LMFDB - Embedded newform 6160.2.a.bx.1.3

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:	$0$
Dimension:	$5$
Coefficient field:	5.5.15785648.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 12x^{3} + 22x^{2} + 24x - 28 $ x^5 - 2x^4 - 12x^3 + 22x^2 + 24x - 28
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 3080)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.840204$ 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)

$f(q)$	$=$	$q-0.840204 q^{3} +1.00000 q^{5} -1.00000 q^{7} -2.29406 q^{9} +O(q^{10})$	$q-0.840204 q^{3} +1.00000 q^{5} -1.00000 q^{7} -2.29406 q^{9} -1.00000 q^{11} +4.71947 q^{13} -0.840204 q^{15} +2.71947 q^{17} -4.56882 q^{19} +0.840204 q^{21} +6.85373 q^{23} +1.00000 q^{25} +4.44809 q^{27} +3.55967 q^{29} -2.57459 q^{31} +0.840204 q^{33} -1.00000 q^{35} +5.73439 q^{37} -3.96531 q^{39} -10.2628 q^{41} -9.42255 q^{43} -2.29406 q^{45} +5.17909 q^{47} +1.00000 q^{49} -2.28491 q^{51} -4.85373 q^{53} -1.00000 q^{55} +3.83874 q^{57} -1.56544 q^{59} +2.57459 q^{61} +2.29406 q^{63} +4.71947 q^{65} -6.97447 q^{67} -5.75853 q^{69} -9.70746 q^{71} -2.08028 q^{73} -0.840204 q^{75} +1.00000 q^{77} +4.72524 q^{79} +3.14488 q^{81} +3.31044 q^{83} +2.71947 q^{85} -2.99085 q^{87} -5.56743 q^{89} -4.71947 q^{91} +2.16318 q^{93} -4.56882 q^{95} +15.1271 q^{97} +2.29406 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 2 q^{3} + 5 q^{5} - 5 q^{7} + 13 q^{9}+O(q^{10}) $ 5 * q - 2 * q^3 + 5 * q^5 - 5 * q^7 + 13 * q^9	$ 5 q - 2 q^{3} + 5 q^{5} - 5 q^{7} + 13 q^{9} - 5 q^{11} + 12 q^{13} - 2 q^{15} + 2 q^{17} - 8 q^{19} + 2 q^{21} - 4 q^{23} + 5 q^{25} - 2 q^{27} + 4 q^{29} + 2 q^{33} - 5 q^{35} + 18 q^{37} + 8 q^{39} + 4 q^{41} + 6 q^{43} + 13 q^{45} - 6 q^{47} + 5 q^{49} + 12 q^{51} + 14 q^{53} - 5 q^{55} - 12 q^{57} + 4 q^{59} - 13 q^{63} + 12 q^{65} - 6 q^{67} - 10 q^{69} + 28 q^{71} + 10 q^{73} - 2 q^{75} + 5 q^{77} + 14 q^{79} + 17 q^{81} + 22 q^{83} + 2 q^{85} - 16 q^{87} + 24 q^{89} - 12 q^{91} + 10 q^{93} - 8 q^{95} + 10 q^{97} - 13 q^{99}+O(q^{100}) $ 5 * q - 2 * q^3 + 5 * q^5 - 5 * q^7 + 13 * q^9 - 5 * q^11 + 12 * q^13 - 2 * q^15 + 2 * q^17 - 8 * q^19 + 2 * q^21 - 4 * q^23 + 5 * q^25 - 2 * q^27 + 4 * q^29 + 2 * q^33 - 5 * q^35 + 18 * q^37 + 8 * q^39 + 4 * q^41 + 6 * q^43 + 13 * q^45 - 6 * q^47 + 5 * q^49 + 12 * q^51 + 14 * q^53 - 5 * q^55 - 12 * q^57 + 4 * q^59 - 13 * q^63 + 12 * q^65 - 6 * q^67 - 10 * q^69 + 28 * q^71 + 10 * q^73 - 2 * q^75 + 5 * q^77 + 14 * q^79 + 17 * q^81 + 22 * q^83 + 2 * q^85 - 16 * q^87 + 24 * q^89 - 12 * q^91 + 10 * q^93 - 8 * q^95 + 10 * q^97 - 13 * 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
$3$	−0.840204	−0.485092	−0.242546	−	0.970140i	$-0.577982\pi$
$3$	−0.840204	−0.485092	−0.242546	+	0.970140i	$0.577982\pi$
$4$	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
$9$	−2.29406	−0.764686
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	4.71947	1.30894	0.654472	−	0.756086i	$-0.272891\pi$
$13$	4.71947	1.30894	0.654472	+	0.756086i	$0.272891\pi$
$14$	0	0
$15$	−0.840204	−0.216940
$16$	0	0
$17$	2.71947	0.659568	0.329784	−	0.944056i	$-0.393024\pi$
$17$	2.71947	0.659568	0.329784	+	0.944056i	$0.393024\pi$
$18$	0	0
$19$	−4.56882	−1.04816	−0.524080	−	0.851669i	$-0.675591\pi$
$19$	−4.56882	−1.04816	−0.524080	+	0.851669i	$0.675591\pi$
$20$	0	0
$21$	0.840204	0.183347
$22$	0	0
$23$	6.85373	1.42910	0.714551	−	0.699584i	$-0.246631\pi$
$23$	6.85373	1.42910	0.714551	+	0.699584i	$0.246631\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	4.44809	0.856035
$28$	0	0
$29$	3.55967	0.661014	0.330507	−	0.943803i	$-0.392780\pi$
$29$	3.55967	0.661014	0.330507	+	0.943803i	$0.392780\pi$
$30$	0	0
$31$	−2.57459	−0.462410	−0.231205	−	0.972905i	$-0.574267\pi$
$31$	−2.57459	−0.462410	−0.231205	+	0.972905i	$0.574267\pi$
$32$	0	0
$33$	0.840204	0.146261
$34$	0	0
$35$	−1.00000	−0.169031
$36$	0	0
$37$	5.73439	0.942727	0.471364	−	0.881939i	$-0.343762\pi$
$37$	5.73439	0.942727	0.471364	+	0.881939i	$0.343762\pi$
$38$	0	0
$39$	−3.96531	−0.634958
$40$	0	0
$41$	−10.2628	−1.60277	−0.801387	−	0.598147i	$-0.795904\pi$
$41$	−10.2628	−1.60277	−0.801387	+	0.598147i	$0.795904\pi$
$42$	0	0
$43$	−9.42255	−1.43693	−0.718463	−	0.695565i	$-0.755154\pi$
$43$	−9.42255	−1.43693	−0.718463	+	0.695565i	$0.755154\pi$
$44$	0	0
$45$	−2.29406	−0.341978
$46$	0	0
$47$	5.17909	0.755448	0.377724	−	0.925918i	$-0.376707\pi$
$47$	5.17909	0.755448	0.377724	+	0.925918i	$0.376707\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−2.28491	−0.319951
$52$	0	0
$53$	−4.85373	−0.666711	−0.333355	−	0.942801i	$-0.608181\pi$
$53$	−4.85373	−0.666711	−0.333355	+	0.942801i	$0.608181\pi$
$54$	0	0
$55$	−1.00000	−0.134840
$56$	0	0
$57$	3.83874	0.508454
$58$	0	0
$59$	−1.56544	−0.203803	−0.101901	−	0.994795i	$-0.532493\pi$
$59$	−1.56544	−0.203803	−0.101901	+	0.994795i	$0.532493\pi$
$60$	0	0
$61$	2.57459	0.329643	0.164821	−	0.986323i	$-0.447295\pi$
$61$	2.57459	0.329643	0.164821	+	0.986323i	$0.447295\pi$
$62$	0	0
$63$	2.29406	0.289024
$64$	0	0
$65$	4.71947	0.585378
$66$	0	0
$67$	−6.97447	−0.852067	−0.426033	−	0.904707i	$-0.640089\pi$
$67$	−6.97447	−0.852067	−0.426033	+	0.904707i	$0.640089\pi$
$68$	0	0
$69$	−5.75853	−0.693245
$70$	0	0
$71$	−9.70746	−1.15206	−0.576032	−	0.817427i	$-0.695399\pi$
$71$	−9.70746	−1.15206	−0.576032	+	0.817427i	$0.695399\pi$
$72$	0	0
$73$	−2.08028	−0.243479	−0.121739	−	0.992562i	$-0.538847\pi$
$73$	−2.08028	−0.243479	−0.121739	+	0.992562i	$0.538847\pi$
$74$	0	0
$75$	−0.840204	−0.0970183
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	4.72524	0.531631	0.265815	−	0.964024i	$-0.414359\pi$
$79$	4.72524	0.531631	0.265815	+	0.964024i	$0.414359\pi$
$80$	0	0
$81$	3.14488	0.349431
$82$	0	0
$83$	3.31044	0.363368	0.181684	−	0.983357i	$-0.441845\pi$
$83$	3.31044	0.363368	0.181684	+	0.983357i	$0.441845\pi$
$84$	0	0
$85$	2.71947	0.294968
$86$	0	0
$87$	−2.99085	−0.320653
$88$	0	0
$89$	−5.56743	−0.590146	−0.295073	−	0.955475i	$-0.595344\pi$
$89$	−5.56743	−0.590146	−0.295073	+	0.955475i	$0.595344\pi$
$90$	0	0
$91$	−4.71947	−0.494735
$92$	0	0
$93$	2.16318	0.224311
$94$	0	0
$95$	−4.56882	−0.468751
$96$	0	0
$97$	15.1271	1.53592	0.767962	−	0.640495i	$-0.221271\pi$
$97$	15.1271	1.53592	0.767962	+	0.640495i	$0.221271\pi$
$98$	0	0
$99$	2.29406	0.230562
$100$	0	0
$101$	12.4616	1.23998	0.619988	−	0.784611i	$-0.287137\pi$
$101$	12.4616	1.23998	0.619988	+	0.784611i	$0.287137\pi$
$102$	0	0
$103$	19.5646	1.92775	0.963877	−	0.266347i	$-0.0858167\pi$
$103$	19.5646	1.92775	0.963877	+	0.266347i	$0.0858167\pi$
$104$	0	0
$105$	0.840204	0.0819955
$106$	0	0
$107$	−13.0362	−1.26026	−0.630129	−	0.776491i	$-0.716998\pi$
$107$	−13.0362	−1.26026	−0.630129	+	0.776491i	$0.716998\pi$
$108$	0	0
$109$	18.9475	1.81484	0.907422	−	0.420220i	$-0.138047\pi$
$109$	18.9475	1.81484	0.907422	+	0.420220i	$0.138047\pi$
$110$	0	0
$111$	−4.81805	−0.457309
$112$	0	0
$113$	1.77006	0.166514	0.0832568	−	0.996528i	$-0.473468\pi$
$113$	1.77006	0.166514	0.0832568	+	0.996528i	$0.473468\pi$
$114$	0	0
$115$	6.85373	0.639113
$116$	0	0
$117$	−10.8267	−1.00093
$118$	0	0
$119$	−2.71947	−0.249293
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	8.62280	0.777492
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−1.18195	−0.104881	−0.0524404	−	0.998624i	$-0.516700\pi$
$127$	−1.18195	−0.104881	−0.0524404	+	0.998624i	$0.516700\pi$
$128$	0	0
$129$	7.91686	0.697041
$130$	0	0
$131$	−8.78697	−0.767721	−0.383861	−	0.923391i	$-0.625406\pi$
$131$	−8.78697	−0.767721	−0.383861	+	0.923391i	$0.625406\pi$
$132$	0	0
$133$	4.56882	0.396167
$134$	0	0
$135$	4.44809	0.382830
$136$	0	0
$137$	6.76283	0.577788	0.288894	−	0.957361i	$-0.406713\pi$
$137$	6.76283	0.577788	0.288894	+	0.957361i	$0.406713\pi$
$138$	0	0
$139$	−0.410489	−0.0348172	−0.0174086	−	0.999848i	$-0.505542\pi$
$139$	−0.410489	−0.0348172	−0.0174086	+	0.999848i	$0.505542\pi$
$140$	0	0
$141$	−4.35149	−0.366462
$142$	0	0
$143$	−4.71947	−0.394662
$144$	0	0
$145$	3.55967	0.295615
$146$	0	0
$147$	−0.840204	−0.0692988
$148$	0	0
$149$	15.8282	1.29670	0.648348	−	0.761344i	$-0.275460\pi$
$149$	15.8282	1.29670	0.648348	+	0.761344i	$0.275460\pi$
$150$	0	0
$151$	−5.59436	−0.455262	−0.227631	−	0.973747i	$-0.573098\pi$
$151$	−5.59436	−0.455262	−0.227631	+	0.973747i	$0.573098\pi$
$152$	0	0
$153$	−6.23862	−0.504362
$154$	0	0
$155$	−2.57459	−0.206796
$156$	0	0
$157$	6.45962	0.515534	0.257767	−	0.966207i	$-0.417013\pi$
$157$	6.45962	0.515534	0.257767	+	0.966207i	$0.417013\pi$
$158$	0	0
$159$	4.07812	0.323416
$160$	0	0
$161$	−6.85373	−0.540149
$162$	0	0
$163$	19.2315	1.50632	0.753162	−	0.657835i	$-0.228528\pi$
$163$	19.2315	1.50632	0.753162	+	0.657835i	$0.228528\pi$
$164$	0	0
$165$	0.840204	0.0654098
$166$	0	0
$167$	15.8360	1.22542	0.612711	−	0.790307i	$-0.290079\pi$
$167$	15.8360	1.22542	0.612711	+	0.790307i	$0.290079\pi$
$168$	0	0
$169$	9.27337	0.713336
$170$	0	0
$171$	10.4811	0.801513
$172$	0	0
$173$	11.6272	0.883998	0.441999	−	0.897016i	$-0.354270\pi$
$173$	11.6272	0.883998	0.441999	+	0.897016i	$0.354270\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	1.31529	0.0988630
$178$	0	0
$179$	17.6754	1.32112	0.660560	−	0.750773i	$-0.270319\pi$
$179$	17.6754	1.32112	0.660560	+	0.750773i	$0.270319\pi$
$180$	0	0
$181$	−21.8360	−1.62305	−0.811527	−	0.584315i	$-0.801363\pi$
$181$	−21.8360	−1.62305	−0.811527	+	0.584315i	$0.801363\pi$
$182$	0	0
$183$	−2.16318	−0.159907
$184$	0	0
$185$	5.73439	0.421601
$186$	0	0
$187$	−2.71947	−0.198867
$188$	0	0
$189$	−4.44809	−0.323551
$190$	0	0
$191$	8.51000	0.615762	0.307881	−	0.951425i	$-0.400380\pi$
$191$	8.51000	0.615762	0.307881	+	0.951425i	$0.400380\pi$
$192$	0	0
$193$	−11.3146	−0.814442	−0.407221	−	0.913330i	$-0.633502\pi$
$193$	−11.3146	−0.814442	−0.407221	+	0.913330i	$0.633502\pi$
$194$	0	0
$195$	−3.96531	−0.283962
$196$	0	0
$197$	−19.9603	−1.42211	−0.711056	−	0.703135i	$-0.751783\pi$
$197$	−19.9603	−1.42211	−0.711056	+	0.703135i	$0.751783\pi$
$198$	0	0
$199$	19.1627	1.35841	0.679204	−	0.733949i	$-0.262325\pi$
$199$	19.1627	1.35841	0.679204	+	0.733949i	$0.262325\pi$
$200$	0	0
$201$	5.85997	0.413330
$202$	0	0
$203$	−3.55967	−0.249840
$204$	0	0
$205$	−10.2628	−0.716782
$206$	0	0
$207$	−15.7229	−1.09281
$208$	0	0
$209$	4.56882	0.316032
$210$	0	0
$211$	12.7997	0.881171	0.440586	−	0.897711i	$-0.354771\pi$
$211$	12.7997	0.881171	0.440586	+	0.897711i	$0.354771\pi$
$212$	0	0
$213$	8.15624	0.558856
$214$	0	0
$215$	−9.42255	−0.642613
$216$	0	0
$217$	2.57459	0.174775
$218$	0	0
$219$	1.74786	0.118109
$220$	0	0
$221$	12.8344	0.863338
$222$	0	0
$223$	−23.4681	−1.57154	−0.785772	−	0.618517i	$-0.787734\pi$
$223$	−23.4681	−1.57154	−0.785772	+	0.618517i	$0.787734\pi$
$224$	0	0
$225$	−2.29406	−0.152937
$226$	0	0
$227$	9.59950	0.637141	0.318570	−	0.947899i	$-0.396797\pi$
$227$	9.59950	0.637141	0.318570	+	0.947899i	$0.396797\pi$
$228$	0	0
$229$	15.8204	1.04544	0.522722	−	0.852503i	$-0.324917\pi$
$229$	15.8204	1.04544	0.522722	+	0.852503i	$0.324917\pi$
$230$	0	0
$231$	−0.840204	−0.0552813
$232$	0	0
$233$	−26.3806	−1.72825	−0.864126	−	0.503275i	$-0.832128\pi$
$233$	−26.3806	−1.72825	−0.864126	+	0.503275i	$0.832128\pi$
$234$	0	0
$235$	5.17909	0.337847
$236$	0	0
$237$	−3.97016	−0.257890
$238$	0	0
$239$	−2.70885	−0.175221	−0.0876106	−	0.996155i	$-0.527923\pi$
$239$	−2.70885	−0.175221	−0.0876106	+	0.996155i	$0.527923\pi$
$240$	0	0
$241$	9.46301	0.609566	0.304783	−	0.952422i	$-0.401416\pi$
$241$	9.46301	0.609566	0.304783	+	0.952422i	$0.401416\pi$
$242$	0	0
$243$	−15.9866	−1.02554
$244$	0	0
$245$	1.00000	0.0638877
$246$	0	0
$247$	−21.5624	−1.37198
$248$	0	0
$249$	−2.78144	−0.176267
$250$	0	0
$251$	−13.5925	−0.857950	−0.428975	−	0.903316i	$-0.641125\pi$
$251$	−13.5925	−0.857950	−0.428975	+	0.903316i	$0.641125\pi$
$252$	0	0
$253$	−6.85373	−0.430890
$254$	0	0
$255$	−2.28491	−0.143086
$256$	0	0
$257$	25.7961	1.60912	0.804559	−	0.593873i	$-0.202402\pi$
$257$	25.7961	1.60912	0.804559	+	0.593873i	$0.202402\pi$
$258$	0	0
$259$	−5.73439	−0.356317
$260$	0	0
$261$	−8.16609	−0.505468
$262$	0	0
$263$	20.3168	1.25279	0.626394	−	0.779507i	$-0.284530\pi$
$263$	20.3168	1.25279	0.626394	+	0.779507i	$0.284530\pi$
$264$	0	0
$265$	−4.85373	−0.298162
$266$	0	0
$267$	4.67777	0.286275
$268$	0	0
$269$	4.55828	0.277923	0.138961	−	0.990298i	$-0.455624\pi$
$269$	4.55828	0.277923	0.138961	+	0.990298i	$0.455624\pi$
$270$	0	0
$271$	17.2982	1.05079	0.525396	−	0.850858i	$-0.323917\pi$
$271$	17.2982	1.05079	0.525396	+	0.850858i	$0.323917\pi$
$272$	0	0
$273$	3.96531	0.239992
$274$	0	0
$275$	−1.00000	−0.0603023
$276$	0	0
$277$	25.2913	1.51960	0.759802	−	0.650154i	$-0.225296\pi$
$277$	25.2913	1.51960	0.759802	+	0.650154i	$0.225296\pi$
$278$	0	0
$279$	5.90626	0.353599
$280$	0	0
$281$	15.3580	0.916183	0.458091	−	0.888905i	$-0.348533\pi$
$281$	15.3580	0.916183	0.458091	+	0.888905i	$0.348533\pi$
$282$	0	0
$283$	−11.8665	−0.705389	−0.352695	−	0.935738i	$-0.614735\pi$
$283$	−11.8665	−0.705389	−0.352695	+	0.935738i	$0.614735\pi$
$284$	0	0
$285$	3.83874	0.227387
$286$	0	0
$287$	10.2628	0.605791
$288$	0	0
$289$	−9.60450	−0.564971
$290$	0	0
$291$	−12.7098	−0.745064
$292$	0	0
$293$	24.0127	1.40283	0.701417	−	0.712751i	$-0.252551\pi$
$293$	24.0127	1.40283	0.701417	+	0.712751i	$0.252551\pi$
$294$	0	0
$295$	−1.56544	−0.0911434
$296$	0	0
$297$	−4.44809	−0.258104
$298$	0	0
$299$	32.3459	1.87061
$300$	0	0
$301$	9.42255	0.543107
$302$	0	0
$303$	−10.4703	−0.601502
$304$	0	0
$305$	2.57459	0.147421
$306$	0	0
$307$	−1.48931	−0.0849993	−0.0424996	−	0.999096i	$-0.513532\pi$
$307$	−1.48931	−0.0849993	−0.0424996	+	0.999096i	$0.513532\pi$
$308$	0	0
$309$	−16.4382	−0.935138
$310$	0	0
$311$	8.97161	0.508733	0.254367	−	0.967108i	$-0.418133\pi$
$311$	8.97161	0.508733	0.254367	+	0.967108i	$0.418133\pi$
$312$	0	0
$313$	20.4558	1.15623	0.578116	−	0.815954i	$-0.303788\pi$
$313$	20.4558	1.15623	0.578116	+	0.815954i	$0.303788\pi$
$314$	0	0
$315$	2.29406	0.129256
$316$	0	0
$317$	18.8877	1.06084	0.530420	−	0.847735i	$-0.322034\pi$
$317$	18.8877	1.06084	0.530420	+	0.847735i	$0.322034\pi$
$318$	0	0
$319$	−3.55967	−0.199303
$320$	0	0
$321$	10.9531	0.611340
$322$	0	0
$323$	−12.4248	−0.691332
$324$	0	0
$325$	4.71947	0.261789
$326$	0	0
$327$	−15.9198	−0.880366
$328$	0	0
$329$	−5.17909	−0.285532
$330$	0	0
$331$	31.5048	1.73166	0.865831	−	0.500337i	$-0.166791\pi$
$331$	31.5048	1.73166	0.865831	+	0.500337i	$0.166791\pi$
$332$	0	0
$333$	−13.1550	−0.720891
$334$	0	0
$335$	−6.97447	−0.381056
$336$	0	0
$337$	8.07829	0.440053	0.220026	−	0.975494i	$-0.429386\pi$
$337$	8.07829	0.440053	0.220026	+	0.975494i	$0.429386\pi$
$338$	0	0
$339$	−1.48721	−0.0807744
$340$	0	0
$341$	2.57459	0.139422
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−5.75853	−0.310029
$346$	0	0
$347$	11.1145	0.596657	0.298329	−	0.954463i	$-0.403571\pi$
$347$	11.1145	0.596657	0.298329	+	0.954463i	$0.403571\pi$
$348$	0	0
$349$	−13.1512	−0.703966	−0.351983	−	0.936006i	$-0.614493\pi$
$349$	−13.1512	−0.703966	−0.351983	+	0.936006i	$0.614493\pi$
$350$	0	0
$351$	20.9926	1.12050
$352$	0	0
$353$	−20.0581	−1.06759	−0.533793	−	0.845615i	$-0.679234\pi$
$353$	−20.0581	−1.06759	−0.533793	+	0.845615i	$0.679234\pi$
$354$	0	0
$355$	−9.70746	−0.515218
$356$	0	0
$357$	2.28491	0.120930
$358$	0	0
$359$	30.0119	1.58397	0.791984	−	0.610542i	$-0.209048\pi$
$359$	30.0119	1.58397	0.791984	+	0.610542i	$0.209048\pi$
$360$	0	0
$361$	1.87414	0.0986390
$362$	0	0
$363$	−0.840204	−0.0440992
$364$	0	0
$365$	−2.08028	−0.108887
$366$	0	0
$367$	−29.7046	−1.55057	−0.775284	−	0.631613i	$-0.782393\pi$
$367$	−29.7046	−1.55057	−0.775284	+	0.631613i	$0.782393\pi$
$368$	0	0
$369$	23.5434	1.22562
$370$	0	0
$371$	4.85373	0.251993
$372$	0	0
$373$	−13.3722	−0.692385	−0.346193	−	0.938163i	$-0.612526\pi$
$373$	−13.3722	−0.692385	−0.346193	+	0.938163i	$0.612526\pi$
$374$	0	0
$375$	−0.840204	−0.0433879
$376$	0	0
$377$	16.7997	0.865231
$378$	0	0
$379$	−7.07106	−0.363216	−0.181608	−	0.983371i	$-0.558130\pi$
$379$	−7.07106	−0.363216	−0.181608	+	0.983371i	$0.558130\pi$
$380$	0	0
$381$	0.993076	0.0508768
$382$	0	0
$383$	−6.09859	−0.311623	−0.155812	−	0.987787i	$-0.549799\pi$
$383$	−6.09859	−0.311623	−0.155812	+	0.987787i	$0.549799\pi$
$384$	0	0
$385$	1.00000	0.0509647
$386$	0	0
$387$	21.6159	1.09880
$388$	0	0
$389$	−18.0426	−0.914794	−0.457397	−	0.889263i	$-0.651218\pi$
$389$	−18.0426	−0.914794	−0.457397	+	0.889263i	$0.651218\pi$
$390$	0	0
$391$	18.6385	0.942589
$392$	0	0
$393$	7.38285	0.372415
$394$	0	0
$395$	4.72524	0.237752
$396$	0	0
$397$	7.87151	0.395060	0.197530	−	0.980297i	$-0.436708\pi$
$397$	7.87151	0.395060	0.197530	+	0.980297i	$0.436708\pi$
$398$	0	0
$399$	−3.83874	−0.192177
$400$	0	0
$401$	−18.9282	−0.945231	−0.472616	−	0.881269i	$-0.656690\pi$
$401$	−18.9282	−0.945231	−0.472616	+	0.881269i	$0.656690\pi$
$402$	0	0
$403$	−12.1507	−0.605269
$404$	0	0
$405$	3.14488	0.156270
$406$	0	0
$407$	−5.73439	−0.284243
$408$	0	0
$409$	−24.2833	−1.20073	−0.600366	−	0.799726i	$-0.704978\pi$
$409$	−24.2833	−1.20073	−0.600366	+	0.799726i	$0.704978\pi$
$410$	0	0
$411$	−5.68216	−0.280280
$412$	0	0
$413$	1.56544	0.0770302
$414$	0	0
$415$	3.31044	0.162503
$416$	0	0
$417$	0.344894	0.0168895
$418$	0	0
$419$	26.0733	1.27377	0.636883	−	0.770960i	$-0.280223\pi$
$419$	26.0733	1.27377	0.636883	+	0.770960i	$0.280223\pi$
$420$	0	0
$421$	28.4012	1.38419	0.692094	−	0.721807i	$-0.256688\pi$
$421$	28.4012	1.38419	0.692094	+	0.721807i	$0.256688\pi$
$422$	0	0
$423$	−11.8811	−0.577680
$424$	0	0
$425$	2.71947	0.131914
$426$	0	0
$427$	−2.57459	−0.124593
$428$	0	0
$429$	3.96531	0.191447
$430$	0	0
$431$	−5.27197	−0.253942	−0.126971	−	0.991906i	$-0.540526\pi$
$431$	−5.27197	−0.253942	−0.126971	+	0.991906i	$0.540526\pi$
$432$	0	0
$433$	−37.1862	−1.78706	−0.893528	−	0.449008i	$-0.851777\pi$
$433$	−37.1862	−1.78706	−0.893528	+	0.449008i	$0.851777\pi$
$434$	0	0
$435$	−2.99085	−0.143400
$436$	0	0
$437$	−31.3135	−1.49793
$438$	0	0
$439$	17.6476	0.842276	0.421138	−	0.906997i	$-0.361631\pi$
$439$	17.6476	0.842276	0.421138	+	0.906997i	$0.361631\pi$
$440$	0	0
$441$	−2.29406	−0.109241
$442$	0	0
$443$	−33.4972	−1.59150	−0.795750	−	0.605626i	$-0.792923\pi$
$443$	−33.4972	−1.59150	−0.795750	+	0.605626i	$0.792923\pi$
$444$	0	0
$445$	−5.56743	−0.263921
$446$	0	0
$447$	−13.2989	−0.629017
$448$	0	0
$449$	20.1121	0.949149	0.474575	−	0.880215i	$-0.342602\pi$
$449$	20.1121	0.949149	0.474575	+	0.880215i	$0.342602\pi$
$450$	0	0
$451$	10.2628	0.483254
$452$	0	0
$453$	4.70040	0.220844
$454$	0	0
$455$	−4.71947	−0.221252
$456$	0	0
$457$	−31.7188	−1.48374	−0.741872	−	0.670542i	$-0.766062\pi$
$457$	−31.7188	−1.48374	−0.741872	+	0.670542i	$0.766062\pi$
$458$	0	0
$459$	12.0964	0.564613
$460$	0	0
$461$	−33.9121	−1.57944	−0.789722	−	0.613465i	$-0.789775\pi$
$461$	−33.9121	−1.57944	−0.789722	+	0.613465i	$0.789775\pi$
$462$	0	0
$463$	−2.10226	−0.0977003	−0.0488501	−	0.998806i	$-0.515556\pi$
$463$	−2.10226	−0.0977003	−0.0488501	+	0.998806i	$0.515556\pi$
$464$	0	0
$465$	2.16318	0.100315
$466$	0	0
$467$	5.97785	0.276622	0.138311	−	0.990389i	$-0.455833\pi$
$467$	5.97785	0.276622	0.138311	+	0.990389i	$0.455833\pi$
$468$	0	0
$469$	6.97447	0.322051
$470$	0	0
$471$	−5.42740	−0.250081
$472$	0	0
$473$	9.42255	0.433249
$474$	0	0
$475$	−4.56882	−0.209632
$476$	0	0
$477$	11.1347	0.509824
$478$	0	0
$479$	−14.6270	−0.668323	−0.334161	−	0.942516i	$-0.608453\pi$
$479$	−14.6270	−0.668323	−0.334161	+	0.942516i	$0.608453\pi$
$480$	0	0
$481$	27.0633	1.23398
$482$	0	0
$483$	5.75853	0.262022
$484$	0	0
$485$	15.1271	0.686886
$486$	0	0
$487$	−5.38774	−0.244142	−0.122071	−	0.992521i	$-0.538954\pi$
$487$	−5.38774	−0.244142	−0.122071	+	0.992521i	$0.538954\pi$
$488$	0	0
$489$	−16.1583	−0.730705
$490$	0	0
$491$	26.3235	1.18796	0.593981	−	0.804479i	$-0.297555\pi$
$491$	26.3235	1.18796	0.593981	+	0.804479i	$0.297555\pi$
$492$	0	0
$493$	9.68041	0.435984
$494$	0	0
$495$	2.29406	0.103110
$496$	0	0
$497$	9.70746	0.435439
$498$	0	0
$499$	13.8320	0.619204	0.309602	−	0.950866i	$-0.399804\pi$
$499$	13.8320	0.619204	0.309602	+	0.950866i	$0.399804\pi$
$500$	0	0
$501$	−13.3054	−0.594442
$502$	0	0
$503$	17.3443	0.773342	0.386671	−	0.922218i	$-0.373625\pi$
$503$	17.3443	0.773342	0.386671	+	0.922218i	$0.373625\pi$
$504$	0	0
$505$	12.4616	0.554534
$506$	0	0
$507$	−7.79152	−0.346033
$508$	0	0
$509$	2.16056	0.0957652	0.0478826	−	0.998853i	$-0.484753\pi$
$509$	2.16056	0.0957652	0.0478826	+	0.998853i	$0.484753\pi$
$510$	0	0
$511$	2.08028	0.0920262
$512$	0	0
$513$	−20.3225	−0.897261
$514$	0	0
$515$	19.5646	0.862118
$516$	0	0
$517$	−5.17909	−0.227776
$518$	0	0
$519$	−9.76919	−0.428820
$520$	0	0
$521$	−6.43655	−0.281990	−0.140995	−	0.990010i	$-0.545030\pi$
$521$	−6.43655	−0.281990	−0.140995	+	0.990010i	$0.545030\pi$
$522$	0	0
$523$	34.1826	1.49470	0.747350	−	0.664430i	$-0.231326\pi$
$523$	34.1826	1.49470	0.747350	+	0.664430i	$0.231326\pi$
$524$	0	0
$525$	0.840204	0.0366695
$526$	0	0
$527$	−7.00152	−0.304991
$528$	0	0
$529$	23.9736	1.04233
$530$	0	0
$531$	3.59121	0.155845
$532$	0	0
$533$	−48.4347	−2.09794
$534$	0	0
$535$	−13.0362	−0.563604
$536$	0	0
$537$	−14.8509	−0.640865
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	−8.95033	−0.384805	−0.192402	−	0.981316i	$-0.561628\pi$
$541$	−8.95033	−0.384805	−0.192402	+	0.981316i	$0.561628\pi$
$542$	0	0
$543$	18.3466	0.787330
$544$	0	0
$545$	18.9475	0.811623
$546$	0	0
$547$	30.5824	1.30761	0.653805	−	0.756664i	$-0.273172\pi$
$547$	30.5824	1.30761	0.653805	+	0.756664i	$0.273172\pi$
$548$	0	0
$549$	−5.90626	−0.252073
$550$	0	0
$551$	−16.2635	−0.692849
$552$	0	0
$553$	−4.72524	−0.200937
$554$	0	0
$555$	−4.81805	−0.204515
$556$	0	0
$557$	45.2971	1.91930	0.959650	−	0.281199i	$-0.0907319\pi$
$557$	45.2971	1.91930	0.959650	+	0.281199i	$0.0907319\pi$
$558$	0	0
$559$	−44.4694	−1.88086
$560$	0	0
$561$	2.28491	0.0964688
$562$	0	0
$563$	8.51000	0.358654	0.179327	−	0.983790i	$-0.442608\pi$
$563$	8.51000	0.358654	0.179327	+	0.983790i	$0.442608\pi$
$564$	0	0
$565$	1.77006	0.0744671
$566$	0	0
$567$	−3.14488	−0.132072
$568$	0	0
$569$	−8.81805	−0.369672	−0.184836	−	0.982769i	$-0.559175\pi$
$569$	−8.81805	−0.369672	−0.184836	+	0.982769i	$0.559175\pi$
$570$	0	0
$571$	−0.121976	−0.00510454	−0.00255227	−	0.999997i	$-0.500812\pi$
$571$	−0.121976	−0.00510454	−0.00255227	+	0.999997i	$0.500812\pi$
$572$	0	0
$573$	−7.15013	−0.298701
$574$	0	0
$575$	6.85373	0.285820
$576$	0	0
$577$	3.18096	0.132425	0.0662125	−	0.997806i	$-0.478908\pi$
$577$	3.18096	0.132425	0.0662125	+	0.997806i	$0.478908\pi$
$578$	0	0
$579$	9.50656	0.395079
$580$	0	0
$581$	−3.31044	−0.137340
$582$	0	0
$583$	4.85373	0.201021
$584$	0	0
$585$	−10.8267	−0.447630
$586$	0	0
$587$	41.4621	1.71133	0.855663	−	0.517534i	$-0.173150\pi$
$587$	41.4621	1.71133	0.855663	+	0.517534i	$0.173150\pi$
$588$	0	0
$589$	11.7628	0.484680
$590$	0	0
$591$	16.7707	0.689855
$592$	0	0
$593$	−18.0087	−0.739528	−0.369764	−	0.929126i	$-0.620561\pi$
$593$	−18.0087	−0.739528	−0.369764	+	0.929126i	$0.620561\pi$
$594$	0	0
$595$	−2.71947	−0.111487
$596$	0	0
$597$	−16.1006	−0.658953
$598$	0	0
$599$	−8.93476	−0.365064	−0.182532	−	0.983200i	$-0.558429\pi$
$599$	−8.93476	−0.365064	−0.182532	+	0.983200i	$0.558429\pi$
$600$	0	0
$601$	−33.3493	−1.36035	−0.680174	−	0.733051i	$-0.738096\pi$
$601$	−33.3493	−1.36035	−0.680174	+	0.733051i	$0.738096\pi$
$602$	0	0
$603$	15.9998	0.651563
$604$	0	0
$605$	1.00000	0.0406558
$606$	0	0
$607$	12.6249	0.512427	0.256214	−	0.966620i	$-0.417525\pi$
$607$	12.6249	0.512427	0.256214	+	0.966620i	$0.417525\pi$
$608$	0	0
$609$	2.99085	0.121195
$610$	0	0
$611$	24.4425	0.988839
$612$	0	0
$613$	9.27575	0.374644	0.187322	−	0.982299i	$-0.440019\pi$
$613$	9.27575	0.374644	0.187322	+	0.982299i	$0.440019\pi$
$614$	0	0
$615$	8.62280	0.347705
$616$	0	0
$617$	18.2399	0.734311	0.367156	−	0.930160i	$-0.380332\pi$
$617$	18.2399	0.734311	0.367156	+	0.930160i	$0.380332\pi$
$618$	0	0
$619$	21.7876	0.875716	0.437858	−	0.899044i	$-0.355737\pi$
$619$	21.7876	0.875716	0.437858	+	0.899044i	$0.355737\pi$
$620$	0	0
$621$	30.4860	1.22336
$622$	0	0
$623$	5.56743	0.223054
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−3.83874	−0.153305
$628$	0	0
$629$	15.5945	0.621793
$630$	0	0
$631$	−2.93755	−0.116942	−0.0584710	−	0.998289i	$-0.518623\pi$
$631$	−2.93755	−0.116942	−0.0584710	+	0.998289i	$0.518623\pi$
$632$	0	0
$633$	−10.7544	−0.427449
$634$	0	0
$635$	−1.18195	−0.0469042
$636$	0	0
$637$	4.71947	0.186992
$638$	0	0
$639$	22.2695	0.880967
$640$	0	0
$641$	24.6132	0.972165	0.486082	−	0.873913i	$-0.338426\pi$
$641$	24.6132	0.972165	0.486082	+	0.873913i	$0.338426\pi$
$642$	0	0
$643$	3.99590	0.157583	0.0787915	−	0.996891i	$-0.474894\pi$
$643$	3.99590	0.157583	0.0787915	+	0.996891i	$0.474894\pi$
$644$	0	0
$645$	7.91686	0.311726
$646$	0	0
$647$	−42.0811	−1.65438	−0.827189	−	0.561924i	$-0.810061\pi$
$647$	−42.0811	−1.65438	−0.827189	+	0.561924i	$0.810061\pi$
$648$	0	0
$649$	1.56544	0.0614488
$650$	0	0
$651$	−2.16318	−0.0847817
$652$	0	0
$653$	7.87647	0.308230	0.154115	−	0.988053i	$-0.450747\pi$
$653$	7.87647	0.308230	0.154115	+	0.988053i	$0.450747\pi$
$654$	0	0
$655$	−8.78697	−0.343335
$656$	0	0
$657$	4.77229	0.186185
$658$	0	0
$659$	48.5776	1.89231	0.946156	−	0.323710i	$-0.104930\pi$
$659$	48.5776	1.89231	0.946156	+	0.323710i	$0.104930\pi$
$660$	0	0
$661$	−4.14003	−0.161028	−0.0805142	−	0.996753i	$-0.525656\pi$
$661$	−4.14003	−0.161028	−0.0805142	+	0.996753i	$0.525656\pi$
$662$	0	0
$663$	−10.7835	−0.418798
$664$	0	0
$665$	4.56882	0.177171
$666$	0	0
$667$	24.3970	0.944656
$668$	0	0
$669$	19.7180	0.762343
$670$	0	0
$671$	−2.57459	−0.0993910
$672$	0	0
$673$	−48.7388	−1.87874	−0.939372	−	0.342901i	$-0.888590\pi$
$673$	−48.7388	−1.87874	−0.939372	+	0.342901i	$0.888590\pi$
$674$	0	0
$675$	4.44809	0.171207
$676$	0	0
$677$	6.45702	0.248163	0.124082	−	0.992272i	$-0.460402\pi$
$677$	6.45702	0.248163	0.124082	+	0.992272i	$0.460402\pi$
$678$	0	0
$679$	−15.1271	−0.580525
$680$	0	0
$681$	−8.06553	−0.309072
$682$	0	0
$683$	12.4943	0.478081	0.239041	−	0.971010i	$-0.423167\pi$
$683$	12.4943	0.478081	0.239041	+	0.971010i	$0.423167\pi$
$684$	0	0
$685$	6.76283	0.258395
$686$	0	0
$687$	−13.2924	−0.507136
$688$	0	0
$689$	−22.9070	−0.872688
$690$	0	0
$691$	19.3445	0.735899	0.367950	−	0.929846i	$-0.380060\pi$
$691$	19.3445	0.735899	0.367950	+	0.929846i	$0.380060\pi$
$692$	0	0
$693$	−2.29406	−0.0871441
$694$	0	0
$695$	−0.410489	−0.0155707
$696$	0	0
$697$	−27.9092	−1.05714
$698$	0	0
$699$	22.1651	0.838361
$700$	0	0
$701$	−16.3980	−0.619344	−0.309672	−	0.950843i	$-0.600219\pi$
$701$	−16.3980	−0.619344	−0.309672	+	0.950843i	$0.600219\pi$
$702$	0	0
$703$	−26.1994	−0.988129
$704$	0	0
$705$	−4.35149	−0.163887
$706$	0	0
$707$	−12.4616	−0.468667
$708$	0	0
$709$	27.8309	1.04521	0.522606	−	0.852574i	$-0.324960\pi$
$709$	27.8309	1.04521	0.522606	+	0.852574i	$0.324960\pi$
$710$	0	0
$711$	−10.8400	−0.406530
$712$	0	0
$713$	−17.6455	−0.660831
$714$	0	0
$715$	−4.71947	−0.176498
$716$	0	0
$717$	2.27599	0.0849983
$718$	0	0
$719$	34.6199	1.29111	0.645553	−	0.763715i	$-0.276627\pi$
$719$	34.6199	1.29111	0.645553	+	0.763715i	$0.276627\pi$
$720$	0	0
$721$	−19.5646	−0.728623
$722$	0	0
$723$	−7.95085	−0.295695
$724$	0	0
$725$	3.55967	0.132203
$726$	0	0
$727$	−49.5596	−1.83806	−0.919031	−	0.394185i	$-0.871027\pi$
$727$	−49.5596	−1.83806	−0.919031	+	0.394185i	$0.871027\pi$
$728$	0	0
$729$	3.99737	0.148051
$730$	0	0
$731$	−25.6243	−0.947750
$732$	0	0
$733$	−6.74070	−0.248973	−0.124487	−	0.992221i	$-0.539728\pi$
$733$	−6.74070	−0.248973	−0.124487	+	0.992221i	$0.539728\pi$
$734$	0	0
$735$	−0.840204	−0.0309914
$736$	0	0
$737$	6.97447	0.256908
$738$	0	0
$739$	5.82546	0.214293	0.107146	−	0.994243i	$-0.465829\pi$
$739$	5.82546	0.214293	0.107146	+	0.994243i	$0.465829\pi$
$740$	0	0
$741$	18.1168	0.665538
$742$	0	0
$743$	29.5667	1.08470	0.542349	−	0.840153i	$-0.317535\pi$
$743$	29.5667	1.08470	0.542349	+	0.840153i	$0.317535\pi$
$744$	0	0
$745$	15.8282	0.579900
$746$	0	0
$747$	−7.59434	−0.277862
$748$	0	0
$749$	13.0362	0.476332
$750$	0	0
$751$	24.0984	0.879363	0.439682	−	0.898154i	$-0.355091\pi$
$751$	24.0984	0.879363	0.439682	+	0.898154i	$0.355091\pi$
$752$	0	0
$753$	11.4205	0.416185
$754$	0	0
$755$	−5.59436	−0.203600
$756$	0	0
$757$	30.4491	1.10669	0.553345	−	0.832952i	$-0.313351\pi$
$757$	30.4491	1.10669	0.553345	+	0.832952i	$0.313351\pi$
$758$	0	0
$759$	5.75853	0.209021
$760$	0	0
$761$	8.64426	0.313354	0.156677	−	0.987650i	$-0.449922\pi$
$761$	8.64426	0.313354	0.156677	+	0.987650i	$0.449922\pi$
$762$	0	0
$763$	−18.9475	−0.685947
$764$	0	0
$765$	−6.23862	−0.225558
$766$	0	0
$767$	−7.38804	−0.266767
$768$	0	0
$769$	−25.3378	−0.913704	−0.456852	−	0.889543i	$-0.651023\pi$
$769$	−25.3378	−0.913704	−0.456852	+	0.889543i	$0.651023\pi$
$770$	0	0
$771$	−21.6740	−0.780570
$772$	0	0
$773$	16.5763	0.596209	0.298105	−	0.954533i	$-0.403646\pi$
$773$	16.5763	0.596209	0.298105	+	0.954533i	$0.403646\pi$
$774$	0	0
$775$	−2.57459	−0.0924820
$776$	0	0
$777$	4.81805	0.172847
$778$	0	0
$779$	46.8887	1.67996
$780$	0	0
$781$	9.70746	0.347360
$782$	0	0
$783$	15.8337	0.565851
$784$	0	0
$785$	6.45962	0.230554
$786$	0	0
$787$	−19.2871	−0.687511	−0.343756	−	0.939059i	$-0.611699\pi$
$787$	−19.2871	−0.687511	−0.343756	+	0.939059i	$0.611699\pi$
$788$	0	0
$789$	−17.0702	−0.607717
$790$	0	0
$791$	−1.77006	−0.0629362
$792$	0	0
$793$	12.1507	0.431484
$794$	0	0
$795$	4.07812	0.144636
$796$	0	0
$797$	−13.1582	−0.466087	−0.233043	−	0.972466i	$-0.574868\pi$
$797$	−13.1582	−0.466087	−0.233043	+	0.972466i	$0.574868\pi$
$798$	0	0
$799$	14.0844	0.498269
$800$	0	0
$801$	12.7720	0.451276
$802$	0	0
$803$	2.08028	0.0734115
$804$	0	0
$805$	−6.85373	−0.241562
$806$	0	0
$807$	−3.82988	−0.134818
$808$	0	0
$809$	8.54953	0.300585	0.150293	−	0.988642i	$-0.451978\pi$
$809$	8.54953	0.300585	0.150293	+	0.988642i	$0.451978\pi$
$810$	0	0
$811$	−47.9567	−1.68399	−0.841994	−	0.539488i	$-0.818618\pi$
$811$	−47.9567	−1.68399	−0.841994	+	0.539488i	$0.818618\pi$
$812$	0	0
$813$	−14.5340	−0.509730
$814$	0	0
$815$	19.2315	0.673648
$816$	0	0
$817$	43.0500	1.50613
$818$	0	0
$819$	10.8267	0.378317
$820$	0	0
$821$	−4.06385	−0.141829	−0.0709146	−	0.997482i	$-0.522592\pi$
$821$	−4.06385	−0.141829	−0.0709146	+	0.997482i	$0.522592\pi$
$822$	0	0
$823$	−7.40618	−0.258163	−0.129082	−	0.991634i	$-0.541203\pi$
$823$	−7.40618	−0.258163	−0.129082	+	0.991634i	$0.541203\pi$
$824$	0	0
$825$	0.840204	0.0292521
$826$	0	0
$827$	7.55955	0.262871	0.131436	−	0.991325i	$-0.458041\pi$
$827$	7.55955	0.262871	0.131436	+	0.991325i	$0.458041\pi$
$828$	0	0
$829$	−29.0662	−1.00951	−0.504755	−	0.863263i	$-0.668417\pi$
$829$	−29.0662	−1.00951	−0.504755	+	0.863263i	$0.668417\pi$
$830$	0	0
$831$	−21.2498	−0.737148
$832$	0	0
$833$	2.71947	0.0942239
$834$	0	0
$835$	15.8360	0.548026
$836$	0	0
$837$	−11.4520	−0.395839
$838$	0	0
$839$	20.7744	0.717213	0.358607	−	0.933489i	$-0.383252\pi$
$839$	20.7744	0.717213	0.358607	+	0.933489i	$0.383252\pi$
$840$	0	0
$841$	−16.3287	−0.563060
$842$	0	0
$843$	−12.9039	−0.444433
$844$	0	0
$845$	9.27337	0.319014
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	9.97026	0.342179
$850$	0	0
$851$	39.3019	1.34725
$852$	0	0
$853$	−6.14940	−0.210552	−0.105276	−	0.994443i	$-0.533573\pi$
$853$	−6.14940	−0.210552	−0.105276	+	0.994443i	$0.533573\pi$
$854$	0	0
$855$	10.4811	0.358448
$856$	0	0
$857$	−16.2451	−0.554923	−0.277462	−	0.960737i	$-0.589493\pi$
$857$	−16.2451	−0.554923	−0.277462	+	0.960737i	$0.589493\pi$
$858$	0	0
$859$	−19.2017	−0.655153	−0.327576	−	0.944825i	$-0.606232\pi$
$859$	−19.2017	−0.655153	−0.327576	+	0.944825i	$0.606232\pi$
$860$	0	0
$861$	−8.62280	−0.293864
$862$	0	0
$863$	46.2514	1.57442	0.787208	−	0.616688i	$-0.211526\pi$
$863$	46.2514	1.57442	0.787208	+	0.616688i	$0.211526\pi$
$864$	0	0
$865$	11.6272	0.395336
$866$	0	0
$867$	8.06973	0.274063
$868$	0	0
$869$	−4.72524	−0.160293
$870$	0	0
$871$	−32.9158	−1.11531
$872$	0	0
$873$	−34.7024	−1.17450
$874$	0	0
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	44.4472	1.50088	0.750438	−	0.660941i	$-0.229843\pi$
$877$	44.4472	1.50088	0.750438	+	0.660941i	$0.229843\pi$
$878$	0	0
$879$	−20.1755	−0.680503
$880$	0	0
$881$	−45.1968	−1.52272	−0.761359	−	0.648331i	$-0.775467\pi$
$881$	−45.1968	−1.52272	−0.761359	+	0.648331i	$0.775467\pi$
$882$	0	0
$883$	10.0244	0.337349	0.168675	−	0.985672i	$-0.446051\pi$
$883$	10.0244	0.337349	0.168675	+	0.985672i	$0.446051\pi$
$884$	0	0
$885$	1.31529	0.0442129
$886$	0	0
$887$	−44.1505	−1.48243	−0.741215	−	0.671268i	$-0.765750\pi$
$887$	−44.1505	−1.48243	−0.741215	+	0.671268i	$0.765750\pi$
$888$	0	0
$889$	1.18195	0.0396412
$890$	0	0
$891$	−3.14488	−0.105357
$892$	0	0
$893$	−23.6623	−0.791830
$894$	0	0
$895$	17.6754	0.590823
$896$	0	0
$897$	−27.1772	−0.907420
$898$	0	0
$899$	−9.16470	−0.305660
$900$	0	0
$901$	−13.1996	−0.439741
$902$	0	0
$903$	−7.91686	−0.263457
$904$	0	0
$905$	−21.8360	−0.725852
$906$	0	0
$907$	−13.5510	−0.449955	−0.224978	−	0.974364i	$-0.572231\pi$
$907$	−13.5510	−0.449955	−0.224978	+	0.974364i	$0.572231\pi$
$908$	0	0
$909$	−28.5877	−0.948193
$910$	0	0
$911$	17.7075	0.586674	0.293337	−	0.956009i	$-0.405234\pi$
$911$	17.7075	0.586674	0.293337	+	0.956009i	$0.405234\pi$
$912$	0	0
$913$	−3.31044	−0.109560
$914$	0	0
$915$	−2.16318	−0.0715125
$916$	0	0
$917$	8.78697	0.290171
$918$	0	0
$919$	36.3469	1.19897	0.599487	−	0.800384i	$-0.295371\pi$
$919$	36.3469	1.19897	0.599487	+	0.800384i	$0.295371\pi$
$920$	0	0
$921$	1.25132	0.0412324
$922$	0	0
$923$	−45.8140	−1.50799
$924$	0	0
$925$	5.73439	0.188545
$926$	0	0
$927$	−44.8823	−1.47413
$928$	0	0
$929$	−19.5762	−0.642273	−0.321137	−	0.947033i	$-0.604065\pi$
$929$	−19.5762	−0.642273	−0.321137	+	0.947033i	$0.604065\pi$
$930$	0	0
$931$	−4.56882	−0.149737
$932$	0	0
$933$	−7.53798	−0.246782
$934$	0	0
$935$	−2.71947	−0.0889361
$936$	0	0
$937$	17.9278	0.585675	0.292837	−	0.956162i	$-0.405401\pi$
$937$	17.9278	0.585675	0.292837	+	0.956162i	$0.405401\pi$
$938$	0	0
$939$	−17.1871	−0.560879
$940$	0	0
$941$	−31.4978	−1.02680	−0.513400	−	0.858150i	$-0.671614\pi$
$941$	−31.4978	−1.02680	−0.513400	+	0.858150i	$0.671614\pi$
$942$	0	0
$943$	−70.3381	−2.29052
$944$	0	0
$945$	−4.44809	−0.144696
$946$	0	0
$947$	26.3468	0.856156	0.428078	−	0.903742i	$-0.359191\pi$
$947$	26.3468	0.856156	0.428078	+	0.903742i	$0.359191\pi$
$948$	0	0
$949$	−9.81782	−0.318700
$950$	0	0
$951$	−15.8695	−0.514605
$952$	0	0
$953$	−43.9711	−1.42436	−0.712182	−	0.701994i	$-0.752293\pi$
$953$	−43.9711	−1.42436	−0.712182	+	0.701994i	$0.752293\pi$
$954$	0	0
$955$	8.51000	0.275377
$956$	0	0
$957$	2.99085	0.0966804
$958$	0	0
$959$	−6.76283	−0.218383
$960$	0	0
$961$	−24.3715	−0.786177
$962$	0	0
$963$	29.9058	0.963701
$964$	0	0
$965$	−11.3146	−0.364230
$966$	0	0
$967$	−8.03246	−0.258306	−0.129153	−	0.991625i	$-0.541226\pi$
$967$	−8.03246	−0.258306	−0.129153	+	0.991625i	$0.541226\pi$
$968$	0	0
$969$	10.4393	0.335360
$970$	0	0
$971$	25.0615	0.804262	0.402131	−	0.915582i	$-0.368270\pi$
$971$	25.0615	0.804262	0.402131	+	0.915582i	$0.368270\pi$
$972$	0	0
$973$	0.410489	0.0131597
$974$	0	0
$975$	−3.96531	−0.126992
$976$	0	0
$977$	13.6138	0.435545	0.217772	−	0.976000i	$-0.430121\pi$
$977$	13.6138	0.435545	0.217772	+	0.976000i	$0.430121\pi$
$978$	0	0
$979$	5.56743	0.177936
$980$	0	0
$981$	−43.4667	−1.38779
$982$	0	0
$983$	−39.8564	−1.27122	−0.635611	−	0.772009i	$-0.719252\pi$
$983$	−39.8564	−1.27122	−0.635611	+	0.772009i	$0.719252\pi$
$984$	0	0
$985$	−19.9603	−0.635988
$986$	0	0
$987$	4.35149	0.138509
$988$	0	0
$989$	−64.5796	−2.05351
$990$	0	0
$991$	38.5824	1.22561	0.612805	−	0.790234i	$-0.290041\pi$
$991$	38.5824	1.22561	0.612805	+	0.790234i	$0.290041\pi$
$992$	0	0
$993$	−26.4705	−0.840015
$994$	0	0
$995$	19.1627	0.607499
$996$	0	0
$997$	−14.1189	−0.447149	−0.223575	−	0.974687i	$-0.571773\pi$
$997$	−14.1189	−0.447149	−0.223575	+	0.974687i	$0.571773\pi$
$998$	0	0
$999$	25.5070	0.807007

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6160.2.a.bx.1.3		5
4.3	odd	2		3080.2.a.u.1.3	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3080.2.a.u.1.3	✓	5	4.3	odd	2
6160.2.a.bx.1.3		5	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)

\(f(q)\)	\(=\)	\(q-0.840204 q^{3} +1.00000 q^{5} -1.00000 q^{7} -2.29406 q^{9} +O(q^{10})\)	\(q-0.840204 q^{3} +1.00000 q^{5} -1.00000 q^{7} -2.29406 q^{9} -1.00000 q^{11} +4.71947 q^{13} -0.840204 q^{15} +2.71947 q^{17} -4.56882 q^{19} +0.840204 q^{21} +6.85373 q^{23} +1.00000 q^{25} +4.44809 q^{27} +3.55967 q^{29} -2.57459 q^{31} +0.840204 q^{33} -1.00000 q^{35} +5.73439 q^{37} -3.96531 q^{39} -10.2628 q^{41} -9.42255 q^{43} -2.29406 q^{45} +5.17909 q^{47} +1.00000 q^{49} -2.28491 q^{51} -4.85373 q^{53} -1.00000 q^{55} +3.83874 q^{57} -1.56544 q^{59} +2.57459 q^{61} +2.29406 q^{63} +4.71947 q^{65} -6.97447 q^{67} -5.75853 q^{69} -9.70746 q^{71} -2.08028 q^{73} -0.840204 q^{75} +1.00000 q^{77} +4.72524 q^{79} +3.14488 q^{81} +3.31044 q^{83} +2.71947 q^{85} -2.99085 q^{87} -5.56743 q^{89} -4.71947 q^{91} +2.16318 q^{93} -4.56882 q^{95} +15.1271 q^{97} +2.29406 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 2 q^{3} + 5 q^{5} - 5 q^{7} + 13 q^{9}+O(q^{10}) \) 5 * q - 2 * q^3 + 5 * q^5 - 5 * q^7 + 13 * q^9	\( 5 q - 2 q^{3} + 5 q^{5} - 5 q^{7} + 13 q^{9} - 5 q^{11} + 12 q^{13} - 2 q^{15} + 2 q^{17} - 8 q^{19} + 2 q^{21} - 4 q^{23} + 5 q^{25} - 2 q^{27} + 4 q^{29} + 2 q^{33} - 5 q^{35} + 18 q^{37} + 8 q^{39} + 4 q^{41} + 6 q^{43} + 13 q^{45} - 6 q^{47} + 5 q^{49} + 12 q^{51} + 14 q^{53} - 5 q^{55} - 12 q^{57} + 4 q^{59} - 13 q^{63} + 12 q^{65} - 6 q^{67} - 10 q^{69} + 28 q^{71} + 10 q^{73} - 2 q^{75} + 5 q^{77} + 14 q^{79} + 17 q^{81} + 22 q^{83} + 2 q^{85} - 16 q^{87} + 24 q^{89} - 12 q^{91} + 10 q^{93} - 8 q^{95} + 10 q^{97} - 13 q^{99}+O(q^{100}) \) 5 * q - 2 * q^3 + 5 * q^5 - 5 * q^7 + 13 * q^9 - 5 * q^11 + 12 * q^13 - 2 * q^15 + 2 * q^17 - 8 * q^19 + 2 * q^21 - 4 * q^23 + 5 * q^25 - 2 * q^27 + 4 * q^29 + 2 * q^33 - 5 * q^35 + 18 * q^37 + 8 * q^39 + 4 * q^41 + 6 * q^43 + 13 * q^45 - 6 * q^47 + 5 * q^49 + 12 * q^51 + 14 * q^53 - 5 * q^55 - 12 * q^57 + 4 * q^59 - 13 * q^63 + 12 * q^65 - 6 * q^67 - 10 * q^69 + 28 * q^71 + 10 * q^73 - 2 * q^75 + 5 * q^77 + 14 * q^79 + 17 * q^81 + 22 * q^83 + 2 * q^85 - 16 * q^87 + 24 * q^89 - 12 * q^91 + 10 * q^93 - 8 * q^95 + 10 * q^97 - 13 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more