LMFDB - Embedded newform 3080.2.a.m.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3080,2,Mod(1,3080)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3080, 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("3080.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3080 = 2^{3} \cdot 5 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3080.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:	$24.5939238226$
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:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.86620$ of defining polynomial
Character	$\chi$	$=$	3080.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.86620 q^{3} -1.00000 q^{5} -1.00000 q^{7} +0.482696 q^{9} +O(q^{10})$	$q-1.86620 q^{3} -1.00000 q^{5} -1.00000 q^{7} +0.482696 q^{9} -1.00000 q^{11} +1.86620 q^{13} +1.86620 q^{15} -2.83159 q^{17} +3.03461 q^{19} +1.86620 q^{21} +5.21509 q^{23} +1.00000 q^{25} +4.69779 q^{27} -2.00000 q^{29} +5.08129 q^{31} +1.86620 q^{33} +1.00000 q^{35} +3.21509 q^{37} -3.48270 q^{39} -11.0813 q^{41} -11.9129 q^{43} -0.482696 q^{45} +10.5640 q^{47} +1.00000 q^{49} +5.28431 q^{51} +8.24970 q^{53} +1.00000 q^{55} -5.66318 q^{57} +10.7445 q^{59} -11.4181 q^{61} -0.482696 q^{63} -1.86620 q^{65} -3.55191 q^{67} -9.73240 q^{69} -1.93078 q^{71} -5.16841 q^{73} -1.86620 q^{75} +1.00000 q^{77} -16.2497 q^{79} -10.2151 q^{81} +8.69779 q^{83} +2.83159 q^{85} +3.73240 q^{87} -3.73240 q^{89} -1.86620 q^{91} -9.48270 q^{93} -3.03461 q^{95} -5.80161 q^{97} -0.482696 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−1.86620	−1.07745	−0.538725	−	0.842482i	$-0.681094\pi$
$3$	−1.86620	−1.07745	−0.538725	+	0.842482i	$0.681094\pi$
$4$	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
$9$	0.482696	0.160899
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	1.86620	0.517590	0.258795	−	0.965932i	$-0.416675\pi$
$13$	1.86620	0.517590	0.258795	+	0.965932i	$0.416675\pi$
$14$	0	0
$15$	1.86620	0.481850
$16$	0	0
$17$	−2.83159	−0.686761	−0.343381	−	0.939196i	$-0.611572\pi$
$17$	−2.83159	−0.686761	−0.343381	+	0.939196i	$0.611572\pi$
$18$	0	0
$19$	3.03461	0.696187	0.348093	−	0.937460i	$-0.386829\pi$
$19$	3.03461	0.696187	0.348093	+	0.937460i	$0.386829\pi$
$20$	0	0
$21$	1.86620	0.407238
$22$	0	0
$23$	5.21509	1.08742	0.543711	−	0.839273i	$-0.317019\pi$
$23$	5.21509	1.08742	0.543711	+	0.839273i	$0.317019\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	4.69779	0.904090
$28$	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
$31$	5.08129	0.912627	0.456313	−	0.889819i	$-0.349170\pi$
$31$	5.08129	0.912627	0.456313	+	0.889819i	$0.349170\pi$
$32$	0	0
$33$	1.86620	0.324863
$34$	0	0
$35$	1.00000	0.169031
$36$	0	0
$37$	3.21509	0.528558	0.264279	−	0.964446i	$-0.414866\pi$
$37$	3.21509	0.528558	0.264279	+	0.964446i	$0.414866\pi$
$38$	0	0
$39$	−3.48270	−0.557678
$40$	0	0
$41$	−11.0813	−1.73061	−0.865303	−	0.501248i	$-0.832874\pi$
$41$	−11.0813	−1.73061	−0.865303	+	0.501248i	$0.832874\pi$
$42$	0	0
$43$	−11.9129	−1.81670	−0.908349	−	0.418214i	$-0.862656\pi$
$43$	−11.9129	−1.81670	−0.908349	+	0.418214i	$0.862656\pi$
$44$	0	0
$45$	−0.482696	−0.0719561
$46$	0	0
$47$	10.5640	1.54092	0.770458	−	0.637491i	$-0.220028\pi$
$47$	10.5640	1.54092	0.770458	+	0.637491i	$0.220028\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	5.28431	0.739951
$52$	0	0
$53$	8.24970	1.13318	0.566592	−	0.823999i	$-0.308262\pi$
$53$	8.24970	1.13318	0.566592	+	0.823999i	$0.308262\pi$
$54$	0	0
$55$	1.00000	0.134840
$56$	0	0
$57$	−5.66318	−0.750107
$58$	0	0
$59$	10.7445	1.39881	0.699405	−	0.714725i	$-0.253448\pi$
$59$	10.7445	1.39881	0.699405	+	0.714725i	$0.253448\pi$
$60$	0	0
$61$	−11.4181	−1.46194	−0.730970	−	0.682410i	$-0.760932\pi$
$61$	−11.4181	−1.46194	−0.730970	+	0.682410i	$0.760932\pi$
$62$	0	0
$63$	−0.482696	−0.0608140
$64$	0	0
$65$	−1.86620	−0.231473
$66$	0	0
$67$	−3.55191	−0.433935	−0.216968	−	0.976179i	$-0.569617\pi$
$67$	−3.55191	−0.433935	−0.216968	+	0.976179i	$0.569617\pi$
$68$	0	0
$69$	−9.73240	−1.17164
$70$	0	0
$71$	−1.93078	−0.229142	−0.114571	−	0.993415i	$-0.536549\pi$
$71$	−1.93078	−0.229142	−0.114571	+	0.993415i	$0.536549\pi$
$72$	0	0
$73$	−5.16841	−0.604917	−0.302458	−	0.953163i	$-0.597807\pi$
$73$	−5.16841	−0.604917	−0.302458	+	0.953163i	$0.597807\pi$
$74$	0	0
$75$	−1.86620	−0.215490
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	−16.2497	−1.82823	−0.914117	−	0.405450i	$-0.867115\pi$
$79$	−16.2497	−1.82823	−0.914117	+	0.405450i	$0.867115\pi$
$80$	0	0
$81$	−10.2151	−1.13501
$82$	0	0
$83$	8.69779	0.954706	0.477353	−	0.878712i	$-0.341596\pi$
$83$	8.69779	0.954706	0.477353	+	0.878712i	$0.341596\pi$
$84$	0	0
$85$	2.83159	0.307129
$86$	0	0
$87$	3.73240	0.400155
$88$	0	0
$89$	−3.73240	−0.395633	−0.197817	−	0.980239i	$-0.563385\pi$
$89$	−3.73240	−0.395633	−0.197817	+	0.980239i	$0.563385\pi$
$90$	0	0
$91$	−1.86620	−0.195631
$92$	0	0
$93$	−9.48270	−0.983310
$94$	0	0
$95$	−3.03461	−0.311344
$96$	0	0
$97$	−5.80161	−0.589065	−0.294532	−	0.955642i	$-0.595164\pi$
$97$	−5.80161	−0.589065	−0.294532	+	0.955642i	$0.595164\pi$
$98$	0	0
$99$	−0.482696	−0.0485128
$100$	0	0
$101$	13.5115	1.34444	0.672221	−	0.740351i	$-0.265340\pi$
$101$	13.5115	1.34444	0.672221	+	0.740351i	$0.265340\pi$
$102$	0	0
$103$	3.52938	0.347760	0.173880	−	0.984767i	$-0.444370\pi$
$103$	3.52938	0.347760	0.173880	+	0.984767i	$0.444370\pi$
$104$	0	0
$105$	−1.86620	−0.182122
$106$	0	0
$107$	10.4302	1.00832	0.504162	−	0.863609i	$-0.331801\pi$
$107$	10.4302	1.00832	0.504162	+	0.863609i	$0.331801\pi$
$108$	0	0
$109$	−5.73240	−0.549064	−0.274532	−	0.961578i	$-0.588523\pi$
$109$	−5.73240	−0.549064	−0.274532	+	0.961578i	$0.588523\pi$
$110$	0	0
$111$	−6.00000	−0.569495
$112$	0	0
$113$	−11.8016	−1.11020	−0.555101	−	0.831783i	$-0.687320\pi$
$113$	−11.8016	−1.11020	−0.555101	+	0.831783i	$0.687320\pi$
$114$	0	0
$115$	−5.21509	−0.486310
$116$	0	0
$117$	0.900806	0.0832796
$118$	0	0
$119$	2.83159	0.259571
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	20.6799	1.86464
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−8.62857	−0.765662	−0.382831	−	0.923818i	$-0.625051\pi$
$127$	−8.62857	−0.765662	−0.382831	+	0.923818i	$0.625051\pi$
$128$	0	0
$129$	22.2318	1.95740
$130$	0	0
$131$	−5.39558	−0.471414	−0.235707	−	0.971824i	$-0.575741\pi$
$131$	−5.39558	−0.471414	−0.235707	+	0.971824i	$0.575741\pi$
$132$	0	0
$133$	−3.03461	−0.263134
$134$	0	0
$135$	−4.69779	−0.404321
$136$	0	0
$137$	9.81952	0.838938	0.419469	−	0.907770i	$-0.362216\pi$
$137$	9.81952	0.838938	0.419469	+	0.907770i	$0.362216\pi$
$138$	0	0
$139$	−15.1972	−1.28901	−0.644504	−	0.764601i	$-0.722936\pi$
$139$	−15.1972	−1.28901	−0.644504	+	0.764601i	$0.722936\pi$
$140$	0	0
$141$	−19.7145	−1.66026
$142$	0	0
$143$	−1.86620	−0.156059
$144$	0	0
$145$	2.00000	0.166091
$146$	0	0
$147$	−1.86620	−0.153921
$148$	0	0
$149$	−24.1626	−1.97948	−0.989738	−	0.142895i	$-0.954359\pi$
$149$	−24.1626	−1.97948	−0.989738	+	0.142895i	$0.954359\pi$
$150$	0	0
$151$	14.0513	1.14348	0.571740	−	0.820435i	$-0.306269\pi$
$151$	14.0513	1.14348	0.571740	+	0.820435i	$0.306269\pi$
$152$	0	0
$153$	−1.36680	−0.110499
$154$	0	0
$155$	−5.08129	−0.408139
$156$	0	0
$157$	2.06922	0.165141	0.0825707	−	0.996585i	$-0.473687\pi$
$157$	2.06922	0.165141	0.0825707	+	0.996585i	$0.473687\pi$
$158$	0	0
$159$	−15.3956	−1.22095
$160$	0	0
$161$	−5.21509	−0.411007
$162$	0	0
$163$	−8.44809	−0.661705	−0.330853	−	0.943682i	$-0.607336\pi$
$163$	−8.44809	−0.661705	−0.330853	+	0.943682i	$0.607336\pi$
$164$	0	0
$165$	−1.86620	−0.145283
$166$	0	0
$167$	−18.8604	−1.45946	−0.729730	−	0.683736i	$-0.760354\pi$
$167$	−18.8604	−1.45946	−0.729730	+	0.683736i	$0.760354\pi$
$168$	0	0
$169$	−9.51730	−0.732100
$170$	0	0
$171$	1.46479	0.112016
$172$	0	0
$173$	5.26178	0.400045	0.200023	−	0.979791i	$-0.435898\pi$
$173$	5.26178	0.400045	0.200023	+	0.979791i	$0.435898\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	−20.0513	−1.50715
$178$	0	0
$179$	−25.3598	−1.89548	−0.947739	−	0.319046i	$-0.896637\pi$
$179$	−25.3598	−1.89548	−0.947739	+	0.319046i	$0.896637\pi$
$180$	0	0
$181$	7.10382	0.528023	0.264012	−	0.964520i	$-0.414954\pi$
$181$	7.10382	0.528023	0.264012	+	0.964520i	$0.414954\pi$
$182$	0	0
$183$	21.3085	1.57517
$184$	0	0
$185$	−3.21509	−0.236378
$186$	0	0
$187$	2.83159	0.207066
$188$	0	0
$189$	−4.69779	−0.341714
$190$	0	0
$191$	−8.26760	−0.598223	−0.299111	−	0.954218i	$-0.596690\pi$
$191$	−8.26760	−0.598223	−0.299111	+	0.954218i	$0.596690\pi$
$192$	0	0
$193$	−18.9475	−1.36387	−0.681935	−	0.731413i	$-0.738861\pi$
$193$	−18.9475	−1.36387	−0.681935	+	0.731413i	$0.738861\pi$
$194$	0	0
$195$	3.48270	0.249401
$196$	0	0
$197$	−10.9475	−0.779976	−0.389988	−	0.920820i	$-0.627521\pi$
$197$	−10.9475	−0.779976	−0.389988	+	0.920820i	$0.627521\pi$
$198$	0	0
$199$	−7.70986	−0.546538	−0.273269	−	0.961938i	$-0.588105\pi$
$199$	−7.70986	−0.546538	−0.273269	+	0.961938i	$0.588105\pi$
$200$	0	0
$201$	6.62857	0.467543
$202$	0	0
$203$	2.00000	0.140372
$204$	0	0
$205$	11.0813	0.773951
$206$	0	0
$207$	2.51730	0.174965
$208$	0	0
$209$	−3.03461	−0.209908
$210$	0	0
$211$	25.4889	1.75473	0.877366	−	0.479823i	$-0.159299\pi$
$211$	25.4889	1.75473	0.877366	+	0.479823i	$0.159299\pi$
$212$	0	0
$213$	3.60323	0.246889
$214$	0	0
$215$	11.9129	0.812452
$216$	0	0
$217$	−5.08129	−0.344940
$218$	0	0
$219$	9.64528	0.651767
$220$	0	0
$221$	−5.28431	−0.355461
$222$	0	0
$223$	−1.33099	−0.0891298	−0.0445649	−	0.999006i	$-0.514190\pi$
$223$	−1.33099	−0.0891298	−0.0445649	+	0.999006i	$0.514190\pi$
$224$	0	0
$225$	0.482696	0.0321797
$226$	0	0
$227$	29.5340	1.96024	0.980121	−	0.198403i	$-0.0635755\pi$
$227$	29.5340	1.96024	0.980121	+	0.198403i	$0.0635755\pi$
$228$	0	0
$229$	−21.5582	−1.42460	−0.712302	−	0.701874i	$-0.752347\pi$
$229$	−21.5582	−1.42460	−0.712302	+	0.701874i	$0.752347\pi$
$230$	0	0
$231$	−1.86620	−0.122787
$232$	0	0
$233$	3.71449	0.243345	0.121672	−	0.992570i	$-0.461174\pi$
$233$	3.71449	0.243345	0.121672	+	0.992570i	$0.461174\pi$
$234$	0	0
$235$	−10.5640	−0.689119
$236$	0	0
$237$	30.3252	1.96983
$238$	0	0
$239$	−14.8362	−0.959675	−0.479838	−	0.877357i	$-0.659304\pi$
$239$	−14.8362	−0.959675	−0.479838	+	0.877357i	$0.659304\pi$
$240$	0	0
$241$	−10.7203	−0.690557	−0.345278	−	0.938500i	$-0.612215\pi$
$241$	−10.7203	−0.690557	−0.345278	+	0.938500i	$0.612215\pi$
$242$	0	0
$243$	4.97002	0.318827
$244$	0	0
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	5.66318	0.360340
$248$	0	0
$249$	−16.2318	−1.02865
$250$	0	0
$251$	24.0467	1.51781	0.758907	−	0.651200i	$-0.225734\pi$
$251$	24.0467	1.51781	0.758907	+	0.651200i	$0.225734\pi$
$252$	0	0
$253$	−5.21509	−0.327870
$254$	0	0
$255$	−5.28431	−0.330916
$256$	0	0
$257$	−15.8258	−0.987184	−0.493592	−	0.869694i	$-0.664316\pi$
$257$	−15.8258	−0.987184	−0.493592	+	0.869694i	$0.664316\pi$
$258$	0	0
$259$	−3.21509	−0.199776
$260$	0	0
$261$	−0.965392	−0.0597563
$262$	0	0
$263$	−27.6966	−1.70784	−0.853922	−	0.520400i	$-0.825783\pi$
$263$	−27.6966	−1.70784	−0.853922	+	0.520400i	$0.825783\pi$
$264$	0	0
$265$	−8.24970	−0.506775
$266$	0	0
$267$	6.96539	0.426275
$268$	0	0
$269$	−12.1384	−0.740093	−0.370047	−	0.929013i	$-0.620658\pi$
$269$	−12.1384	−0.740093	−0.370047	+	0.929013i	$0.620658\pi$
$270$	0	0
$271$	15.0588	0.914754	0.457377	−	0.889273i	$-0.348789\pi$
$271$	15.0588	0.914754	0.457377	+	0.889273i	$0.348789\pi$
$272$	0	0
$273$	3.48270	0.210782
$274$	0	0
$275$	−1.00000	−0.0603023
$276$	0	0
$277$	11.2151	0.673850	0.336925	−	0.941532i	$-0.390613\pi$
$277$	11.2151	0.673850	0.336925	+	0.941532i	$0.390613\pi$
$278$	0	0
$279$	2.45272	0.146840
$280$	0	0
$281$	−10.3610	−0.618084	−0.309042	−	0.951048i	$-0.600008\pi$
$281$	−10.3610	−0.618084	−0.309042	+	0.951048i	$0.600008\pi$
$282$	0	0
$283$	−17.8950	−1.06375	−0.531873	−	0.846824i	$-0.678512\pi$
$283$	−17.8950	−1.06375	−0.531873	+	0.846824i	$0.678512\pi$
$284$	0	0
$285$	5.66318	0.335458
$286$	0	0
$287$	11.0813	0.654108
$288$	0	0
$289$	−8.98210	−0.528359
$290$	0	0
$291$	10.8270	0.634688
$292$	0	0
$293$	−12.4014	−0.724498	−0.362249	−	0.932081i	$-0.617991\pi$
$293$	−12.4014	−0.724498	−0.362249	+	0.932081i	$0.617991\pi$
$294$	0	0
$295$	−10.7445	−0.625567
$296$	0	0
$297$	−4.69779	−0.272593
$298$	0	0
$299$	9.73240	0.562839
$300$	0	0
$301$	11.9129	0.686647
$302$	0	0
$303$	−25.2151	−1.44857
$304$	0	0
$305$	11.4181	0.653799
$306$	0	0
$307$	−17.2906	−0.986824	−0.493412	−	0.869796i	$-0.664251\pi$
$307$	−17.2906	−0.986824	−0.493412	+	0.869796i	$0.664251\pi$
$308$	0	0
$309$	−6.58652	−0.374694
$310$	0	0
$311$	−11.1747	−0.633657	−0.316828	−	0.948483i	$-0.602618\pi$
$311$	−11.1747	−0.633657	−0.316828	+	0.948483i	$0.602618\pi$
$312$	0	0
$313$	23.3598	1.32037	0.660186	−	0.751102i	$-0.270477\pi$
$313$	23.3598	1.32037	0.660186	+	0.751102i	$0.270477\pi$
$314$	0	0
$315$	0.482696	0.0271968
$316$	0	0
$317$	15.3956	0.864702	0.432351	−	0.901705i	$-0.357684\pi$
$317$	15.3956	0.864702	0.432351	+	0.901705i	$0.357684\pi$
$318$	0	0
$319$	2.00000	0.111979
$320$	0	0
$321$	−19.4648	−1.08642
$322$	0	0
$323$	−8.59277	−0.478114
$324$	0	0
$325$	1.86620	0.103518
$326$	0	0
$327$	10.6978	0.591589
$328$	0	0
$329$	−10.5640	−0.582411
$330$	0	0
$331$	8.23180	0.452461	0.226230	−	0.974074i	$-0.427360\pi$
$331$	8.23180	0.452461	0.226230	+	0.974074i	$0.427360\pi$
$332$	0	0
$333$	1.55191	0.0850443
$334$	0	0
$335$	3.55191	0.194062
$336$	0	0
$337$	−3.12173	−0.170051	−0.0850257	−	0.996379i	$-0.527097\pi$
$337$	−3.12173	−0.170051	−0.0850257	+	0.996379i	$0.527097\pi$
$338$	0	0
$339$	22.0241	1.19619
$340$	0	0
$341$	−5.08129	−0.275167
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	9.73240	0.523975
$346$	0	0
$347$	13.4827	0.723789	0.361895	−	0.932219i	$-0.382130\pi$
$347$	13.4827	0.723789	0.361895	+	0.932219i	$0.382130\pi$
$348$	0	0
$349$	−8.31429	−0.445054	−0.222527	−	0.974927i	$-0.571430\pi$
$349$	−8.31429	−0.445054	−0.222527	+	0.974927i	$0.571430\pi$
$350$	0	0
$351$	8.76700	0.467948
$352$	0	0
$353$	27.8950	1.48470	0.742350	−	0.670012i	$-0.233711\pi$
$353$	27.8950	1.48470	0.742350	+	0.670012i	$0.233711\pi$
$354$	0	0
$355$	1.93078	0.102475
$356$	0	0
$357$	−5.28431	−0.279675
$358$	0	0
$359$	−6.54145	−0.345245	−0.172622	−	0.984988i	$-0.555224\pi$
$359$	−6.54145	−0.345245	−0.172622	+	0.984988i	$0.555224\pi$
$360$	0	0
$361$	−9.79115	−0.515324
$362$	0	0
$363$	−1.86620	−0.0979500
$364$	0	0
$365$	5.16841	0.270527
$366$	0	0
$367$	25.5294	1.33262	0.666311	−	0.745674i	$-0.267872\pi$
$367$	25.5294	1.33262	0.666311	+	0.745674i	$0.267872\pi$
$368$	0	0
$369$	−5.34889	−0.278452
$370$	0	0
$371$	−8.24970	−0.428303
$372$	0	0
$373$	2.41228	0.124903	0.0624516	−	0.998048i	$-0.480108\pi$
$373$	2.41228	0.124903	0.0624516	+	0.998048i	$0.480108\pi$
$374$	0	0
$375$	1.86620	0.0963701
$376$	0	0
$377$	−3.73240	−0.192228
$378$	0	0
$379$	11.4648	0.588907	0.294453	−	0.955666i	$-0.404862\pi$
$379$	11.4648	0.588907	0.294453	+	0.955666i	$0.404862\pi$
$380$	0	0
$381$	16.1026	0.824963
$382$	0	0
$383$	26.5040	1.35429	0.677146	−	0.735848i	$-0.263216\pi$
$383$	26.5040	1.35429	0.677146	+	0.735848i	$0.263216\pi$
$384$	0	0
$385$	−1.00000	−0.0509647
$386$	0	0
$387$	−5.75030	−0.292304
$388$	0	0
$389$	26.0576	1.32117	0.660585	−	0.750751i	$-0.270308\pi$
$389$	26.0576	1.32117	0.660585	+	0.750751i	$0.270308\pi$
$390$	0	0
$391$	−14.7670	−0.746800
$392$	0	0
$393$	10.0692	0.507925
$394$	0	0
$395$	16.2497	0.817611
$396$	0	0
$397$	4.49940	0.225818	0.112909	−	0.993605i	$-0.463983\pi$
$397$	4.49940	0.225818	0.112909	+	0.993605i	$0.463983\pi$
$398$	0	0
$399$	5.66318	0.283514
$400$	0	0
$401$	−32.9296	−1.64443	−0.822213	−	0.569181i	$-0.807261\pi$
$401$	−32.9296	−1.64443	−0.822213	+	0.569181i	$0.807261\pi$
$402$	0	0
$403$	9.48270	0.472367
$404$	0	0
$405$	10.2151	0.507592
$406$	0	0
$407$	−3.21509	−0.159366
$408$	0	0
$409$	−9.44226	−0.466890	−0.233445	−	0.972370i	$-0.575000\pi$
$409$	−9.44226	−0.466890	−0.233445	+	0.972370i	$0.575000\pi$
$410$	0	0
$411$	−18.3252	−0.903914
$412$	0	0
$413$	−10.7445	−0.528701
$414$	0	0
$415$	−8.69779	−0.426958
$416$	0	0
$417$	28.3610	1.38884
$418$	0	0
$419$	−38.4169	−1.87679	−0.938394	−	0.345566i	$-0.887687\pi$
$419$	−38.4169	−1.87679	−0.938394	+	0.345566i	$0.887687\pi$
$420$	0	0
$421$	−18.2497	−0.889436	−0.444718	−	0.895671i	$-0.646696\pi$
$421$	−18.2497	−0.889436	−0.444718	+	0.895671i	$0.646696\pi$
$422$	0	0
$423$	5.09919	0.247931
$424$	0	0
$425$	−2.83159	−0.137352
$426$	0	0
$427$	11.4181	0.552561
$428$	0	0
$429$	3.48270	0.168146
$430$	0	0
$431$	−8.92334	−0.429822	−0.214911	−	0.976634i	$-0.568946\pi$
$431$	−8.92334	−0.429822	−0.214911	+	0.976634i	$0.568946\pi$
$432$	0	0
$433$	−0.872027	−0.0419069	−0.0209535	−	0.999780i	$-0.506670\pi$
$433$	−0.872027	−0.0419069	−0.0209535	+	0.999780i	$0.506670\pi$
$434$	0	0
$435$	−3.73240	−0.178955
$436$	0	0
$437$	15.8258	0.757049
$438$	0	0
$439$	−24.9054	−1.18867	−0.594336	−	0.804217i	$-0.702585\pi$
$439$	−24.9054	−1.18867	−0.594336	+	0.804217i	$0.702585\pi$
$440$	0	0
$441$	0.482696	0.0229855
$442$	0	0
$443$	13.2151	0.627868	0.313934	−	0.949445i	$-0.398353\pi$
$443$	13.2151	0.627868	0.313934	+	0.949445i	$0.398353\pi$
$444$	0	0
$445$	3.73240	0.176933
$446$	0	0
$447$	45.0922	2.13279
$448$	0	0
$449$	−3.64528	−0.172031	−0.0860156	−	0.996294i	$-0.527414\pi$
$449$	−3.64528	−0.172031	−0.0860156	+	0.996294i	$0.527414\pi$
$450$	0	0
$451$	11.0813	0.521798
$452$	0	0
$453$	−26.2225	−1.23204
$454$	0	0
$455$	1.86620	0.0874887
$456$	0	0
$457$	−30.4123	−1.42263	−0.711313	−	0.702875i	$-0.751899\pi$
$457$	−30.4123	−1.42263	−0.711313	+	0.702875i	$0.751899\pi$
$458$	0	0
$459$	−13.3022	−0.620894
$460$	0	0
$461$	11.7099	0.545383	0.272691	−	0.962102i	$-0.412086\pi$
$461$	11.7099	0.545383	0.272691	+	0.962102i	$0.412086\pi$
$462$	0	0
$463$	−32.5507	−1.51276	−0.756380	−	0.654132i	$-0.773034\pi$
$463$	−32.5507	−1.51276	−0.756380	+	0.654132i	$0.773034\pi$
$464$	0	0
$465$	9.48270	0.439749
$466$	0	0
$467$	23.9354	1.10760	0.553799	−	0.832650i	$-0.313177\pi$
$467$	23.9354	1.10760	0.553799	+	0.832650i	$0.313177\pi$
$468$	0	0
$469$	3.55191	0.164012
$470$	0	0
$471$	−3.86157	−0.177932
$472$	0	0
$473$	11.9129	0.547755
$474$	0	0
$475$	3.03461	0.139237
$476$	0	0
$477$	3.98210	0.182328
$478$	0	0
$479$	−2.53521	−0.115837	−0.0579183	−	0.998321i	$-0.518446\pi$
$479$	−2.53521	−0.115837	−0.0579183	+	0.998321i	$0.518446\pi$
$480$	0	0
$481$	6.00000	0.273576
$482$	0	0
$483$	9.73240	0.442839
$484$	0	0
$485$	5.80161	0.263438
$486$	0	0
$487$	−42.5749	−1.92925	−0.964626	−	0.263624i	$-0.915082\pi$
$487$	−42.5749	−1.92925	−0.964626	+	0.263624i	$0.915082\pi$
$488$	0	0
$489$	15.7658	0.712954
$490$	0	0
$491$	42.6441	1.92450	0.962250	−	0.272166i	$-0.0877401\pi$
$491$	42.6441	1.92450	0.962250	+	0.272166i	$0.0877401\pi$
$492$	0	0
$493$	5.66318	0.255057
$494$	0	0
$495$	0.482696	0.0216956
$496$	0	0
$497$	1.93078	0.0866075
$498$	0	0
$499$	11.9642	0.535591	0.267795	−	0.963476i	$-0.413705\pi$
$499$	11.9642	0.535591	0.267795	+	0.963476i	$0.413705\pi$
$500$	0	0
$501$	35.1972	1.57249
$502$	0	0
$503$	21.2330	0.946732	0.473366	−	0.880866i	$-0.343039\pi$
$503$	21.2330	0.946732	0.473366	+	0.880866i	$0.343039\pi$
$504$	0	0
$505$	−13.5115	−0.601253
$506$	0	0
$507$	17.7612	0.788802
$508$	0	0
$509$	13.8708	0.614814	0.307407	−	0.951578i	$-0.400539\pi$
$509$	13.8708	0.614814	0.307407	+	0.951578i	$0.400539\pi$
$510$	0	0
$511$	5.16841	0.228637
$512$	0	0
$513$	14.2559	0.629415
$514$	0	0
$515$	−3.52938	−0.155523
$516$	0	0
$517$	−10.5640	−0.464604
$518$	0	0
$519$	−9.81952	−0.431029
$520$	0	0
$521$	32.6861	1.43201	0.716003	−	0.698097i	$-0.245970\pi$
$521$	32.6861	1.43201	0.716003	+	0.698097i	$0.245970\pi$
$522$	0	0
$523$	19.3598	0.846544	0.423272	−	0.906003i	$-0.360882\pi$
$523$	19.3598	0.846544	0.423272	+	0.906003i	$0.360882\pi$
$524$	0	0
$525$	1.86620	0.0814476
$526$	0	0
$527$	−14.3881	−0.626757
$528$	0	0
$529$	4.19719	0.182487
$530$	0	0
$531$	5.18631	0.225067
$532$	0	0
$533$	−20.6799	−0.895745
$534$	0	0
$535$	−10.4302	−0.450936
$536$	0	0
$537$	47.3264	2.04228
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	4.69779	0.201974	0.100987	−	0.994888i	$-0.467800\pi$
$541$	4.69779	0.201974	0.100987	+	0.994888i	$0.467800\pi$
$542$	0	0
$543$	−13.2571	−0.568919
$544$	0	0
$545$	5.73240	0.245549
$546$	0	0
$547$	25.8016	1.10320	0.551599	−	0.834110i	$-0.314018\pi$
$547$	25.8016	1.10320	0.551599	+	0.834110i	$0.314018\pi$
$548$	0	0
$549$	−5.51148	−0.235224
$550$	0	0
$551$	−6.06922	−0.258557
$552$	0	0
$553$	16.2497	0.691008
$554$	0	0
$555$	6.00000	0.254686
$556$	0	0
$557$	−5.28431	−0.223903	−0.111952	−	0.993714i	$-0.535710\pi$
$557$	−5.28431	−0.223903	−0.111952	+	0.993714i	$0.535710\pi$
$558$	0	0
$559$	−22.2318	−0.940305
$560$	0	0
$561$	−5.28431	−0.223104
$562$	0	0
$563$	35.6966	1.50443	0.752216	−	0.658917i	$-0.228985\pi$
$563$	35.6966	1.50443	0.752216	+	0.658917i	$0.228985\pi$
$564$	0	0
$565$	11.8016	0.496498
$566$	0	0
$567$	10.2151	0.428994
$568$	0	0
$569$	7.12797	0.298820	0.149410	−	0.988775i	$-0.452263\pi$
$569$	7.12797	0.298820	0.149410	+	0.988775i	$0.452263\pi$
$570$	0	0
$571$	−23.4197	−0.980085	−0.490043	−	0.871699i	$-0.663019\pi$
$571$	−23.4197	−0.980085	−0.490043	+	0.871699i	$0.663019\pi$
$572$	0	0
$573$	15.4290	0.644555
$574$	0	0
$575$	5.21509	0.217484
$576$	0	0
$577$	34.1626	1.42221	0.711103	−	0.703087i	$-0.248196\pi$
$577$	34.1626	1.42221	0.711103	+	0.703087i	$0.248196\pi$
$578$	0	0
$579$	35.3598	1.46950
$580$	0	0
$581$	−8.69779	−0.360845
$582$	0	0
$583$	−8.24970	−0.341668
$584$	0	0
$585$	−0.900806	−0.0372438
$586$	0	0
$587$	−27.4936	−1.13478	−0.567391	−	0.823449i	$-0.692047\pi$
$587$	−27.4936	−1.13478	−0.567391	+	0.823449i	$0.692047\pi$
$588$	0	0
$589$	15.4197	0.635359
$590$	0	0
$591$	20.4302	0.840386
$592$	0	0
$593$	−9.59859	−0.394167	−0.197084	−	0.980387i	$-0.563147\pi$
$593$	−9.59859	−0.394167	−0.197084	+	0.980387i	$0.563147\pi$
$594$	0	0
$595$	−2.83159	−0.116084
$596$	0	0
$597$	14.3881	0.588867
$598$	0	0
$599$	6.96539	0.284598	0.142299	−	0.989824i	$-0.454551\pi$
$599$	6.96539	0.284598	0.142299	+	0.989824i	$0.454551\pi$
$600$	0	0
$601$	14.8137	0.604263	0.302131	−	0.953266i	$-0.402302\pi$
$601$	14.8137	0.604263	0.302131	+	0.953266i	$0.402302\pi$
$602$	0	0
$603$	−1.71449	−0.0698196
$604$	0	0
$605$	−1.00000	−0.0406558
$606$	0	0
$607$	−9.86157	−0.400269	−0.200134	−	0.979768i	$-0.564138\pi$
$607$	−9.86157	−0.400269	−0.200134	+	0.979768i	$0.564138\pi$
$608$	0	0
$609$	−3.73240	−0.151244
$610$	0	0
$611$	19.7145	0.797563
$612$	0	0
$613$	39.0409	1.57685	0.788423	−	0.615134i	$-0.210898\pi$
$613$	39.0409	1.57685	0.788423	+	0.615134i	$0.210898\pi$
$614$	0	0
$615$	−20.6799	−0.833893
$616$	0	0
$617$	−9.64528	−0.388304	−0.194152	−	0.980971i	$-0.562196\pi$
$617$	−9.64528	−0.388304	−0.194152	+	0.980971i	$0.562196\pi$
$618$	0	0
$619$	21.9175	0.880939	0.440470	−	0.897768i	$-0.354812\pi$
$619$	21.9175	0.880939	0.440470	+	0.897768i	$0.354812\pi$
$620$	0	0
$621$	24.4994	0.983127
$622$	0	0
$623$	3.73240	0.149535
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	5.66318	0.226166
$628$	0	0
$629$	−9.10382	−0.362993
$630$	0	0
$631$	−10.7429	−0.427666	−0.213833	−	0.976870i	$-0.568595\pi$
$631$	−10.7429	−0.427666	−0.213833	+	0.976870i	$0.568595\pi$
$632$	0	0
$633$	−47.5674	−1.89064
$634$	0	0
$635$	8.62857	0.342414
$636$	0	0
$637$	1.86620	0.0739415
$638$	0	0
$639$	−0.931982	−0.0368686
$640$	0	0
$641$	1.71449	0.0677184	0.0338592	−	0.999427i	$-0.489220\pi$
$641$	1.71449	0.0677184	0.0338592	+	0.999427i	$0.489220\pi$
$642$	0	0
$643$	−38.3897	−1.51394	−0.756972	−	0.653447i	$-0.773322\pi$
$643$	−38.3897	−1.51394	−0.756972	+	0.653447i	$0.773322\pi$
$644$	0	0
$645$	−22.2318	−0.875376
$646$	0	0
$647$	−8.99417	−0.353597	−0.176799	−	0.984247i	$-0.556574\pi$
$647$	−8.99417	−0.353597	−0.176799	+	0.984247i	$0.556574\pi$
$648$	0	0
$649$	−10.7445	−0.421757
$650$	0	0
$651$	9.48270	0.371656
$652$	0	0
$653$	−30.5928	−1.19719	−0.598594	−	0.801053i	$-0.704274\pi$
$653$	−30.5928	−1.19719	−0.598594	+	0.801053i	$0.704274\pi$
$654$	0	0
$655$	5.39558	0.210823
$656$	0	0
$657$	−2.49477	−0.0973303
$658$	0	0
$659$	−45.7028	−1.78033	−0.890165	−	0.455639i	$-0.849411\pi$
$659$	−45.7028	−1.78033	−0.890165	+	0.455639i	$0.849411\pi$
$660$	0	0
$661$	−28.9537	−1.12617	−0.563085	−	0.826399i	$-0.690386\pi$
$661$	−28.9537	−1.12617	−0.563085	+	0.826399i	$0.690386\pi$
$662$	0	0
$663$	9.86157	0.382992
$664$	0	0
$665$	3.03461	0.117677
$666$	0	0
$667$	−10.4302	−0.403858
$668$	0	0
$669$	2.48389	0.0960329
$670$	0	0
$671$	11.4181	0.440791
$672$	0	0
$673$	−15.7052	−0.605392	−0.302696	−	0.953087i	$-0.597887\pi$
$673$	−15.7052	−0.605392	−0.302696	+	0.953087i	$0.597887\pi$
$674$	0	0
$675$	4.69779	0.180818
$676$	0	0
$677$	−5.42436	−0.208475	−0.104237	−	0.994552i	$-0.533240\pi$
$677$	−5.42436	−0.208475	−0.104237	+	0.994552i	$0.533240\pi$
$678$	0	0
$679$	5.80161	0.222645
$680$	0	0
$681$	−55.1163	−2.11206
$682$	0	0
$683$	−28.7374	−1.09961	−0.549804	−	0.835294i	$-0.685298\pi$
$683$	−28.7374	−1.09961	−0.549804	+	0.835294i	$0.685298\pi$
$684$	0	0
$685$	−9.81952	−0.375184
$686$	0	0
$687$	40.2318	1.53494
$688$	0	0
$689$	15.3956	0.586525
$690$	0	0
$691$	−27.3131	−1.03904	−0.519519	−	0.854459i	$-0.673889\pi$
$691$	−27.3131	−1.03904	−0.519519	+	0.854459i	$0.673889\pi$
$692$	0	0
$693$	0.482696	0.0183361
$694$	0	0
$695$	15.1972	0.576462
$696$	0	0
$697$	31.3777	1.18851
$698$	0	0
$699$	−6.93198	−0.262192
$700$	0	0
$701$	42.2318	1.59507	0.797536	−	0.603271i	$-0.206136\pi$
$701$	42.2318	1.59507	0.797536	+	0.603271i	$0.206136\pi$
$702$	0	0
$703$	9.75655	0.367975
$704$	0	0
$705$	19.7145	0.742491
$706$	0	0
$707$	−13.5115	−0.508151
$708$	0	0
$709$	47.5823	1.78699	0.893496	−	0.449072i	$-0.148245\pi$
$709$	47.5823	1.78699	0.893496	+	0.449072i	$0.148245\pi$
$710$	0	0
$711$	−7.84366	−0.294160
$712$	0	0
$713$	26.4994	0.992410
$714$	0	0
$715$	1.86620	0.0697919
$716$	0	0
$717$	27.6873	1.03400
$718$	0	0
$719$	9.55654	0.356399	0.178199	−	0.983994i	$-0.442973\pi$
$719$	9.55654	0.356399	0.178199	+	0.983994i	$0.442973\pi$
$720$	0	0
$721$	−3.52938	−0.131441
$722$	0	0
$723$	20.0062	0.744040
$724$	0	0
$725$	−2.00000	−0.0742781
$726$	0	0
$727$	−4.66901	−0.173164	−0.0865820	−	0.996245i	$-0.527594\pi$
$727$	−4.66901	−0.173164	−0.0865820	+	0.996245i	$0.527594\pi$
$728$	0	0
$729$	21.3702	0.791490
$730$	0	0
$731$	33.7324	1.24764
$732$	0	0
$733$	21.8903	0.808538	0.404269	−	0.914640i	$-0.367526\pi$
$733$	21.8903	0.808538	0.404269	+	0.914640i	$0.367526\pi$
$734$	0	0
$735$	1.86620	0.0688358
$736$	0	0
$737$	3.55191	0.130836
$738$	0	0
$739$	−46.2559	−1.70155	−0.850776	−	0.525528i	$-0.823868\pi$
$739$	−46.2559	−1.70155	−0.850776	+	0.525528i	$0.823868\pi$
$740$	0	0
$741$	−10.5686	−0.388248
$742$	0	0
$743$	20.8962	0.766606	0.383303	−	0.923623i	$-0.374786\pi$
$743$	20.8962	0.766606	0.383303	+	0.923623i	$0.374786\pi$
$744$	0	0
$745$	24.1626	0.885248
$746$	0	0
$747$	4.19839	0.153611
$748$	0	0
$749$	−10.4302	−0.381111
$750$	0	0
$751$	−13.9400	−0.508679	−0.254340	−	0.967115i	$-0.581858\pi$
$751$	−13.9400	−0.508679	−0.254340	+	0.967115i	$0.581858\pi$
$752$	0	0
$753$	−44.8759	−1.63537
$754$	0	0
$755$	−14.0513	−0.511380
$756$	0	0
$757$	42.0934	1.52991	0.764955	−	0.644084i	$-0.222761\pi$
$757$	42.0934	1.52991	0.764955	+	0.644084i	$0.222761\pi$
$758$	0	0
$759$	9.73240	0.353264
$760$	0	0
$761$	31.1054	1.12757	0.563786	−	0.825921i	$-0.309344\pi$
$761$	31.1054	1.12757	0.563786	+	0.825921i	$0.309344\pi$
$762$	0	0
$763$	5.73240	0.207527
$764$	0	0
$765$	1.36680	0.0494167
$766$	0	0
$767$	20.0513	0.724011
$768$	0	0
$769$	−40.7445	−1.46928	−0.734642	−	0.678455i	$-0.762650\pi$
$769$	−40.7445	−1.46928	−0.734642	+	0.678455i	$0.762650\pi$
$770$	0	0
$771$	29.5340	1.06364
$772$	0	0
$773$	−19.3356	−0.695454	−0.347727	−	0.937596i	$-0.613046\pi$
$773$	−19.3356	−0.695454	−0.347727	+	0.937596i	$0.613046\pi$
$774$	0	0
$775$	5.08129	0.182525
$776$	0	0
$777$	6.00000	0.215249
$778$	0	0
$779$	−33.6274	−1.20483
$780$	0	0
$781$	1.93078	0.0690889
$782$	0	0
$783$	−9.39558	−0.335771
$784$	0	0
$785$	−2.06922	−0.0738535
$786$	0	0
$787$	19.4197	0.692238	0.346119	−	0.938191i	$-0.387499\pi$
$787$	19.4197	0.692238	0.346119	+	0.938191i	$0.387499\pi$
$788$	0	0
$789$	51.6873	1.84012
$790$	0	0
$791$	11.8016	0.419617
$792$	0	0
$793$	−21.3085	−0.756686
$794$	0	0
$795$	15.3956	0.546025
$796$	0	0
$797$	−18.0692	−0.640044	−0.320022	−	0.947410i	$-0.603690\pi$
$797$	−18.0692	−0.640044	−0.320022	+	0.947410i	$0.603690\pi$
$798$	0	0
$799$	−29.9129	−1.05824
$800$	0	0
$801$	−1.80161	−0.0636569
$802$	0	0
$803$	5.16841	0.182389
$804$	0	0
$805$	5.21509	0.183808
$806$	0	0
$807$	22.6527	0.797414
$808$	0	0
$809$	28.5686	1.00442	0.502210	−	0.864746i	$-0.332521\pi$
$809$	28.5686	1.00442	0.502210	+	0.864746i	$0.332521\pi$
$810$	0	0
$811$	−5.73240	−0.201292	−0.100646	−	0.994922i	$-0.532091\pi$
$811$	−5.73240	−0.201292	−0.100646	+	0.994922i	$0.532091\pi$
$812$	0	0
$813$	−28.1026	−0.985602
$814$	0	0
$815$	8.44809	0.295924
$816$	0	0
$817$	−36.1509	−1.26476
$818$	0	0
$819$	−0.900806	−0.0314767
$820$	0	0
$821$	−56.5236	−1.97269	−0.986343	−	0.164706i	$-0.947333\pi$
$821$	−56.5236	−1.97269	−0.986343	+	0.164706i	$0.947333\pi$
$822$	0	0
$823$	29.1342	1.01556	0.507778	−	0.861488i	$-0.330467\pi$
$823$	29.1342	1.01556	0.507778	+	0.861488i	$0.330467\pi$
$824$	0	0
$825$	1.86620	0.0649727
$826$	0	0
$827$	21.5853	0.750595	0.375298	−	0.926904i	$-0.377541\pi$
$827$	21.5853	0.750595	0.375298	+	0.926904i	$0.377541\pi$
$828$	0	0
$829$	13.6032	0.472460	0.236230	−	0.971697i	$-0.424088\pi$
$829$	13.6032	0.472460	0.236230	+	0.971697i	$0.424088\pi$
$830$	0	0
$831$	−20.9296	−0.726039
$832$	0	0
$833$	−2.83159	−0.0981088
$834$	0	0
$835$	18.8604	0.652690
$836$	0	0
$837$	23.8708	0.825097
$838$	0	0
$839$	−23.7191	−0.818875	−0.409438	−	0.912338i	$-0.634275\pi$
$839$	−23.7191	−0.818875	−0.409438	+	0.912338i	$0.634275\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	19.3356	0.665954
$844$	0	0
$845$	9.51730	0.327405
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	33.3956	1.14613
$850$	0	0
$851$	16.7670	0.574766
$852$	0	0
$853$	−28.5282	−0.976786	−0.488393	−	0.872624i	$-0.662417\pi$
$853$	−28.5282	−0.976786	−0.488393	+	0.872624i	$0.662417\pi$
$854$	0	0
$855$	−1.46479	−0.0500949
$856$	0	0
$857$	−49.1809	−1.67999	−0.839994	−	0.542596i	$-0.817441\pi$
$857$	−49.1809	−1.67999	−0.839994	+	0.542596i	$0.817441\pi$
$858$	0	0
$859$	38.8020	1.32391	0.661954	−	0.749544i	$-0.269727\pi$
$859$	38.8020	1.32391	0.661954	+	0.749544i	$0.269727\pi$
$860$	0	0
$861$	−20.6799	−0.704769
$862$	0	0
$863$	−17.7145	−0.603008	−0.301504	−	0.953465i	$-0.597489\pi$
$863$	−17.7145	−0.603008	−0.301504	+	0.953465i	$0.597489\pi$
$864$	0	0
$865$	−5.26178	−0.178906
$866$	0	0
$867$	16.7624	0.569280
$868$	0	0
$869$	16.2497	0.551233
$870$	0	0
$871$	−6.62857	−0.224601
$872$	0	0
$873$	−2.80041	−0.0947797
$874$	0	0
$875$	1.00000	0.0338062
$876$	0	0
$877$	−4.33380	−0.146342	−0.0731711	−	0.997319i	$-0.523312\pi$
$877$	−4.33380	−0.146342	−0.0731711	+	0.997319i	$0.523312\pi$
$878$	0	0
$879$	23.1435	0.780610
$880$	0	0
$881$	−20.3610	−0.685978	−0.342989	−	0.939339i	$-0.611439\pi$
$881$	−20.3610	−0.685978	−0.342989	+	0.939339i	$0.611439\pi$
$882$	0	0
$883$	−31.5161	−1.06060	−0.530301	−	0.847810i	$-0.677921\pi$
$883$	−31.5161	−1.06060	−0.530301	+	0.847810i	$0.677921\pi$
$884$	0	0
$885$	20.0513	0.674018
$886$	0	0
$887$	29.2664	0.982670	0.491335	−	0.870971i	$-0.336509\pi$
$887$	29.2664	0.982670	0.491335	+	0.870971i	$0.336509\pi$
$888$	0	0
$889$	8.62857	0.289393
$890$	0	0
$891$	10.2151	0.342218
$892$	0	0
$893$	32.0576	1.07277
$894$	0	0
$895$	25.3598	0.847684
$896$	0	0
$897$	−18.1626	−0.606431
$898$	0	0
$899$	−10.1626	−0.338941
$900$	0	0
$901$	−23.3598	−0.778227
$902$	0	0
$903$	−22.2318	−0.739828
$904$	0	0
$905$	−7.10382	−0.236139
$906$	0	0
$907$	−29.1793	−0.968882	−0.484441	−	0.874824i	$-0.660977\pi$
$907$	−29.1793	−0.968882	−0.484441	+	0.874824i	$0.660977\pi$
$908$	0	0
$909$	6.52193	0.216319
$910$	0	0
$911$	46.7195	1.54789	0.773944	−	0.633254i	$-0.218281\pi$
$911$	46.7195	1.54789	0.773944	+	0.633254i	$0.218281\pi$
$912$	0	0
$913$	−8.69779	−0.287855
$914$	0	0
$915$	−21.3085	−0.704436
$916$	0	0
$917$	5.39558	0.178178
$918$	0	0
$919$	−38.8362	−1.28109	−0.640544	−	0.767921i	$-0.721291\pi$
$919$	−38.8362	−1.28109	−0.640544	+	0.767921i	$0.721291\pi$
$920$	0	0
$921$	32.2676	1.06325
$922$	0	0
$923$	−3.60323	−0.118602
$924$	0	0
$925$	3.21509	0.105712
$926$	0	0
$927$	1.70362	0.0559541
$928$	0	0
$929$	46.0817	1.51189	0.755946	−	0.654634i	$-0.227177\pi$
$929$	46.0817	1.51189	0.755946	+	0.654634i	$0.227177\pi$
$930$	0	0
$931$	3.03461	0.0994553
$932$	0	0
$933$	20.8541	0.682733
$934$	0	0
$935$	−2.83159	−0.0926029
$936$	0	0
$937$	9.05413	0.295785	0.147893	−	0.989003i	$-0.452751\pi$
$937$	9.05413	0.295785	0.147893	+	0.989003i	$0.452751\pi$
$938$	0	0
$939$	−43.5940	−1.42264
$940$	0	0
$941$	−13.3489	−0.435162	−0.217581	−	0.976042i	$-0.569817\pi$
$941$	−13.3489	−0.435162	−0.217581	+	0.976042i	$0.569817\pi$
$942$	0	0
$943$	−57.7900	−1.88190
$944$	0	0
$945$	4.69779	0.152819
$946$	0	0
$947$	−33.4018	−1.08541	−0.542707	−	0.839922i	$-0.682600\pi$
$947$	−33.4018	−1.08541	−0.542707	+	0.839922i	$0.682600\pi$
$948$	0	0
$949$	−9.64528	−0.313099
$950$	0	0
$951$	−28.7312	−0.931673
$952$	0	0
$953$	−22.3672	−0.724545	−0.362273	−	0.932072i	$-0.617999\pi$
$953$	−22.3672	−0.724545	−0.362273	+	0.932072i	$0.617999\pi$
$954$	0	0
$955$	8.26760	0.267533
$956$	0	0
$957$	−3.73240	−0.120651
$958$	0	0
$959$	−9.81952	−0.317089
$960$	0	0
$961$	−5.18048	−0.167112
$962$	0	0
$963$	5.03461	0.162238
$964$	0	0
$965$	18.9475	0.609941
$966$	0	0
$967$	20.9475	0.673626	0.336813	−	0.941572i	$-0.390651\pi$
$967$	20.9475	0.673626	0.336813	+	0.941572i	$0.390651\pi$
$968$	0	0
$969$	16.0358	0.515144
$970$	0	0
$971$	31.3489	1.00603	0.503017	−	0.864277i	$-0.332223\pi$
$971$	31.3489	1.00603	0.503017	+	0.864277i	$0.332223\pi$
$972$	0	0
$973$	15.1972	0.487200
$974$	0	0
$975$	−3.48270	−0.111536
$976$	0	0
$977$	−9.91913	−0.317341	−0.158670	−	0.987332i	$-0.550721\pi$
$977$	−9.91913	−0.317341	−0.158670	+	0.987332i	$0.550721\pi$
$978$	0	0
$979$	3.73240	0.119288
$980$	0	0
$981$	−2.76700	−0.0883437
$982$	0	0
$983$	−46.2248	−1.47434	−0.737171	−	0.675707i	$-0.763839\pi$
$983$	−46.2248	−1.47434	−0.737171	+	0.675707i	$0.763839\pi$
$984$	0	0
$985$	10.9475	0.348816
$986$	0	0
$987$	19.7145	0.627519
$988$	0	0
$989$	−62.1268	−1.97552
$990$	0	0
$991$	−15.7682	−0.500893	−0.250447	−	0.968130i	$-0.580577\pi$
$991$	−15.7682	−0.500893	−0.250447	+	0.968130i	$0.580577\pi$
$992$	0	0
$993$	−15.3622	−0.487504
$994$	0	0
$995$	7.70986	0.244419
$996$	0	0
$997$	−14.4256	−0.456862	−0.228431	−	0.973560i	$-0.573360\pi$
$997$	−14.4256	−0.456862	−0.228431	+	0.973560i	$0.573360\pi$
$998$	0	0
$999$	15.1038	0.477864

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3080.2.a.m.1.1	✓	3	1.1	even	1	trivial
6160.2.a.bd.1.3		3	4.3	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:	\( N \)	\(=\)	\( 3080 = 2^{3} \cdot 5 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3080.a (trivial)

\(f(q)\)	\(=\)	\(q-1.86620 q^{3} -1.00000 q^{5} -1.00000 q^{7} +0.482696 q^{9} +O(q^{10})\)	\(q-1.86620 q^{3} -1.00000 q^{5} -1.00000 q^{7} +0.482696 q^{9} -1.00000 q^{11} +1.86620 q^{13} +1.86620 q^{15} -2.83159 q^{17} +3.03461 q^{19} +1.86620 q^{21} +5.21509 q^{23} +1.00000 q^{25} +4.69779 q^{27} -2.00000 q^{29} +5.08129 q^{31} +1.86620 q^{33} +1.00000 q^{35} +3.21509 q^{37} -3.48270 q^{39} -11.0813 q^{41} -11.9129 q^{43} -0.482696 q^{45} +10.5640 q^{47} +1.00000 q^{49} +5.28431 q^{51} +8.24970 q^{53} +1.00000 q^{55} -5.66318 q^{57} +10.7445 q^{59} -11.4181 q^{61} -0.482696 q^{63} -1.86620 q^{65} -3.55191 q^{67} -9.73240 q^{69} -1.93078 q^{71} -5.16841 q^{73} -1.86620 q^{75} +1.00000 q^{77} -16.2497 q^{79} -10.2151 q^{81} +8.69779 q^{83} +2.83159 q^{85} +3.73240 q^{87} -3.73240 q^{89} -1.86620 q^{91} -9.48270 q^{93} -3.03461 q^{95} -5.80161 q^{97} -0.482696 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{3} - 3 q^{5} - 3 q^{7} + 3 q^{9}+O(q^{10}) \) 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

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