LMFDB - Embedded newform 2610.2.e.e.2089.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2610,2,Mod(2089,2610)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2610, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2610.2089");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2610 = 2 \cdot 3^{2} \cdot 5 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2610.e (of order $2$, degree $1$, 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:	$20.8409549276$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 290)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2089.2
Root			$-1.61803i$ of defining polynomial
Character	$\chi$	$=$	2610.2089
Dual form			2610.2.e.e.2089.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{4} +2.23607i q^{5} -2.61803i q^{7} +1.00000i q^{8} +O(q^{10})$	\(q-1.00000i q^{2} -1.00000 q^{4} +2.23607i q^{5} -2.61803i q^{7} +1.00000i q^{8} +2.23607 q^{10} -2.00000 q^{11} +0.381966i q^{13} -2.61803 q^{14} +1.00000 q^{16} +3.38197i q^{17} +2.00000 q^{19} -2.23607i q^{20} +2.00000i q^{22} -4.85410i q^{23} -5.00000 q^{25} +0.381966 q^{26} +2.61803i q^{28} -1.00000 q^{29} -3.85410 q^{31} -1.00000i q^{32} +3.38197 q^{34} +5.85410 q^{35} -8.94427i q^{37} -2.00000i q^{38} -2.23607 q^{40} -7.70820 q^{41} +3.38197i q^{43} +2.00000 q^{44} -4.85410 q^{46} +1.52786i q^{47} +0.145898 q^{49} +5.00000i q^{50} -0.381966i q^{52} -9.38197i q^{53} -4.47214i q^{55} +2.61803 q^{56} +1.00000i q^{58} +2.85410 q^{59} -10.8541 q^{61} +3.85410i q^{62} -1.00000 q^{64} -0.854102 q^{65} +8.94427i q^{67} -3.38197i q^{68} -5.85410i q^{70} +5.70820 q^{71} -5.56231i q^{73} -8.94427 q^{74} -2.00000 q^{76} +5.23607i q^{77} -6.56231 q^{79} +2.23607i q^{80} +7.70820i q^{82} -14.9443i q^{83} -7.56231 q^{85} +3.38197 q^{86} -2.00000i q^{88} +2.00000 q^{89} +1.00000 q^{91} +4.85410i q^{92} +1.52786 q^{94} +4.47214i q^{95} -7.14590i q^{97} -0.145898i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4}+O(q^{10}) $ 4 * q - 4 * q^4	$ 4 q - 4 q^{4} - 8 q^{11} - 6 q^{14} + 4 q^{16} + 8 q^{19} - 20 q^{25} + 6 q^{26} - 4 q^{29} - 2 q^{31} + 18 q^{34} + 10 q^{35} - 4 q^{41} + 8 q^{44} - 6 q^{46} + 14 q^{49} + 6 q^{56} - 2 q^{59} - 30 q^{61} - 4 q^{64} + 10 q^{65} - 4 q^{71} - 8 q^{76} + 14 q^{79} + 10 q^{85} + 18 q^{86} + 8 q^{89} + 4 q^{91} + 24 q^{94}+O(q^{100}) $ 4 * q - 4 * q^4 - 8 * q^11 - 6 * q^14 + 4 * q^16 + 8 * q^19 - 20 * q^25 + 6 * q^26 - 4 * q^29 - 2 * q^31 + 18 * q^34 + 10 * q^35 - 4 * q^41 + 8 * q^44 - 6 * q^46 + 14 * q^49 + 6 * q^56 - 2 * q^59 - 30 * q^61 - 4 * q^64 + 10 * q^65 - 4 * q^71 - 8 * q^76 + 14 * q^79 + 10 * q^85 + 18 * q^86 + 8 * q^89 + 4 * q^91 + 24 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2610\mathbb{Z}\right)^\times$.

$n$	$901$	$1451$	$1567$
$\chi(n)$	$1$	$1$	$-1$

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 $

$2$	− 1.00000i	− 0.707107i
$2$	− 1.00000i	− 0.707107i
$3$	0	0
$3$	0	0
$4$	−1.00000	−0.500000
$5$	2.23607i	1.00000i
$5$	2.23607i	1.00000i
$6$	0	0
$7$	− 2.61803i	− 0.989524i	−0.869029	−	0.494762i	$-0.835255\pi$
$7$	− 2.61803i	− 0.989524i	0.869029	−	0.494762i	$-0.164745\pi$
$8$	1.00000i	0.353553i
$9$	0	0
$10$	2.23607	0.707107
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	0	0
$13$	0.381966i	0.105938i	0.998596	+	0.0529692i	$0.0168685\pi$
$13$	0.381966i	0.105938i	−0.998596	+	0.0529692i	$0.983131\pi$
$14$	−2.61803	−0.699699
$15$	0	0
$16$	1.00000	0.250000
$17$	3.38197i	0.820247i	0.912030	+	0.410124i	$0.134514\pi$
$17$	3.38197i	0.820247i	−0.912030	+	0.410124i	$0.865486\pi$
$18$	0	0
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	− 2.23607i	− 0.500000i
$21$	0	0
$22$	2.00000i	0.426401i
$23$	− 4.85410i	− 1.01215i	−0.862489	−	0.506075i	$-0.831096\pi$
$23$	− 4.85410i	− 1.01215i	0.862489	−	0.506075i	$-0.168904\pi$
$24$	0	0
$25$	−5.00000	−1.00000
$26$	0.381966	0.0749097
$27$	0	0
$28$	2.61803i	0.494762i
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	−3.85410	−0.692217	−0.346109	−	0.938194i	$-0.612497\pi$
$31$	−3.85410	−0.692217	−0.346109	+	0.938194i	$0.612497\pi$
$32$	− 1.00000i	− 0.176777i
$33$	0	0
$34$	3.38197	0.580002
$35$	5.85410	0.989524
$36$	0	0
$37$	− 8.94427i	− 1.47043i	−0.677834	−	0.735215i	$-0.737081\pi$
$37$	− 8.94427i	− 1.47043i	0.677834	−	0.735215i	$-0.262919\pi$
$38$	− 2.00000i	− 0.324443i
$39$	0	0
$40$	−2.23607	−0.353553
$41$	−7.70820	−1.20382	−0.601910	−	0.798564i	$-0.705593\pi$
$41$	−7.70820	−1.20382	−0.601910	+	0.798564i	$0.705593\pi$
$42$	0	0
$43$	3.38197i	0.515745i	0.966179	+	0.257872i	$0.0830214\pi$
$43$	3.38197i	0.515745i	−0.966179	+	0.257872i	$0.916979\pi$
$44$	2.00000	0.301511
$45$	0	0
$46$	−4.85410	−0.715698
$47$	1.52786i	0.222862i	0.993772	+	0.111431i	$0.0355434\pi$
$47$	1.52786i	0.222862i	−0.993772	+	0.111431i	$0.964457\pi$
$48$	0	0
$49$	0.145898	0.0208426
$50$	5.00000i	0.707107i
$51$	0	0
$52$	− 0.381966i	− 0.0529692i
$53$	− 9.38197i	− 1.28871i	−0.764726	−	0.644356i	$-0.777125\pi$
$53$	− 9.38197i	− 1.28871i	0.764726	−	0.644356i	$-0.222875\pi$
$54$	0	0
$55$	− 4.47214i	− 0.603023i
$56$	2.61803	0.349850
$57$	0	0
$58$	1.00000i	0.131306i
$59$	2.85410	0.371572	0.185786	−	0.982590i	$-0.440517\pi$
$59$	2.85410	0.371572	0.185786	+	0.982590i	$0.440517\pi$
$60$	0	0
$61$	−10.8541	−1.38973	−0.694863	−	0.719142i	$-0.744535\pi$
$61$	−10.8541	−1.38973	−0.694863	+	0.719142i	$0.744535\pi$
$62$	3.85410i	0.489471i
$63$	0	0
$64$	−1.00000	−0.125000
$65$	−0.854102	−0.105938
$66$	0	0
$67$	8.94427i	1.09272i	0.837552	+	0.546358i	$0.183986\pi$
$67$	8.94427i	1.09272i	−0.837552	+	0.546358i	$0.816014\pi$
$68$	− 3.38197i	− 0.410124i
$69$	0	0
$70$	− 5.85410i	− 0.699699i
$71$	5.70820	0.677439	0.338720	−	0.940887i	$-0.390006\pi$
$71$	5.70820	0.677439	0.338720	+	0.940887i	$0.390006\pi$
$72$	0	0
$73$	− 5.56231i	− 0.651019i	−0.945539	−	0.325509i	$-0.894464\pi$
$73$	− 5.56231i	− 0.651019i	0.945539	−	0.325509i	$-0.105536\pi$
$74$	−8.94427	−1.03975
$75$	0	0
$76$	−2.00000	−0.229416
$77$	5.23607i	0.596705i
$78$	0	0
$79$	−6.56231	−0.738317	−0.369159	−	0.929366i	$-0.620354\pi$
$79$	−6.56231	−0.738317	−0.369159	+	0.929366i	$0.620354\pi$
$80$	2.23607i	0.250000i
$81$	0	0
$82$	7.70820i	0.851229i
$83$	− 14.9443i	− 1.64035i	−0.572115	−	0.820173i	$-0.693877\pi$
$83$	− 14.9443i	− 1.64035i	0.572115	−	0.820173i	$-0.306123\pi$
$84$	0	0
$85$	−7.56231	−0.820247
$86$	3.38197	0.364687
$87$	0	0
$88$	− 2.00000i	− 0.213201i
$89$	2.00000	0.212000	0.106000	−	0.994366i	$-0.466196\pi$
$89$	2.00000	0.212000	0.106000	+	0.994366i	$0.466196\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	4.85410i	0.506075i
$93$	0	0
$94$	1.52786	0.157587
$95$	4.47214i	0.458831i
$96$	0	0
$97$	− 7.14590i	− 0.725556i	−0.931876	−	0.362778i	$-0.881828\pi$
$97$	− 7.14590i	− 0.725556i	0.931876	−	0.362778i	$-0.118172\pi$
$98$	− 0.145898i	− 0.0147379i
$99$	0	0
$100$	5.00000	0.500000
$101$	−10.8541	−1.08002	−0.540012	−	0.841657i	$-0.681580\pi$
$101$	−10.8541	−1.08002	−0.540012	+	0.841657i	$0.681580\pi$
$102$	0	0
$103$	7.41641i	0.730760i	0.930858	+	0.365380i	$0.119061\pi$
$103$	7.41641i	0.730760i	−0.930858	+	0.365380i	$0.880939\pi$
$104$	−0.381966	−0.0374548
$105$	0	0
$106$	−9.38197	−0.911257
$107$	− 14.1803i	− 1.37087i	−0.728136	−	0.685433i	$-0.759613\pi$
$107$	− 14.1803i	− 1.37087i	0.728136	−	0.685433i	$-0.240387\pi$
$108$	0	0
$109$	−19.4164	−1.85975	−0.929877	−	0.367870i	$-0.880087\pi$
$109$	−19.4164	−1.85975	−0.929877	+	0.367870i	$0.880087\pi$
$110$	−4.47214	−0.426401
$111$	0	0
$112$	− 2.61803i	− 0.247381i
$113$	− 4.09017i	− 0.384771i	−0.981319	−	0.192385i	$-0.938378\pi$
$113$	− 4.09017i	− 0.384771i	0.981319	−	0.192385i	$-0.0616224\pi$
$114$	0	0
$115$	10.8541	1.01215
$116$	1.00000	0.0928477
$117$	0	0
$118$	− 2.85410i	− 0.262741i
$119$	8.85410	0.811654
$120$	0	0
$121$	−7.00000	−0.636364
$122$	10.8541i	0.982684i
$123$	0	0
$124$	3.85410	0.346109
$125$	− 11.1803i	− 1.00000i
$126$	0	0
$127$	8.29180i	0.735778i	0.929870	+	0.367889i	$0.119919\pi$
$127$	8.29180i	0.735778i	−0.929870	+	0.367889i	$0.880081\pi$
$128$	1.00000i	0.0883883i
$129$	0	0
$130$	0.854102i	0.0749097i
$131$	−21.4164	−1.87116	−0.935580	−	0.353114i	$-0.885123\pi$
$131$	−21.4164	−1.87116	−0.935580	+	0.353114i	$0.885123\pi$
$132$	0	0
$133$	− 5.23607i	− 0.454025i
$134$	8.94427	0.772667
$135$	0	0
$136$	−3.38197	−0.290001
$137$	− 4.85410i	− 0.414714i	−0.978265	−	0.207357i	$-0.933514\pi$
$137$	− 4.85410i	− 0.414714i	0.978265	−	0.207357i	$-0.0664862\pi$
$138$	0	0
$139$	5.56231	0.471789	0.235894	−	0.971779i	$-0.424198\pi$
$139$	5.56231	0.471789	0.235894	+	0.971779i	$0.424198\pi$
$140$	−5.85410	−0.494762
$141$	0	0
$142$	− 5.70820i	− 0.479022i
$143$	− 0.763932i	− 0.0638832i
$144$	0	0
$145$	− 2.23607i	− 0.185695i
$146$	−5.56231	−0.460340
$147$	0	0
$148$	8.94427i	0.735215i
$149$	−21.4164	−1.75450	−0.877250	−	0.480033i	$-0.840625\pi$
$149$	−21.4164	−1.75450	−0.877250	+	0.480033i	$0.840625\pi$
$150$	0	0
$151$	4.29180	0.349261	0.174631	−	0.984634i	$-0.444127\pi$
$151$	4.29180	0.349261	0.174631	+	0.984634i	$0.444127\pi$
$152$	2.00000i	0.162221i
$153$	0	0
$154$	5.23607	0.421934
$155$	− 8.61803i	− 0.692217i
$156$	0	0
$157$	− 9.70820i	− 0.774799i	−0.921912	−	0.387400i	$-0.873373\pi$
$157$	− 9.70820i	− 0.774799i	0.921912	−	0.387400i	$-0.126627\pi$
$158$	6.56231i	0.522069i
$159$	0	0
$160$	2.23607	0.176777
$161$	−12.7082	−1.00155
$162$	0	0
$163$	7.41641i	0.580898i	0.956890	+	0.290449i	$0.0938046\pi$
$163$	7.41641i	0.580898i	−0.956890	+	0.290449i	$0.906195\pi$
$164$	7.70820	0.601910
$165$	0	0
$166$	−14.9443	−1.15990
$167$	2.61803i	0.202590i	0.994856	+	0.101295i	$0.0322985\pi$
$167$	2.61803i	0.202590i	−0.994856	+	0.101295i	$0.967701\pi$
$168$	0	0
$169$	12.8541	0.988777
$170$	7.56231i	0.580002i
$171$	0	0
$172$	− 3.38197i	− 0.257872i
$173$	− 17.5066i	− 1.33100i	−0.746398	−	0.665500i	$-0.768218\pi$
$173$	− 17.5066i	− 1.33100i	0.746398	−	0.665500i	$-0.231782\pi$
$174$	0	0
$175$	13.0902i	0.989524i
$176$	−2.00000	−0.150756
$177$	0	0
$178$	− 2.00000i	− 0.149906i
$179$	9.14590	0.683597	0.341798	−	0.939773i	$-0.388964\pi$
$179$	9.14590	0.683597	0.341798	+	0.939773i	$0.388964\pi$
$180$	0	0
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	− 1.00000i	− 0.0741249i
$183$	0	0
$184$	4.85410	0.357849
$185$	20.0000	1.47043
$186$	0	0
$187$	− 6.76393i	− 0.494628i
$188$	− 1.52786i	− 0.111431i
$189$	0	0
$190$	4.47214	0.324443
$191$	−23.2705	−1.68379	−0.841897	−	0.539637i	$-0.818561\pi$
$191$	−23.2705	−1.68379	−0.841897	+	0.539637i	$0.818561\pi$
$192$	0	0
$193$	− 2.56231i	− 0.184439i	−0.995739	−	0.0922194i	$-0.970604\pi$
$193$	− 2.56231i	− 0.184439i	0.995739	−	0.0922194i	$-0.0293961\pi$
$194$	−7.14590	−0.513046
$195$	0	0
$196$	−0.145898	−0.0104213
$197$	9.38197i	0.668437i	0.942496	+	0.334219i	$0.108472\pi$
$197$	9.38197i	0.668437i	−0.942496	+	0.334219i	$0.891528\pi$
$198$	0	0
$199$	−19.1246	−1.35571	−0.677854	−	0.735197i	$-0.737090\pi$
$199$	−19.1246	−1.35571	−0.677854	+	0.735197i	$0.737090\pi$
$200$	− 5.00000i	− 0.353553i
$201$	0	0
$202$	10.8541i	0.763692i
$203$	2.61803i	0.183750i
$204$	0	0
$205$	− 17.2361i	− 1.20382i
$206$	7.41641	0.516726
$207$	0	0
$208$	0.381966i	0.0264846i
$209$	−4.00000	−0.276686
$210$	0	0
$211$	−11.4164	−0.785938	−0.392969	−	0.919552i	$-0.628552\pi$
$211$	−11.4164	−0.785938	−0.392969	+	0.919552i	$0.628552\pi$
$212$	9.38197i	0.644356i
$213$	0	0
$214$	−14.1803	−0.969348
$215$	−7.56231	−0.515745
$216$	0	0
$217$	10.0902i	0.684965i
$218$	19.4164i	1.31505i
$219$	0	0
$220$	4.47214i	0.301511i
$221$	−1.29180	−0.0868956
$222$	0	0
$223$	− 26.5623i	− 1.77874i	−0.457185	−	0.889372i	$-0.651142\pi$
$223$	− 26.5623i	− 1.77874i	0.457185	−	0.889372i	$-0.348858\pi$
$224$	−2.61803	−0.174925
$225$	0	0
$226$	−4.09017	−0.272074
$227$	16.4721i	1.09329i	0.837363	+	0.546647i	$0.184096\pi$
$227$	16.4721i	1.09329i	−0.837363	+	0.546647i	$0.815904\pi$
$228$	0	0
$229$	8.14590	0.538296	0.269148	−	0.963099i	$-0.413258\pi$
$229$	8.14590	0.538296	0.269148	+	0.963099i	$0.413258\pi$
$230$	− 10.8541i	− 0.715698i
$231$	0	0
$232$	− 1.00000i	− 0.0656532i
$233$	11.2361i	0.736099i	0.929806	+	0.368050i	$0.119974\pi$
$233$	11.2361i	0.736099i	−0.929806	+	0.368050i	$0.880026\pi$
$234$	0	0
$235$	−3.41641	−0.222862
$236$	−2.85410	−0.185786
$237$	0	0
$238$	− 8.85410i	− 0.573926i
$239$	5.70820	0.369233	0.184617	−	0.982811i	$-0.440896\pi$
$239$	5.70820	0.369233	0.184617	+	0.982811i	$0.440896\pi$
$240$	0	0
$241$	−8.85410	−0.570343	−0.285171	−	0.958477i	$-0.592051\pi$
$241$	−8.85410	−0.570343	−0.285171	+	0.958477i	$0.592051\pi$
$242$	7.00000i	0.449977i
$243$	0	0
$244$	10.8541	0.694863
$245$	0.326238i	0.0208426i
$246$	0	0
$247$	0.763932i	0.0486078i
$248$	− 3.85410i	− 0.244736i
$249$	0	0
$250$	−11.1803	−0.707107
$251$	29.1246	1.83833	0.919165	−	0.393874i	$-0.128865\pi$
$251$	29.1246	1.83833	0.919165	+	0.393874i	$0.128865\pi$
$252$	0	0
$253$	9.70820i	0.610350i
$254$	8.29180	0.520274
$255$	0	0
$256$	1.00000	0.0625000
$257$	2.29180i	0.142958i	0.997442	+	0.0714792i	$0.0227719\pi$
$257$	2.29180i	0.142958i	−0.997442	+	0.0714792i	$0.977228\pi$
$258$	0	0
$259$	−23.4164	−1.45502
$260$	0.854102	0.0529692
$261$	0	0
$262$	21.4164i	1.32311i
$263$	21.5967i	1.33171i	0.746080	+	0.665856i	$0.231934\pi$
$263$	21.5967i	1.33171i	−0.746080	+	0.665856i	$0.768066\pi$
$264$	0	0
$265$	20.9787	1.28871
$266$	−5.23607	−0.321044
$267$	0	0
$268$	− 8.94427i	− 0.546358i
$269$	3.27051	0.199407	0.0997033	−	0.995017i	$-0.468211\pi$
$269$	3.27051	0.199407	0.0997033	+	0.995017i	$0.468211\pi$
$270$	0	0
$271$	23.4164	1.42245	0.711223	−	0.702967i	$-0.248142\pi$
$271$	23.4164	1.42245	0.711223	+	0.702967i	$0.248142\pi$
$272$	3.38197i	0.205062i
$273$	0	0
$274$	−4.85410	−0.293247
$275$	10.0000	0.603023
$276$	0	0
$277$	30.0000i	1.80253i	0.433273	+	0.901263i	$0.357359\pi$
$277$	30.0000i	1.80253i	−0.433273	+	0.901263i	$0.642641\pi$
$278$	− 5.56231i	− 0.333605i
$279$	0	0
$280$	5.85410i	0.349850i
$281$	17.5623	1.04768	0.523840	−	0.851817i	$-0.324499\pi$
$281$	17.5623	1.04768	0.523840	+	0.851817i	$0.324499\pi$
$282$	0	0
$283$	− 6.76393i	− 0.402074i	−0.979584	−	0.201037i	$-0.935569\pi$
$283$	− 6.76393i	− 0.402074i	0.979584	−	0.201037i	$-0.0644312\pi$
$284$	−5.70820	−0.338720
$285$	0	0
$286$	−0.763932	−0.0451722
$287$	20.1803i	1.19121i
$288$	0	0
$289$	5.56231	0.327194
$290$	−2.23607	−0.131306
$291$	0	0
$292$	5.56231i	0.325509i
$293$	19.4164i	1.13432i	0.823608	+	0.567159i	$0.191958\pi$
$293$	19.4164i	1.13432i	−0.823608	+	0.567159i	$0.808042\pi$
$294$	0	0
$295$	6.38197i	0.371572i
$296$	8.94427	0.519875
$297$	0	0
$298$	21.4164i	1.24062i
$299$	1.85410	0.107225
$300$	0	0
$301$	8.85410	0.510342
$302$	− 4.29180i	− 0.246965i
$303$	0	0
$304$	2.00000	0.114708
$305$	− 24.2705i	− 1.38973i
$306$	0	0
$307$	24.0000i	1.36975i	0.728659	+	0.684876i	$0.240144\pi$
$307$	24.0000i	1.36975i	−0.728659	+	0.684876i	$0.759856\pi$
$308$	− 5.23607i	− 0.298353i
$309$	0	0
$310$	−8.61803	−0.489471
$311$	34.5623	1.95985	0.979924	−	0.199370i	$-0.0638896\pi$
$311$	34.5623	1.95985	0.979924	+	0.199370i	$0.0638896\pi$
$312$	0	0
$313$	− 12.7639i	− 0.721460i	−0.932670	−	0.360730i	$-0.882528\pi$
$313$	− 12.7639i	− 0.721460i	0.932670	−	0.360730i	$-0.117472\pi$
$314$	−9.70820	−0.547866
$315$	0	0
$316$	6.56231	0.369159
$317$	− 20.1803i	− 1.13344i	−0.823910	−	0.566720i	$-0.808212\pi$
$317$	− 20.1803i	− 1.13344i	0.823910	−	0.566720i	$-0.191788\pi$
$318$	0	0
$319$	2.00000	0.111979
$320$	− 2.23607i	− 0.125000i
$321$	0	0
$322$	12.7082i	0.708201i
$323$	6.76393i	0.376355i
$324$	0	0
$325$	− 1.90983i	− 0.105938i
$326$	7.41641	0.410757
$327$	0	0
$328$	− 7.70820i	− 0.425614i
$329$	4.00000	0.220527
$330$	0	0
$331$	27.1246	1.49090	0.745452	−	0.666560i	$-0.232234\pi$
$331$	27.1246	1.49090	0.745452	+	0.666560i	$0.232234\pi$
$332$	14.9443i	0.820173i
$333$	0	0
$334$	2.61803	0.143252
$335$	−20.0000	−1.09272
$336$	0	0
$337$	− 8.61803i	− 0.469454i	−0.972061	−	0.234727i	$-0.924580\pi$
$337$	− 8.61803i	− 0.469454i	0.972061	−	0.234727i	$-0.0754197\pi$
$338$	− 12.8541i	− 0.699171i
$339$	0	0
$340$	7.56231	0.410124
$341$	7.70820	0.417423
$342$	0	0
$343$	− 18.7082i	− 1.01015i
$344$	−3.38197	−0.182343
$345$	0	0
$346$	−17.5066	−0.941159
$347$	− 6.00000i	− 0.322097i	−0.986947	−	0.161048i	$-0.948512\pi$
$347$	− 6.00000i	− 0.322097i	0.986947	−	0.161048i	$-0.0514875\pi$
$348$	0	0
$349$	24.8328	1.32927	0.664635	−	0.747168i	$-0.268587\pi$
$349$	24.8328	1.32927	0.664635	+	0.747168i	$0.268587\pi$
$350$	13.0902	0.699699
$351$	0	0
$352$	2.00000i	0.106600i
$353$	3.05573i	0.162640i	0.996688	+	0.0813200i	$0.0259136\pi$
$353$	3.05573i	0.162640i	−0.996688	+	0.0813200i	$0.974086\pi$
$354$	0	0
$355$	12.7639i	0.677439i
$356$	−2.00000	−0.106000
$357$	0	0
$358$	− 9.14590i	− 0.483376i
$359$	0.270510	0.0142770	0.00713848	−	0.999975i	$-0.497728\pi$
$359$	0.270510	0.0142770	0.00713848	+	0.999975i	$0.497728\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	10.0000i	0.525588i
$363$	0	0
$364$	−1.00000	−0.0524142
$365$	12.4377	0.651019
$366$	0	0
$367$	− 35.1246i	− 1.83349i	−0.399473	−	0.916745i	$-0.630807\pi$
$367$	− 35.1246i	− 1.83349i	0.399473	−	0.916745i	$-0.369193\pi$
$368$	− 4.85410i	− 0.253038i
$369$	0	0
$370$	− 20.0000i	− 1.03975i
$371$	−24.5623	−1.27521
$372$	0	0
$373$	15.2705i	0.790677i	0.918535	+	0.395339i	$0.129373\pi$
$373$	15.2705i	0.790677i	−0.918535	+	0.395339i	$0.870627\pi$
$374$	−6.76393	−0.349755
$375$	0	0
$376$	−1.52786	−0.0787936
$377$	− 0.381966i	− 0.0196723i
$378$	0	0
$379$	−26.0000	−1.33553	−0.667765	−	0.744372i	$-0.732749\pi$
$379$	−26.0000	−1.33553	−0.667765	+	0.744372i	$0.732749\pi$
$380$	− 4.47214i	− 0.229416i
$381$	0	0
$382$	23.2705i	1.19062i
$383$	7.85410i	0.401326i	0.979660	+	0.200663i	$0.0643096\pi$
$383$	7.85410i	0.401326i	−0.979660	+	0.200663i	$0.935690\pi$
$384$	0	0
$385$	−11.7082	−0.596705
$386$	−2.56231	−0.130418
$387$	0	0
$388$	7.14590i	0.362778i
$389$	−20.8328	−1.05627	−0.528133	−	0.849162i	$-0.677108\pi$
$389$	−20.8328	−1.05627	−0.528133	+	0.849162i	$0.677108\pi$
$390$	0	0
$391$	16.4164	0.830213
$392$	0.145898i	0.00736896i
$393$	0	0
$394$	9.38197	0.472657
$395$	− 14.6738i	− 0.738317i
$396$	0	0
$397$	1.96556i	0.0986485i	0.998783	+	0.0493243i	$0.0157068\pi$
$397$	1.96556i	0.0986485i	−0.998783	+	0.0493243i	$0.984293\pi$
$398$	19.1246i	0.958630i
$399$	0	0
$400$	−5.00000	−0.250000
$401$	−21.9787	−1.09756	−0.548782	−	0.835965i	$-0.684908\pi$
$401$	−21.9787	−1.09756	−0.548782	+	0.835965i	$0.684908\pi$
$402$	0	0
$403$	− 1.47214i	− 0.0733323i
$404$	10.8541	0.540012
$405$	0	0
$406$	2.61803	0.129931
$407$	17.8885i	0.886702i
$408$	0	0
$409$	7.12461	0.352289	0.176145	−	0.984364i	$-0.443637\pi$
$409$	7.12461	0.352289	0.176145	+	0.984364i	$0.443637\pi$
$410$	−17.2361	−0.851229
$411$	0	0
$412$	− 7.41641i	− 0.365380i
$413$	− 7.47214i	− 0.367680i
$414$	0	0
$415$	33.4164	1.64035
$416$	0.381966	0.0187274
$417$	0	0
$418$	4.00000i	0.195646i
$419$	−21.8541	−1.06764	−0.533821	−	0.845597i	$-0.679245\pi$
$419$	−21.8541	−1.06764	−0.533821	+	0.845597i	$0.679245\pi$
$420$	0	0
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	11.4164i	0.555742i
$423$	0	0
$424$	9.38197	0.455629
$425$	− 16.9098i	− 0.820247i
$426$	0	0
$427$	28.4164i	1.37517i
$428$	14.1803i	0.685433i
$429$	0	0
$430$	7.56231i	0.364687i
$431$	33.4164	1.60961	0.804806	−	0.593538i	$-0.202269\pi$
$431$	33.4164	1.60961	0.804806	+	0.593538i	$0.202269\pi$
$432$	0	0
$433$	− 16.4721i	− 0.791600i	−0.918337	−	0.395800i	$-0.870467\pi$
$433$	− 16.4721i	− 0.791600i	0.918337	−	0.395800i	$-0.129533\pi$
$434$	10.0902	0.484344
$435$	0	0
$436$	19.4164	0.929877
$437$	− 9.70820i	− 0.464406i
$438$	0	0
$439$	37.1246	1.77186	0.885931	−	0.463818i	$-0.153521\pi$
$439$	37.1246	1.77186	0.885931	+	0.463818i	$0.153521\pi$
$440$	4.47214	0.213201
$441$	0	0
$442$	1.29180i	0.0614445i
$443$	1.90983i	0.0907388i	0.998970	+	0.0453694i	$0.0144465\pi$
$443$	1.90983i	0.0907388i	−0.998970	+	0.0453694i	$0.985554\pi$
$444$	0	0
$445$	4.47214i	0.212000i
$446$	−26.5623	−1.25776
$447$	0	0
$448$	2.61803i	0.123690i
$449$	−4.58359	−0.216313	−0.108157	−	0.994134i	$-0.534495\pi$
$449$	−4.58359	−0.216313	−0.108157	+	0.994134i	$0.534495\pi$
$450$	0	0
$451$	15.4164	0.725930
$452$	4.09017i	0.192385i
$453$	0	0
$454$	16.4721	0.773076
$455$	2.23607i	0.104828i
$456$	0	0
$457$	14.2918i	0.668542i	0.942477	+	0.334271i	$0.108490\pi$
$457$	14.2918i	0.668542i	−0.942477	+	0.334271i	$0.891510\pi$
$458$	− 8.14590i	− 0.380633i
$459$	0	0
$460$	−10.8541	−0.506075
$461$	24.1459	1.12459	0.562293	−	0.826938i	$-0.309919\pi$
$461$	24.1459	1.12459	0.562293	+	0.826938i	$0.309919\pi$
$462$	0	0
$463$	− 17.8885i	− 0.831351i	−0.909513	−	0.415676i	$-0.863545\pi$
$463$	− 17.8885i	− 0.831351i	0.909513	−	0.415676i	$-0.136455\pi$
$464$	−1.00000	−0.0464238
$465$	0	0
$466$	11.2361	0.520501
$467$	34.7984i	1.61028i	0.593087	+	0.805138i	$0.297909\pi$
$467$	34.7984i	1.61028i	−0.593087	+	0.805138i	$0.702091\pi$
$468$	0	0
$469$	23.4164	1.08127
$470$	3.41641i	0.157587i
$471$	0	0
$472$	2.85410i	0.131371i
$473$	− 6.76393i	− 0.311006i
$474$	0	0
$475$	−10.0000	−0.458831
$476$	−8.85410	−0.405827
$477$	0	0
$478$	− 5.70820i	− 0.261087i
$479$	−18.9787	−0.867160	−0.433580	−	0.901115i	$-0.642750\pi$
$479$	−18.9787	−0.867160	−0.433580	+	0.901115i	$0.642750\pi$
$480$	0	0
$481$	3.41641	0.155775
$482$	8.85410i	0.403293i
$483$	0	0
$484$	7.00000	0.318182
$485$	15.9787	0.725556
$486$	0	0
$487$	− 38.5623i	− 1.74742i	−0.486443	−	0.873712i	$-0.661706\pi$
$487$	− 38.5623i	− 1.74742i	0.486443	−	0.873712i	$-0.338294\pi$
$488$	− 10.8541i	− 0.491342i
$489$	0	0
$490$	0.326238	0.0147379
$491$	6.29180	0.283945	0.141972	−	0.989871i	$-0.454656\pi$
$491$	6.29180	0.283945	0.141972	+	0.989871i	$0.454656\pi$
$492$	0	0
$493$	− 3.38197i	− 0.152316i
$494$	0.763932	0.0343709
$495$	0	0
$496$	−3.85410	−0.173054
$497$	− 14.9443i	− 0.670342i
$498$	0	0
$499$	−28.5623	−1.27862	−0.639312	−	0.768947i	$-0.720781\pi$
$499$	−28.5623	−1.27862	−0.639312	+	0.768947i	$0.720781\pi$
$500$	11.1803i	0.500000i
$501$	0	0
$502$	− 29.1246i	− 1.29990i
$503$	− 32.9443i	− 1.46891i	−0.678656	−	0.734456i	$-0.737437\pi$
$503$	− 32.9443i	− 1.46891i	0.678656	−	0.734456i	$-0.262563\pi$
$504$	0	0
$505$	− 24.2705i	− 1.08002i
$506$	9.70820	0.431582
$507$	0	0
$508$	− 8.29180i	− 0.367889i
$509$	−21.1246	−0.936332	−0.468166	−	0.883641i	$-0.655085\pi$
$509$	−21.1246	−0.936332	−0.468166	+	0.883641i	$0.655085\pi$
$510$	0	0
$511$	−14.5623	−0.644198
$512$	− 1.00000i	− 0.0441942i
$513$	0	0
$514$	2.29180	0.101087
$515$	−16.5836	−0.730760
$516$	0	0
$517$	− 3.05573i	− 0.134391i
$518$	23.4164i	1.02886i
$519$	0	0
$520$	− 0.854102i	− 0.0374548i
$521$	36.3951	1.59450	0.797250	−	0.603650i	$-0.206287\pi$
$521$	36.3951	1.59450	0.797250	+	0.603650i	$0.206287\pi$
$522$	0	0
$523$	− 11.1246i	− 0.486445i	−0.969970	−	0.243223i	$-0.921795\pi$
$523$	− 11.1246i	− 0.486445i	0.969970	−	0.243223i	$-0.0782046\pi$
$524$	21.4164	0.935580
$525$	0	0
$526$	21.5967	0.941663
$527$	− 13.0344i	− 0.567789i
$528$	0	0
$529$	−0.562306	−0.0244481
$530$	− 20.9787i	− 0.911257i
$531$	0	0
$532$	5.23607i	0.227012i
$533$	− 2.94427i	− 0.127531i
$534$	0	0
$535$	31.7082	1.37087
$536$	−8.94427	−0.386334
$537$	0	0
$538$	− 3.27051i	− 0.141002i
$539$	−0.291796	−0.0125685
$540$	0	0
$541$	−27.5623	−1.18500	−0.592498	−	0.805572i	$-0.701858\pi$
$541$	−27.5623	−1.18500	−0.592498	+	0.805572i	$0.701858\pi$
$542$	− 23.4164i	− 1.00582i
$543$	0	0
$544$	3.38197	0.145001
$545$	− 43.4164i	− 1.85975i
$546$	0	0
$547$	14.8328i	0.634205i	0.948391	+	0.317103i	$0.102710\pi$
$547$	14.8328i	0.634205i	−0.948391	+	0.317103i	$0.897290\pi$
$548$	4.85410i	0.207357i
$549$	0	0
$550$	− 10.0000i	− 0.426401i
$551$	−2.00000	−0.0852029
$552$	0	0
$553$	17.1803i	0.730582i
$554$	30.0000	1.27458
$555$	0	0
$556$	−5.56231	−0.235894
$557$	− 7.79837i	− 0.330428i	−0.986258	−	0.165214i	$-0.947169\pi$
$557$	− 7.79837i	− 0.330428i	0.986258	−	0.165214i	$-0.0528314\pi$
$558$	0	0
$559$	−1.29180	−0.0546372
$560$	5.85410	0.247381
$561$	0	0
$562$	− 17.5623i	− 0.740821i
$563$	− 33.3262i	− 1.40453i	−0.711914	−	0.702267i	$-0.752171\pi$
$563$	− 33.3262i	− 1.40453i	0.711914	−	0.702267i	$-0.247829\pi$
$564$	0	0
$565$	9.14590	0.384771
$566$	−6.76393	−0.284309
$567$	0	0
$568$	5.70820i	0.239511i
$569$	12.0000	0.503066	0.251533	−	0.967849i	$-0.419065\pi$
$569$	12.0000	0.503066	0.251533	+	0.967849i	$0.419065\pi$
$570$	0	0
$571$	−6.85410	−0.286835	−0.143418	−	0.989662i	$-0.545809\pi$
$571$	−6.85410	−0.286835	−0.143418	+	0.989662i	$0.545809\pi$
$572$	0.763932i	0.0319416i
$573$	0	0
$574$	20.1803	0.842311
$575$	24.2705i	1.01215i
$576$	0	0
$577$	28.0344i	1.16709i	0.812081	+	0.583545i	$0.198335\pi$
$577$	28.0344i	1.16709i	−0.812081	+	0.583545i	$0.801665\pi$
$578$	− 5.56231i	− 0.231361i
$579$	0	0
$580$	2.23607i	0.0928477i
$581$	−39.1246	−1.62316
$582$	0	0
$583$	18.7639i	0.777123i
$584$	5.56231	0.230170
$585$	0	0
$586$	19.4164	0.802084
$587$	12.6525i	0.522224i	0.965309	+	0.261112i	$0.0840891\pi$
$587$	12.6525i	0.522224i	−0.965309	+	0.261112i	$0.915911\pi$
$588$	0	0
$589$	−7.70820	−0.317611
$590$	6.38197	0.262741
$591$	0	0
$592$	− 8.94427i	− 0.367607i
$593$	− 43.3050i	− 1.77832i	−0.457595	−	0.889161i	$-0.651289\pi$
$593$	− 43.3050i	− 1.77832i	0.457595	−	0.889161i	$-0.348711\pi$
$594$	0	0
$595$	19.7984i	0.811654i
$596$	21.4164	0.877250
$597$	0	0
$598$	− 1.85410i	− 0.0758199i
$599$	23.5623	0.962730	0.481365	−	0.876520i	$-0.340141\pi$
$599$	23.5623	0.962730	0.481365	+	0.876520i	$0.340141\pi$
$600$	0	0
$601$	5.70820	0.232842	0.116421	−	0.993200i	$-0.462858\pi$
$601$	5.70820	0.232842	0.116421	+	0.993200i	$0.462858\pi$
$602$	− 8.85410i	− 0.360866i
$603$	0	0
$604$	−4.29180	−0.174631
$605$	− 15.6525i	− 0.636364i
$606$	0	0
$607$	− 0.763932i	− 0.0310070i	−0.999880	−	0.0155035i	$-0.995065\pi$
$607$	− 0.763932i	− 0.0310070i	0.999880	−	0.0155035i	$-0.00493512\pi$
$608$	− 2.00000i	− 0.0811107i
$609$	0	0
$610$	−24.2705	−0.982684
$611$	−0.583592	−0.0236096
$612$	0	0
$613$	13.7426i	0.555060i	0.960717	+	0.277530i	$0.0895158\pi$
$613$	13.7426i	0.555060i	−0.960717	+	0.277530i	$0.910484\pi$
$614$	24.0000	0.968561
$615$	0	0
$616$	−5.23607	−0.210967
$617$	32.6180i	1.31315i	0.754260	+	0.656576i	$0.227996\pi$
$617$	32.6180i	1.31315i	−0.754260	+	0.656576i	$0.772004\pi$
$618$	0	0
$619$	7.41641	0.298091	0.149045	−	0.988830i	$-0.452380\pi$
$619$	7.41641	0.298091	0.149045	+	0.988830i	$0.452380\pi$
$620$	8.61803i	0.346109i
$621$	0	0
$622$	− 34.5623i	− 1.38582i
$623$	− 5.23607i	− 0.209779i
$624$	0	0
$625$	25.0000	1.00000
$626$	−12.7639	−0.510149
$627$	0	0
$628$	9.70820i	0.387400i
$629$	30.2492	1.20612
$630$	0	0
$631$	−34.8328	−1.38667	−0.693336	−	0.720614i	$-0.743860\pi$
$631$	−34.8328	−1.38667	−0.693336	+	0.720614i	$0.743860\pi$
$632$	− 6.56231i	− 0.261035i
$633$	0	0
$634$	−20.1803	−0.801464
$635$	−18.5410	−0.735778
$636$	0	0
$637$	0.0557281i	0.00220803i
$638$	− 2.00000i	− 0.0791808i
$639$	0	0
$640$	−2.23607	−0.0883883
$641$	−0.291796	−0.0115253	−0.00576263	−	0.999983i	$-0.501834\pi$
$641$	−0.291796	−0.0115253	−0.00576263	+	0.999983i	$0.501834\pi$
$642$	0	0
$643$	− 16.5836i	− 0.653993i	−0.945026	−	0.326997i	$-0.893963\pi$
$643$	− 16.5836i	− 0.653993i	0.945026	−	0.326997i	$-0.106037\pi$
$644$	12.7082	0.500773
$645$	0	0
$646$	6.76393	0.266123
$647$	13.5279i	0.531835i	0.963996	+	0.265918i	$0.0856749\pi$
$647$	13.5279i	0.531835i	−0.963996	+	0.265918i	$0.914325\pi$
$648$	0	0
$649$	−5.70820	−0.224067
$650$	−1.90983	−0.0749097
$651$	0	0
$652$	− 7.41641i	− 0.290449i
$653$	− 37.3050i	− 1.45986i	−0.683524	−	0.729928i	$-0.739554\pi$
$653$	− 37.3050i	− 1.45986i	0.683524	−	0.729928i	$-0.260446\pi$
$654$	0	0
$655$	− 47.8885i	− 1.87116i
$656$	−7.70820	−0.300955
$657$	0	0
$658$	− 4.00000i	− 0.155936i
$659$	40.5410	1.57925	0.789627	−	0.613587i	$-0.210274\pi$
$659$	40.5410	1.57925	0.789627	+	0.613587i	$0.210274\pi$
$660$	0	0
$661$	28.8328	1.12147	0.560733	−	0.827996i	$-0.310519\pi$
$661$	28.8328	1.12147	0.560733	+	0.827996i	$0.310519\pi$
$662$	− 27.1246i	− 1.05423i
$663$	0	0
$664$	14.9443	0.579950
$665$	11.7082	0.454025
$666$	0	0
$667$	4.85410i	0.187952i
$668$	− 2.61803i	− 0.101295i
$669$	0	0
$670$	20.0000i	0.772667i
$671$	21.7082	0.838036
$672$	0	0
$673$	26.9443i	1.03863i	0.854584	+	0.519313i	$0.173812\pi$
$673$	26.9443i	1.03863i	−0.854584	+	0.519313i	$0.826188\pi$
$674$	−8.61803	−0.331954
$675$	0	0
$676$	−12.8541	−0.494389
$677$	32.9443i	1.26615i	0.774090	+	0.633076i	$0.218208\pi$
$677$	32.9443i	1.26615i	−0.774090	+	0.633076i	$0.781792\pi$
$678$	0	0
$679$	−18.7082	−0.717955
$680$	− 7.56231i	− 0.290001i
$681$	0	0
$682$	− 7.70820i	− 0.295162i
$683$	23.7771i	0.909805i	0.890541	+	0.454902i	$0.150326\pi$
$683$	23.7771i	0.909805i	−0.890541	+	0.454902i	$0.849674\pi$
$684$	0	0
$685$	10.8541	0.414714
$686$	−18.7082	−0.714283
$687$	0	0
$688$	3.38197i	0.128936i
$689$	3.58359	0.136524
$690$	0	0
$691$	−13.5623	−0.515934	−0.257967	−	0.966154i	$-0.583053\pi$
$691$	−13.5623	−0.515934	−0.257967	+	0.966154i	$0.583053\pi$
$692$	17.5066i	0.665500i
$693$	0	0
$694$	−6.00000	−0.227757
$695$	12.4377i	0.471789i
$696$	0	0
$697$	− 26.0689i	− 0.987429i
$698$	− 24.8328i	− 0.939936i
$699$	0	0
$700$	− 13.0902i	− 0.494762i
$701$	9.12461	0.344632	0.172316	−	0.985042i	$-0.444875\pi$
$701$	9.12461	0.344632	0.172316	+	0.985042i	$0.444875\pi$
$702$	0	0
$703$	− 17.8885i	− 0.674679i
$704$	2.00000	0.0753778
$705$	0	0
$706$	3.05573	0.115004
$707$	28.4164i	1.06871i
$708$	0	0
$709$	−12.2918	−0.461628	−0.230814	−	0.972998i	$-0.574139\pi$
$709$	−12.2918	−0.461628	−0.230814	+	0.972998i	$0.574139\pi$
$710$	12.7639	0.479022
$711$	0	0
$712$	2.00000i	0.0749532i
$713$	18.7082i	0.700628i
$714$	0	0
$715$	1.70820	0.0638832
$716$	−9.14590	−0.341798
$717$	0	0
$718$	− 0.270510i	− 0.0100953i
$719$	8.29180	0.309232	0.154616	−	0.987975i	$-0.450586\pi$
$719$	8.29180	0.309232	0.154616	+	0.987975i	$0.450586\pi$
$720$	0	0
$721$	19.4164	0.723105
$722$	15.0000i	0.558242i
$723$	0	0
$724$	10.0000	0.371647
$725$	5.00000	0.185695
$726$	0	0
$727$	32.9443i	1.22184i	0.791694	+	0.610918i	$0.209199\pi$
$727$	32.9443i	1.22184i	−0.791694	+	0.610918i	$0.790801\pi$
$728$	1.00000i	0.0370625i
$729$	0	0
$730$	− 12.4377i	− 0.460340i
$731$	−11.4377	−0.423038
$732$	0	0
$733$	12.0000i	0.443230i	0.975134	+	0.221615i	$0.0711328\pi$
$733$	12.0000i	0.443230i	−0.975134	+	0.221615i	$0.928867\pi$
$734$	−35.1246	−1.29647
$735$	0	0
$736$	−4.85410	−0.178925
$737$	− 17.8885i	− 0.658933i
$738$	0	0
$739$	−28.2918	−1.04073	−0.520365	−	0.853944i	$-0.674204\pi$
$739$	−28.2918	−1.04073	−0.520365	+	0.853944i	$0.674204\pi$
$740$	−20.0000	−0.735215
$741$	0	0
$742$	24.5623i	0.901711i
$743$	25.4164i	0.932438i	0.884669	+	0.466219i	$0.154384\pi$
$743$	25.4164i	0.932438i	−0.884669	+	0.466219i	$0.845616\pi$
$744$	0	0
$745$	− 47.8885i	− 1.75450i
$746$	15.2705	0.559093
$747$	0	0
$748$	6.76393i	0.247314i
$749$	−37.1246	−1.35650
$750$	0	0
$751$	−4.00000	−0.145962	−0.0729810	−	0.997333i	$-0.523251\pi$
$751$	−4.00000	−0.145962	−0.0729810	+	0.997333i	$0.523251\pi$
$752$	1.52786i	0.0557155i
$753$	0	0
$754$	−0.381966	−0.0139104
$755$	9.59675i	0.349261i
$756$	0	0
$757$	− 32.0689i	− 1.16556i	−0.812629	−	0.582782i	$-0.801964\pi$
$757$	− 32.0689i	− 1.16556i	0.812629	−	0.582782i	$-0.198036\pi$
$758$	26.0000i	0.944363i
$759$	0	0
$760$	−4.47214	−0.162221
$761$	9.85410	0.357211	0.178605	−	0.983921i	$-0.442841\pi$
$761$	9.85410	0.357211	0.178605	+	0.983921i	$0.442841\pi$
$762$	0	0
$763$	50.8328i	1.84027i
$764$	23.2705	0.841897
$765$	0	0
$766$	7.85410	0.283780
$767$	1.09017i	0.0393638i
$768$	0	0
$769$	18.5836	0.670141	0.335071	−	0.942193i	$-0.391240\pi$
$769$	18.5836	0.670141	0.335071	+	0.942193i	$0.391240\pi$
$770$	11.7082i	0.421934i
$771$	0	0
$772$	2.56231i	0.0922194i
$773$	− 6.76393i	− 0.243282i	−0.992574	−	0.121641i	$-0.961184\pi$
$773$	− 6.76393i	− 0.243282i	0.992574	−	0.121641i	$-0.0388156\pi$
$774$	0	0
$775$	19.2705	0.692217
$776$	7.14590	0.256523
$777$	0	0
$778$	20.8328i	0.746893i
$779$	−15.4164	−0.552350
$780$	0	0
$781$	−11.4164	−0.408511
$782$	− 16.4164i	− 0.587050i
$783$	0	0
$784$	0.145898	0.00521064
$785$	21.7082	0.774799
$786$	0	0
$787$	41.7771i	1.48919i	0.667515	+	0.744596i	$0.267358\pi$
$787$	41.7771i	1.48919i	−0.667515	+	0.744596i	$0.732642\pi$
$788$	− 9.38197i	− 0.334219i
$789$	0	0
$790$	−14.6738	−0.522069
$791$	−10.7082	−0.380740
$792$	0	0
$793$	− 4.14590i	− 0.147225i
$794$	1.96556	0.0697550
$795$	0	0
$796$	19.1246	0.677854
$797$	− 35.2361i	− 1.24813i	−0.781374	−	0.624063i	$-0.785481\pi$
$797$	− 35.2361i	− 1.24813i	0.781374	−	0.624063i	$-0.214519\pi$
$798$	0	0
$799$	−5.16718	−0.182802
$800$	5.00000i	0.176777i
$801$	0	0
$802$	21.9787i	0.776095i
$803$	11.1246i	0.392579i
$804$	0	0
$805$	− 28.4164i	− 1.00155i
$806$	−1.47214	−0.0518538
$807$	0	0
$808$	− 10.8541i	− 0.381846i
$809$	−39.4164	−1.38581	−0.692904	−	0.721030i	$-0.743669\pi$
$809$	−39.4164	−1.38581	−0.692904	+	0.721030i	$0.743669\pi$
$810$	0	0
$811$	48.3951	1.69938	0.849691	−	0.527280i	$-0.176788\pi$
$811$	48.3951	1.69938	0.849691	+	0.527280i	$0.176788\pi$
$812$	− 2.61803i	− 0.0918750i
$813$	0	0
$814$	17.8885	0.626993
$815$	−16.5836	−0.580898
$816$	0	0
$817$	6.76393i	0.236640i
$818$	− 7.12461i	− 0.249106i
$819$	0	0
$820$	17.2361i	0.601910i
$821$	8.00000	0.279202	0.139601	−	0.990208i	$-0.455418\pi$
$821$	8.00000	0.279202	0.139601	+	0.990208i	$0.455418\pi$
$822$	0	0
$823$	5.23607i	0.182518i	0.995827	+	0.0912589i	$0.0290891\pi$
$823$	5.23607i	0.182518i	−0.995827	+	0.0912589i	$0.970911\pi$
$824$	−7.41641	−0.258363
$825$	0	0
$826$	−7.47214	−0.259989
$827$	25.2016i	0.876346i	0.898891	+	0.438173i	$0.144374\pi$
$827$	25.2016i	0.876346i	−0.898891	+	0.438173i	$0.855626\pi$
$828$	0	0
$829$	0.729490	0.0253362	0.0126681	−	0.999920i	$-0.495968\pi$
$829$	0.729490	0.0253362	0.0126681	+	0.999920i	$0.495968\pi$
$830$	− 33.4164i	− 1.15990i
$831$	0	0
$832$	− 0.381966i	− 0.0132423i
$833$	0.493422i	0.0170961i
$834$	0	0
$835$	−5.85410	−0.202590
$836$	4.00000	0.138343
$837$	0	0
$838$	21.8541i	0.754937i
$839$	−38.8328	−1.34066	−0.670329	−	0.742064i	$-0.733847\pi$
$839$	−38.8328	−1.34066	−0.670329	+	0.742064i	$0.733847\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	10.0000i	0.344623i
$843$	0	0
$844$	11.4164	0.392969
$845$	28.7426i	0.988777i
$846$	0	0
$847$	18.3262i	0.629697i
$848$	− 9.38197i	− 0.322178i
$849$	0	0
$850$	−16.9098	−0.580002
$851$	−43.4164	−1.48830
$852$	0	0
$853$	− 8.83282i	− 0.302430i	−0.988501	−	0.151215i	$-0.951681\pi$
$853$	− 8.83282i	− 0.302430i	0.988501	−	0.151215i	$-0.0483186\pi$
$854$	28.4164	0.972389
$855$	0	0
$856$	14.1803	0.484674
$857$	21.8197i	0.745345i	0.927963	+	0.372673i	$0.121559\pi$
$857$	21.8197i	0.745345i	−0.927963	+	0.372673i	$0.878441\pi$
$858$	0	0
$859$	−27.7082	−0.945392	−0.472696	−	0.881226i	$-0.656719\pi$
$859$	−27.7082	−0.945392	−0.472696	+	0.881226i	$0.656719\pi$
$860$	7.56231	0.257872
$861$	0	0
$862$	− 33.4164i	− 1.13817i
$863$	− 30.9230i	− 1.05263i	−0.850289	−	0.526315i	$-0.823573\pi$
$863$	− 30.9230i	− 1.05263i	0.850289	−	0.526315i	$-0.176427\pi$
$864$	0	0
$865$	39.1459	1.33100
$866$	−16.4721	−0.559746
$867$	0	0
$868$	− 10.0902i	− 0.342483i
$869$	13.1246	0.445222
$870$	0	0
$871$	−3.41641	−0.115761
$872$	− 19.4164i	− 0.657523i
$873$	0	0
$874$	−9.70820	−0.328385
$875$	−29.2705	−0.989524
$876$	0	0
$877$	42.2705i	1.42737i	0.700465	+	0.713687i	$0.252976\pi$
$877$	42.2705i	1.42737i	−0.700465	+	0.713687i	$0.747024\pi$
$878$	− 37.1246i	− 1.25289i
$879$	0	0
$880$	− 4.47214i	− 0.150756i
$881$	14.5836	0.491334	0.245667	−	0.969354i	$-0.420993\pi$
$881$	14.5836	0.491334	0.245667	+	0.969354i	$0.420993\pi$
$882$	0	0
$883$	− 33.7082i	− 1.13437i	−0.823590	−	0.567186i	$-0.808032\pi$
$883$	− 33.7082i	− 1.13437i	0.823590	−	0.567186i	$-0.191968\pi$
$884$	1.29180	0.0434478
$885$	0	0
$886$	1.90983	0.0641620
$887$	24.0000i	0.805841i	0.915235	+	0.402921i	$0.132005\pi$
$887$	24.0000i	0.805841i	−0.915235	+	0.402921i	$0.867995\pi$
$888$	0	0
$889$	21.7082	0.728070
$890$	4.47214	0.149906
$891$	0	0
$892$	26.5623i	0.889372i
$893$	3.05573i	0.102256i
$894$	0	0
$895$	20.4508i	0.683597i
$896$	2.61803	0.0874624
$897$	0	0
$898$	4.58359i	0.152956i
$899$	3.85410	0.128541
$900$	0	0
$901$	31.7295	1.05706
$902$	− 15.4164i	− 0.513310i
$903$	0	0
$904$	4.09017	0.136037
$905$	− 22.3607i	− 0.743294i
$906$	0	0
$907$	7.74265i	0.257090i	0.991704	+	0.128545i	$0.0410307\pi$
$907$	7.74265i	0.257090i	−0.991704	+	0.128545i	$0.958969\pi$
$908$	− 16.4721i	− 0.546647i
$909$	0	0
$910$	2.23607	0.0741249
$911$	−12.8541	−0.425875	−0.212938	−	0.977066i	$-0.568303\pi$
$911$	−12.8541	−0.425875	−0.212938	+	0.977066i	$0.568303\pi$
$912$	0	0
$913$	29.8885i	0.989166i
$914$	14.2918	0.472731
$915$	0	0
$916$	−8.14590	−0.269148
$917$	56.0689i	1.85156i
$918$	0	0
$919$	52.5410	1.73317	0.866584	−	0.499031i	$-0.166311\pi$
$919$	52.5410	1.73317	0.866584	+	0.499031i	$0.166311\pi$
$920$	10.8541i	0.357849i
$921$	0	0
$922$	− 24.1459i	− 0.795203i
$923$	2.18034i	0.0717668i
$924$	0	0
$925$	44.7214i	1.47043i
$926$	−17.8885	−0.587854
$927$	0	0
$928$	1.00000i	0.0328266i
$929$	34.1459	1.12029	0.560145	−	0.828394i	$-0.310745\pi$
$929$	34.1459	1.12029	0.560145	+	0.828394i	$0.310745\pi$
$930$	0	0
$931$	0.291796	0.00956323
$932$	− 11.2361i	− 0.368050i
$933$	0	0
$934$	34.7984	1.13864
$935$	15.1246	0.494628
$936$	0	0
$937$	− 44.0689i	− 1.43967i	−0.694146	−	0.719834i	$-0.744218\pi$
$937$	− 44.0689i	− 1.43967i	0.694146	−	0.719834i	$-0.255782\pi$
$938$	− 23.4164i	− 0.764573i
$939$	0	0
$940$	3.41641	0.111431
$941$	21.7082	0.707667	0.353834	−	0.935308i	$-0.384878\pi$
$941$	21.7082	0.707667	0.353834	+	0.935308i	$0.384878\pi$
$942$	0	0
$943$	37.4164i	1.21845i
$944$	2.85410	0.0928931
$945$	0	0
$946$	−6.76393	−0.219914
$947$	60.1591i	1.95491i	0.211152	+	0.977453i	$0.432279\pi$
$947$	60.1591i	1.95491i	−0.211152	+	0.977453i	$0.567721\pi$
$948$	0	0
$949$	2.12461	0.0689678
$950$	10.0000i	0.324443i
$951$	0	0
$952$	8.85410i	0.286963i
$953$	2.83282i	0.0917639i	0.998947	+	0.0458820i	$0.0146098\pi$
$953$	2.83282i	0.0917639i	−0.998947	+	0.0458820i	$0.985390\pi$
$954$	0	0
$955$	− 52.0344i	− 1.68379i
$956$	−5.70820	−0.184617
$957$	0	0
$958$	18.9787i	0.613174i
$959$	−12.7082	−0.410369
$960$	0	0
$961$	−16.1459	−0.520835
$962$	− 3.41641i	− 0.110149i
$963$	0	0
$964$	8.85410	0.285171
$965$	5.72949	0.184439
$966$	0	0
$967$	− 33.7082i	− 1.08398i	−0.840384	−	0.541991i	$-0.817671\pi$
$967$	− 33.7082i	− 1.08398i	0.840384	−	0.541991i	$-0.182329\pi$
$968$	− 7.00000i	− 0.224989i
$969$	0	0
$970$	− 15.9787i	− 0.513046i
$971$	−1.70820	−0.0548189	−0.0274094	−	0.999624i	$-0.508726\pi$
$971$	−1.70820	−0.0548189	−0.0274094	+	0.999624i	$0.508726\pi$
$972$	0	0
$973$	− 14.5623i	− 0.466846i
$974$	−38.5623	−1.23562
$975$	0	0
$976$	−10.8541	−0.347431
$977$	− 33.0557i	− 1.05755i	−0.848763	−	0.528773i	$-0.822652\pi$
$977$	− 33.0557i	− 1.05755i	0.848763	−	0.528773i	$-0.177348\pi$
$978$	0	0
$979$	−4.00000	−0.127841
$980$	− 0.326238i	− 0.0104213i
$981$	0	0
$982$	− 6.29180i	− 0.200779i
$983$	3.59675i	0.114718i	0.998354	+	0.0573592i	$0.0182680\pi$
$983$	3.59675i	0.114718i	−0.998354	+	0.0573592i	$0.981732\pi$
$984$	0	0
$985$	−20.9787	−0.668437
$986$	−3.38197	−0.107704
$987$	0	0
$988$	− 0.763932i	− 0.0243039i
$989$	16.4164	0.522011
$990$	0	0
$991$	−41.4164	−1.31564	−0.657818	−	0.753177i	$-0.728520\pi$
$991$	−41.4164	−1.31564	−0.657818	+	0.753177i	$0.728520\pi$
$992$	3.85410i	0.122368i
$993$	0	0
$994$	−14.9443	−0.474004
$995$	− 42.7639i	− 1.35571i
$996$	0	0
$997$	− 17.1246i	− 0.542342i	−0.962531	−	0.271171i	$-0.912589\pi$
$997$	− 17.1246i	− 0.542342i	0.962531	−	0.271171i	$-0.0874109\pi$
$998$	28.5623i	0.904124i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2610.2.e.e.2089.2		4
3.2	odd	2		290.2.b.a.59.3	yes	4
5.4	even	2	inner	2610.2.e.e.2089.3		4
12.11	even	2		2320.2.d.d.929.2		4
15.2	even	4		1450.2.a.k.1.1		2
15.8	even	4		1450.2.a.l.1.2		2
15.14	odd	2		290.2.b.a.59.2	✓	4
60.59	even	2		2320.2.d.d.929.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
290.2.b.a.59.2	✓	4	15.14	odd	2
290.2.b.a.59.3	yes	4	3.2	odd	2
1450.2.a.k.1.1		2	15.2	even	4
1450.2.a.l.1.2		2	15.8	even	4
2320.2.d.d.929.2		4	12.11	even	2
2320.2.d.d.929.3		4	60.59	even	2
2610.2.e.e.2089.2		4	1.1	even	1	trivial
2610.2.e.e.2089.3		4	5.4	even	2	inner

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:	\( N \)	\(=\)	\( 2610 = 2 \cdot 3^{2} \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2610.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} +2.23607i q^{5} -2.61803i q^{7} +1.00000i q^{8} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000 q^{4} +2.23607i q^{5} -2.61803i q^{7} +1.00000i q^{8} +2.23607 q^{10} -2.00000 q^{11} +0.381966i q^{13} -2.61803 q^{14} +1.00000 q^{16} +3.38197i q^{17} +2.00000 q^{19} -2.23607i q^{20} +2.00000i q^{22} -4.85410i q^{23} -5.00000 q^{25} +0.381966 q^{26} +2.61803i q^{28} -1.00000 q^{29} -3.85410 q^{31} -1.00000i q^{32} +3.38197 q^{34} +5.85410 q^{35} -8.94427i q^{37} -2.00000i q^{38} -2.23607 q^{40} -7.70820 q^{41} +3.38197i q^{43} +2.00000 q^{44} -4.85410 q^{46} +1.52786i q^{47} +0.145898 q^{49} +5.00000i q^{50} -0.381966i q^{52} -9.38197i q^{53} -4.47214i q^{55} +2.61803 q^{56} +1.00000i q^{58} +2.85410 q^{59} -10.8541 q^{61} +3.85410i q^{62} -1.00000 q^{64} -0.854102 q^{65} +8.94427i q^{67} -3.38197i q^{68} -5.85410i q^{70} +5.70820 q^{71} -5.56231i q^{73} -8.94427 q^{74} -2.00000 q^{76} +5.23607i q^{77} -6.56231 q^{79} +2.23607i q^{80} +7.70820i q^{82} -14.9443i q^{83} -7.56231 q^{85} +3.38197 q^{86} -2.00000i q^{88} +2.00000 q^{89} +1.00000 q^{91} +4.85410i q^{92} +1.52786 q^{94} +4.47214i q^{95} -7.14590i q^{97} -0.145898i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4}+O(q^{10}) \) 4 * q - 4 * q^4	\( 4 q - 4 q^{4} - 8 q^{11} - 6 q^{14} + 4 q^{16} + 8 q^{19} - 20 q^{25} + 6 q^{26} - 4 q^{29} - 2 q^{31} + 18 q^{34} + 10 q^{35} - 4 q^{41} + 8 q^{44} - 6 q^{46} + 14 q^{49} + 6 q^{56} - 2 q^{59} - 30 q^{61} - 4 q^{64} + 10 q^{65} - 4 q^{71} - 8 q^{76} + 14 q^{79} + 10 q^{85} + 18 q^{86} + 8 q^{89} + 4 q^{91} + 24 q^{94}+O(q^{100}) \) 4 * q - 4 * q^4 - 8 * q^11 - 6 * q^14 + 4 * q^16 + 8 * q^19 - 20 * q^25 + 6 * q^26 - 4 * q^29 - 2 * q^31 + 18 * q^34 + 10 * q^35 - 4 * q^41 + 8 * q^44 - 6 * q^46 + 14 * q^49 + 6 * q^56 - 2 * q^59 - 30 * q^61 - 4 * q^64 + 10 * q^65 - 4 * q^71 - 8 * q^76 + 14 * q^79 + 10 * q^85 + 18 * q^86 + 8 * q^89 + 4 * q^91 + 24 * q^94

Introduction

L-functions

Modular forms

Varieties

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Database

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