$996$	0	0
$997$	−36.5474	−1.15747	−0.578734	−	0.815516i	$-0.696453\pi$
$997$	−36.5474	−1.15747	−0.578734	+	0.815516i	$0.696453\pi$
$998$	0	0
$999$	6.30825	0.199584

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6160.2.a.bd.1.1		3
4.3	odd	2		3080.2.a.m.1.3	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3080.2.a.m.1.3	✓	3	4.3	odd	2
6160.2.a.bd.1.1		3	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:	\( N \)	\(=\)	\( 6160 = 2^{4} \cdot 5 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6160.a (trivial)

\(f(q)\)	\(=\)	\(q-2.65544 q^{3} -1.00000 q^{5} +1.00000 q^{7} +4.05137 q^{9} +O(q^{10})\)	\(q-2.65544 q^{3} -1.00000 q^{5} +1.00000 q^{7} +4.05137 q^{9} +1.00000 q^{11} -2.65544 q^{13} +2.65544 q^{15} -5.44731 q^{17} +4.10275 q^{19} -2.65544 q^{21} +0.259511 q^{23} +1.00000 q^{25} -2.79186 q^{27} -2.00000 q^{29} +4.91495 q^{31} -2.65544 q^{33} -1.00000 q^{35} -2.25951 q^{37} +7.05137 q^{39} -1.08505 q^{41} +4.53235 q^{43} -4.05137 q^{45} -4.13642 q^{47} +1.00000 q^{49} +14.4650 q^{51} -4.36226 q^{53} -1.00000 q^{55} -10.8946 q^{57} -5.97966 q^{59} +3.80957 q^{61} +4.05137 q^{63} +2.65544 q^{65} -7.15412 q^{67} -0.689115 q^{69} +16.2055 q^{71} -2.55269 q^{73} -2.65544 q^{75} +1.00000 q^{77} +3.63774 q^{79} -4.74049 q^{81} -6.79186 q^{83} +5.44731 q^{85} +5.31088 q^{87} +5.31088 q^{89} -2.65544 q^{91} -13.0514 q^{93} -4.10275 q^{95} +17.5164 q^{97} +4.05137 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 2 q^{3} - 3 q^{5} + 3 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q - 2 * q^3 - 3 * q^5 + 3 * q^7 + 3 * q^9	\( 3 q - 2 q^{3} - 3 q^{5} + 3 q^{7} + 3 q^{9} + 3 q^{11} - 2 q^{13} + 2 q^{15} - 4 q^{17} - 6 q^{19} - 2 q^{21} - 2 q^{23} + 3 q^{25} - 2 q^{27} - 6 q^{29} + 6 q^{31} - 2 q^{33} - 3 q^{35} - 4 q^{37} + 12 q^{39} - 12 q^{41} + 10 q^{43} - 3 q^{45} - 12 q^{47} + 3 q^{49} + 4 q^{51} + 8 q^{53} - 3 q^{55} - 8 q^{57} - 2 q^{59} - 22 q^{61} + 3 q^{63} + 2 q^{65} + 6 q^{67} - 14 q^{69} + 12 q^{71} - 20 q^{73} - 2 q^{75} + 3 q^{77} + 32 q^{79} - 17 q^{81} - 14 q^{83} + 4 q^{85} + 4 q^{87} + 4 q^{89} - 2 q^{91} - 30 q^{93} + 6 q^{95} + 4 q^{97} + 3 q^{99}+O(q^{100}) \) 3 * q - 2 * q^3 - 3 * q^5 + 3 * q^7 + 3 * q^9 + 3 * q^11 - 2 * q^13 + 2 * q^15 - 4 * q^17 - 6 * q^19 - 2 * q^21 - 2 * q^23 + 3 * q^25 - 2 * q^27 - 6 * q^29 + 6 * q^31 - 2 * q^33 - 3 * q^35 - 4 * q^37 + 12 * q^39 - 12 * q^41 + 10 * q^43 - 3 * q^45 - 12 * q^47 + 3 * q^49 + 4 * q^51 + 8 * q^53 - 3 * q^55 - 8 * q^57 - 2 * q^59 - 22 * q^61 + 3 * q^63 + 2 * q^65 + 6 * q^67 - 14 * q^69 + 12 * q^71 - 20 * q^73 - 2 * q^75 + 3 * q^77 + 32 * q^79 - 17 * q^81 - 14 * q^83 + 4 * q^85 + 4 * q^87 + 4 * q^89 - 2 * q^91 - 30 * q^93 + 6 * q^95 + 4 * q^97 + 3 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more