LMFDB - Embedded newform 2475.2.c.t.199.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2475,2,Mod(199,2475)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2475, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2475.199"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2475 = 3^{2} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2475.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,-16,0,0,0,0,0,0,8,0,0,-16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(14)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$19.7629745003$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.9488318464.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 16x^{6} + 82x^{4} + 136x^{2} + 9 $ x^8 + 16x^6 + 82x^4 + 136*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{7} $
Twist minimal:	no (minimal twist has level 495)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.3
Root			$-1.85206i$ of defining polynomial
Character	$\chi$	$=$	2475.199
Dual form			2475.2.c.t.199.6

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.85206i q^{2} -1.43013 q^{4} +4.90749i q^{7} -1.05543i q^{8} +1.00000 q^{11} +4.61162i q^{13} +9.08898 q^{14} -4.81499 q^{16} -3.76776i q^{17} -4.84386 q^{19} -1.85206i q^{22} -0.860262i q^{23} +8.54100 q^{26} -7.01836i q^{28} -10.3794 q^{29} +7.81499 q^{31} +6.80679i q^{32} -6.97811 q^{34} -4.67525i q^{37} +8.97113i q^{38} -2.97113 q^{41} +0.907494i q^{43} -1.43013 q^{44} -1.59326 q^{46} +13.2232i q^{47} -17.0835 q^{49} -6.59522i q^{52} +5.13974i q^{53} +5.17953 q^{56} +19.2232i q^{58} -12.5480 q^{59} -11.2232 q^{61} -14.4738i q^{62} +2.97661 q^{64} +4.26851i q^{67} +5.38838i q^{68} +5.13974 q^{71} +4.61162i q^{73} -8.65885 q^{74} +6.92735 q^{76} +4.90749i q^{77} +0.843861 q^{79} +5.50271i q^{82} -5.75135i q^{83} +1.68073 q^{86} -1.05543i q^{88} +1.40825 q^{89} -22.6315 q^{91} +1.23029i q^{92} +24.4902 q^{94} +8.54798i q^{97} +31.6397i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 16 q^{4} + 8 q^{11} - 16 q^{14} + 24 q^{16} - 8 q^{19} + 32 q^{26} - 8 q^{29} - 8 q^{34} + 8 q^{41} - 16 q^{44} - 32 q^{46} - 40 q^{49} + 96 q^{56} - 48 q^{59} + 16 q^{61} + 32 q^{71} + 24 q^{74}+ \cdots + 80 q^{94}+O(q^{100}) $ 8 * q - 16 * q^4 + 8 * q^11 - 16 * q^14 + 24 * q^16 - 8 * q^19 + 32 * q^26 - 8 * q^29 - 8 * q^34 + 8 * q^41 - 16 * q^44 - 32 * q^46 - 40 * q^49 + 96 * q^56 - 48 * q^59 + 16 * q^61 + 32 * q^71 + 24 * q^74 - 72 * q^76 - 24 * q^79 - 32 * q^86 - 32 * q^89 - 32 * q^91 + 80 * q^94

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2475\mathbb{Z}\right)^\times$.

$n$	$551$	$2026$	$2377$
$\chi(n)$	$1$	$1$	$-1$

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 $

$2$	− 1.85206i	− 1.30961i	−0.755800	−	0.654803i	$-0.772752\pi$
$2$	− 1.85206i	− 1.30961i	0.755800	−	0.654803i	$-0.227248\pi$
$3$	0	0
$3$	0	0
$4$	−1.43013	−0.715065
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.90749i	1.85486i	0.373999	+	0.927429i	$0.377986\pi$
$7$	4.90749i	1.85486i	−0.373999	+	0.927429i	$0.622014\pi$
$8$	− 1.05543i	− 0.373152i
$9$	0	0
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	4.61162i	1.27903i	0.768778	+	0.639516i	$0.220865\pi$
$13$	4.61162i	1.27903i	−0.768778	+	0.639516i	$0.779135\pi$
$14$	9.08898	2.42913
$15$	0	0
$16$	−4.81499	−1.20375
$17$	− 3.76776i	− 0.913815i	−0.889514	−	0.456907i	$-0.848957\pi$
$17$	− 3.76776i	− 0.913815i	0.889514	−	0.456907i	$-0.151043\pi$
$18$	0	0
$19$	−4.84386	−1.11126	−0.555629	−	0.831430i	$-0.687522\pi$
$19$	−4.84386	−1.11126	−0.555629	+	0.831430i	$0.687522\pi$
$20$	0	0
$21$	0	0
$22$	− 1.85206i	− 0.394861i
$23$	− 0.860262i	− 0.179377i	−0.995970	−	0.0896885i	$-0.971413\pi$
$23$	− 0.860262i	− 0.179377i	0.995970	−	0.0896885i	$-0.0285871\pi$
$24$	0	0
$25$	0	0
$26$	8.54100	1.67503
$27$	0	0
$28$	− 7.01836i	− 1.32635i
$29$	−10.3794	−1.92740	−0.963700	−	0.266986i	$-0.913972\pi$
$29$	−10.3794	−1.92740	−0.963700	+	0.266986i	$0.913972\pi$
$30$	0	0
$31$	7.81499	1.40361	0.701807	−	0.712368i	$-0.252377\pi$
$31$	7.81499	1.40361	0.701807	+	0.712368i	$0.252377\pi$
$32$	6.80679i	1.20328i
$33$	0	0
$34$	−6.97811	−1.19674
$35$	0	0
$36$	0	0
$37$	− 4.67525i	− 0.768606i	−0.923207	−	0.384303i	$-0.874442\pi$
$37$	− 4.67525i	− 0.768606i	0.923207	−	0.384303i	$-0.125558\pi$
$38$	8.97113i	1.45531i
$39$	0	0
$40$	0	0
$41$	−2.97113	−0.464012	−0.232006	−	0.972714i	$-0.574529\pi$
$41$	−2.97113	−0.464012	−0.232006	+	0.972714i	$0.574529\pi$
$42$	0	0
$43$	0.907494i	0.138391i	0.997603	+	0.0691957i	$0.0220433\pi$
$43$	0.907494i	0.138391i	−0.997603	+	0.0691957i	$0.977957\pi$
$44$	−1.43013	−0.215600
$45$	0	0
$46$	−1.59326	−0.234913
$47$	13.2232i	1.92881i	0.264436	+	0.964403i	$0.414814\pi$
$47$	13.2232i	1.92881i	−0.264436	+	0.964403i	$0.585186\pi$
$48$	0	0
$49$	−17.0835	−2.44050
$50$	0	0
$51$	0	0
$52$	− 6.59522i	− 0.914592i
$53$	5.13974i	0.705997i	0.935624	+	0.352999i	$0.114838\pi$
$53$	5.13974i	0.705997i	−0.935624	+	0.352999i	$0.885162\pi$
$54$	0	0
$55$	0	0
$56$	5.17953	0.692144
$57$	0	0
$58$	19.2232i	2.52413i
$59$	−12.5480	−1.63361	−0.816804	−	0.576915i	$-0.804256\pi$
$59$	−12.5480	−1.63361	−0.816804	+	0.576915i	$0.804256\pi$
$60$	0	0
$61$	−11.2232	−1.43699	−0.718494	−	0.695533i	$-0.755168\pi$
$61$	−11.2232	−1.43699	−0.718494	+	0.695533i	$0.755168\pi$
$62$	− 14.4738i	− 1.83818i
$63$	0	0
$64$	2.97661	0.372076
$65$	0	0
$66$	0	0
$67$	4.26851i	0.521481i	0.965409	+	0.260741i	$0.0839667\pi$
$67$	4.26851i	0.521481i	−0.965409	+	0.260741i	$0.916033\pi$
$68$	5.38838i	0.653438i
$69$	0	0
$70$	0	0
$71$	5.13974	0.609975	0.304987	−	0.952356i	$-0.401348\pi$
$71$	5.13974	0.609975	0.304987	+	0.952356i	$0.401348\pi$
$72$	0	0
$73$	4.61162i	0.539749i	0.962896	+	0.269874i	$0.0869822\pi$
$73$	4.61162i	0.539749i	−0.962896	+	0.269874i	$0.913018\pi$
$74$	−8.65885	−1.00657
$75$	0	0
$76$	6.92735	0.794622
$77$	4.90749i	0.559261i
$78$	0	0
$79$	0.843861	0.0949417	0.0474709	−	0.998873i	$-0.484884\pi$
$79$	0.843861	0.0949417	0.0474709	+	0.998873i	$0.484884\pi$
$80$	0	0
$81$	0	0
$82$	5.50271i	0.607673i
$83$	− 5.75135i	− 0.631293i	−0.948877	−	0.315647i	$-0.897779\pi$
$83$	− 5.75135i	− 0.631293i	0.948877	−	0.315647i	$-0.102221\pi$
$84$	0	0
$85$	0	0
$86$	1.68073	0.181238
$87$	0	0
$88$	− 1.05543i	− 0.112509i
$89$	1.40825	0.149274	0.0746368	−	0.997211i	$-0.476220\pi$
$89$	1.40825	0.149274	0.0746368	+	0.997211i	$0.476220\pi$
$90$	0	0
$91$	−22.6315	−2.37242
$92$	1.23029i	0.128266i
$93$	0	0
$94$	24.4902	2.52598
$95$	0	0
$96$	0	0
$97$	8.54798i	0.867916i	0.900933	+	0.433958i	$0.142883\pi$
$97$	8.54798i	0.867916i	−0.900933	+	0.433958i	$0.857117\pi$
$98$	31.6397i	3.19609i
$99$	0	0
$100$	0	0
$101$	4.56438	0.454173	0.227087	−	0.973875i	$-0.427080\pi$
$101$	4.56438	0.454173	0.227087	+	0.973875i	$0.427080\pi$
$102$	0	0
$103$	4.73300i	0.466356i	0.972434	+	0.233178i	$0.0749125\pi$
$103$	4.73300i	0.466356i	−0.972434	+	0.233178i	$0.925088\pi$
$104$	4.86725	0.477273
$105$	0	0
$106$	9.51911	0.924578
$107$	− 7.34461i	− 0.710030i	−0.934861	−	0.355015i	$-0.884476\pi$
$107$	− 7.34461i	− 0.710030i	0.934861	−	0.355015i	$-0.115524\pi$
$108$	0	0
$109$	−9.40825	−0.901146	−0.450573	−	0.892739i	$-0.648780\pi$
$109$	−9.40825	−0.901146	−0.450573	+	0.892739i	$0.648780\pi$
$110$	0	0
$111$	0	0
$112$	− 23.6295i	− 2.23278i
$113$	10.9547i	1.03053i	0.857030	+	0.515267i	$0.172307\pi$
$113$	10.9547i	1.03053i	−0.857030	+	0.515267i	$0.827693\pi$
$114$	0	0
$115$	0	0
$116$	14.8439	1.37822
$117$	0	0
$118$	23.2396i	2.13938i
$119$	18.4902	1.69500
$120$	0	0
$121$	1.00000	0.0909091
$122$	20.7861i	1.88189i
$123$	0	0
$124$	−11.1765	−1.00368
$125$	0	0
$126$	0	0
$127$	6.62802i	0.588141i	0.955784	+	0.294071i	$0.0950101\pi$
$127$	6.62802i	0.588141i	−0.955784	+	0.294071i	$0.904990\pi$
$128$	8.10071i	0.716008i
$129$	0	0
$130$	0	0
$131$	−13.2232	−1.15532	−0.577660	−	0.816278i	$-0.696034\pi$
$131$	−13.2232	−1.15532	−0.577660	+	0.816278i	$0.696034\pi$
$132$	0	0
$133$	− 23.7712i	− 2.06123i
$134$	7.90554	0.682934
$135$	0	0
$136$	−3.97661	−0.340992
$137$	6.00000i	0.512615i	0.966595	+	0.256307i	$0.0825059\pi$
$137$	6.00000i	0.512615i	−0.966595	+	0.256307i	$0.917494\pi$
$138$	0	0
$139$	12.9711	1.10020	0.550098	−	0.835100i	$-0.314590\pi$
$139$	12.9711	1.10020	0.550098	+	0.835100i	$0.314590\pi$
$140$	0	0
$141$	0	0
$142$	− 9.51911i	− 0.798826i
$143$	4.61162i	0.385643i
$144$	0	0
$145$	0	0
$146$	8.54100	0.706858
$147$	0	0
$148$	6.68622i	0.549604i
$149$	−0.0273705	−0.00224228	−0.00112114	−	0.999999i	$-0.500357\pi$
$149$	−0.0273705	−0.00224228	−0.00112114	+	0.999999i	$0.500357\pi$
$150$	0	0
$151$	4.84386	0.394188	0.197094	−	0.980385i	$-0.436850\pi$
$151$	4.84386	0.394188	0.197094	+	0.980385i	$0.436850\pi$
$152$	5.11237i	0.414668i
$153$	0	0
$154$	9.08898	0.732411
$155$	0	0
$156$	0	0
$157$	2.12727i	0.169774i	0.996391	+	0.0848872i	$0.0270530\pi$
$157$	2.12727i	0.169774i	−0.996391	+	0.0848872i	$0.972947\pi$
$158$	− 1.56288i	− 0.124336i
$159$	0	0
$160$	0	0
$161$	4.22173	0.332719
$162$	0	0
$163$	− 10.0835i	− 0.789800i	−0.918724	−	0.394900i	$-0.870779\pi$
$163$	− 10.0835i	− 0.789800i	0.918724	−	0.394900i	$-0.129221\pi$
$164$	4.24910	0.331799
$165$	0	0
$166$	−10.6519	−0.826745
$167$	− 17.7514i	− 1.37364i	−0.726827	−	0.686821i	$-0.759006\pi$
$167$	− 17.7514i	− 1.37364i	0.726827	−	0.686821i	$-0.240994\pi$
$168$	0	0
$169$	−8.26700	−0.635923
$170$	0	0
$171$	0	0
$172$	− 1.29783i	− 0.0989590i
$173$	15.7678i	1.19880i	0.800450	+	0.599400i	$0.204594\pi$
$173$	15.7678i	1.19880i	−0.800450	+	0.599400i	$0.795406\pi$
$174$	0	0
$175$	0	0
$176$	−4.81499	−0.362943
$177$	0	0
$178$	− 2.60816i	− 0.195490i
$179$	7.40825	0.553718	0.276859	−	0.960911i	$-0.410706\pi$
$179$	7.40825	0.553718	0.276859	+	0.960911i	$0.410706\pi$
$180$	0	0
$181$	7.26700	0.540152	0.270076	−	0.962839i	$-0.412951\pi$
$181$	7.26700	0.540152	0.270076	+	0.962839i	$0.412951\pi$
$182$	41.9149i	3.10694i
$183$	0	0
$184$	−0.907948	−0.0669348
$185$	0	0
$186$	0	0
$187$	− 3.76776i	− 0.275526i
$188$	− 18.9110i	− 1.37922i
$189$	0	0
$190$	0	0
$191$	2.87123	0.207755	0.103877	−	0.994590i	$-0.466875\pi$
$191$	2.87123	0.207755	0.103877	+	0.994590i	$0.466875\pi$
$192$	0	0
$193$	− 18.8911i	− 1.35981i	−0.733300	−	0.679905i	$-0.762021\pi$
$193$	− 18.8911i	− 1.35981i	0.733300	−	0.679905i	$-0.237979\pi$
$194$	15.8314	1.13663
$195$	0	0
$196$	24.4316	1.74512
$197$	− 14.0472i	− 1.00082i	−0.865787	−	0.500412i	$-0.833182\pi$
$197$	− 14.0472i	− 1.00082i	0.865787	−	0.500412i	$-0.166818\pi$
$198$	0	0
$199$	−3.40825	−0.241604	−0.120802	−	0.992677i	$-0.538547\pi$
$199$	−3.40825	−0.241604	−0.120802	+	0.992677i	$0.538547\pi$
$200$	0	0
$201$	0	0
$202$	− 8.45352i	− 0.594788i
$203$	− 50.9367i	− 3.57506i
$204$	0	0
$205$	0	0
$206$	8.76580	0.610742
$207$	0	0
$208$	− 22.2049i	− 1.53963i
$209$	−4.84386	−0.335057
$210$	0	0
$211$	−24.4738	−1.68485	−0.842424	−	0.538815i	$-0.818872\pi$
$211$	−24.4738	−1.68485	−0.842424	+	0.538815i	$0.818872\pi$
$212$	− 7.35050i	− 0.504834i
$213$	0	0
$214$	−13.6027	−0.929859
$215$	0	0
$216$	0	0
$217$	38.3520i	2.60350i
$218$	17.4246i	1.18015i
$219$	0	0
$220$	0	0
$221$	17.3754	1.16880
$222$	0	0
$223$	11.5355i	0.772475i	0.922399	+	0.386237i	$0.126225\pi$
$223$	11.5355i	0.772475i	−0.922399	+	0.386237i	$0.873775\pi$
$224$	−33.4043	−2.23192
$225$	0	0
$226$	20.2888	1.34959
$227$	1.15960i	0.0769653i	0.999259	+	0.0384827i	$0.0122524\pi$
$227$	1.15960i	0.0769653i	−0.999259	+	0.0384827i	$0.987748\pi$
$228$	0	0
$229$	24.4465	1.61547	0.807734	−	0.589547i	$-0.200694\pi$
$229$	24.4465	1.61547	0.807734	+	0.589547i	$0.200694\pi$
$230$	0	0
$231$	0	0
$232$	10.9547i	0.719213i
$233$	9.45548i	0.619449i	0.950826	+	0.309724i	$0.100237\pi$
$233$	9.45548i	0.619449i	−0.950826	+	0.309724i	$0.899763\pi$
$234$	0	0
$235$	0	0
$236$	17.9453	1.16814
$237$	0	0
$238$	− 34.2451i	− 2.21978i
$239$	−14.9438	−0.966631	−0.483316	−	0.875446i	$-0.660568\pi$
$239$	−14.9438	−0.966631	−0.483316	+	0.875446i	$0.660568\pi$
$240$	0	0
$241$	7.81499	0.503408	0.251704	−	0.967804i	$-0.419009\pi$
$241$	7.81499	0.503408	0.251704	+	0.967804i	$0.419009\pi$
$242$	− 1.85206i	− 0.119055i
$243$	0	0
$244$	16.0507	1.02754
$245$	0	0
$246$	0	0
$247$	− 22.3380i	− 1.42133i
$248$	− 8.24819i	− 0.523761i
$249$	0	0
$250$	0	0
$251$	−3.41921	−0.215819	−0.107909	−	0.994161i	$-0.534416\pi$
$251$	−3.41921	−0.215819	−0.107909	+	0.994161i	$0.534416\pi$
$252$	0	0
$253$	− 0.860262i	− 0.0540842i
$254$	12.2755	0.770233
$255$	0	0
$256$	20.9562	1.30976
$257$	− 1.53551i	− 0.0957826i	−0.998853	−	0.0478913i	$-0.984750\pi$
$257$	− 1.53551i	− 0.0957826i	0.998853	−	0.0478913i	$-0.0152501\pi$
$258$	0	0
$259$	22.9438	1.42566
$260$	0	0
$261$	0	0
$262$	24.4902i	1.51301i
$263$	− 0.190899i	− 0.0117713i	−0.999983	−	0.00588567i	$-0.998127\pi$
$263$	− 0.190899i	− 0.0117713i	0.999983	−	0.00588567i	$-0.00187348\pi$
$264$	0	0
$265$	0	0
$266$	−44.0257	−2.69939
$267$	0	0
$268$	− 6.10452i	− 0.372893i
$269$	14.4465	0.880817	0.440408	−	0.897798i	$-0.354834\pi$
$269$	14.4465	0.880817	0.440408	+	0.897798i	$0.354834\pi$
$270$	0	0
$271$	−9.97263	−0.605794	−0.302897	−	0.953023i	$-0.597954\pi$
$271$	−9.97263	−0.605794	−0.302897	+	0.953023i	$0.597954\pi$
$272$	18.1417i	1.10000i
$273$	0	0
$274$	11.1124	0.671323
$275$	0	0
$276$	0	0
$277$	12.5788i	0.755788i	0.925849	+	0.377894i	$0.123352\pi$
$277$	12.5788i	0.755788i	−0.925849	+	0.377894i	$0.876648\pi$
$278$	− 24.0233i	− 1.44082i
$279$	0	0
$280$	0	0
$281$	16.6917	0.995740	0.497870	−	0.867252i	$-0.334116\pi$
$281$	16.6917	0.995740	0.497870	+	0.867252i	$0.334116\pi$
$282$	0	0
$283$	21.5937i	1.28361i	0.766867	+	0.641806i	$0.221815\pi$
$283$	21.5937i	1.28361i	−0.766867	+	0.641806i	$0.778185\pi$
$284$	−7.35050	−0.436172
$285$	0	0
$286$	8.54100	0.505040
$287$	− 14.5808i	− 0.860677i
$288$	0	0
$289$	2.80402	0.164942
$290$	0	0
$291$	0	0
$292$	− 6.59522i	− 0.385956i
$293$	33.0855i	1.93287i	0.256906	+	0.966436i	$0.417297\pi$
$293$	33.0855i	1.93287i	−0.256906	+	0.966436i	$0.582703\pi$
$294$	0	0
$295$	0	0
$296$	−4.93441	−0.286807
$297$	0	0
$298$	0.0506919i	0.00293650i
$299$	3.96720	0.229429
$300$	0	0
$301$	−4.45352	−0.256697
$302$	− 8.97113i	− 0.516230i
$303$	0	0
$304$	23.3231	1.33767
$305$	0	0
$306$	0	0
$307$	7.72398i	0.440831i	0.975406	+	0.220416i	$0.0707413\pi$
$307$	7.72398i	0.440831i	−0.975406	+	0.220416i	$0.929259\pi$
$308$	− 7.01836i	− 0.399908i
$309$	0	0
$310$	0	0
$311$	12.1780	0.690549	0.345274	−	0.938502i	$-0.387786\pi$
$311$	12.1780	0.690549	0.345274	+	0.938502i	$0.387786\pi$
$312$	0	0
$313$	2.95473i	0.167011i	0.996507	+	0.0835055i	$0.0266116\pi$
$313$	2.95473i	0.167011i	−0.996507	+	0.0835055i	$0.973388\pi$
$314$	3.93983	0.222337
$315$	0	0
$316$	−1.20683	−0.0678896
$317$	− 27.4917i	− 1.54409i	−0.635568	−	0.772045i	$-0.719234\pi$
$317$	− 27.4917i	− 1.54409i	0.635568	−	0.772045i	$-0.280766\pi$
$318$	0	0
$319$	−10.3794	−0.581133
$320$	0	0
$321$	0	0
$322$	− 7.81890i	− 0.435730i
$323$	18.2505i	1.01548i
$324$	0	0
$325$	0	0
$326$	−18.6752	−1.03433
$327$	0	0
$328$	3.13582i	0.173147i
$329$	−64.8929	−3.57766
$330$	0	0
$331$	16.5737	0.910975	0.455487	−	0.890242i	$-0.349465\pi$
$331$	16.5737	0.910975	0.455487	+	0.890242i	$0.349465\pi$
$332$	8.22519i	0.451416i
$333$	0	0
$334$	−32.8766	−1.79893
$335$	0	0
$336$	0	0
$337$	4.64442i	0.252998i	0.991967	+	0.126499i	$0.0403740\pi$
$337$	4.64442i	0.252998i	−0.991967	+	0.126499i	$0.959626\pi$
$338$	15.3110i	0.832809i
$339$	0	0
$340$	0	0
$341$	7.81499	0.423205
$342$	0	0
$343$	− 49.4847i	− 2.67192i
$344$	0.957798	0.0516410
$345$	0	0
$346$	29.2028	1.56995
$347$	− 23.6936i	− 1.27194i	−0.771714	−	0.635970i	$-0.780600\pi$
$347$	− 23.6936i	− 1.27194i	0.771714	−	0.635970i	$-0.219400\pi$
$348$	0	0
$349$	21.7572	1.16464	0.582319	−	0.812960i	$-0.302145\pi$
$349$	21.7572	1.16464	0.582319	+	0.812960i	$0.302145\pi$
$350$	0	0
$351$	0	0
$352$	6.80679i	0.362803i
$353$	6.49024i	0.345440i	0.984971	+	0.172720i	$0.0552556\pi$
$353$	6.49024i	0.345440i	−0.984971	+	0.172720i	$0.944744\pi$
$354$	0	0
$355$	0	0
$356$	−2.01397	−0.106740
$357$	0	0
$358$	− 13.7205i	− 0.725152i
$359$	19.1655	1.01152	0.505758	−	0.862676i	$-0.331213\pi$
$359$	19.1655	1.01152	0.505758	+	0.862676i	$0.331213\pi$
$360$	0	0
$361$	4.46299	0.234894
$362$	− 13.4589i	− 0.707386i
$363$	0	0
$364$	32.3660	1.69644
$365$	0	0
$366$	0	0
$367$	− 10.1850i	− 0.531653i	−0.964021	−	0.265827i	$-0.914355\pi$
$367$	− 10.1850i	− 0.531653i	0.964021	−	0.265827i	$-0.0856449\pi$
$368$	4.14215i	0.215924i
$369$	0	0
$370$	0	0
$371$	−25.2232	−1.30952
$372$	0	0
$373$	− 19.0184i	− 0.984733i	−0.870388	−	0.492367i	$-0.836132\pi$
$373$	− 19.0184i	− 0.984733i	0.870388	−	0.492367i	$-0.163868\pi$
$374$	−6.97811	−0.360830
$375$	0	0
$376$	13.9562	0.719738
$377$	− 47.8657i	− 2.46521i
$378$	0	0
$379$	−11.0710	−0.568680	−0.284340	−	0.958723i	$-0.591774\pi$
$379$	−11.0710	−0.568680	−0.284340	+	0.958723i	$0.591774\pi$
$380$	0	0
$381$	0	0
$382$	− 5.31770i	− 0.272077i
$383$	− 12.7330i	− 0.650626i	−0.945606	−	0.325313i	$-0.894530\pi$
$383$	− 12.7330i	− 0.650626i	0.945606	−	0.325313i	$-0.105470\pi$
$384$	0	0
$385$	0	0
$386$	−34.9875	−1.78081
$387$	0	0
$388$	− 12.2247i	− 0.620617i
$389$	−32.7587	−1.66093	−0.830467	−	0.557068i	$-0.811926\pi$
$389$	−32.7587	−1.66093	−0.830467	+	0.557068i	$0.811926\pi$
$390$	0	0
$391$	−3.24126	−0.163917
$392$	18.0305i	0.910676i
$393$	0	0
$394$	−26.0163	−1.31068
$395$	0	0
$396$	0	0
$397$	− 15.0820i	− 0.756943i	−0.925613	−	0.378472i	$-0.876450\pi$
$397$	− 15.0820i	− 0.756943i	0.925613	−	0.378472i	$-0.123550\pi$
$398$	6.31228i	0.316406i
$399$	0	0
$400$	0	0
$401$	−18.7259	−0.935129	−0.467564	−	0.883959i	$-0.654868\pi$
$401$	−18.7259	−0.935129	−0.467564	+	0.883959i	$0.654868\pi$
$402$	0	0
$403$	36.0397i	1.79527i
$404$	−6.52767	−0.324764
$405$	0	0
$406$	−94.3379	−4.68191
$407$	− 4.67525i	− 0.231744i
$408$	0	0
$409$	−24.4793	−1.21042	−0.605211	−	0.796065i	$-0.706911\pi$
$409$	−24.4793	−1.21042	−0.605211	+	0.796065i	$0.706911\pi$
$410$	0	0
$411$	0	0
$412$	− 6.76880i	− 0.333475i
$413$	− 61.5791i	− 3.03011i
$414$	0	0
$415$	0	0
$416$	−31.3903	−1.53904
$417$	0	0
$418$	8.97113i	0.438792i
$419$	9.12877	0.445970	0.222985	−	0.974822i	$-0.428420\pi$
$419$	9.12877	0.445970	0.222985	+	0.974822i	$0.428420\pi$
$420$	0	0
$421$	−13.5465	−0.660215	−0.330108	−	0.943943i	$-0.607085\pi$
$421$	−13.5465	−0.660215	−0.330108	+	0.943943i	$0.607085\pi$
$422$	45.3270i	2.20649i
$423$	0	0
$424$	5.42465	0.263444
$425$	0	0
$426$	0	0
$427$	− 55.0779i	− 2.66541i
$428$	10.5038i	0.507718i
$429$	0	0
$430$	0	0
$431$	26.4465	1.27388	0.636941	−	0.770913i	$-0.280200\pi$
$431$	26.4465	1.27388	0.636941	+	0.770913i	$0.280200\pi$
$432$	0	0
$433$	16.1780i	0.777463i	0.921351	+	0.388732i	$0.127087\pi$
$433$	16.1780i	0.777463i	−0.921351	+	0.388732i	$0.872913\pi$
$434$	71.0303	3.40956
$435$	0	0
$436$	13.4550	0.644379
$437$	4.16699i	0.199334i
$438$	0	0
$439$	7.02887	0.335470	0.167735	−	0.985832i	$-0.446355\pi$
$439$	7.02887	0.335470	0.167735	+	0.985832i	$0.446355\pi$
$440$	0	0
$441$	0	0
$442$	− 32.1804i	− 1.53066i
$443$	− 24.3630i	− 1.15752i	−0.815498	−	0.578760i	$-0.803537\pi$
$443$	− 24.3630i	− 1.15752i	0.815498	−	0.578760i	$-0.196463\pi$
$444$	0	0
$445$	0	0
$446$	21.3645	1.01164
$447$	0	0
$448$	14.6077i	0.690149i
$449$	−38.3887	−1.81168	−0.905838	−	0.423625i	$-0.860758\pi$
$449$	−38.3887	−1.81168	−0.905838	+	0.423625i	$0.860758\pi$
$450$	0	0
$451$	−2.97113	−0.139905
$452$	− 15.6667i	− 0.736899i
$453$	0	0
$454$	2.14765	0.100794
$455$	0	0
$456$	0	0
$457$	21.9621i	1.02734i	0.857987	+	0.513672i	$0.171715\pi$
$457$	21.9621i	1.02734i	−0.857987	+	0.513672i	$0.828285\pi$
$458$	− 45.2764i	− 2.11562i
$459$	0	0
$460$	0	0
$461$	−27.5698	−1.28405	−0.642027	−	0.766682i	$-0.721906\pi$
$461$	−27.5698	−1.28405	−0.642027	+	0.766682i	$0.721906\pi$
$462$	0	0
$463$	24.8860i	1.15655i	0.815842	+	0.578275i	$0.196274\pi$
$463$	24.8860i	1.15655i	−0.815842	+	0.578275i	$0.803726\pi$
$464$	49.9765	2.32010
$465$	0	0
$466$	17.5121	0.811233
$467$	29.0273i	1.34322i	0.740904	+	0.671610i	$0.234397\pi$
$467$	29.0273i	1.34322i	−0.740904	+	0.671610i	$0.765603\pi$
$468$	0	0
$469$	−20.9477	−0.967274
$470$	0	0
$471$	0	0
$472$	13.2435i	0.609584i
$473$	0.907494i	0.0417266i
$474$	0	0
$475$	0	0
$476$	−26.4435	−1.21203
$477$	0	0
$478$	27.6768i	1.26591i
$479$	29.4450	1.34537	0.672687	−	0.739927i	$-0.265140\pi$
$479$	29.4450	1.34537	0.672687	+	0.739927i	$0.265140\pi$
$480$	0	0
$481$	21.5605	0.983072
$482$	− 14.4738i	− 0.659265i
$483$	0	0
$484$	−1.43013	−0.0650060
$485$	0	0
$486$	0	0
$487$	27.5285i	1.24743i	0.781650	+	0.623717i	$0.214378\pi$
$487$	27.5285i	1.24743i	−0.781650	+	0.623717i	$0.785622\pi$
$488$	11.8454i	0.536214i
$489$	0	0
$490$	0	0
$491$	−22.2795	−1.00546	−0.502729	−	0.864444i	$-0.667671\pi$
$491$	−22.2795	−1.00546	−0.502729	+	0.864444i	$0.667671\pi$
$492$	0	0
$493$	39.1069i	1.76129i
$494$	−41.3714	−1.86139
$495$	0	0
$496$	−37.6291	−1.68959
$497$	25.2232i	1.13142i
$498$	0	0
$499$	35.0382	1.56853	0.784263	−	0.620428i	$-0.213041\pi$
$499$	35.0382	1.56853	0.784263	+	0.620428i	$0.213041\pi$
$500$	0	0
$501$	0	0
$502$	6.33259i	0.282638i
$503$	− 33.9731i	− 1.51478i	−0.652960	−	0.757392i	$-0.726473\pi$
$503$	− 33.9731i	− 1.51478i	0.652960	−	0.757392i	$-0.273527\pi$
$504$	0	0
$505$	0	0
$506$	−1.59326	−0.0708289
$507$	0	0
$508$	− 9.47893i	− 0.420560i
$509$	13.3505	0.591750	0.295875	−	0.955227i	$-0.404389\pi$
$509$	13.3505	0.591750	0.295875	+	0.955227i	$0.404389\pi$
$510$	0	0
$511$	−22.6315	−1.00116
$512$	− 22.6108i	− 0.999266i
$513$	0	0
$514$	−2.84386	−0.125437
$515$	0	0
$516$	0	0
$517$	13.2232i	0.581557i
$518$	− 42.4932i	− 1.86705i
$519$	0	0
$520$	0	0
$521$	7.46599	0.327091	0.163546	−	0.986536i	$-0.447707\pi$
$521$	7.46599	0.327091	0.163546	+	0.986536i	$0.447707\pi$
$522$	0	0
$523$	15.9785i	0.698692i	0.936994	+	0.349346i	$0.113596\pi$
$523$	15.9785i	0.698692i	−0.936994	+	0.349346i	$0.886404\pi$
$524$	18.9110	0.826129
$525$	0	0
$526$	−0.353557	−0.0154158
$527$	− 29.4450i	− 1.28264i
$528$	0	0
$529$	22.2599	0.967824
$530$	0	0
$531$	0	0
$532$	33.9960i	1.47391i
$533$	− 13.7017i	− 0.593486i
$534$	0	0
$535$	0	0
$536$	4.50512	0.194592
$537$	0	0
$538$	− 26.7557i	− 1.15352i
$539$	−17.0835	−0.735838
$540$	0	0
$541$	−5.46299	−0.234872	−0.117436	−	0.993080i	$-0.537468\pi$
$541$	−5.46299	−0.234872	−0.117436	+	0.993080i	$0.537468\pi$
$542$	18.4699i	0.793351i
$543$	0	0
$544$	25.6463	1.09958
$545$	0	0
$546$	0	0
$547$	38.8895i	1.66279i	0.555679	+	0.831397i	$0.312458\pi$
$547$	38.8895i	1.66279i	−0.555679	+	0.831397i	$0.687542\pi$
$548$	− 8.58079i	− 0.366553i
$549$	0	0
$550$	0	0
$551$	50.2762	2.14184
$552$	0	0
$553$	4.14124i	0.176103i
$554$	23.2967	0.989783
$555$	0	0
$556$	−18.5504	−0.786713
$557$	6.58425i	0.278983i	0.990223	+	0.139492i	$0.0445469\pi$
$557$	6.58425i	0.278983i	−0.990223	+	0.139492i	$0.955453\pi$
$558$	0	0
$559$	−4.18501	−0.177007
$560$	0	0
$561$	0	0
$562$	− 30.9140i	− 1.30403i
$563$	− 25.0323i	− 1.05499i	−0.849559	−	0.527494i	$-0.823132\pi$
$563$	− 25.0323i	− 1.05499i	0.849559	−	0.527494i	$-0.176868\pi$
$564$	0	0
$565$	0	0
$566$	39.9929	1.68103
$567$	0	0
$568$	− 5.42465i	− 0.227613i
$569$	34.5066	1.44659	0.723297	−	0.690537i	$-0.242626\pi$
$569$	34.5066	1.44659	0.723297	+	0.690537i	$0.242626\pi$
$570$	0	0
$571$	−18.6588	−0.780848	−0.390424	−	0.920635i	$-0.627672\pi$
$571$	−18.6588	−0.780848	−0.390424	+	0.920635i	$0.627672\pi$
$572$	− 6.59522i	− 0.275760i
$573$	0	0
$574$	−27.0045	−1.12715
$575$	0	0
$576$	0	0
$577$	− 37.6697i	− 1.56821i	−0.620628	−	0.784105i	$-0.713122\pi$
$577$	− 37.6697i	− 1.56821i	0.620628	−	0.784105i	$-0.286878\pi$
$578$	− 5.19321i	− 0.216009i
$579$	0	0
$580$	0	0
$581$	28.2247	1.17096
$582$	0	0
$583$	5.13974i	0.212866i
$584$	4.86725	0.201408
$585$	0	0
$586$	61.2763	2.53130
$587$	11.0242i	0.455019i	0.973776	+	0.227510i	$0.0730583\pi$
$587$	11.0242i	0.455019i	−0.973776	+	0.227510i	$0.926942\pi$
$588$	0	0
$589$	−37.8547	−1.55978
$590$	0	0
$591$	0	0
$592$	22.5113i	0.925207i
$593$	39.4703i	1.62085i	0.585843	+	0.810425i	$0.300764\pi$
$593$	39.4703i	1.62085i	−0.585843	+	0.810425i	$0.699236\pi$
$594$	0	0
$595$	0	0
$596$	0.0391434	0.00160338
$597$	0	0
$598$	− 7.34749i	− 0.300461i
$599$	2.44646	0.0999598	0.0499799	−	0.998750i	$-0.484084\pi$
$599$	2.44646	0.0999598	0.0499799	+	0.998750i	$0.484084\pi$
$600$	0	0
$601$	−37.6697	−1.53658	−0.768290	−	0.640103i	$-0.778892\pi$
$601$	−37.6697	−1.53658	−0.768290	+	0.640103i	$0.778892\pi$
$602$	8.24819i	0.336171i
$603$	0	0
$604$	−6.92735	−0.281870
$605$	0	0
$606$	0	0
$607$	6.62802i	0.269023i	0.990912	+	0.134511i	$0.0429465\pi$
$607$	6.62802i	0.269023i	−0.990912	+	0.134511i	$0.957053\pi$
$608$	− 32.9711i	− 1.33716i
$609$	0	0
$610$	0	0
$611$	−60.9805	−2.46701
$612$	0	0
$613$	− 15.6498i	− 0.632091i	−0.948744	−	0.316045i	$-0.897645\pi$
$613$	− 15.6498i	− 0.632091i	0.948744	−	0.316045i	$-0.102355\pi$
$614$	14.3053	0.577315
$615$	0	0
$616$	5.17953	0.208689
$617$	27.7463i	1.11702i	0.829497	+	0.558511i	$0.188627\pi$
$617$	27.7463i	1.11702i	−0.829497	+	0.558511i	$0.811373\pi$
$618$	0	0
$619$	−16.5737	−0.666154	−0.333077	−	0.942900i	$-0.608087\pi$
$619$	−16.5737	−0.666154	−0.333077	+	0.942900i	$0.608087\pi$
$620$	0	0
$621$	0	0
$622$	− 22.5543i	− 0.904346i
$623$	6.91095i	0.276882i
$624$	0	0
$625$	0	0
$626$	5.47233	0.218718
$627$	0	0
$628$	− 3.04227i	− 0.121400i
$629$	−17.6152	−0.702364
$630$	0	0
$631$	−37.8547	−1.50697	−0.753486	−	0.657464i	$-0.771629\pi$
$631$	−37.8547	−1.50697	−0.753486	+	0.657464i	$0.771629\pi$
$632$	− 0.890638i	− 0.0354277i
$633$	0	0
$634$	−50.9164	−2.02215
$635$	0	0
$636$	0	0
$637$	− 78.7825i	− 3.12148i
$638$	19.2232i	0.761055i
$639$	0	0
$640$	0	0
$641$	−11.9423	−0.471691	−0.235845	−	0.971791i	$-0.575786\pi$
$641$	−11.9423	−0.471691	−0.235845	+	0.971791i	$0.575786\pi$
$642$	0	0
$643$	− 10.3380i	− 0.407692i	−0.979003	−	0.203846i	$-0.934656\pi$
$643$	− 10.3380i	− 0.407692i	0.979003	−	0.203846i	$-0.0653442\pi$
$644$	−6.03763	−0.237916
$645$	0	0
$646$	33.8010	1.32988
$647$	− 11.0125i	− 0.432945i	−0.976289	−	0.216472i	$-0.930545\pi$
$647$	− 11.0125i	− 0.432945i	0.976289	−	0.216472i	$-0.0694552\pi$
$648$	0	0
$649$	−12.5480	−0.492551
$650$	0	0
$651$	0	0
$652$	14.4207i	0.564759i
$653$	13.2810i	0.519725i	0.965646	+	0.259862i	$0.0836772\pi$
$653$	13.2810i	0.519725i	−0.965646	+	0.259862i	$0.916323\pi$
$654$	0	0
$655$	0	0
$656$	14.3059	0.558553
$657$	0	0
$658$	120.186i	4.68533i
$659$	−10.2247	−0.398299	−0.199150	−	0.979969i	$-0.563818\pi$
$659$	−10.2247	−0.398299	−0.199150	+	0.979969i	$0.563818\pi$
$660$	0	0
$661$	17.0710	0.663986	0.331993	−	0.943282i	$-0.392279\pi$
$661$	17.0710	0.663986	0.331993	+	0.943282i	$0.392279\pi$
$662$	− 30.6956i	− 1.19302i
$663$	0	0
$664$	−6.07017	−0.235568
$665$	0	0
$666$	0	0
$667$	8.92898i	0.345731i
$668$	25.3868i	0.982243i
$669$	0	0
$670$	0	0
$671$	−11.2232	−0.433268
$672$	0	0
$673$	− 27.8926i	− 1.07518i	−0.843206	−	0.537590i	$-0.819335\pi$
$673$	− 27.8926i	− 1.07518i	0.843206	−	0.537590i	$-0.180665\pi$
$674$	8.60175	0.331327
$675$	0	0
$676$	11.8229	0.454727
$677$	19.4922i	0.749146i	0.927197	+	0.374573i	$0.122211\pi$
$677$	19.4922i	0.749146i	−0.927197	+	0.374573i	$0.877789\pi$
$678$	0	0
$679$	−41.9492	−1.60986
$680$	0	0
$681$	0	0
$682$	− 14.4738i	− 0.554232i
$683$	− 26.9438i	− 1.03097i	−0.856897	−	0.515487i	$-0.827611\pi$
$683$	− 26.9438i	− 1.03097i	0.856897	−	0.515487i	$-0.172389\pi$
$684$	0	0
$685$	0	0
$686$	−91.6487	−3.49916
$687$	0	0
$688$	− 4.36957i	− 0.166588i
$689$	−23.7025	−0.902993
$690$	0	0
$691$	41.2849	1.57055	0.785276	−	0.619146i	$-0.212521\pi$
$691$	41.2849	1.57055	0.785276	+	0.619146i	$0.212521\pi$
$692$	− 22.5500i	− 0.857221i
$693$	0	0
$694$	−43.8820	−1.66574
$695$	0	0
$696$	0	0
$697$	11.1945i	0.424021i
$698$	− 40.2957i	− 1.52522i
$699$	0	0
$700$	0	0
$701$	−41.1631	−1.55471	−0.777354	−	0.629064i	$-0.783438\pi$
$701$	−41.1631	−1.55471	−0.777354	+	0.629064i	$0.783438\pi$
$702$	0	0
$703$	22.6463i	0.854120i
$704$	2.97661	0.112185
$705$	0	0
$706$	12.0203	0.452391
$707$	22.3997i	0.842427i
$708$	0	0
$709$	19.8040	0.743756	0.371878	−	0.928282i	$-0.378714\pi$
$709$	19.8040	0.743756	0.371878	+	0.928282i	$0.378714\pi$
$710$	0	0
$711$	0	0
$712$	− 1.48631i	− 0.0557017i
$713$	− 6.72294i	− 0.251776i
$714$	0	0
$715$	0	0
$716$	−10.5948	−0.395945
$717$	0	0
$718$	− 35.4957i	− 1.32469i
$719$	−42.6682	−1.59126	−0.795628	−	0.605786i	$-0.792859\pi$
$719$	−42.6682	−1.59126	−0.795628	+	0.605786i	$0.792859\pi$
$720$	0	0
$721$	−23.2271	−0.865024
$722$	− 8.26572i	− 0.307618i
$723$	0	0
$724$	−10.3928	−0.386244
$725$	0	0
$726$	0	0
$727$	− 39.0054i	− 1.44663i	−0.690518	−	0.723315i	$-0.742617\pi$
$727$	− 39.0054i	− 1.44663i	0.690518	−	0.723315i	$-0.257383\pi$
$728$	23.8860i	0.885274i
$729$	0	0
$730$	0	0
$731$	3.41921	0.126464
$732$	0	0
$733$	12.2744i	0.453365i	0.973969	+	0.226683i	$0.0727880\pi$
$733$	12.2744i	0.453365i	−0.973969	+	0.226683i	$0.927212\pi$
$734$	−18.8633	−0.696256
$735$	0	0
$736$	5.85562	0.215841
$737$	4.26851i	0.157232i
$738$	0	0
$739$	−34.0343	−1.25197	−0.625986	−	0.779834i	$-0.715303\pi$
$739$	−34.0343	−1.25197	−0.625986	+	0.779834i	$0.715303\pi$
$740$	0	0
$741$	0	0
$742$	46.7150i	1.71496i
$743$	16.2854i	0.597452i	0.954339	+	0.298726i	$0.0965617\pi$
$743$	16.2854i	0.597452i	−0.954339	+	0.298726i	$0.903438\pi$
$744$	0	0
$745$	0	0
$746$	−35.2232	−1.28961
$747$	0	0
$748$	5.38838i	0.197019i
$749$	36.0436	1.31701
$750$	0	0
$751$	14.0577	0.512974	0.256487	−	0.966548i	$-0.417435\pi$
$751$	14.0577	0.512974	0.256487	+	0.966548i	$0.417435\pi$
$752$	− 63.6697i	− 2.32179i
$753$	0	0
$754$	−88.6502	−3.22845
$755$	0	0
$756$	0	0
$757$	38.2286i	1.38944i	0.719278	+	0.694722i	$0.244473\pi$
$757$	38.2286i	1.38944i	−0.719278	+	0.694722i	$0.755527\pi$
$758$	20.5042i	0.744746i
$759$	0	0
$760$	0	0
$761$	9.52616	0.345323	0.172662	−	0.984981i	$-0.444763\pi$
$761$	9.52616	0.345323	0.172662	+	0.984981i	$0.444763\pi$
$762$	0	0
$763$	− 46.1709i	− 1.67150i
$764$	−4.10624	−0.148558
$765$	0	0
$766$	−23.5823	−0.852063
$767$	− 57.8665i	− 2.08944i
$768$	0	0
$769$	−2.24276	−0.0808760	−0.0404380	−	0.999182i	$-0.512875\pi$
$769$	−2.24276	−0.0808760	−0.0404380	+	0.999182i	$0.512875\pi$
$770$	0	0
$771$	0	0
$772$	27.0167i	0.972354i
$773$	− 17.7025i	− 0.636715i	−0.947971	−	0.318357i	$-0.896869\pi$
$773$	− 17.7025i	− 0.636715i	0.947971	−	0.318357i	$-0.103131\pi$
$774$	0	0
$775$	0	0
$776$	9.02182	0.323864
$777$	0	0
$778$	60.6712i	2.17517i
$779$	14.3917	0.515637
$780$	0	0
$781$	5.13974	0.183914
$782$	6.00301i	0.214667i
$783$	0	0
$784$	82.2568	2.93774
$785$	0	0
$786$	0	0
$787$	− 17.4445i	− 0.621830i	−0.950438	−	0.310915i	$-0.899365\pi$
$787$	− 17.4445i	− 0.621830i	0.950438	−	0.310915i	$-0.100635\pi$
$788$	20.0894i	0.715655i
$789$	0	0
$790$	0	0
$791$	−53.7602	−1.91149
$792$	0	0
$793$	− 51.7572i	− 1.83795i
$794$	−27.9328	−0.991297
$795$	0	0
$796$	4.87424	0.172763
$797$	− 11.2093i	− 0.397052i	−0.980096	−	0.198526i	$-0.936385\pi$
$797$	− 11.2093i	− 0.397052i	0.980096	−	0.198526i	$-0.0636155\pi$
$798$	0	0
$799$	49.8219	1.76257
$800$	0	0
$801$	0	0
$802$	34.6816i	1.22465i
$803$	4.61162i	0.162740i
$804$	0	0
$805$	0	0
$806$	66.7478	2.35109
$807$	0	0
$808$	− 4.81740i	− 0.169476i
$809$	43.1381	1.51666	0.758328	−	0.651874i	$-0.226017\pi$
$809$	43.1381	1.51666	0.758328	+	0.651874i	$0.226017\pi$
$810$	0	0
$811$	11.0289	0.387276	0.193638	−	0.981073i	$-0.437971\pi$
$811$	11.0289	0.387276	0.193638	+	0.981073i	$0.437971\pi$
$812$	72.8462i	2.55640i
$813$	0	0
$814$	−8.65885	−0.303492
$815$	0	0
$816$	0	0
$817$	− 4.39577i	− 0.153789i
$818$	45.3371i	1.58517i
$819$	0	0
$820$	0	0
$821$	−14.9164	−0.520585	−0.260293	−	0.965530i	$-0.583819\pi$
$821$	−14.9164	−0.520585	−0.260293	+	0.965530i	$0.583819\pi$
$822$	0	0
$823$	41.4589i	1.44517i	0.691283	+	0.722584i	$0.257046\pi$
$823$	41.4589i	1.44517i	−0.691283	+	0.722584i	$0.742954\pi$
$824$	4.99536	0.174022
$825$	0	0
$826$	−114.048	−3.96825
$827$	− 19.4171i	− 0.675200i	−0.941290	−	0.337600i	$-0.890385\pi$
$827$	− 19.4171i	− 0.675200i	0.941290	−	0.337600i	$-0.109615\pi$
$828$	0	0
$829$	17.8290	0.619225	0.309613	−	0.950863i	$-0.399801\pi$
$829$	17.8290	0.619225	0.309613	+	0.950863i	$0.399801\pi$
$830$	0	0
$831$	0	0
$832$	13.7270i	0.475898i
$833$	64.3664i	2.23016i
$834$	0	0
$835$	0	0
$836$	6.92735	0.239588
$837$	0	0
$838$	− 16.9070i	− 0.584044i
$839$	12.2935	0.424417	0.212209	−	0.977224i	$-0.431934\pi$
$839$	12.2935	0.424417	0.212209	+	0.977224i	$0.431934\pi$
$840$	0	0
$841$	78.7314	2.71487
$842$	25.0889i	0.864621i
$843$	0	0
$844$	35.0008	1.20478
$845$	0	0
$846$	0	0
$847$	4.90749i	0.168623i
$848$	− 24.7478i	− 0.849842i
$849$	0	0
$850$	0	0
$851$	−4.02194	−0.137870
$852$	0	0
$853$	18.0776i	0.618966i	0.950905	+	0.309483i	$0.100156\pi$
$853$	18.0776i	0.618966i	−0.950905	+	0.309483i	$0.899844\pi$
$854$	−102.008	−3.49063
$855$	0	0
$856$	−7.75174	−0.264949
$857$	55.3649i	1.89123i	0.325288	+	0.945615i	$0.394539\pi$
$857$	55.3649i	1.89123i	−0.325288	+	0.945615i	$0.605461\pi$
$858$	0	0
$859$	0.943756	0.0322005	0.0161003	−	0.999870i	$-0.494875\pi$
$859$	0.943756	0.0322005	0.0161003	+	0.999870i	$0.494875\pi$
$860$	0	0
$861$	0	0
$862$	− 48.9805i	− 1.66828i
$863$	− 23.6370i	− 0.804614i	−0.915505	−	0.402307i	$-0.868208\pi$
$863$	− 23.6370i	− 0.804614i	0.915505	−	0.402307i	$-0.131792\pi$
$864$	0	0
$865$	0	0
$866$	29.9626	1.01817
$867$	0	0
$868$	− 54.8484i	− 1.86168i
$869$	0.843861	0.0286260
$870$	0	0
$871$	−19.6847	−0.666991
$872$	9.92977i	0.336264i
$873$	0	0
$874$	7.71752	0.261049
$875$	0	0
$876$	0	0
$877$	− 16.6841i	− 0.563383i	−0.959505	−	0.281692i	$-0.909104\pi$
$877$	− 16.6841i	− 0.563383i	0.959505	−	0.281692i	$-0.0908955\pi$
$878$	− 13.0179i	− 0.439333i
$879$	0	0
$880$	0	0
$881$	16.2217	0.546524	0.273262	−	0.961940i	$-0.411897\pi$
$881$	16.2217	0.546524	0.273262	+	0.961940i	$0.411897\pi$
$882$	0	0
$883$	− 35.0710i	− 1.18023i	−0.807318	−	0.590117i	$-0.799082\pi$
$883$	− 35.0710i	− 1.18023i	0.807318	−	0.590117i	$-0.200918\pi$
$884$	−24.8492	−0.835768
$885$	0	0
$886$	−45.1217	−1.51589
$887$	− 16.1581i	− 0.542536i	−0.962504	−	0.271268i	$-0.912557\pi$
$887$	− 16.1581i	− 0.542536i	0.962504	−	0.271268i	$-0.0874429\pi$
$888$	0	0
$889$	−32.5270	−1.09092
$890$	0	0
$891$	0	0
$892$	− 16.4973i	− 0.552370i
$893$	− 64.0515i	− 2.14340i
$894$	0	0
$895$	0	0
$896$	−39.7542	−1.32809
$897$	0	0
$898$	71.0983i	2.37258i
$899$	−81.1147	−2.70533
$900$	0	0
$901$	19.3653	0.645151
$902$	5.50271i	0.183220i
$903$	0	0
$904$	11.5620	0.384545
$905$	0	0
$906$	0	0
$907$	41.9820i	1.39399i	0.717077	+	0.696994i	$0.245480\pi$
$907$	41.9820i	1.39399i	−0.717077	+	0.696994i	$0.754520\pi$
$908$	− 1.65838i	− 0.0550352i
$909$	0	0
$910$	0	0
$911$	57.2302	1.89612	0.948060	−	0.318092i	$-0.103042\pi$
$911$	57.2302	1.89612	0.948060	+	0.318092i	$0.103042\pi$
$912$	0	0
$913$	− 5.75135i	− 0.190342i
$914$	40.6752	1.34542
$915$	0	0
$916$	−34.9616	−1.15517
$917$	− 64.8929i	− 2.14295i
$918$	0	0
$919$	0.346569	0.0114323	0.00571613	−	0.999984i	$-0.498180\pi$
$919$	0.346569	0.0114323	0.00571613	+	0.999984i	$0.498180\pi$
$920$	0	0
$921$	0	0
$922$	51.0610i	1.68160i
$923$	23.7025i	0.780177i
$924$	0	0
$925$	0	0
$926$	46.0904	1.51462
$927$	0	0
$928$	− 70.6502i	− 2.31921i
$929$	−7.66278	−0.251408	−0.125704	−	0.992068i	$-0.540119\pi$
$929$	−7.66278	−0.251408	−0.125704	+	0.992068i	$0.540119\pi$
$930$	0	0
$931$	82.7501	2.71202
$932$	− 13.5226i	− 0.442947i
$933$	0	0
$934$	53.7602	1.75909
$935$	0	0
$936$	0	0
$937$	34.0894i	1.11365i	0.830629	+	0.556826i	$0.187981\pi$
$937$	34.0894i	1.11365i	−0.830629	+	0.556826i	$0.812019\pi$
$938$	38.7964i	1.26675i
$939$	0	0
$940$	0	0
$941$	28.1944	0.919110	0.459555	−	0.888149i	$-0.348009\pi$
$941$	28.1944	0.919110	0.459555	+	0.888149i	$0.348009\pi$
$942$	0	0
$943$	2.55595i	0.0832331i
$944$	60.4184	1.96645
$945$	0	0
$946$	1.68073	0.0546454
$947$	− 29.7602i	− 0.967078i	−0.875323	−	0.483539i	$-0.839351\pi$
$947$	− 29.7602i	− 0.967078i	0.875323	−	0.483539i	$-0.160649\pi$
$948$	0	0
$949$	−21.2670	−0.690356
$950$	0	0
$951$	0	0
$952$	− 19.5152i	− 0.632491i
$953$	− 20.8309i	− 0.674780i	−0.941365	−	0.337390i	$-0.890456\pi$
$953$	− 20.8309i	− 0.674780i	0.941365	−	0.337390i	$-0.109544\pi$
$954$	0	0
$955$	0	0
$956$	21.3715	0.691205
$957$	0	0
$958$	− 54.5339i	− 1.76191i
$959$	−29.4450	−0.950827
$960$	0	0
$961$	30.0740	0.970130
$962$	− 39.9313i	− 1.28744i
$963$	0	0
$964$	−11.1765	−0.359969
$965$	0	0
$966$	0	0
$967$	− 30.7225i	− 0.987968i	−0.869471	−	0.493984i	$-0.835540\pi$
$967$	− 30.7225i	− 0.987968i	0.869471	−	0.493984i	$-0.164460\pi$
$968$	− 1.05543i	− 0.0339229i
$969$	0	0
$970$	0	0
$971$	−21.9095	−0.703108	−0.351554	−	0.936168i	$-0.614347\pi$
$971$	−21.9095	−0.703108	−0.351554	+	0.936168i	$0.614347\pi$
$972$	0	0
$973$	63.6557i	2.04071i
$974$	50.9844	1.63365
$975$	0	0
$976$	54.0397	1.72977
$977$	14.1889i	0.453944i	0.973901	+	0.226972i	$0.0728826\pi$
$977$	14.1889i	0.453944i	−0.973901	+	0.226972i	$0.927117\pi$
$978$	0	0
$979$	1.40825	0.0450077
$980$	0	0
$981$	0	0
$982$	41.2630i	1.31675i
$983$	− 5.52454i	− 0.176206i	−0.996111	−	0.0881028i	$-0.971920\pi$
$983$	− 5.52454i	− 0.176206i	0.996111	−	0.0881028i	$-0.0280804\pi$
$984$	0	0
$985$	0	0
$986$	72.4284	2.30659
$987$	0	0
$988$	31.9463i	1.01635i
$989$	0.780682	0.0248243
$990$	0	0
$991$	7.93048	0.251920	0.125960	−	0.992035i	$-0.459799\pi$
$991$	7.93048	0.251920	0.125960	+	0.992035i	$0.459799\pi$
$992$	53.1950i	1.68894i
$993$	0	0
$994$	46.7150	1.48171
$995$	0	0
$996$	0	0
$997$	− 52.0596i	− 1.64874i	−0.566049	−	0.824372i	$-0.691529\pi$
$997$	− 52.0596i	− 1.64874i	0.566049	−	0.824372i	$-0.308471\pi$
$998$	− 64.8929i	− 2.05415i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2475.2.c.t.199.3		8
3.2	odd	2		2475.2.c.s.199.6		8
5.2	odd	4		2475.2.a.bj.1.3		4
5.3	odd	4		495.2.a.f.1.2	✓	4
5.4	even	2	inner	2475.2.c.t.199.6		8
15.2	even	4		2475.2.a.bf.1.2		4
15.8	even	4		495.2.a.g.1.3	yes	4
15.14	odd	2		2475.2.c.s.199.3		8
20.3	even	4		7920.2.a.cm.1.1		4
55.43	even	4		5445.2.a.bs.1.3		4
60.23	odd	4		7920.2.a.cn.1.1		4
165.98	odd	4		5445.2.a.bh.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
495.2.a.f.1.2	✓	4	5.3	odd	4
495.2.a.g.1.3	yes	4	15.8	even	4
2475.2.a.bf.1.2		4	15.2	even	4
2475.2.a.bj.1.3		4	5.2	odd	4
2475.2.c.s.199.3		8	15.14	odd	2
2475.2.c.s.199.6		8	3.2	odd	2
2475.2.c.t.199.3		8	1.1	even	1	trivial
2475.2.c.t.199.6		8	5.4	even	2	inner
5445.2.a.bh.1.2		4	165.98	odd	4
5445.2.a.bs.1.3		4	55.43	even	4
7920.2.a.cm.1.1		4	20.3	even	4
7920.2.a.cn.1.1		4	60.23	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2475.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.85206i q^{2} -1.43013 q^{4} +4.90749i q^{7} -1.05543i q^{8} +1.00000 q^{11} +4.61162i q^{13} +9.08898 q^{14} -4.81499 q^{16} -3.76776i q^{17} -4.84386 q^{19} -1.85206i q^{22} -0.860262i q^{23} +8.54100 q^{26} -7.01836i q^{28} -10.3794 q^{29} +7.81499 q^{31} +6.80679i q^{32} -6.97811 q^{34} -4.67525i q^{37} +8.97113i q^{38} -2.97113 q^{41} +0.907494i q^{43} -1.43013 q^{44} -1.59326 q^{46} +13.2232i q^{47} -17.0835 q^{49} -6.59522i q^{52} +5.13974i q^{53} +5.17953 q^{56} +19.2232i q^{58} -12.5480 q^{59} -11.2232 q^{61} -14.4738i q^{62} +2.97661 q^{64} +4.26851i q^{67} +5.38838i q^{68} +5.13974 q^{71} +4.61162i q^{73} -8.65885 q^{74} +6.92735 q^{76} +4.90749i q^{77} +0.843861 q^{79} +5.50271i q^{82} -5.75135i q^{83} +1.68073 q^{86} -1.05543i q^{88} +1.40825 q^{89} -22.6315 q^{91} +1.23029i q^{92} +24.4902 q^{94} +8.54798i q^{97} +31.6397i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 16 q^{4} + 8 q^{11} - 16 q^{14} + 24 q^{16} - 8 q^{19} + 32 q^{26} - 8 q^{29} - 8 q^{34} + 8 q^{41} - 16 q^{44} - 32 q^{46} - 40 q^{49} + 96 q^{56} - 48 q^{59} + 16 q^{61} + 32 q^{71} + 24 q^{74}+ \cdots + 80 q^{94}+O(q^{100}) \) 8 * q - 16 * q^4 + 8 * q^11 - 16 * q^14 + 24 * q^16 - 8 * q^19 + 32 * q^26 - 8 * q^29 - 8 * q^34 + 8 * q^41 - 16 * q^44 - 32 * q^46 - 40 * q^49 + 96 * q^56 - 48 * q^59 + 16 * q^61 + 32 * q^71 + 24 * q^74 - 72 * q^76 - 24 * q^79 - 32 * q^86 - 32 * q^89 - 32 * q^91 + 80 * q^94

Introduction

L-functions

Modular forms

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Database

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