LMFDB - Embedded newform 6000.2.f.o.1249.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6000,2,Mod(1249,6000)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6000, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6000.1249");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6000 = 2^{4} \cdot 3 \cdot 5^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6000.f (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$47.9102412128$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.1632160000.5
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 12x^{6} + 46x^{4} + 65x^{2} + 25 $ x^8 + 12x^6 + 46x^4 + 65*x^2 + 25
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 375)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1249.5
Root			$-1.46673i$ of defining polynomial
Character	$\chi$	$=$	6000.1249
Dual form			6000.2.f.o.1249.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} -2.93346i q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} -2.93346i q^{7} -1.00000 q^{9} -5.04905 q^{11} +3.69740i q^{13} -2.85410i q^{17} +4.43101 q^{19} +2.93346 q^{21} +8.36448i q^{23} -1.00000i q^{27} -1.88442 q^{29} -0.266381 q^{31} -5.04905i q^{33} -8.35655i q^{37} -3.69740 q^{39} -0.169532 q^{41} -9.59262i q^{43} -2.43101i q^{47} -1.60521 q^{49} +2.85410 q^{51} +7.24889i q^{53} +4.43101i q^{57} +5.89522 q^{59} -6.48496 q^{61} +2.93346i q^{63} +8.55942i q^{67} -8.36448 q^{69} +16.2676 q^{71} +1.81298i q^{73} +14.8112i q^{77} +2.67198 q^{79} +1.00000 q^{81} +12.9032i q^{83} -1.88442i q^{87} -12.6309 q^{89} +10.8462 q^{91} -0.266381i q^{93} -6.89032i q^{97} +5.04905 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{9}+O(q^{10}) $ 8 * q - 8 * q^9	$ 8 q - 8 q^{9} - 12 q^{11} + 16 q^{19} - 8 q^{21} - 12 q^{29} - 8 q^{31} - 16 q^{39} + 48 q^{41} - 40 q^{49} - 4 q^{51} + 4 q^{59} + 20 q^{61} - 16 q^{69} + 24 q^{71} - 40 q^{79} + 8 q^{81} - 56 q^{89} + 72 q^{91} + 12 q^{99}+O(q^{100}) $ 8 * q - 8 * q^9 - 12 * q^11 + 16 * q^19 - 8 * q^21 - 12 * q^29 - 8 * q^31 - 16 * q^39 + 48 * q^41 - 40 * q^49 - 4 * q^51 + 4 * q^59 + 20 * q^61 - 16 * q^69 + 24 * q^71 - 40 * q^79 + 8 * q^81 - 56 * q^89 + 72 * q^91 + 12 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/6000\mathbb{Z}\right)^\times$.

$n$	$751$	$4001$	$4501$	$5377$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 2.93346i	− 1.10875i	−0.832269	−	0.554373i	$-0.812958\pi$
$7$	− 2.93346i	− 1.10875i	0.832269	−	0.554373i	$-0.187042\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−5.04905	−1.52235	−0.761173	−	0.648549i	$-0.775376\pi$
$11$	−5.04905	−1.52235	−0.761173	+	0.648549i	$0.775376\pi$
$12$	0	0
$13$	3.69740i	1.02547i	0.858546	+	0.512737i	$0.171368\pi$
$13$	3.69740i	1.02547i	−0.858546	+	0.512737i	$0.828632\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 2.85410i	− 0.692221i	−0.938194	−	0.346111i	$-0.887502\pi$
$17$	− 2.85410i	− 0.692221i	0.938194	−	0.346111i	$-0.112498\pi$
$18$	0	0
$19$	4.43101	1.01654	0.508272	−	0.861196i	$-0.330284\pi$
$19$	4.43101	1.01654	0.508272	+	0.861196i	$0.330284\pi$
$20$	0	0
$21$	2.93346	0.640134
$22$	0	0
$23$	8.36448i	1.74411i	0.489404	+	0.872057i	$0.337214\pi$
$23$	8.36448i	1.74411i	−0.489404	+	0.872057i	$0.662786\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 1.00000i	− 0.192450i
$28$	0	0
$29$	−1.88442	−0.349927	−0.174964	−	0.984575i	$-0.555981\pi$
$29$	−1.88442	−0.349927	−0.174964	+	0.984575i	$0.555981\pi$
$30$	0	0
$31$	−0.266381	−0.0478435	−0.0239218	−	0.999714i	$-0.507615\pi$
$31$	−0.266381	−0.0478435	−0.0239218	+	0.999714i	$0.507615\pi$
$32$	0	0
$33$	− 5.04905i	− 0.878926i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 8.35655i	− 1.37381i	−0.726748	−	0.686904i	$-0.758969\pi$
$37$	− 8.35655i	− 1.37381i	0.726748	−	0.686904i	$-0.241031\pi$
$38$	0	0
$39$	−3.69740	−0.592057
$40$	0	0
$41$	−0.169532	−0.0264764	−0.0132382	−	0.999912i	$-0.504214\pi$
$41$	−0.169532	−0.0264764	−0.0132382	+	0.999912i	$0.504214\pi$
$42$	0	0
$43$	− 9.59262i	− 1.46286i	−0.681916	−	0.731430i	$-0.738853\pi$
$43$	− 9.59262i	− 1.46286i	0.681916	−	0.731430i	$-0.261147\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 2.43101i	− 0.354600i	−0.984157	−	0.177300i	$-0.943264\pi$
$47$	− 2.43101i	− 0.354600i	0.984157	−	0.177300i	$-0.0567363\pi$
$48$	0	0
$49$	−1.60521	−0.229316
$50$	0	0
$51$	2.85410	0.399654
$52$	0	0
$53$	7.24889i	0.995712i	0.867260	+	0.497856i	$0.165879\pi$
$53$	7.24889i	0.995712i	−0.867260	+	0.497856i	$0.834121\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	4.43101i	0.586902i
$58$	0	0
$59$	5.89522	0.767493	0.383746	−	0.923439i	$-0.374634\pi$
$59$	5.89522	0.767493	0.383746	+	0.923439i	$0.374634\pi$
$60$	0	0
$61$	−6.48496	−0.830314	−0.415157	−	0.909750i	$-0.636273\pi$
$61$	−6.48496	−0.830314	−0.415157	+	0.909750i	$0.636273\pi$
$62$	0	0
$63$	2.93346i	0.369582i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	8.55942i	1.04570i	0.852425	+	0.522850i	$0.175131\pi$
$67$	8.55942i	1.04570i	−0.852425	+	0.522850i	$0.824869\pi$
$68$	0	0
$69$	−8.36448	−1.00696
$70$	0	0
$71$	16.2676	1.93061	0.965306	−	0.261121i	$-0.0840921\pi$
$71$	16.2676	1.93061	0.965306	+	0.261121i	$0.0840921\pi$
$72$	0	0
$73$	1.81298i	0.212193i	0.994356	+	0.106097i	$0.0338353\pi$
$73$	1.81298i	0.212193i	−0.994356	+	0.106097i	$0.966165\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	14.8112i	1.68789i
$78$	0	0
$79$	2.67198	0.300621	0.150311	−	0.988639i	$-0.451973\pi$
$79$	2.67198	0.300621	0.150311	+	0.988639i	$0.451973\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	12.9032i	1.41630i	0.706060	+	0.708152i	$0.250471\pi$
$83$	12.9032i	1.41630i	−0.706060	+	0.708152i	$0.749529\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 1.88442i	− 0.202031i
$88$	0	0
$89$	−12.6309	−1.33887	−0.669434	−	0.742871i	$-0.733463\pi$
$89$	−12.6309	−1.33887	−0.669434	+	0.742871i	$0.733463\pi$
$90$	0	0
$91$	10.8462	1.13699
$92$	0	0
$93$	− 0.266381i	− 0.0276225i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 6.89032i	− 0.699606i	−0.936823	−	0.349803i	$-0.886248\pi$
$97$	− 6.89032i	− 0.699606i	0.936823	−	0.349803i	$-0.113752\pi$
$98$	0	0
$99$	5.04905	0.507448
$100$	0	0
$101$	9.77964	0.973110	0.486555	−	0.873650i	$-0.338253\pi$
$101$	9.77964	0.973110	0.486555	+	0.873650i	$0.338253\pi$
$102$	0	0
$103$	− 14.8620i	− 1.46440i	−0.681090	−	0.732200i	$-0.738494\pi$
$103$	− 14.8620i	− 1.46440i	0.681090	−	0.732200i	$-0.261506\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 3.63552i	− 0.351459i	−0.984438	−	0.175730i	$-0.943772\pi$
$107$	− 3.63552i	− 0.351459i	0.984438	−	0.175730i	$-0.0562284\pi$
$108$	0	0
$109$	4.62293	0.442797	0.221398	−	0.975183i	$-0.428938\pi$
$109$	4.62293	0.442797	0.221398	+	0.975183i	$0.428938\pi$
$110$	0	0
$111$	8.35655	0.793169
$112$	0	0
$113$	− 0.776757i	− 0.0730712i	−0.999332	−	0.0365356i	$-0.988368\pi$
$113$	− 0.776757i	− 0.0730712i	0.999332	−	0.0365356i	$-0.0116322\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 3.69740i	− 0.341824i
$118$	0	0
$119$	−8.37240	−0.767497
$120$	0	0
$121$	14.4929	1.31754
$122$	0	0
$123$	− 0.169532i	− 0.0152861i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 6.26763i	− 0.556162i	−0.960558	−	0.278081i	$-0.910302\pi$
$127$	− 6.26763i	− 0.556162i	0.960558	−	0.278081i	$-0.0896984\pi$
$128$	0	0
$129$	9.59262	0.844583
$130$	0	0
$131$	16.1346	1.40968	0.704841	−	0.709365i	$-0.251018\pi$
$131$	16.1346	1.40968	0.704841	+	0.709365i	$0.251018\pi$
$132$	0	0
$133$	− 12.9982i	− 1.12709i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 17.9650i	− 1.53486i	−0.641135	−	0.767428i	$-0.721536\pi$
$137$	− 17.9650i	− 1.53486i	0.641135	−	0.767428i	$-0.278464\pi$
$138$	0	0
$139$	8.83661	0.749512	0.374756	−	0.927124i	$-0.377727\pi$
$139$	8.83661	0.749512	0.374756	+	0.927124i	$0.377727\pi$
$140$	0	0
$141$	2.43101	0.204728
$142$	0	0
$143$	− 18.6683i	− 1.56112i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 1.60521i	− 0.132395i
$148$	0	0
$149$	17.3881	1.42449	0.712245	−	0.701931i	$-0.247679\pi$
$149$	17.3881	1.42449	0.712245	+	0.701931i	$0.247679\pi$
$150$	0	0
$151$	−2.96378	−0.241189	−0.120594	−	0.992702i	$-0.538480\pi$
$151$	−2.96378	−0.241189	−0.120594	+	0.992702i	$0.538480\pi$
$152$	0	0
$153$	2.85410i	0.230740i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	3.06654i	0.244736i	0.992485	+	0.122368i	$0.0390489\pi$
$157$	3.06654i	0.244736i	−0.992485	+	0.122368i	$0.960951\pi$
$158$	0	0
$159$	−7.24889	−0.574875
$160$	0	0
$161$	24.5369	1.93378
$162$	0	0
$163$	10.1803i	0.797386i	0.917085	+	0.398693i	$0.130536\pi$
$163$	10.1803i	0.797386i	−0.917085	+	0.398693i	$0.869464\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 20.2373i	− 1.56601i	−0.622015	−	0.783005i	$-0.713686\pi$
$167$	− 20.2373i	− 1.56601i	0.622015	−	0.783005i	$-0.286314\pi$
$168$	0	0
$169$	−0.670734	−0.0515950
$170$	0	0
$171$	−4.43101	−0.338848
$172$	0	0
$173$	5.07734i	0.386023i	0.981196	+	0.193012i	$0.0618255\pi$
$173$	5.07734i	0.386023i	−0.981196	+	0.193012i	$0.938174\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	5.89522i	0.443112i
$178$	0	0
$179$	12.3216	0.920958	0.460479	−	0.887671i	$-0.347678\pi$
$179$	12.3216	0.920958	0.460479	+	0.887671i	$0.347678\pi$
$180$	0	0
$181$	18.4243	1.36947	0.684735	−	0.728792i	$-0.259918\pi$
$181$	18.4243	1.36947	0.684735	+	0.728792i	$0.259918\pi$
$182$	0	0
$183$	− 6.48496i	− 0.479382i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	14.4105i	1.05380i
$188$	0	0
$189$	−2.93346	−0.213378
$190$	0	0
$191$	−1.60847	−0.116385	−0.0581925	−	0.998305i	$-0.518534\pi$
$191$	−1.60847	−0.116385	−0.0581925	+	0.998305i	$0.518534\pi$
$192$	0	0
$193$	− 15.3167i	− 1.10252i	−0.834334	−	0.551259i	$-0.814148\pi$
$193$	− 15.3167i	− 1.10252i	0.834334	−	0.551259i	$-0.185852\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 5.48496i	− 0.390787i	−0.980725	−	0.195394i	$-0.937402\pi$
$197$	− 5.48496i	− 0.390787i	0.980725	−	0.195394i	$-0.0625984\pi$
$198$	0	0
$199$	12.0885	0.856933	0.428467	−	0.903558i	$-0.359054\pi$
$199$	12.0885	0.856933	0.428467	+	0.903558i	$0.359054\pi$
$200$	0	0
$201$	−8.55942	−0.603735
$202$	0	0
$203$	5.52786i	0.387980i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 8.36448i	− 0.581371i
$208$	0	0
$209$	−22.3724	−1.54753
$210$	0	0
$211$	5.26148	0.362215	0.181108	−	0.983463i	$-0.442032\pi$
$211$	5.26148	0.362215	0.181108	+	0.983463i	$0.442032\pi$
$212$	0	0
$213$	16.2676i	1.11464i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0.781420i	0.0530463i
$218$	0	0
$219$	−1.81298	−0.122510
$220$	0	0
$221$	10.5527	0.709854
$222$	0	0
$223$	1.41228i	0.0945732i	0.998881	+	0.0472866i	$0.0150574\pi$
$223$	1.41228i	0.0945732i	−0.998881	+	0.0472866i	$0.984943\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	19.5656i	1.29861i	0.760527	+	0.649306i	$0.224941\pi$
$227$	19.5656i	1.29861i	−0.760527	+	0.649306i	$0.775059\pi$
$228$	0	0
$229$	21.3735	1.41240	0.706200	−	0.708012i	$-0.250408\pi$
$229$	21.3735	1.41240	0.706200	+	0.708012i	$0.250408\pi$
$230$	0	0
$231$	−14.8112	−0.974505
$232$	0	0
$233$	3.15382i	0.206614i	0.994650	+	0.103307i	$0.0329424\pi$
$233$	3.15382i	0.206614i	−0.994650	+	0.103307i	$0.967058\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	2.67198i	0.173564i
$238$	0	0
$239$	−19.2725	−1.24664	−0.623318	−	0.781968i	$-0.714216\pi$
$239$	−19.2725	−1.24664	−0.623318	+	0.781968i	$0.714216\pi$
$240$	0	0
$241$	22.7210	1.46359	0.731795	−	0.681525i	$-0.238683\pi$
$241$	22.7210	1.46359	0.731795	+	0.681525i	$0.238683\pi$
$242$	0	0
$243$	1.00000i	0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	16.3832i	1.04244i
$248$	0	0
$249$	−12.9032	−0.817704
$250$	0	0
$251$	−8.14124	−0.513870	−0.256935	−	0.966429i	$-0.582713\pi$
$251$	−8.14124	−0.513870	−0.256935	+	0.966429i	$0.582713\pi$
$252$	0	0
$253$	− 42.2327i	− 2.65514i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	22.6653i	1.41382i	0.707302	+	0.706911i	$0.249912\pi$
$257$	22.6653i	1.41382i	−0.707302	+	0.706911i	$0.750088\pi$
$258$	0	0
$259$	−24.5136	−1.52320
$260$	0	0
$261$	1.88442	0.116642
$262$	0	0
$263$	− 6.81500i	− 0.420231i	−0.977677	−	0.210115i	$-0.932616\pi$
$263$	− 6.81500i	− 0.420231i	0.977677	−	0.210115i	$-0.0673840\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 12.6309i	− 0.772996i
$268$	0	0
$269$	20.1089	1.22606	0.613031	−	0.790059i	$-0.289950\pi$
$269$	20.1089	1.22606	0.613031	+	0.790059i	$0.289950\pi$
$270$	0	0
$271$	16.8366	1.02275	0.511376	−	0.859357i	$-0.329136\pi$
$271$	16.8366	1.02275	0.511376	+	0.859357i	$0.329136\pi$
$272$	0	0
$273$	10.8462i	0.656440i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 25.7015i	− 1.54425i	−0.635468	−	0.772127i	$-0.719193\pi$
$277$	− 25.7015i	− 1.54425i	0.635468	−	0.772127i	$-0.280807\pi$
$278$	0	0
$279$	0.266381	0.0159478
$280$	0	0
$281$	−27.7097	−1.65302	−0.826511	−	0.562921i	$-0.809677\pi$
$281$	−27.7097	−1.65302	−0.826511	+	0.562921i	$0.809677\pi$
$282$	0	0
$283$	11.6466i	0.692317i	0.938176	+	0.346158i	$0.112514\pi$
$283$	11.6466i	0.692317i	−0.938176	+	0.346158i	$0.887486\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0.497315i	0.0293556i
$288$	0	0
$289$	8.85410	0.520830
$290$	0	0
$291$	6.89032	0.403918
$292$	0	0
$293$	7.72593i	0.451354i	0.974202	+	0.225677i	$0.0724593\pi$
$293$	7.72593i	0.451354i	−0.974202	+	0.225677i	$0.927541\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	5.04905i	0.292975i
$298$	0	0
$299$	−30.9268	−1.78854
$300$	0	0
$301$	−28.1396	−1.62194
$302$	0	0
$303$	9.77964i	0.561826i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	12.3674i	0.705842i	0.935653	+	0.352921i	$0.114812\pi$
$307$	12.3674i	0.705842i	−0.935653	+	0.352921i	$0.885188\pi$
$308$	0	0
$309$	14.8620	0.845471
$310$	0	0
$311$	4.53276	0.257029	0.128515	−	0.991708i	$-0.458979\pi$
$311$	4.53276	0.257029	0.128515	+	0.991708i	$0.458979\pi$
$312$	0	0
$313$	− 2.33090i	− 0.131750i	−0.997828	−	0.0658752i	$-0.979016\pi$
$313$	− 2.33090i	− 0.131750i	0.997828	−	0.0658752i	$-0.0209839\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	7.86693i	0.441851i	0.975291	+	0.220925i	$0.0709077\pi$
$317$	7.86693i	0.441851i	−0.975291	+	0.220925i	$0.929092\pi$
$318$	0	0
$319$	9.51450	0.532710
$320$	0	0
$321$	3.63552	0.202915
$322$	0	0
$323$	− 12.6466i	− 0.703674i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	4.62293i	0.255649i
$328$	0	0
$329$	−7.13129	−0.393161
$330$	0	0
$331$	20.3744	1.11988	0.559940	−	0.828533i	$-0.310824\pi$
$331$	20.3744	1.11988	0.559940	+	0.828533i	$0.310824\pi$
$332$	0	0
$333$	8.35655i	0.457936i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	17.0141i	0.926816i	0.886145	+	0.463408i	$0.153373\pi$
$337$	17.0141i	0.926816i	−0.886145	+	0.463408i	$0.846627\pi$
$338$	0	0
$339$	0.776757	0.0421877
$340$	0	0
$341$	1.34497	0.0728344
$342$	0	0
$343$	− 15.8254i	− 0.854493i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 33.7908i	− 1.81399i	−0.421145	−	0.906993i	$-0.638372\pi$
$347$	− 33.7908i	− 1.81399i	0.421145	−	0.906993i	$-0.361628\pi$
$348$	0	0
$349$	−18.8798	−1.01061	−0.505305	−	0.862941i	$-0.668620\pi$
$349$	−18.8798	−1.01061	−0.505305	+	0.862941i	$0.668620\pi$
$350$	0	0
$351$	3.69740	0.197352
$352$	0	0
$353$	− 14.2330i	− 0.757548i	−0.925489	−	0.378774i	$-0.876346\pi$
$353$	− 14.2330i	− 0.757548i	0.925489	−	0.378774i	$-0.123654\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 8.37240i	− 0.443115i
$358$	0	0
$359$	11.1014	0.585907	0.292954	−	0.956127i	$-0.405362\pi$
$359$	11.1014	0.585907	0.292954	+	0.956127i	$0.405362\pi$
$360$	0	0
$361$	0.633887	0.0333625
$362$	0	0
$363$	14.4929i	0.760679i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 32.1387i	− 1.67763i	−0.544420	−	0.838813i	$-0.683250\pi$
$367$	− 32.1387i	− 1.67763i	0.544420	−	0.838813i	$-0.316750\pi$
$368$	0	0
$369$	0.169532	0.00882546
$370$	0	0
$371$	21.2644	1.10399
$372$	0	0
$373$	15.4754i	0.801286i	0.916234	+	0.400643i	$0.131213\pi$
$373$	15.4754i	0.801286i	−0.916234	+	0.400643i	$0.868787\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 6.96743i	− 0.358841i
$378$	0	0
$379$	−23.4164	−1.20282	−0.601410	−	0.798941i	$-0.705394\pi$
$379$	−23.4164	−1.20282	−0.601410	+	0.798941i	$0.705394\pi$
$380$	0	0
$381$	6.26763	0.321100
$382$	0	0
$383$	− 2.64042i	− 0.134919i	−0.997722	−	0.0674596i	$-0.978511\pi$
$383$	− 2.64042i	− 0.134919i	0.997722	−	0.0674596i	$-0.0214894\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	9.59262i	0.487620i
$388$	0	0
$389$	3.98415	0.202004	0.101002	−	0.994886i	$-0.467795\pi$
$389$	3.98415	0.202004	0.101002	+	0.994886i	$0.467795\pi$
$390$	0	0
$391$	23.8731	1.20731
$392$	0	0
$393$	16.1346i	0.813881i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 7.25768i	− 0.364253i	−0.983275	−	0.182126i	$-0.941702\pi$
$397$	− 7.25768i	− 0.364253i	0.983275	−	0.182126i	$-0.0582980\pi$
$398$	0	0
$399$	12.9982	0.650725
$400$	0	0
$401$	4.83373	0.241385	0.120693	−	0.992690i	$-0.461488\pi$
$401$	4.83373	0.241385	0.120693	+	0.992690i	$0.461488\pi$
$402$	0	0
$403$	− 0.984918i	− 0.0490622i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	42.1926i	2.09141i
$408$	0	0
$409$	−15.1508	−0.749159	−0.374579	−	0.927195i	$-0.622213\pi$
$409$	−15.1508	−0.749159	−0.374579	+	0.927195i	$0.622213\pi$
$410$	0	0
$411$	17.9650	0.886149
$412$	0	0
$413$	− 17.2934i	− 0.850954i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	8.83661i	0.432731i
$418$	0	0
$419$	16.2070	0.791764	0.395882	−	0.918301i	$-0.370439\pi$
$419$	16.2070	0.791764	0.395882	+	0.918301i	$0.370439\pi$
$420$	0	0
$421$	12.4085	0.604752	0.302376	−	0.953189i	$-0.402220\pi$
$421$	12.4085	0.604752	0.302376	+	0.953189i	$0.402220\pi$
$422$	0	0
$423$	2.43101i	0.118200i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	19.0234i	0.920607i
$428$	0	0
$429$	18.6683	0.901315
$430$	0	0
$431$	1.68255	0.0810457	0.0405229	−	0.999179i	$-0.487098\pi$
$431$	1.68255	0.0810457	0.0405229	+	0.999179i	$0.487098\pi$
$432$	0	0
$433$	14.9052i	0.716297i	0.933665	+	0.358148i	$0.116592\pi$
$433$	14.9052i	0.716297i	−0.933665	+	0.358148i	$0.883408\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	37.0631i	1.77297i
$438$	0	0
$439$	−22.0631	−1.05302	−0.526508	−	0.850170i	$-0.676499\pi$
$439$	−22.0631	−1.05302	−0.526508	+	0.850170i	$0.676499\pi$
$440$	0	0
$441$	1.60521	0.0764385
$442$	0	0
$443$	20.0061i	0.950521i	0.879845	+	0.475260i	$0.157646\pi$
$443$	20.0061i	0.950521i	−0.879845	+	0.475260i	$0.842354\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	17.3881i	0.822429i
$448$	0	0
$449$	16.5517	0.781125	0.390562	−	0.920576i	$-0.372281\pi$
$449$	16.5517	0.781125	0.390562	+	0.920576i	$0.372281\pi$
$450$	0	0
$451$	0.855973	0.0403062
$452$	0	0
$453$	− 2.96378i	− 0.139250i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	9.29771i	0.434928i	0.976068	+	0.217464i	$0.0697785\pi$
$457$	9.29771i	0.434928i	−0.976068	+	0.217464i	$0.930221\pi$
$458$	0	0
$459$	−2.85410	−0.133218
$460$	0	0
$461$	18.8179	0.876436	0.438218	−	0.898869i	$-0.355610\pi$
$461$	18.8179	0.876436	0.438218	+	0.898869i	$0.355610\pi$
$462$	0	0
$463$	− 7.50533i	− 0.348802i	−0.984675	−	0.174401i	$-0.944201\pi$
$463$	− 7.50533i	− 0.348802i	0.984675	−	0.174401i	$-0.0557990\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	4.50749i	0.208582i	0.994547	+	0.104291i	$0.0332573\pi$
$467$	4.50749i	0.208582i	−0.994547	+	0.104291i	$0.966743\pi$
$468$	0	0
$469$	25.1088	1.15941
$470$	0	0
$471$	−3.06654	−0.141299
$472$	0	0
$473$	48.4336i	2.22698i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 7.24889i	− 0.331904i
$478$	0	0
$479$	21.3291	0.974552	0.487276	−	0.873248i	$-0.337991\pi$
$479$	21.3291	0.974552	0.487276	+	0.873248i	$0.337991\pi$
$480$	0	0
$481$	30.8975	1.40880
$482$	0	0
$483$	24.5369i	1.11647i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	18.5203i	0.839236i	0.907701	+	0.419618i	$0.137836\pi$
$487$	18.5203i	0.839236i	−0.907701	+	0.419618i	$0.862164\pi$
$488$	0	0
$489$	−10.1803	−0.460371
$490$	0	0
$491$	−9.44369	−0.426188	−0.213094	−	0.977032i	$-0.568354\pi$
$491$	−9.44369	−0.426188	−0.213094	+	0.977032i	$0.568354\pi$
$492$	0	0
$493$	5.37831i	0.242227i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 47.7205i	− 2.14056i
$498$	0	0
$499$	−24.5706	−1.09993	−0.549966	−	0.835187i	$-0.685359\pi$
$499$	−24.5706	−1.09993	−0.549966	+	0.835187i	$0.685359\pi$
$500$	0	0
$501$	20.2373	0.904137
$502$	0	0
$503$	25.7970i	1.15023i	0.818072	+	0.575115i	$0.195043\pi$
$503$	25.7970i	1.15023i	−0.818072	+	0.575115i	$0.804957\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 0.670734i	− 0.0297884i
$508$	0	0
$509$	27.4970	1.21878	0.609392	−	0.792869i	$-0.291414\pi$
$509$	27.4970	1.21878	0.609392	+	0.792869i	$0.291414\pi$
$510$	0	0
$511$	5.31831	0.235268
$512$	0	0
$513$	− 4.43101i	− 0.195634i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	12.2743i	0.539823i
$518$	0	0
$519$	−5.07734	−0.222871
$520$	0	0
$521$	−9.23443	−0.404568	−0.202284	−	0.979327i	$-0.564836\pi$
$521$	−9.23443	−0.404568	−0.202284	+	0.979327i	$0.564836\pi$
$522$	0	0
$523$	− 34.1612i	− 1.49377i	−0.664956	−	0.746883i	$-0.731550\pi$
$523$	− 34.1612i	− 1.49377i	0.664956	−	0.746883i	$-0.268450\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0.760280i	0.0331183i
$528$	0	0
$529$	−46.9645	−2.04193
$530$	0	0
$531$	−5.89522	−0.255831
$532$	0	0
$533$	− 0.626825i	− 0.0271508i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	12.3216i	0.531715i
$538$	0	0
$539$	8.10478	0.349097
$540$	0	0
$541$	16.4858	0.708781	0.354391	−	0.935098i	$-0.384688\pi$
$541$	16.4858	0.708781	0.354391	+	0.935098i	$0.384688\pi$
$542$	0	0
$543$	18.4243i	0.790664i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 17.6399i	− 0.754227i	−0.926167	−	0.377114i	$-0.876917\pi$
$547$	− 17.6399i	− 0.754227i	0.926167	−	0.377114i	$-0.123083\pi$
$548$	0	0
$549$	6.48496	0.276771
$550$	0	0
$551$	−8.34987	−0.355716
$552$	0	0
$553$	− 7.83816i	− 0.333312i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	23.9964i	1.01676i	0.861132	+	0.508381i	$0.169756\pi$
$557$	23.9964i	1.01676i	−0.861132	+	0.508381i	$0.830244\pi$
$558$	0	0
$559$	35.4677	1.50012
$560$	0	0
$561$	−14.4105	−0.608412
$562$	0	0
$563$	3.56884i	0.150409i	0.997168	+	0.0752043i	$0.0239609\pi$
$563$	3.56884i	0.150409i	−0.997168	+	0.0752043i	$0.976039\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 2.93346i	− 0.123194i
$568$	0	0
$569$	11.4585	0.480367	0.240183	−	0.970728i	$-0.422792\pi$
$569$	11.4585	0.480367	0.240183	+	0.970728i	$0.422792\pi$
$570$	0	0
$571$	−35.5027	−1.48574	−0.742871	−	0.669435i	$-0.766536\pi$
$571$	−35.5027	−1.48574	−0.742871	+	0.669435i	$0.766536\pi$
$572$	0	0
$573$	− 1.60847i	− 0.0671949i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	0.225114i	0.00937161i	0.999989	+	0.00468581i	$0.00149154\pi$
$577$	0.225114i	0.00937161i	−0.999989	+	0.00468581i	$0.998508\pi$
$578$	0	0
$579$	15.3167	0.636539
$580$	0	0
$581$	37.8509	1.57032
$582$	0	0
$583$	− 36.6000i	− 1.51582i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	13.8981i	0.573636i	0.957985	+	0.286818i	$0.0925975\pi$
$587$	13.8981i	0.573636i	−0.957985	+	0.286818i	$0.907402\pi$
$588$	0	0
$589$	−1.18034	−0.0486351
$590$	0	0
$591$	5.48496	0.225621
$592$	0	0
$593$	− 9.45488i	− 0.388266i	−0.980975	−	0.194133i	$-0.937811\pi$
$593$	− 9.45488i	− 0.388266i	0.980975	−	0.194133i	$-0.0621893\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	12.0885i	0.494751i
$598$	0	0
$599$	8.07648	0.329996	0.164998	−	0.986294i	$-0.447238\pi$
$599$	8.07648	0.329996	0.164998	+	0.986294i	$0.447238\pi$
$600$	0	0
$601$	28.6038	1.16677	0.583387	−	0.812194i	$-0.301727\pi$
$601$	28.6038	1.16677	0.583387	+	0.812194i	$0.301727\pi$
$602$	0	0
$603$	− 8.55942i	− 0.348567i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	1.42208i	0.0577203i	0.999583	+	0.0288602i	$0.00918775\pi$
$607$	1.42208i	0.0577203i	−0.999583	+	0.0288602i	$0.990812\pi$
$608$	0	0
$609$	−5.52786	−0.224000
$610$	0	0
$611$	8.98842	0.363633
$612$	0	0
$613$	14.7556i	0.595974i	0.954570	+	0.297987i	$0.0963152\pi$
$613$	14.7556i	0.595974i	−0.954570	+	0.297987i	$0.903685\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	31.3519i	1.26218i	0.775710	+	0.631090i	$0.217392\pi$
$617$	31.3519i	1.26218i	−0.775710	+	0.631090i	$0.782608\pi$
$618$	0	0
$619$	12.6563	0.508699	0.254349	−	0.967112i	$-0.418139\pi$
$619$	12.6563	0.508699	0.254349	+	0.967112i	$0.418139\pi$
$620$	0	0
$621$	8.36448	0.335655
$622$	0	0
$623$	37.0522i	1.48446i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 22.3724i	− 0.893468i
$628$	0	0
$629$	−23.8504	−0.950980
$630$	0	0
$631$	−19.0741	−0.759327	−0.379663	−	0.925125i	$-0.623960\pi$
$631$	−19.0741	−0.759327	−0.379663	+	0.925125i	$0.623960\pi$
$632$	0	0
$633$	5.26148i	0.209125i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 5.93509i	− 0.235157i
$638$	0	0
$639$	−16.2676	−0.643537
$640$	0	0
$641$	−3.58594	−0.141636	−0.0708180	−	0.997489i	$-0.522561\pi$
$641$	−3.58594	−0.141636	−0.0708180	+	0.997489i	$0.522561\pi$
$642$	0	0
$643$	− 37.6409i	− 1.48441i	−0.670172	−	0.742206i	$-0.733780\pi$
$643$	− 37.6409i	− 1.48441i	0.670172	−	0.742206i	$-0.266220\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	2.21935i	0.0872517i	0.999048	+	0.0436258i	$0.0138909\pi$
$647$	2.21935i	0.0872517i	−0.999048	+	0.0436258i	$0.986109\pi$
$648$	0	0
$649$	−29.7653	−1.16839
$650$	0	0
$651$	−0.781420	−0.0306263
$652$	0	0
$653$	− 3.13494i	− 0.122680i	−0.998117	−	0.0613399i	$-0.980463\pi$
$653$	− 3.13494i	− 0.122680i	0.998117	−	0.0613399i	$-0.0195374\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 1.81298i	− 0.0707311i
$658$	0	0
$659$	−21.6691	−0.844108	−0.422054	−	0.906571i	$-0.638691\pi$
$659$	−21.6691	−0.844108	−0.422054	+	0.906571i	$0.638691\pi$
$660$	0	0
$661$	−21.9730	−0.854649	−0.427324	−	0.904098i	$-0.640544\pi$
$661$	−21.9730	−0.854649	−0.427324	+	0.904098i	$0.640544\pi$
$662$	0	0
$663$	10.5527i	0.409835i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 15.7622i	− 0.610313i
$668$	0	0
$669$	−1.41228	−0.0546019
$670$	0	0
$671$	32.7429	1.26402
$672$	0	0
$673$	− 32.7911i	− 1.26400i	−0.774967	−	0.632001i	$-0.782234\pi$
$673$	− 32.7911i	− 1.26400i	0.774967	−	0.632001i	$-0.217766\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 25.8289i	− 0.992686i	−0.868126	−	0.496343i	$-0.834676\pi$
$677$	− 25.8289i	− 0.992686i	0.868126	−	0.496343i	$-0.165324\pi$
$678$	0	0
$679$	−20.2125	−0.775685
$680$	0	0
$681$	−19.5656	−0.749754
$682$	0	0
$683$	23.4044i	0.895543i	0.894148	+	0.447771i	$0.147782\pi$
$683$	23.4044i	0.895543i	−0.894148	+	0.447771i	$0.852218\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	21.3735i	0.815450i
$688$	0	0
$689$	−26.8020	−1.02108
$690$	0	0
$691$	−9.91799	−0.377298	−0.188649	−	0.982045i	$-0.560411\pi$
$691$	−9.91799	−0.377298	−0.188649	+	0.982045i	$0.560411\pi$
$692$	0	0
$693$	− 14.8112i	− 0.562631i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0.483860i	0.0183275i
$698$	0	0
$699$	−3.15382	−0.119289
$700$	0	0
$701$	−36.6906	−1.38578	−0.692892	−	0.721042i	$-0.743664\pi$
$701$	−36.6906	−1.38578	−0.692892	+	0.721042i	$0.743664\pi$
$702$	0	0
$703$	− 37.0280i	− 1.39654i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 28.6882i	− 1.07893i
$708$	0	0
$709$	33.0393	1.24082	0.620409	−	0.784278i	$-0.286967\pi$
$709$	33.0393	1.24082	0.620409	+	0.784278i	$0.286967\pi$
$710$	0	0
$711$	−2.67198	−0.100207
$712$	0	0
$713$	− 2.22814i	− 0.0834446i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 19.2725i	− 0.719746i
$718$	0	0
$719$	30.0339	1.12008	0.560038	−	0.828467i	$-0.310787\pi$
$719$	30.0339	1.12008	0.560038	+	0.828467i	$0.310787\pi$
$720$	0	0
$721$	−43.5972	−1.62365
$722$	0	0
$723$	22.7210i	0.845004i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 29.5053i	− 1.09429i	−0.837037	−	0.547146i	$-0.815714\pi$
$727$	− 29.5053i	− 1.09429i	0.837037	−	0.547146i	$-0.184286\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−27.3783	−1.01262
$732$	0	0
$733$	− 39.2642i	− 1.45026i	−0.688614	−	0.725128i	$-0.741780\pi$
$733$	− 39.2642i	− 1.45026i	0.688614	−	0.725128i	$-0.258220\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 43.2169i	− 1.59192i
$738$	0	0
$739$	7.12841	0.262223	0.131111	−	0.991368i	$-0.458145\pi$
$739$	7.12841	0.262223	0.131111	+	0.991368i	$0.458145\pi$
$740$	0	0
$741$	−16.3832	−0.601852
$742$	0	0
$743$	− 10.2060i	− 0.374421i	−0.982320	−	0.187211i	$-0.940055\pi$
$743$	− 10.2060i	− 0.374421i	0.982320	−	0.187211i	$-0.0599447\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 12.9032i	− 0.472102i
$748$	0	0
$749$	−10.6647	−0.389679
$750$	0	0
$751$	−5.89220	−0.215009	−0.107505	−	0.994205i	$-0.534286\pi$
$751$	−5.89220	−0.215009	−0.107505	+	0.994205i	$0.534286\pi$
$752$	0	0
$753$	− 8.14124i	− 0.296683i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	28.9643i	1.05272i	0.850261	+	0.526362i	$0.176444\pi$
$757$	28.9643i	1.05272i	−0.850261	+	0.526362i	$0.823556\pi$
$758$	0	0
$759$	42.2327	1.53295
$760$	0	0
$761$	23.8720	0.865358	0.432679	−	0.901548i	$-0.357568\pi$
$761$	23.8720	0.865358	0.432679	+	0.901548i	$0.357568\pi$
$762$	0	0
$763$	− 13.5612i	− 0.490949i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	21.7970i	0.787043i
$768$	0	0
$769$	−32.1167	−1.15816	−0.579079	−	0.815272i	$-0.696588\pi$
$769$	−32.1167	−1.15816	−0.579079	+	0.815272i	$0.696588\pi$
$770$	0	0
$771$	−22.6653	−0.816271
$772$	0	0
$773$	22.9098i	0.824009i	0.911182	+	0.412005i	$0.135171\pi$
$773$	22.9098i	0.824009i	−0.911182	+	0.412005i	$0.864829\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 24.5136i	− 0.879422i
$778$	0	0
$779$	−0.751197	−0.0269144
$780$	0	0
$781$	−82.1360	−2.93906
$782$	0	0
$783$	1.88442i	0.0673435i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 0.790593i	− 0.0281816i	−0.999901	−	0.0140908i	$-0.995515\pi$
$787$	− 0.790593i	− 0.0281816i	0.999901	−	0.0140908i	$-0.00448539\pi$
$788$	0	0
$789$	6.81500	0.242620
$790$	0	0
$791$	−2.27859	−0.0810173
$792$	0	0
$793$	− 23.9775i	− 0.851465i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 31.1524i	− 1.10348i	−0.834018	−	0.551738i	$-0.813965\pi$
$797$	− 31.1524i	− 1.10348i	0.834018	−	0.551738i	$-0.186035\pi$
$798$	0	0
$799$	−6.93836	−0.245462
$800$	0	0
$801$	12.6309	0.446289
$802$	0	0
$803$	− 9.15382i	− 0.323031i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	20.1089i	0.707867i
$808$	0	0
$809$	0.120629	0.00424110	0.00212055	−	0.999998i	$-0.499325\pi$
$809$	0.120629	0.00424110	0.00212055	+	0.999998i	$0.499325\pi$
$810$	0	0
$811$	−1.55313	−0.0545379	−0.0272689	−	0.999628i	$-0.508681\pi$
$811$	−1.55313	−0.0545379	−0.0272689	+	0.999628i	$0.508681\pi$
$812$	0	0
$813$	16.8366i	0.590486i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 42.5050i	− 1.48706i
$818$	0	0
$819$	−10.8462	−0.378996
$820$	0	0
$821$	−14.6965	−0.512912	−0.256456	−	0.966556i	$-0.582555\pi$
$821$	−14.6965	−0.512912	−0.256456	+	0.966556i	$0.582555\pi$
$822$	0	0
$823$	9.71652i	0.338697i	0.985556	+	0.169348i	$0.0541663\pi$
$823$	9.71652i	0.338697i	−0.985556	+	0.169348i	$0.945834\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 32.6997i	− 1.13708i	−0.822656	−	0.568539i	$-0.807509\pi$
$827$	− 32.6997i	− 1.13708i	0.822656	−	0.568539i	$-0.192491\pi$
$828$	0	0
$829$	37.5623	1.30459	0.652296	−	0.757964i	$-0.273806\pi$
$829$	37.5623	1.30459	0.652296	+	0.757964i	$0.273806\pi$
$830$	0	0
$831$	25.7015	0.891576
$832$	0	0
$833$	4.58143i	0.158737i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0.266381i	0.00920749i
$838$	0	0
$839$	23.6116	0.815163	0.407581	−	0.913169i	$-0.366372\pi$
$839$	23.6116	0.815163	0.407581	+	0.913169i	$0.366372\pi$
$840$	0	0
$841$	−25.4490	−0.877551
$842$	0	0
$843$	− 27.7097i	− 0.954372i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 42.5144i	− 1.46081i
$848$	0	0
$849$	−11.6466	−0.399709
$850$	0	0
$851$	69.8982	2.39608
$852$	0	0
$853$	1.72305i	0.0589960i	0.999565	+	0.0294980i	$0.00939086\pi$
$853$	1.72305i	0.0589960i	−0.999565	+	0.0294980i	$0.990609\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	25.9461i	0.886303i	0.896447	+	0.443152i	$0.146140\pi$
$857$	25.9461i	0.886303i	−0.896447	+	0.443152i	$0.853860\pi$
$858$	0	0
$859$	33.1188	1.13000	0.565000	−	0.825091i	$-0.308876\pi$
$859$	33.1188	1.13000	0.565000	+	0.825091i	$0.308876\pi$
$860$	0	0
$861$	−0.497315	−0.0169484
$862$	0	0
$863$	− 29.3696i	− 0.999753i	−0.866097	−	0.499877i	$-0.833379\pi$
$863$	− 29.3696i	− 0.999753i	0.866097	−	0.499877i	$-0.166621\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	8.85410i	0.300701i
$868$	0	0
$869$	−13.4910	−0.457649
$870$	0	0
$871$	−31.6476	−1.07234
$872$	0	0
$873$	6.89032i	0.233202i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	51.6306i	1.74344i	0.490004	+	0.871720i	$0.336995\pi$
$877$	51.6306i	1.74344i	−0.490004	+	0.871720i	$0.663005\pi$
$878$	0	0
$879$	−7.72593	−0.260589
$880$	0	0
$881$	−17.6132	−0.593405	−0.296702	−	0.954970i	$-0.595887\pi$
$881$	−17.6132	−0.593405	−0.296702	+	0.954970i	$0.595887\pi$
$882$	0	0
$883$	1.65829i	0.0558059i	0.999611	+	0.0279030i	$0.00888294\pi$
$883$	1.65829i	0.0558059i	−0.999611	+	0.0279030i	$0.991117\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	20.1490i	0.676538i	0.941050	+	0.338269i	$0.109841\pi$
$887$	20.1490i	0.676538i	−0.941050	+	0.338269i	$0.890159\pi$
$888$	0	0
$889$	−18.3859	−0.616642
$890$	0	0
$891$	−5.04905	−0.169149
$892$	0	0
$893$	− 10.7719i	− 0.360466i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 30.9268i	− 1.03262i
$898$	0	0
$899$	0.501973	0.0167417
$900$	0	0
$901$	20.6891	0.689253
$902$	0	0
$903$	− 28.1396i	− 0.936427i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	31.8504i	1.05758i	0.848754	+	0.528788i	$0.177353\pi$
$907$	31.8504i	1.05758i	−0.848754	+	0.528788i	$0.822647\pi$
$908$	0	0
$909$	−9.77964	−0.324370
$910$	0	0
$911$	−30.7213	−1.01784	−0.508920	−	0.860814i	$-0.669955\pi$
$911$	−30.7213	−1.01784	−0.508920	+	0.860814i	$0.669955\pi$
$912$	0	0
$913$	− 65.1486i	− 2.15610i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 47.3301i	− 1.56298i
$918$	0	0
$919$	−36.0752	−1.19001	−0.595005	−	0.803722i	$-0.702850\pi$
$919$	−36.0752	−1.19001	−0.595005	+	0.803722i	$0.702850\pi$
$920$	0	0
$921$	−12.3674	−0.407518
$922$	0	0
$923$	60.1479i	1.97979i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	14.8620i	0.488133i
$928$	0	0
$929$	−13.7648	−0.451608	−0.225804	−	0.974173i	$-0.572501\pi$
$929$	−13.7648	−0.451608	−0.225804	+	0.974173i	$0.572501\pi$
$930$	0	0
$931$	−7.11270	−0.233109
$932$	0	0
$933$	4.53276i	0.148396i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	40.1571i	1.31187i	0.754815	+	0.655937i	$0.227726\pi$
$937$	40.1571i	1.31187i	−0.754815	+	0.655937i	$0.772274\pi$
$938$	0	0
$939$	2.33090	0.0760661
$940$	0	0
$941$	13.7005	0.446624	0.223312	−	0.974747i	$-0.428313\pi$
$941$	13.7005	0.446624	0.223312	+	0.974747i	$0.428313\pi$
$942$	0	0
$943$	− 1.41804i	− 0.0461778i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 27.1382i	− 0.881873i	−0.897538	−	0.440937i	$-0.854646\pi$
$947$	− 27.1382i	− 0.881873i	0.897538	−	0.440937i	$-0.145354\pi$
$948$	0	0
$949$	−6.70331	−0.217598
$950$	0	0
$951$	−7.86693	−0.255103
$952$	0	0
$953$	47.9437i	1.55305i	0.630086	+	0.776525i	$0.283019\pi$
$953$	47.9437i	1.55305i	−0.630086	+	0.776525i	$0.716981\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	9.51450i	0.307560i
$958$	0	0
$959$	−52.6997	−1.70176
$960$	0	0
$961$	−30.9290	−0.997711
$962$	0	0
$963$	3.63552i	0.117153i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	2.33998i	0.0752487i	0.999292	+	0.0376243i	$0.0119790\pi$
$967$	2.33998i	0.0752487i	−0.999292	+	0.0376243i	$0.988021\pi$
$968$	0	0
$969$	12.6466	0.406266
$970$	0	0
$971$	−50.9343	−1.63456	−0.817280	−	0.576241i	$-0.804519\pi$
$971$	−50.9343	−1.63456	−0.817280	+	0.576241i	$0.804519\pi$
$972$	0	0
$973$	− 25.9219i	− 0.831017i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	58.0048i	1.85574i	0.372905	+	0.927870i	$0.378362\pi$
$977$	58.0048i	1.85574i	−0.372905	+	0.927870i	$0.621638\pi$
$978$	0	0
$979$	63.7738	2.03822
$980$	0	0
$981$	−4.62293	−0.147599
$982$	0	0
$983$	35.2324i	1.12374i	0.827226	+	0.561870i	$0.189918\pi$
$983$	35.2324i	1.12374i	−0.827226	+	0.561870i	$0.810082\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 7.13129i	− 0.226992i
$988$	0	0
$989$	80.2373	2.55140
$990$	0	0
$991$	23.7908	0.755741	0.377870	−	0.925859i	$-0.376657\pi$
$991$	23.7908	0.755741	0.377870	+	0.925859i	$0.376657\pi$
$992$	0	0
$993$	20.3744i	0.646563i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	8.78958i	0.278369i	0.990266	+	0.139184i	$0.0444481\pi$
$997$	8.78958i	0.278369i	−0.990266	+	0.139184i	$0.955552\pi$
$998$	0	0
$999$	−8.35655	−0.264390

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6000.2.f.o.1249.5		8
4.3	odd	2		375.2.b.c.124.5		8
5.2	odd	4		6000.2.a.bh.1.4		4
5.3	odd	4		6000.2.a.bg.1.1		4
5.4	even	2	inner	6000.2.f.o.1249.4		8
12.11	even	2		1125.2.b.g.874.4		8
20.3	even	4		375.2.a.f.1.2	yes	4
20.7	even	4		375.2.a.e.1.3	✓	4
20.19	odd	2		375.2.b.c.124.4		8
60.23	odd	4		1125.2.a.h.1.3		4
60.47	odd	4		1125.2.a.l.1.2		4
60.59	even	2		1125.2.b.g.874.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
375.2.a.e.1.3	✓	4	20.7	even	4
375.2.a.f.1.2	yes	4	20.3	even	4
375.2.b.c.124.4		8	20.19	odd	2
375.2.b.c.124.5		8	4.3	odd	2
1125.2.a.h.1.3		4	60.23	odd	4
1125.2.a.l.1.2		4	60.47	odd	4
1125.2.b.g.874.4		8	12.11	even	2
1125.2.b.g.874.5		8	60.59	even	2
6000.2.a.bg.1.1		4	5.3	odd	4
6000.2.a.bh.1.4		4	5.2	odd	4
6000.2.f.o.1249.4		8	5.4	even	2	inner
6000.2.f.o.1249.5		8	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:	\( N \)	\(=\)	\( 6000 = 2^{4} \cdot 3 \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6000.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} -2.93346i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} -2.93346i q^{7} -1.00000 q^{9} -5.04905 q^{11} +3.69740i q^{13} -2.85410i q^{17} +4.43101 q^{19} +2.93346 q^{21} +8.36448i q^{23} -1.00000i q^{27} -1.88442 q^{29} -0.266381 q^{31} -5.04905i q^{33} -8.35655i q^{37} -3.69740 q^{39} -0.169532 q^{41} -9.59262i q^{43} -2.43101i q^{47} -1.60521 q^{49} +2.85410 q^{51} +7.24889i q^{53} +4.43101i q^{57} +5.89522 q^{59} -6.48496 q^{61} +2.93346i q^{63} +8.55942i q^{67} -8.36448 q^{69} +16.2676 q^{71} +1.81298i q^{73} +14.8112i q^{77} +2.67198 q^{79} +1.00000 q^{81} +12.9032i q^{83} -1.88442i q^{87} -12.6309 q^{89} +10.8462 q^{91} -0.266381i q^{93} -6.89032i q^{97} +5.04905 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{9}+O(q^{10}) \) 8 * q - 8 * q^9	\( 8 q - 8 q^{9} - 12 q^{11} + 16 q^{19} - 8 q^{21} - 12 q^{29} - 8 q^{31} - 16 q^{39} + 48 q^{41} - 40 q^{49} - 4 q^{51} + 4 q^{59} + 20 q^{61} - 16 q^{69} + 24 q^{71} - 40 q^{79} + 8 q^{81} - 56 q^{89} + 72 q^{91} + 12 q^{99}+O(q^{100}) \) 8 * q - 8 * q^9 - 12 * q^11 + 16 * q^19 - 8 * q^21 - 12 * q^29 - 8 * q^31 - 16 * q^39 + 48 * q^41 - 40 * q^49 - 4 * q^51 + 4 * q^59 + 20 * q^61 - 16 * q^69 + 24 * q^71 - 40 * q^79 + 8 * q^81 - 56 * q^89 + 72 * q^91 + 12 * q^99

Introduction

L-functions

Modular forms

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