LMFDB - Embedded newform 1840.2.e.g.369.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1840,2,Mod(369,1840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1840, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1840.369");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1840 = 2^{4} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1840.e (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:	$14.6924739719$
Analytic rank:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 2 x^{11} + 39 x^{10} - 10 x^{9} + 2 x^{8} - 26 x^{7} + 297 x^{6} - 116 x^{5} + 24 x^{4} + \cdots + 8 $ x^14 - 2x^11 + 39x^10 - 10x^9 + 2x^8 - 26x^7 + 297x^6 - 116x^5 + 24x^4 - 20x^3 + 64x^2 - 32*x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 920)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			369.1
Root			$-0.503254 - 0.503254i$ of defining polynomial
Character	$\chi$	$=$	1840.369
Dual form			1840.2.e.g.369.14

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.98707i q^{3} +(0.274289 - 2.21918i) q^{5} -0.980560i q^{7} -5.92257 q^{9} +O(q^{10})$	$q-2.98707i q^{3} +(0.274289 - 2.21918i) q^{5} -0.980560i q^{7} -5.92257 q^{9} +6.14721 q^{11} -6.37400i q^{13} +(-6.62885 - 0.819321i) q^{15} -3.36098i q^{17} +1.08276 q^{19} -2.92900 q^{21} +1.00000i q^{23} +(-4.84953 - 1.21740i) q^{25} +8.72993i q^{27} +0.271042 q^{29} +8.77792 q^{31} -18.3621i q^{33} +(-2.17604 - 0.268957i) q^{35} +8.84665i q^{37} -19.0396 q^{39} -4.85308 q^{41} -1.87756i q^{43} +(-1.62450 + 13.1433i) q^{45} +0.196089i q^{47} +6.03850 q^{49} -10.0395 q^{51} -1.93157i q^{53} +(1.68612 - 13.6418i) q^{55} -3.23429i q^{57} +13.0036 q^{59} +6.00189 q^{61} +5.80744i q^{63} +(-14.1451 - 1.74832i) q^{65} +2.26847i q^{67} +2.98707 q^{69} -10.2677 q^{71} +1.38188i q^{73} +(-3.63644 + 14.4859i) q^{75} -6.02771i q^{77} -4.67459 q^{79} +8.30917 q^{81} +15.7171i q^{83} +(-7.45862 - 0.921881i) q^{85} -0.809622i q^{87} +11.1548 q^{89} -6.25008 q^{91} -26.2202i q^{93} +(0.296991 - 2.40285i) q^{95} +10.2868i q^{97} -36.4073 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q + 2 q^{5} - 4 q^{9}+O(q^{10}) $ 14 * q + 2 * q^5 - 4 * q^9	$ 14 q + 2 q^{5} - 4 q^{9} + 14 q^{11} + 6 q^{15} - 14 q^{19} - 12 q^{21} - 14 q^{25} + 22 q^{29} + 20 q^{31} + 2 q^{35} - 48 q^{39} - 32 q^{41} - 26 q^{45} + 34 q^{49} + 14 q^{51} + 38 q^{55} - 22 q^{59} + 10 q^{61} - 38 q^{65} + 6 q^{69} + 28 q^{71} + 24 q^{75} - 64 q^{79} - 10 q^{81} - 50 q^{85} + 48 q^{89} + 14 q^{91} + 30 q^{95} - 122 q^{99}+O(q^{100}) $ 14 * q + 2 * q^5 - 4 * q^9 + 14 * q^11 + 6 * q^15 - 14 * q^19 - 12 * q^21 - 14 * q^25 + 22 * q^29 + 20 * q^31 + 2 * q^35 - 48 * q^39 - 32 * q^41 - 26 * q^45 + 34 * q^49 + 14 * q^51 + 38 * q^55 - 22 * q^59 + 10 * q^61 - 38 * q^65 + 6 * q^69 + 28 * q^71 + 24 * q^75 - 64 * q^79 - 10 * q^81 - 50 * q^85 + 48 * q^89 + 14 * q^91 + 30 * q^95 - 122 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1840\mathbb{Z}\right)^\times$.

$n$	$737$	$1151$	$1201$	$1381$
$\chi(n)$	$-1$	$1$	$1$	$1$

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
$3$		−	2.98707i		−	1.72458i	−0.506411	−	0.862292i	$-0.669028\pi$
$3$		−	2.98707i		−	1.72458i	0.506411	−	0.862292i	$-0.330972\pi$
$4$		0			0
$5$	0.274289	−	2.21918i	0.122666	−	0.992448i
$5$	0.274289	−	2.21918i	0.122666	−	0.992448i
$6$		0			0
$7$		−	0.980560i		−	0.370617i	−0.982680	−	0.185308i	$-0.940672\pi$
$7$		−	0.980560i		−	0.370617i	0.982680	−	0.185308i	$-0.0593284\pi$
$8$		0			0
$9$	−5.92257			−1.97419
$10$		0			0
$11$	6.14721			1.85345			0.926727	−	0.375734i	$-0.122609\pi$
$11$	6.14721			1.85345			0.926727	+	0.375734i	$0.122609\pi$
$12$		0			0
$13$		−	6.37400i		−	1.76783i	−0.467649	−	0.883914i	$-0.654899\pi$
$13$		−	6.37400i		−	1.76783i	0.467649	−	0.883914i	$-0.345101\pi$
$14$		0			0
$15$	−6.62885	−	0.819321i	−1.71156	−	0.211548i
$16$		0			0
$17$		−	3.36098i		−	0.815157i	−0.913170	−	0.407579i	$-0.866373\pi$
$17$		−	3.36098i		−	0.815157i	0.913170	−	0.407579i	$-0.133627\pi$
$18$		0			0
$19$	1.08276			0.248403			0.124202	−	0.992257i	$-0.460363\pi$
$19$	1.08276			0.248403			0.124202	+	0.992257i	$0.460363\pi$
$20$		0			0
$21$	−2.92900			−0.639160
$22$		0			0
$23$			1.00000i			0.208514i
$23$			1.00000i			0.208514i
$24$		0			0
$25$	−4.84953	−	1.21740i	−0.969906	−	0.243479i
$26$		0			0
$27$			8.72993i			1.68008i
$28$		0			0
$29$	0.271042			0.0503313			0.0251657	−	0.999683i	$-0.491989\pi$
$29$	0.271042			0.0503313			0.0251657	+	0.999683i	$0.491989\pi$
$30$		0			0
$31$	8.77792			1.57656			0.788280	−	0.615316i	$-0.210972\pi$
$31$	8.77792			1.57656			0.788280	+	0.615316i	$0.210972\pi$
$32$		0			0
$33$		−	18.3621i		−	3.19644i
$34$		0			0
$35$	−2.17604	−	0.268957i	−0.367818	−	0.0454621i
$36$		0			0
$37$			8.84665i			1.45438i	0.686436	+	0.727190i	$0.259174\pi$
$37$			8.84665i			1.45438i	−0.686436	+	0.727190i	$0.740826\pi$
$38$		0			0
$39$	−19.0396			−3.04877
$40$		0			0
$41$	−4.85308			−0.757923			−0.378962	−	0.925412i	$-0.623719\pi$
$41$	−4.85308			−0.757923			−0.378962	+	0.925412i	$0.623719\pi$
$42$		0			0
$43$		−	1.87756i		−	0.286326i	−0.989699	−	0.143163i	$-0.954273\pi$
$43$		−	1.87756i		−	0.286326i	0.989699	−	0.143163i	$-0.0457273\pi$
$44$		0			0
$45$	−1.62450	+	13.1433i	−0.242166	+	1.95928i
$46$		0			0
$47$			0.196089i			0.0286026i	0.999898	+	0.0143013i	$0.00455239\pi$
$47$			0.196089i			0.0286026i	−0.999898	+	0.0143013i	$0.995448\pi$
$48$		0			0
$49$	6.03850			0.862643
$50$		0			0
$51$	−10.0395			−1.40581
$52$		0			0
$53$		−	1.93157i		−	0.265321i	−0.991162	−	0.132661i	$-0.957648\pi$
$53$		−	1.93157i		−	0.265321i	0.991162	−	0.132661i	$-0.0423521\pi$
$54$		0			0
$55$	1.68612	−	13.6418i	0.227356	−	1.83946i
$56$		0			0
$57$		−	3.23429i		−	0.428392i
$58$		0			0
$59$	13.0036			1.69293			0.846465	−	0.532444i	$-0.178726\pi$
$59$	13.0036			1.69293			0.846465	+	0.532444i	$0.178726\pi$
$60$		0			0
$61$	6.00189			0.768464			0.384232	−	0.923237i	$-0.374466\pi$
$61$	6.00189			0.768464			0.384232	+	0.923237i	$0.374466\pi$
$62$		0			0
$63$			5.80744i			0.731668i
$64$		0			0
$65$	−14.1451	−	1.74832i	−1.75448	−	0.216852i
$66$		0			0
$67$			2.26847i			0.277138i	0.990353	+	0.138569i	$0.0442503\pi$
$67$			2.26847i			0.277138i	−0.990353	+	0.138569i	$0.955750\pi$
$68$		0			0
$69$	2.98707			0.359601
$70$		0			0
$71$	−10.2677			−1.21855			−0.609277	−	0.792958i	$-0.708540\pi$
$71$	−10.2677			−1.21855			−0.609277	+	0.792958i	$0.708540\pi$
$72$		0			0
$73$			1.38188i			0.161737i	0.996725	+	0.0808685i	$0.0257694\pi$
$73$			1.38188i			0.161737i	−0.996725	+	0.0808685i	$0.974231\pi$
$74$		0			0
$75$	−3.63644	+	14.4859i	−0.419900	+	1.67269i
$76$		0			0
$77$		−	6.02771i		−	0.686921i
$78$		0			0
$79$	−4.67459			−0.525932			−0.262966	−	0.964805i	$-0.584701\pi$
$79$	−4.67459			−0.525932			−0.262966	+	0.964805i	$0.584701\pi$
$80$		0			0
$81$	8.30917			0.923241
$82$		0			0
$83$			15.7171i			1.72517i	0.505909	+	0.862587i	$0.331157\pi$
$83$			15.7171i			1.72517i	−0.505909	+	0.862587i	$0.668843\pi$
$84$		0			0
$85$	−7.45862	−	0.921881i	−0.809001	−	0.0999921i
$86$		0			0
$87$		−	0.809622i		−	0.0868006i
$88$		0			0
$89$	11.1548			1.18241			0.591206	−	0.806521i	$-0.298652\pi$
$89$	11.1548			1.18241			0.591206	+	0.806521i	$0.298652\pi$
$90$		0			0
$91$	−6.25008			−0.655187
$92$		0			0
$93$		−	26.2202i		−	2.71891i
$94$		0			0
$95$	0.296991	−	2.40285i	0.0304706	−	0.246527i
$96$		0			0
$97$			10.2868i			1.04447i	0.852802	+	0.522234i	$0.174901\pi$
$97$			10.2868i			1.04447i	−0.852802	+	0.522234i	$0.825099\pi$
$98$		0			0
$99$	−36.4073			−3.65908
$100$		0			0
$101$	−15.2625			−1.51868			−0.759339	−	0.650695i	$-0.774478\pi$
$101$	−15.2625			−1.51868			−0.759339	+	0.650695i	$0.774478\pi$
$102$		0			0
$103$			7.60795i			0.749634i	0.927099	+	0.374817i	$0.122294\pi$
$103$			7.60795i			0.749634i	−0.927099	+	0.374817i	$0.877706\pi$
$104$		0			0
$105$	−0.803394	+	6.49998i	−0.0784032	+	0.634333i
$106$		0			0
$107$			11.3081i			1.09320i	0.837395	+	0.546599i	$0.184078\pi$
$107$			11.3081i			1.09320i	−0.837395	+	0.546599i	$0.815922\pi$
$108$		0			0
$109$	−3.45336			−0.330772			−0.165386	−	0.986229i	$-0.552887\pi$
$109$	−3.45336			−0.330772			−0.165386	+	0.986229i	$0.552887\pi$
$110$		0			0
$111$	26.4256			2.50820
$112$		0			0
$113$			14.6889i			1.38181i	0.722944	+	0.690907i	$0.242788\pi$
$113$			14.6889i			1.38181i	−0.722944	+	0.690907i	$0.757212\pi$
$114$		0			0
$115$	2.21918	+	0.274289i	0.206940	+	0.0255776i
$116$		0			0
$117$			37.7505i			3.49003i
$118$		0			0
$119$	−3.29564			−0.302111
$120$		0			0
$121$	26.7882			2.43530
$122$		0			0
$123$			14.4965i			1.30710i
$124$		0			0
$125$	−4.03180	+	10.4281i	−0.360615	+	0.932715i
$126$		0			0
$127$		−	1.19312i		−	0.105873i	−0.998598	−	0.0529363i	$-0.983142\pi$
$127$		−	1.19312i		−	0.105873i	0.998598	−	0.0529363i	$-0.0168580\pi$
$128$		0			0
$129$	−5.60841			−0.493793
$130$		0			0
$131$	11.7326			1.02509			0.512543	−	0.858662i	$-0.328704\pi$
$131$	11.7326			1.02509			0.512543	+	0.858662i	$0.328704\pi$
$132$		0			0
$133$		−	1.06171i		−	0.0920623i
$134$		0			0
$135$	19.3733	+	2.39453i	1.66739	+	0.206088i
$136$		0			0
$137$			10.4665i			0.894210i	0.894481	+	0.447105i	$0.147545\pi$
$137$			10.4665i			0.894210i	−0.894481	+	0.447105i	$0.852455\pi$
$138$		0			0
$139$	4.11349			0.348902			0.174451	−	0.984666i	$-0.444185\pi$
$139$	4.11349			0.348902			0.174451	+	0.984666i	$0.444185\pi$
$140$		0			0
$141$	0.585732			0.0493275
$142$		0			0
$143$		−	39.1823i		−	3.27659i
$144$		0			0
$145$	0.0743441	−	0.601492i	0.00617394	−	0.0499512i
$146$		0			0
$147$		−	18.0374i		−	1.48770i
$148$		0			0
$149$	−1.37793			−0.112885			−0.0564424	−	0.998406i	$-0.517976\pi$
$149$	−1.37793			−0.112885			−0.0564424	+	0.998406i	$0.517976\pi$
$150$		0			0
$151$	3.49384			0.284325			0.142162	−	0.989843i	$-0.454594\pi$
$151$	3.49384			0.284325			0.142162	+	0.989843i	$0.454594\pi$
$152$		0			0
$153$			19.9057i			1.60928i
$154$		0			0
$155$	2.40769	−	19.4798i	0.193390	−	1.56465i
$156$		0			0
$157$			3.32373i			0.265262i	0.991165	+	0.132631i	$0.0423426\pi$
$157$			3.32373i			0.265262i	−0.991165	+	0.132631i	$0.957657\pi$
$158$		0			0
$159$	−5.76973			−0.457569
$160$		0			0
$161$	0.980560			0.0772789
$162$		0			0
$163$			8.52075i			0.667397i	0.942680	+	0.333698i	$0.108297\pi$
$163$			8.52075i			0.667397i	−0.942680	+	0.333698i	$0.891703\pi$
$164$		0			0
$165$	−40.7489	−	5.03654i	−3.17230	−	0.392094i
$166$		0			0
$167$			15.6190i			1.20864i	0.796742	+	0.604319i	$0.206555\pi$
$167$			15.6190i			1.20864i	−0.796742	+	0.604319i	$0.793445\pi$
$168$		0			0
$169$	−27.6278			−2.12522
$170$		0			0
$171$	−6.41275			−0.490395
$172$		0			0
$173$		−	8.42727i		−	0.640713i	−0.947297	−	0.320357i	$-0.896197\pi$
$173$		−	8.42727i		−	0.640713i	0.947297	−	0.320357i	$-0.103803\pi$
$174$		0			0
$175$	−1.19373	+	4.75525i	−0.0902375	+	0.359463i
$176$		0			0
$177$		−	38.8428i		−	2.91960i
$178$		0			0
$179$	10.8047			0.807580			0.403790	−	0.914852i	$-0.367693\pi$
$179$	10.8047			0.807580			0.403790	+	0.914852i	$0.367693\pi$
$180$		0			0
$181$	−15.4282			−1.14677			−0.573385	−	0.819286i	$-0.694370\pi$
$181$	−15.4282			−1.14677			−0.573385	+	0.819286i	$0.694370\pi$
$182$		0			0
$183$		−	17.9281i		−	1.32528i
$184$		0			0
$185$	19.6323	+	2.42654i	1.44340	+	0.178403i
$186$		0			0
$187$		−	20.6607i		−	1.51086i
$188$		0			0
$189$	8.56022			0.622664
$190$		0			0
$191$	22.2634			1.61092			0.805462	−	0.592648i	$-0.201917\pi$
$191$	22.2634			1.61092			0.805462	+	0.592648i	$0.201917\pi$
$192$		0			0
$193$			16.1399i			1.16178i	0.813983	+	0.580889i	$0.197295\pi$
$193$			16.1399i			1.16178i	−0.813983	+	0.580889i	$0.802705\pi$
$194$		0			0
$195$	−5.22235	+	42.2522i	−0.373980	+	3.02575i
$196$		0			0
$197$		−	13.9936i		−	0.997004i	−0.866889	−	0.498502i	$-0.833884\pi$
$197$		−	13.9936i		−	0.997004i	0.866889	−	0.498502i	$-0.166116\pi$
$198$		0			0
$199$	−6.91926			−0.490493			−0.245246	−	0.969461i	$-0.578869\pi$
$199$	−6.91926			−0.490493			−0.245246	+	0.969461i	$0.578869\pi$
$200$		0			0
$201$	6.77608			0.477948
$202$		0			0
$203$		−	0.265773i		−	0.0186536i
$204$		0			0
$205$	−1.33115	+	10.7699i	−0.0929714	+	0.752200i
$206$		0			0
$207$		−	5.92257i		−	0.411647i
$208$		0			0
$209$	6.65598			0.460404
$210$		0			0
$211$	−12.9170			−0.889246			−0.444623	−	0.895718i	$-0.646662\pi$
$211$	−12.9170			−0.889246			−0.444623	+	0.895718i	$0.646662\pi$
$212$		0			0
$213$			30.6704i			2.10150i
$214$		0			0
$215$	−4.16665	−	0.514996i	−0.284164	−	0.0351224i
$216$		0			0
$217$		−	8.60728i		−	0.584300i
$218$		0			0
$219$	4.12777			0.278929
$220$		0			0
$221$	−21.4229			−1.44106
$222$		0			0
$223$		−	26.6579i		−	1.78514i	−0.450905	−	0.892572i	$-0.648899\pi$
$223$		−	26.6579i		−	1.78514i	0.450905	−	0.892572i	$-0.351101\pi$
$224$		0			0
$225$	28.7217	+	7.21012i	1.91478	+	0.480675i
$226$		0			0
$227$		−	9.44407i		−	0.626825i	−0.949617	−	0.313412i	$-0.898528\pi$
$227$		−	9.44407i		−	0.626825i	0.949617	−	0.313412i	$-0.101472\pi$
$228$		0			0
$229$	−14.1994			−0.938322			−0.469161	−	0.883113i	$-0.655444\pi$
$229$	−14.1994			−0.938322			−0.469161	+	0.883113i	$0.655444\pi$
$230$		0			0
$231$	−18.0052			−1.18465
$232$		0			0
$233$		−	9.41134i		−	0.616557i	−0.951296	−	0.308279i	$-0.900247\pi$
$233$		−	9.41134i		−	0.616557i	0.951296	−	0.308279i	$-0.0997529\pi$
$234$		0			0
$235$	0.435158	+	0.0537852i	0.0283866	+	0.00350856i
$236$		0			0
$237$			13.9633i			0.907014i
$238$		0			0
$239$	−15.4781			−1.00119			−0.500597	−	0.865680i	$-0.666886\pi$
$239$	−15.4781			−1.00119			−0.500597	+	0.865680i	$0.666886\pi$
$240$		0			0
$241$	−3.51052			−0.226133			−0.113066	−	0.993587i	$-0.536067\pi$
$241$	−3.51052			−0.226133			−0.113066	+	0.993587i	$0.536067\pi$
$242$		0			0
$243$			1.36974i			0.0878689i
$244$		0			0
$245$	1.65630	−	13.4005i	0.105817	−	0.856129i
$246$		0			0
$247$		−	6.90153i		−	0.439134i
$248$		0			0
$249$	46.9480			2.97521
$250$		0			0
$251$	23.6649			1.49372			0.746859	−	0.664982i	$-0.231561\pi$
$251$	23.6649			1.49372			0.746859	+	0.664982i	$0.231561\pi$
$252$		0			0
$253$			6.14721i			0.386472i
$254$		0			0
$255$	−2.75372	+	22.2794i	−0.172445	+	1.39519i
$256$		0			0
$257$		−	30.8326i		−	1.92328i	−0.274311	−	0.961641i	$-0.588450\pi$
$257$		−	30.8326i		−	1.92328i	0.274311	−	0.961641i	$-0.411550\pi$
$258$		0			0
$259$	8.67467			0.539018
$260$		0			0
$261$	−1.60527			−0.0993636
$262$		0			0
$263$		−	20.5880i		−	1.26951i	−0.772714	−	0.634755i	$-0.781101\pi$
$263$		−	20.5880i		−	1.26951i	0.772714	−	0.634755i	$-0.218899\pi$
$264$		0			0
$265$	−4.28650	−	0.529809i	−0.263318	−	0.0325459i
$266$		0			0
$267$		−	33.3203i		−	2.03917i
$268$		0			0
$269$	17.0277			1.03820			0.519099	−	0.854714i	$-0.326268\pi$
$269$	17.0277			1.03820			0.519099	+	0.854714i	$0.326268\pi$
$270$		0			0
$271$	−13.6709			−0.830450			−0.415225	−	0.909719i	$-0.636297\pi$
$271$	−13.6709			−0.830450			−0.415225	+	0.909719i	$0.636297\pi$
$272$		0			0
$273$			18.6694i			1.12993i
$274$		0			0
$275$	−29.8111	−	7.48360i	−1.79768	−	0.451278i
$276$		0			0
$277$		−	16.2380i		−	0.975647i	−0.872942	−	0.487824i	$-0.837791\pi$
$277$		−	16.2380i		−	0.975647i	0.872942	−	0.487824i	$-0.162209\pi$
$278$		0			0
$279$	−51.9879			−3.11243
$280$		0			0
$281$	−0.362407			−0.0216194			−0.0108097	−	0.999942i	$-0.503441\pi$
$281$	−0.362407			−0.0216194			−0.0108097	+	0.999942i	$0.503441\pi$
$282$		0			0
$283$		−	6.80039i		−	0.404241i	−0.979361	−	0.202120i	$-0.935217\pi$
$283$		−	6.80039i		−	0.404241i	0.979361	−	0.202120i	$-0.0647833\pi$
$284$		0			0
$285$	−7.17747	−	0.887131i	−0.425157	−	0.0525491i
$286$		0			0
$287$			4.75873i			0.280899i
$288$		0			0
$289$	5.70382			0.335519
$290$		0			0
$291$	30.7274			1.80127
$292$		0			0
$293$		−	26.2172i		−	1.53162i	−0.643065	−	0.765812i	$-0.722337\pi$
$293$		−	26.2172i		−	1.53162i	0.643065	−	0.765812i	$-0.277663\pi$
$294$		0			0
$295$	3.56676	−	28.8574i	0.207665	−	1.68015i
$296$		0			0
$297$			53.6647i			3.11394i
$298$		0			0
$299$	6.37400			0.368618
$300$		0			0
$301$	−1.84106			−0.106117
$302$		0			0
$303$			45.5902i			2.61909i
$304$		0			0
$305$	1.64626	−	13.3193i	0.0942644	−	0.762660i
$306$		0			0
$307$			19.7146i			1.12517i	0.826738	+	0.562587i	$0.190194\pi$
$307$			19.7146i			1.12517i	−0.826738	+	0.562587i	$0.809806\pi$
$308$		0			0
$309$	22.7255			1.29281
$310$		0			0
$311$	−15.1003			−0.856258			−0.428129	−	0.903718i	$-0.640827\pi$
$311$	−15.1003			−0.856258			−0.428129	+	0.903718i	$0.640827\pi$
$312$		0			0
$313$			12.9813i			0.733747i	0.930271	+	0.366874i	$0.119572\pi$
$313$			12.9813i			0.733747i	−0.930271	+	0.366874i	$0.880428\pi$
$314$		0			0
$315$	12.8878	+	1.59292i	0.726143	+	0.0897508i
$316$		0			0
$317$		−	17.3503i		−	0.974489i	−0.873266	−	0.487244i	$-0.838002\pi$
$317$		−	17.3503i		−	0.974489i	0.873266	−	0.487244i	$-0.161998\pi$
$318$		0			0
$319$	1.66616			0.0932868
$320$		0			0
$321$	33.7781			1.88531
$322$		0			0
$323$		−	3.63915i		−	0.202488i
$324$		0			0
$325$	−7.75968	+	30.9109i	−0.430429	+	1.71463i
$326$		0			0
$327$			10.3154i			0.570444i
$328$		0			0
$329$	0.192277			0.0106006
$330$		0			0
$331$	−0.984756			−0.0541271			−0.0270635	−	0.999634i	$-0.508616\pi$
$331$	−0.984756			−0.0541271			−0.0270635	+	0.999634i	$0.508616\pi$
$332$		0			0
$333$		−	52.3950i		−	2.87123i
$334$		0			0
$335$	5.03415	+	0.622218i	0.275045	+	0.0339954i
$336$		0			0
$337$		−	23.5575i		−	1.28326i	−0.767015	−	0.641629i	$-0.778259\pi$
$337$		−	23.5575i		−	1.28326i	0.767015	−	0.641629i	$-0.221741\pi$
$338$		0			0
$339$	43.8767			2.38305
$340$		0			0
$341$	53.9598			2.92208
$342$		0			0
$343$		−	12.7850i		−	0.690327i
$344$		0			0
$345$	0.819321	−	6.62885i	0.0441108	−	0.356885i
$346$		0			0
$347$			26.0916i			1.40067i	0.713813	+	0.700337i	$0.246967\pi$
$347$			26.0916i			1.40067i	−0.713813	+	0.700337i	$0.753033\pi$
$348$		0			0
$349$	17.8609			0.956070			0.478035	−	0.878341i	$-0.341349\pi$
$349$	17.8609			0.956070			0.478035	+	0.878341i	$0.341349\pi$
$350$		0			0
$351$	55.6445			2.97009
$352$		0			0
$353$		−	13.1080i		−	0.697667i	−0.937185	−	0.348833i	$-0.886578\pi$
$353$		−	13.1080i		−	0.697667i	0.937185	−	0.348833i	$-0.113422\pi$
$354$		0			0
$355$	−2.81633	+	22.7859i	−0.149475	+	1.20935i
$356$		0			0
$357$			9.84430i			0.521016i
$358$		0			0
$359$	22.1119			1.16702			0.583510	−	0.812106i	$-0.301679\pi$
$359$	22.1119			1.16702			0.583510	+	0.812106i	$0.301679\pi$
$360$		0			0
$361$	−17.8276			−0.938296
$362$		0			0
$363$		−	80.0183i		−	4.19987i
$364$		0			0
$365$	3.06665	+	0.379036i	0.160516	+	0.0198396i
$366$		0			0
$367$			7.98495i			0.416811i	0.978043	+	0.208406i	$0.0668274\pi$
$367$			7.98495i			0.416811i	−0.978043	+	0.208406i	$0.933173\pi$
$368$		0			0
$369$	28.7427			1.49629
$370$		0			0
$371$	−1.89402			−0.0983326
$372$		0			0
$373$			23.8712i			1.23600i	0.786177	+	0.618002i	$0.212058\pi$
$373$			23.8712i			1.23600i	−0.786177	+	0.618002i	$0.787942\pi$
$374$		0			0
$375$	31.1493	+	12.0433i	1.60855	+	0.621911i
$376$		0			0
$377$		−	1.72762i		−	0.0889771i
$378$		0			0
$379$	−34.5647			−1.77547			−0.887735	−	0.460355i	$-0.847722\pi$
$379$	−34.5647			−1.77547			−0.887735	+	0.460355i	$0.847722\pi$
$380$		0			0
$381$	−3.56394			−0.182586
$382$		0			0
$383$		−	12.3139i		−	0.629211i	−0.949223	−	0.314606i	$-0.898128\pi$
$383$		−	12.3139i		−	0.629211i	0.949223	−	0.314606i	$-0.101872\pi$
$384$		0			0
$385$	−13.3766	−	1.65334i	−0.681734	−	0.0842619i
$386$		0			0
$387$			11.1200i			0.565262i
$388$		0			0
$389$	27.0447			1.37122			0.685611	−	0.727968i	$-0.259535\pi$
$389$	27.0447			1.37122			0.685611	+	0.727968i	$0.259535\pi$
$390$		0			0
$391$	3.36098			0.169972
$392$		0			0
$393$		−	35.0462i		−	1.76785i
$394$		0			0
$395$	−1.28219	+	10.3738i	−0.0645140	+	0.521960i
$396$		0			0
$397$			9.07631i			0.455527i	0.973717	+	0.227763i	$0.0731413\pi$
$397$			9.07631i			0.455527i	−0.973717	+	0.227763i	$0.926859\pi$
$398$		0			0
$399$	−3.17141			−0.158769
$400$		0			0
$401$	6.73762			0.336461			0.168230	−	0.985748i	$-0.446195\pi$
$401$	6.73762			0.336461			0.168230	+	0.985748i	$0.446195\pi$
$402$		0			0
$403$		−	55.9504i		−	2.78709i
$404$		0			0
$405$	2.27912	−	18.4395i	0.113250	−	0.916268i
$406$		0			0
$407$			54.3823i			2.69563i
$408$		0			0
$409$	−25.7205			−1.27180			−0.635898	−	0.771773i	$-0.719370\pi$
$409$	−25.7205			−1.27180			−0.635898	+	0.771773i	$0.719370\pi$
$410$		0			0
$411$	31.2640			1.54214
$412$		0			0
$413$		−	12.7509i		−	0.627428i
$414$		0			0
$415$	34.8791	+	4.31103i	1.71215	+	0.211620i
$416$		0			0
$417$		−	12.2873i		−	0.601711i
$418$		0			0
$419$	−16.2666			−0.794676			−0.397338	−	0.917672i	$-0.630066\pi$
$419$	−16.2666			−0.794676			−0.397338	+	0.917672i	$0.630066\pi$
$420$		0			0
$421$	−18.9106			−0.921645			−0.460822	−	0.887492i	$-0.652445\pi$
$421$	−18.9106			−0.921645			−0.460822	+	0.887492i	$0.652445\pi$
$422$		0			0
$423$		−	1.16135i		−	0.0564669i
$424$		0			0
$425$	−4.09164	+	16.2992i	−0.198474	+	0.790626i
$426$		0			0
$427$		−	5.88522i		−	0.284806i
$428$		0			0
$429$	−117.040			−5.65076
$430$		0			0
$431$	11.4086			0.549535			0.274767	−	0.961511i	$-0.411399\pi$
$431$	11.4086			0.549535			0.274767	+	0.961511i	$0.411399\pi$
$432$		0			0
$433$		−	23.2756i		−	1.11856i	−0.828980	−	0.559278i	$-0.811078\pi$
$433$		−	23.2756i		−	1.11856i	0.828980	−	0.559278i	$-0.188922\pi$
$434$		0			0
$435$	−1.79670	−	0.222071i	−0.0861451	−	0.0106475i
$436$		0			0
$437$			1.08276i			0.0517956i
$438$		0			0
$439$	−13.0581			−0.623231			−0.311615	−	0.950208i	$-0.600870\pi$
$439$	−13.0581			−0.623231			−0.311615	+	0.950208i	$0.600870\pi$
$440$		0			0
$441$	−35.7635			−1.70302
$442$		0			0
$443$			0.113844i			0.00540891i	0.999996	+	0.00270446i	$0.000860856\pi$
$443$			0.113844i			0.00540891i	−0.999996	+	0.00270446i	$0.999139\pi$
$444$		0			0
$445$	3.05966	−	24.7546i	0.145042	−	1.17348i
$446$		0			0
$447$			4.11598i			0.194679i
$448$		0			0
$449$	11.0371			0.520874			0.260437	−	0.965491i	$-0.416133\pi$
$449$	11.0371			0.520874			0.260437	+	0.965491i	$0.416133\pi$
$450$		0			0
$451$	−29.8329			−1.40478
$452$		0			0
$453$		−	10.4363i		−	0.490342i
$454$		0			0
$455$	−1.71433	+	13.8701i	−0.0803691	+	0.650239i
$456$		0			0
$457$			30.1638i			1.41100i	0.708709	+	0.705501i	$0.249278\pi$
$457$			30.1638i			1.41100i	−0.708709	+	0.705501i	$0.750722\pi$
$458$		0			0
$459$	29.3411			1.36953
$460$		0			0
$461$	−37.3908			−1.74146			−0.870732	−	0.491759i	$-0.836354\pi$
$461$	−37.3908			−1.74146			−0.870732	+	0.491759i	$0.836354\pi$
$462$		0			0
$463$			16.7365i			0.777810i	0.921278	+	0.388905i	$0.127147\pi$
$463$			16.7365i			0.777810i	−0.921278	+	0.388905i	$0.872853\pi$
$464$		0			0
$465$	−58.1875	−	7.19194i	−2.69838	−	0.333518i
$466$		0			0
$467$		−	18.2260i		−	0.843399i	−0.906736	−	0.421699i	$-0.861434\pi$
$467$		−	18.2260i		−	0.843399i	0.906736	−	0.421699i	$-0.138566\pi$
$468$		0			0
$469$	2.22437			0.102712
$470$		0			0
$471$	9.92820			0.457467
$472$		0			0
$473$		−	11.5418i		−	0.530692i
$474$		0			0
$475$	−5.25090	−	1.31815i	−0.240928	−	0.0604810i
$476$		0			0
$477$			11.4399i			0.523795i
$478$		0			0
$479$	3.31948			0.151671			0.0758355	−	0.997120i	$-0.475838\pi$
$479$	3.31948			0.151671			0.0758355	+	0.997120i	$0.475838\pi$
$480$		0			0
$481$	56.3885			2.57110
$482$		0			0
$483$		−	2.92900i		−	0.133274i
$484$		0			0
$485$	22.8283	+	2.82157i	1.03658	+	0.128121i
$486$		0			0
$487$		−	19.7412i		−	0.894558i	−0.894395	−	0.447279i	$-0.852393\pi$
$487$		−	19.7412i		−	0.894558i	0.894395	−	0.447279i	$-0.147607\pi$
$488$		0			0
$489$	25.4521			1.15098
$490$		0			0
$491$	−18.1353			−0.818432			−0.409216	−	0.912437i	$-0.634198\pi$
$491$	−18.1353			−0.818432			−0.409216	+	0.912437i	$0.634198\pi$
$492$		0			0
$493$		−	0.910968i		−	0.0410279i
$494$		0			0
$495$	−9.98615	+	80.7945i	−0.448844	+	3.63144i
$496$		0			0
$497$			10.0681i			0.451616i
$498$		0			0
$499$	14.9797			0.670582			0.335291	−	0.942115i	$-0.391165\pi$
$499$	14.9797			0.670582			0.335291	+	0.942115i	$0.391165\pi$
$500$		0			0
$501$	46.6552			2.08440
$502$		0			0
$503$		−	7.90238i		−	0.352350i	−0.984359	−	0.176175i	$-0.943628\pi$
$503$		−	7.90238i		−	0.352350i	0.984359	−	0.176175i	$-0.0563724\pi$
$504$		0			0
$505$	−4.18635	+	33.8703i	−0.186290	+	1.50721i
$506$		0			0
$507$			82.5262i			3.66512i
$508$		0			0
$509$	21.1882			0.939152			0.469576	−	0.882892i	$-0.344407\pi$
$509$	21.1882			0.939152			0.469576	+	0.882892i	$0.344407\pi$
$510$		0			0
$511$	1.35502			0.0599425
$512$		0			0
$513$			9.45245i			0.417336i
$514$		0			0
$515$	16.8834	+	2.08678i	0.743973	+	0.0919546i
$516$		0			0
$517$			1.20540i			0.0530136i
$518$		0			0
$519$	−25.1728			−1.10496
$520$		0			0
$521$	−11.1210			−0.487219			−0.243610	−	0.969873i	$-0.578332\pi$
$521$	−11.1210			−0.487219			−0.243610	+	0.969873i	$0.578332\pi$
$522$		0			0
$523$		−	16.2670i		−	0.711304i	−0.934618	−	0.355652i	$-0.884259\pi$
$523$		−	16.2670i		−	0.711304i	0.934618	−	0.355652i	$-0.115741\pi$
$524$		0			0
$525$	14.2043	+	3.56575i	0.619925	+	0.155622i
$526$		0			0
$527$		−	29.5024i		−	1.28515i
$528$		0			0
$529$	−1.00000			−0.0434783
$530$		0			0
$531$	−77.0151			−3.34217
$532$		0			0
$533$			30.9335i			1.33988i
$534$		0			0
$535$	25.0948	+	3.10170i	1.08494	+	0.134098i
$536$		0			0
$537$		−	32.2743i		−	1.39274i
$538$		0			0
$539$	37.1200			1.59887
$540$		0			0
$541$	−13.8797			−0.596733			−0.298367	−	0.954451i	$-0.596442\pi$
$541$	−13.8797			−0.596733			−0.298367	+	0.954451i	$0.596442\pi$
$542$		0			0
$543$			46.0851i			1.97770i
$544$		0			0
$545$	−0.947220	+	7.66363i	−0.0405745	+	0.328274i
$546$		0			0
$547$			5.91647i			0.252970i	0.991969	+	0.126485i	$0.0403696\pi$
$547$			5.91647i			0.252970i	−0.991969	+	0.126485i	$0.959630\pi$
$548$		0			0
$549$	−35.5467			−1.51709
$550$		0			0
$551$	0.293475			0.0125025
$552$		0			0
$553$			4.58371i			0.194919i
$554$		0			0
$555$	7.24825	−	58.6431i	0.307671	−	2.48926i
$556$		0			0
$557$			28.2857i			1.19850i	0.800560	+	0.599252i	$0.204535\pi$
$557$			28.2857i			1.19850i	−0.800560	+	0.599252i	$0.795465\pi$
$558$		0			0
$559$	−11.9676			−0.506175
$560$		0			0
$561$	−61.7148			−2.60560
$562$		0			0
$563$			31.7025i			1.33610i	0.744117	+	0.668050i	$0.232871\pi$
$563$			31.7025i			1.33610i	−0.744117	+	0.668050i	$0.767129\pi$
$564$		0			0
$565$	32.5973	+	4.02900i	1.37138	+	0.169502i
$566$		0			0
$567$		−	8.14763i		−	0.342169i
$568$		0			0
$569$	24.0702			1.00907			0.504537	−	0.863390i	$-0.331663\pi$
$569$	24.0702			1.00907			0.504537	+	0.863390i	$0.331663\pi$
$570$		0			0
$571$	−14.3935			−0.602349			−0.301175	−	0.953569i	$-0.597379\pi$
$571$	−14.3935			−0.602349			−0.301175	+	0.953569i	$0.597379\pi$
$572$		0			0
$573$		−	66.5023i		−	2.77817i
$574$		0			0
$575$	1.21740	−	4.84953i	0.0507689	−	0.202239i
$576$		0			0
$577$			5.77998i			0.240624i	0.992736	+	0.120312i	$0.0383895\pi$
$577$			5.77998i			0.240624i	−0.992736	+	0.120312i	$0.961611\pi$
$578$		0			0
$579$	48.2111			2.00358
$580$		0			0
$581$	15.4115			0.639378
$582$		0			0
$583$		−	11.8738i		−	0.491761i
$584$		0			0
$585$	83.7751	+	10.3546i	3.46368	+	0.428108i
$586$		0			0
$587$			27.7897i			1.14700i	0.819204	+	0.573502i	$0.194416\pi$
$587$			27.7897i			1.14700i	−0.819204	+	0.573502i	$0.805584\pi$
$588$		0			0
$589$	9.50441			0.391623
$590$		0			0
$591$	−41.7999			−1.71942
$592$		0			0
$593$		−	22.1872i		−	0.911118i	−0.890205	−	0.455559i	$-0.849439\pi$
$593$		−	22.1872i		−	0.911118i	0.890205	−	0.455559i	$-0.150561\pi$
$594$		0			0
$595$	−0.903960	+	7.31363i	−0.0370587	+	0.299829i
$596$		0			0
$597$			20.6683i			0.845897i
$598$		0			0
$599$	15.1380			0.618521			0.309260	−	0.950977i	$-0.399919\pi$
$599$	15.1380			0.618521			0.309260	+	0.950977i	$0.399919\pi$
$600$		0			0
$601$	20.5353			0.837652			0.418826	−	0.908067i	$-0.362442\pi$
$601$	20.5353			0.837652			0.418826	+	0.908067i	$0.362442\pi$
$602$		0			0
$603$		−	13.4352i		−	0.547123i
$604$		0			0
$605$	7.34773	−	59.4480i	0.298728	−	2.41690i
$606$		0			0
$607$			28.9906i			1.17669i	0.808610	+	0.588345i	$0.200220\pi$
$607$			28.9906i			1.17669i	−0.808610	+	0.588345i	$0.799780\pi$
$608$		0			0
$609$	−0.793883			−0.0321698
$610$		0			0
$611$	1.24987			0.0505644
$612$		0			0
$613$		−	9.03621i		−	0.364969i	−0.983209	−	0.182485i	$-0.941586\pi$
$613$		−	9.03621i		−	0.364969i	0.983209	−	0.182485i	$-0.0584139\pi$
$614$		0			0
$615$	32.1703	+	3.97623i	1.29723	+	0.160337i
$616$		0			0
$617$			4.42741i			0.178241i	0.996021	+	0.0891203i	$0.0284056\pi$
$617$			4.42741i			0.178241i	−0.996021	+	0.0891203i	$0.971594\pi$
$618$		0			0
$619$	14.8403			0.596483			0.298241	−	0.954490i	$-0.403600\pi$
$619$	14.8403			0.596483			0.298241	+	0.954490i	$0.403600\pi$
$620$		0			0
$621$	−8.72993			−0.350320
$622$		0			0
$623$		−	10.9380i		−	0.438222i
$624$		0			0
$625$	22.0359	+	11.8076i	0.881436	+	0.472304i
$626$		0			0
$627$		−	19.8819i		−	0.794005i
$628$		0			0
$629$	29.7334			1.18555
$630$		0			0
$631$	−41.8369			−1.66550			−0.832751	−	0.553648i	$-0.813235\pi$
$631$	−41.8369			−1.66550			−0.832751	+	0.553648i	$0.813235\pi$
$632$		0			0
$633$			38.5841i			1.53358i
$634$		0			0
$635$	−2.64776	−	0.327261i	−0.105073	−	0.0129870i
$636$		0			0
$637$		−	38.4894i		−	1.52501i
$638$		0			0
$639$	60.8113			2.40566
$640$		0			0
$641$	5.86478			0.231645			0.115822	−	0.993270i	$-0.463050\pi$
$641$	5.86478			0.231645			0.115822	+	0.993270i	$0.463050\pi$
$642$		0			0
$643$			0.460745i			0.0181700i	0.999959	+	0.00908500i	$0.00289188\pi$
$643$			0.460745i			0.0181700i	−0.999959	+	0.00908500i	$0.997108\pi$
$644$		0			0
$645$	−1.53833	+	12.4461i	−0.0605716	+	0.490064i
$646$		0			0
$647$			15.5690i			0.612079i	0.952019	+	0.306040i	$0.0990040\pi$
$647$			15.5690i			0.612079i	−0.952019	+	0.306040i	$0.900996\pi$
$648$		0			0
$649$	79.9362			3.13777
$650$		0			0
$651$	−25.7105			−1.00767
$652$		0			0
$653$			29.4538i			1.15262i	0.817233	+	0.576308i	$0.195507\pi$
$653$			29.4538i			1.15262i	−0.817233	+	0.576308i	$0.804493\pi$
$654$		0			0
$655$	3.21814	−	26.0369i	0.125743	−	1.01734i
$656$		0			0
$657$		−	8.18430i		−	0.319300i
$658$		0			0
$659$	22.5604			0.878829			0.439415	−	0.898284i	$-0.355186\pi$
$659$	22.5604			0.878829			0.439415	+	0.898284i	$0.355186\pi$
$660$		0			0
$661$	−39.7530			−1.54621			−0.773106	−	0.634277i	$-0.781298\pi$
$661$	−39.7530			−1.54621			−0.773106	+	0.634277i	$0.781298\pi$
$662$		0			0
$663$			63.9916i			2.48523i
$664$		0			0
$665$	−2.35614	−	0.291217i	−0.0913671	−	0.0112929i
$666$		0			0
$667$			0.271042i			0.0104948i
$668$		0			0
$669$	−79.6289			−3.07863
$670$		0			0
$671$	36.8949			1.42431
$672$		0			0
$673$		−	11.5047i		−	0.443475i	−0.975106	−	0.221737i	$-0.928827\pi$
$673$		−	11.5047i		−	0.443475i	0.975106	−	0.221737i	$-0.0711728\pi$
$674$		0			0
$675$	10.6278	−	42.3361i	0.409064	−	1.62952i
$676$		0			0
$677$			31.7703i			1.22103i	0.792005	+	0.610515i	$0.209038\pi$
$677$			31.7703i			1.22103i	−0.792005	+	0.610515i	$0.790962\pi$
$678$		0			0
$679$	10.0868			0.387098
$680$		0			0
$681$	−28.2101			−1.08101
$682$		0			0
$683$			7.22683i			0.276527i	0.990395	+	0.138264i	$0.0441521\pi$
$683$			7.22683i			0.276527i	−0.990395	+	0.138264i	$0.955848\pi$
$684$		0			0
$685$	23.2270	+	2.87084i	0.887457	+	0.109689i
$686$		0			0
$687$			42.4145i			1.61822i
$688$		0			0
$689$	−12.3118			−0.469043
$690$		0			0
$691$	−3.29788			−0.125457			−0.0627286	−	0.998031i	$-0.519980\pi$
$691$	−3.29788			−0.125457			−0.0627286	+	0.998031i	$0.519980\pi$
$692$		0			0
$693$			35.6996i			1.35611i
$694$		0			0
$695$	1.12829	−	9.12859i	0.0427984	−	0.346267i
$696$		0			0
$697$			16.3111i			0.617827i
$698$		0			0
$699$	−28.1123			−1.06331
$700$		0			0
$701$	18.7936			0.709824			0.354912	−	0.934900i	$-0.384511\pi$
$701$	18.7936			0.709824			0.354912	+	0.934900i	$0.384511\pi$
$702$		0			0
$703$			9.57884i			0.361273i
$704$		0			0
$705$	0.160660	−	1.29985i	0.00605081	−	0.0489550i
$706$		0			0
$707$			14.9658i			0.562848i
$708$		0			0
$709$	−14.1039			−0.529685			−0.264842	−	0.964292i	$-0.585320\pi$
$709$	−14.1039			−0.529685			−0.264842	+	0.964292i	$0.585320\pi$
$710$		0			0
$711$	27.6856			1.03829
$712$		0			0
$713$			8.77792i			0.328736i
$714$		0			0
$715$	−86.9527	−	10.7473i	−3.25185	−	0.401926i
$716$		0			0
$717$			46.2341i			1.72664i
$718$		0			0
$719$	−23.5631			−0.878754			−0.439377	−	0.898303i	$-0.644801\pi$
$719$	−23.5631			−0.878754			−0.439377	+	0.898303i	$0.644801\pi$
$720$		0			0
$721$	7.46005			0.277827
$722$		0			0
$723$			10.4862i			0.389985i
$724$		0			0
$725$	−1.31443	−	0.329966i	−0.0488166	−	0.0122546i
$726$		0			0
$727$			3.57021i			0.132412i	0.997806	+	0.0662059i	$0.0210894\pi$
$727$			3.57021i			0.132412i	−0.997806	+	0.0662059i	$0.978911\pi$
$728$		0			0
$729$	29.0190			1.07478
$730$		0			0
$731$	−6.31045			−0.233401
$732$		0			0
$733$		−	10.3158i		−	0.381022i	−0.981685	−	0.190511i	$-0.938985\pi$
$733$		−	10.3158i		−	0.381022i	0.981685	−	0.190511i	$-0.0610145\pi$
$734$		0			0
$735$	−40.0283	−	4.94747i	−1.47647	−	0.182490i
$736$		0			0
$737$			13.9448i			0.513663i
$738$		0			0
$739$	−36.2056			−1.33185			−0.665923	−	0.746021i	$-0.731962\pi$
$739$	−36.2056			−1.33185			−0.665923	+	0.746021i	$0.731962\pi$
$740$		0			0
$741$	−20.6153			−0.757324
$742$		0			0
$743$			3.24466i			0.119035i	0.998227	+	0.0595174i	$0.0189562\pi$
$743$			3.24466i			0.119035i	−0.998227	+	0.0595174i	$0.981044\pi$
$744$		0			0
$745$	−0.377953	+	3.05788i	−0.0138471	+	0.112032i
$746$		0			0
$747$		−	93.0856i		−	3.40582i
$748$		0			0
$749$	11.0883			0.405157
$750$		0			0
$751$	−26.9796			−0.984499			−0.492250	−	0.870454i	$-0.663825\pi$
$751$	−26.9796			−0.984499			−0.492250	+	0.870454i	$0.663825\pi$
$752$		0			0
$753$		−	70.6888i		−	2.57604i
$754$		0			0
$755$	0.958324	−	7.75347i	0.0348770	−	0.282178i
$756$		0			0
$757$		−	11.7249i		−	0.426148i	−0.977036	−	0.213074i	$-0.931652\pi$
$757$		−	11.7249i		−	0.426148i	0.977036	−	0.213074i	$-0.0683476\pi$
$758$		0			0
$759$	18.3621			0.666504
$760$		0			0
$761$	−24.4303			−0.885599			−0.442800	−	0.896621i	$-0.646015\pi$
$761$	−24.4303			−0.885599			−0.442800	+	0.896621i	$0.646015\pi$
$762$		0			0
$763$			3.38623i			0.122590i
$764$		0			0
$765$	44.1742	+	5.45991i	1.59712	+	0.197403i
$766$		0			0
$767$		−	82.8852i		−	2.99281i
$768$		0			0
$769$	−0.804588			−0.0290142			−0.0145071	−	0.999895i	$-0.504618\pi$
$769$	−0.804588			−0.0290142			−0.0145071	+	0.999895i	$0.504618\pi$
$770$		0			0
$771$	−92.0989			−3.31686
$772$		0			0
$773$		−	6.02477i		−	0.216696i	−0.994113	−	0.108348i	$-0.965444\pi$
$773$		−	6.02477i		−	0.216696i	0.994113	−	0.108348i	$-0.0345561\pi$
$774$		0			0
$775$	−42.5688	−	10.6862i	−1.52912	−	0.383860i
$776$		0			0
$777$		−	25.9118i		−	0.929582i
$778$		0			0
$779$	−5.25474			−0.188270
$780$		0			0
$781$	−63.1179			−2.25853
$782$		0			0
$783$			2.36618i			0.0845604i
$784$		0			0
$785$	7.37596	+	0.911664i	0.263259	+	0.0325387i
$786$		0			0
$787$			12.9326i			0.460996i	0.973073	+	0.230498i	$0.0740356\pi$
$787$			12.9326i			0.460996i	−0.973073	+	0.230498i	$0.925964\pi$
$788$		0			0
$789$	−61.4977			−2.18938
$790$		0			0
$791$	14.4033			0.512123
$792$		0			0
$793$		−	38.2561i		−	1.35851i
$794$		0			0
$795$	−1.58258	+	12.8041i	−0.0561282	+	0.454114i
$796$		0			0
$797$			36.3660i			1.28815i	0.764962	+	0.644075i	$0.222758\pi$
$797$			36.3660i			1.28815i	−0.764962	+	0.644075i	$0.777242\pi$
$798$		0			0
$799$	0.659052			0.0233156
$800$		0			0
$801$	−66.0654			−2.33431
$802$		0			0
$803$			8.49472i			0.299772i
$804$		0			0
$805$	0.268957	−	2.17604i	0.00947950	−	0.0766953i
$806$		0			0
$807$		−	50.8629i		−	1.79046i
$808$		0			0
$809$	−26.5840			−0.934642			−0.467321	−	0.884088i	$-0.654781\pi$
$809$	−26.5840			−0.934642			−0.467321	+	0.884088i	$0.654781\pi$
$810$		0			0
$811$	44.7712			1.57213			0.786064	−	0.618145i	$-0.212116\pi$
$811$	44.7712			1.57213			0.786064	+	0.618145i	$0.212116\pi$
$812$		0			0
$813$			40.8360i			1.43218i
$814$		0			0
$815$	18.9091	+	2.33715i	0.662357	+	0.0818669i
$816$		0			0
$817$		−	2.03296i		−	0.0711242i
$818$		0			0
$819$	37.0166			1.29346
$820$		0			0
$821$	22.3949			0.781587			0.390794	−	0.920478i	$-0.372201\pi$
$821$	22.3949			0.781587			0.390794	+	0.920478i	$0.372201\pi$
$822$		0			0
$823$			2.18913i			0.0763081i	0.999272	+	0.0381541i	$0.0121478\pi$
$823$			2.18913i			0.0763081i	−0.999272	+	0.0381541i	$0.987852\pi$
$824$		0			0
$825$	−22.3540	+	89.0478i	−0.778267	+	3.10025i
$826$		0			0
$827$		−	39.4367i		−	1.37135i	−0.727909	−	0.685674i	$-0.759508\pi$
$827$		−	39.4367i		−	1.37135i	0.727909	−	0.685674i	$-0.240492\pi$
$828$		0			0
$829$	11.0629			0.384229			0.192115	−	0.981373i	$-0.438465\pi$
$829$	11.0629			0.384229			0.192115	+	0.981373i	$0.438465\pi$
$830$		0			0
$831$	−48.5040			−1.68259
$832$		0			0
$833$		−	20.2953i		−	0.703190i
$834$		0			0
$835$	34.6615	+	4.28414i	1.19951	+	0.148259i
$836$		0			0
$837$			76.6306i			2.64874i
$838$		0			0
$839$	−37.8510			−1.30676			−0.653381	−	0.757030i	$-0.726650\pi$
$839$	−37.8510			−1.30676			−0.653381	+	0.757030i	$0.726650\pi$
$840$		0			0
$841$	−28.9265			−0.997467
$842$		0			0
$843$			1.08254i			0.0372845i
$844$		0			0
$845$	−7.57802	+	61.3111i	−0.260692	+	2.10917i
$846$		0			0
$847$		−	26.2675i		−	0.902561i
$848$		0			0
$849$	−20.3132			−0.697148
$850$		0			0
$851$	−8.84665			−0.303259
$852$		0			0
$853$		−	28.9770i		−	0.992155i	−0.868278	−	0.496078i	$-0.834773\pi$
$853$		−	28.9770i		−	0.992155i	0.868278	−	0.496078i	$-0.165227\pi$
$854$		0			0
$855$	−1.75895	+	14.2311i	−0.0601548	+	0.486692i
$856$		0			0
$857$		−	56.0462i		−	1.91450i	−0.289261	−	0.957250i	$-0.593410\pi$
$857$		−	56.0462i		−	1.91450i	0.289261	−	0.957250i	$-0.406590\pi$
$858$		0			0
$859$	43.8866			1.49739			0.748695	−	0.662914i	$-0.230681\pi$
$859$	43.8866			1.49739			0.748695	+	0.662914i	$0.230681\pi$
$860$		0			0
$861$	14.2147			0.484434
$862$		0			0
$863$			22.1955i			0.755542i	0.925899	+	0.377771i	$0.123309\pi$
$863$			22.1955i			0.755542i	−0.925899	+	0.377771i	$0.876691\pi$
$864$		0			0
$865$	−18.7016	−	2.31151i	−0.635875	−	0.0785937i
$866$		0			0
$867$		−	17.0377i		−	0.578630i
$868$		0			0
$869$	−28.7357			−0.974791
$870$		0			0
$871$	14.4592			0.489932
$872$		0			0
$873$		−	60.9245i		−	2.06198i
$874$		0			0
$875$	10.2253	+	3.95342i	0.345680	+	0.133650i
$876$		0			0
$877$		−	5.44979i		−	0.184026i	−0.995758	−	0.0920131i	$-0.970670\pi$
$877$		−	5.44979i		−	0.184026i	0.995758	−	0.0920131i	$-0.0293302\pi$
$878$		0			0
$879$	−78.3125			−2.64141
$880$		0			0
$881$	−11.7854			−0.397060			−0.198530	−	0.980095i	$-0.563617\pi$
$881$	−11.7854			−0.397060			−0.198530	+	0.980095i	$0.563617\pi$
$882$		0			0
$883$			23.7945i			0.800748i	0.916352	+	0.400374i	$0.131120\pi$
$883$			23.7945i			0.800748i	−0.916352	+	0.400374i	$0.868880\pi$
$884$		0			0
$885$	−86.1991	−	10.6542i	−2.89755	−	0.358136i
$886$		0			0
$887$			44.4300i			1.49181i	0.666050	+	0.745907i	$0.267984\pi$
$887$			44.4300i			1.49181i	−0.666050	+	0.745907i	$0.732016\pi$
$888$		0			0
$889$	−1.16993			−0.0392381
$890$		0			0
$891$	51.0782			1.71119
$892$		0			0
$893$			0.212318i			0.00710496i
$894$		0			0
$895$	2.96361	−	23.9775i	0.0990626	−	0.801481i
$896$		0			0
$897$		−	19.0396i		−	0.635712i
$898$		0			0
$899$	2.37919			0.0793504
$900$		0			0
$901$	−6.49196			−0.216279
$902$		0			0
$903$			5.49938i			0.183008i
$904$		0			0
$905$	−4.23180	+	34.2380i	−0.140670	+	1.13811i
$906$		0			0
$907$			40.7542i			1.35322i	0.736341	+	0.676611i	$0.236552\pi$
$907$			40.7542i			1.35322i	−0.736341	+	0.676611i	$0.763448\pi$
$908$		0			0
$909$	90.3934			2.99816
$910$		0			0
$911$	2.71520			0.0899587			0.0449794	−	0.998988i	$-0.485678\pi$
$911$	2.71520			0.0899587			0.0449794	+	0.998988i	$0.485678\pi$
$912$		0			0
$913$			96.6163i			3.19753i
$914$		0			0
$915$	−39.7856	−	4.91748i	−1.31527	−	0.162567i
$916$		0			0
$917$		−	11.5046i		−	0.379914i
$918$		0			0
$919$	10.7227			0.353708			0.176854	−	0.984237i	$-0.443408\pi$
$919$	10.7227			0.353708			0.176854	+	0.984237i	$0.443408\pi$
$920$		0			0
$921$	58.8890			1.94046
$922$		0			0
$923$			65.4464i			2.15419i
$924$		0			0
$925$	10.7699	−	42.9021i	0.354112	−	1.41061i
$926$		0			0
$927$		−	45.0587i		−	1.47992i
$928$		0			0
$929$	−38.8670			−1.27518			−0.637592	−	0.770374i	$-0.720070\pi$
$929$	−38.8670			−1.27518			−0.637592	+	0.770374i	$0.720070\pi$
$930$		0			0
$931$	6.53827			0.214283
$932$		0			0
$933$			45.1055i			1.47669i
$934$		0			0
$935$	−45.8498	−	5.66700i	−1.49945	−	0.185331i
$936$		0			0
$937$		−	38.6117i		−	1.26139i	−0.776032	−	0.630694i	$-0.782770\pi$
$937$		−	38.6117i		−	1.26139i	0.776032	−	0.630694i	$-0.217230\pi$
$938$		0			0
$939$	38.7761			1.26541
$940$		0			0
$941$	26.3928			0.860381			0.430191	−	0.902738i	$-0.358446\pi$
$941$	26.3928			0.860381			0.430191	+	0.902738i	$0.358446\pi$
$942$		0			0
$943$		−	4.85308i		−	0.158038i
$944$		0			0
$945$	2.34798	−	18.9967i	0.0763797	−	0.617962i
$946$		0			0
$947$		−	22.3777i		−	0.727179i	−0.931559	−	0.363589i	$-0.881551\pi$
$947$		−	22.3777i		−	0.727179i	0.931559	−	0.363589i	$-0.118449\pi$
$948$		0			0
$949$	8.80811			0.285923
$950$		0			0
$951$	−51.8265			−1.68059
$952$		0			0
$953$		−	17.4362i		−	0.564815i	−0.959295	−	0.282407i	$-0.908867\pi$
$953$		−	17.4362i		−	0.564815i	0.959295	−	0.282407i	$-0.0911330\pi$
$954$		0			0
$955$	6.10662	−	49.4065i	0.197605	−	1.59876i
$956$		0			0
$957$		−	4.97692i		−	0.160881i
$958$		0			0
$959$	10.2630			0.331409
$960$		0			0
$961$	46.0519			1.48554
$962$		0			0
$963$		−	66.9732i		−	2.15818i
$964$		0			0
$965$	35.8174	+	4.42702i	1.15300	+	0.142511i
$966$		0			0
$967$		−	45.1587i		−	1.45221i	−0.687586	−	0.726103i	$-0.741330\pi$
$967$		−	45.1587i		−	1.45221i	0.687586	−	0.726103i	$-0.258670\pi$
$968$		0			0
$969$	−10.8704			−0.349207
$970$		0			0
$971$	34.3123			1.10113			0.550567	−	0.834791i	$-0.314412\pi$
$971$	34.3123			1.10113			0.550567	+	0.834791i	$0.314412\pi$
$972$		0			0
$973$		−	4.03353i		−	0.129309i
$974$		0			0
$975$	92.3329	+	23.1787i	2.95702	+	0.742312i
$976$		0			0
$977$			1.51320i			0.0484115i	0.999707	+	0.0242058i	$0.00770569\pi$
$977$			1.51320i			0.0484115i	−0.999707	+	0.0242058i	$0.992294\pi$
$978$		0			0
$979$	68.5713			2.19155
$980$		0			0
$981$	20.4528			0.653007
$982$		0			0
$983$			15.1715i			0.483895i	0.970289	+	0.241947i	$0.0777861\pi$
$983$			15.1715i			0.483895i	−0.970289	+	0.241947i	$0.922214\pi$
$984$		0			0
$985$	−31.0544	−	3.83830i	−0.989475	−	0.122298i
$986$		0			0
$987$		−	0.574345i		−	0.0182816i
$988$		0			0
$989$	1.87756			0.0597031
$990$		0			0
$991$	39.4470			1.25307			0.626537	−	0.779392i	$-0.284472\pi$
$991$	39.4470			1.25307			0.626537	+	0.779392i	$0.284472\pi$
$992$		0			0
$993$			2.94153i			0.0933467i
$994$		0			0
$995$	−1.89788	+	15.3551i	−0.0601668	+	0.486789i
$996$		0			0
$997$		−	2.33319i		−	0.0738928i	−0.999317	−	0.0369464i	$-0.988237\pi$
$997$		−	2.33319i		−	0.0738928i	0.999317	−	0.0369464i	$-0.0117631\pi$
$998$		0			0
$999$	−77.2307			−2.44347

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1840.2.e.g.369.1		14
4.3	odd	2		920.2.e.b.369.14	yes	14
5.2	odd	4		9200.2.a.cz.1.1		7
5.3	odd	4		9200.2.a.dc.1.7		7
5.4	even	2	inner	1840.2.e.g.369.14		14
20.3	even	4		4600.2.a.bh.1.1		7
20.7	even	4		4600.2.a.bi.1.7		7
20.19	odd	2		920.2.e.b.369.1	✓	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.2.e.b.369.1	✓	14	20.19	odd	2
920.2.e.b.369.14	yes	14	4.3	odd	2
1840.2.e.g.369.1		14	1.1	even	1	trivial
1840.2.e.g.369.14		14	5.4	even	2	inner
4600.2.a.bh.1.1		7	20.3	even	4
4600.2.a.bi.1.7		7	20.7	even	4
9200.2.a.cz.1.1		7	5.2	odd	4
9200.2.a.dc.1.7		7	5.3	odd	4

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:	\( N \)	\(=\)	\( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1840.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.98707i q^{3} +(0.274289 - 2.21918i) q^{5} -0.980560i q^{7} -5.92257 q^{9} +O(q^{10})\)	\(q-2.98707i q^{3} +(0.274289 - 2.21918i) q^{5} -0.980560i q^{7} -5.92257 q^{9} +6.14721 q^{11} -6.37400i q^{13} +(-6.62885 - 0.819321i) q^{15} -3.36098i q^{17} +1.08276 q^{19} -2.92900 q^{21} +1.00000i q^{23} +(-4.84953 - 1.21740i) q^{25} +8.72993i q^{27} +0.271042 q^{29} +8.77792 q^{31} -18.3621i q^{33} +(-2.17604 - 0.268957i) q^{35} +8.84665i q^{37} -19.0396 q^{39} -4.85308 q^{41} -1.87756i q^{43} +(-1.62450 + 13.1433i) q^{45} +0.196089i q^{47} +6.03850 q^{49} -10.0395 q^{51} -1.93157i q^{53} +(1.68612 - 13.6418i) q^{55} -3.23429i q^{57} +13.0036 q^{59} +6.00189 q^{61} +5.80744i q^{63} +(-14.1451 - 1.74832i) q^{65} +2.26847i q^{67} +2.98707 q^{69} -10.2677 q^{71} +1.38188i q^{73} +(-3.63644 + 14.4859i) q^{75} -6.02771i q^{77} -4.67459 q^{79} +8.30917 q^{81} +15.7171i q^{83} +(-7.45862 - 0.921881i) q^{85} -0.809622i q^{87} +11.1548 q^{89} -6.25008 q^{91} -26.2202i q^{93} +(0.296991 - 2.40285i) q^{95} +10.2868i q^{97} -36.4073 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + 2 q^{5} - 4 q^{9}+O(q^{10}) \) 14 * q + 2 * q^5 - 4 * q^9	\( 14 q + 2 q^{5} - 4 q^{9} + 14 q^{11} + 6 q^{15} - 14 q^{19} - 12 q^{21} - 14 q^{25} + 22 q^{29} + 20 q^{31} + 2 q^{35} - 48 q^{39} - 32 q^{41} - 26 q^{45} + 34 q^{49} + 14 q^{51} + 38 q^{55} - 22 q^{59} + 10 q^{61} - 38 q^{65} + 6 q^{69} + 28 q^{71} + 24 q^{75} - 64 q^{79} - 10 q^{81} - 50 q^{85} + 48 q^{89} + 14 q^{91} + 30 q^{95} - 122 q^{99}+O(q^{100}) \) 14 * q + 2 * q^5 - 4 * q^9 + 14 * q^11 + 6 * q^15 - 14 * q^19 - 12 * q^21 - 14 * q^25 + 22 * q^29 + 20 * q^31 + 2 * q^35 - 48 * q^39 - 32 * q^41 - 26 * q^45 + 34 * q^49 + 14 * q^51 + 38 * q^55 - 22 * q^59 + 10 * q^61 - 38 * q^65 + 6 * q^69 + 28 * q^71 + 24 * q^75 - 64 * q^79 - 10 * q^81 - 50 * q^85 + 48 * q^89 + 14 * q^91 + 30 * q^95 - 122 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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