LMFDB - Embedded newform 1075.2.b.b.474.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1075,2,Mod(474,1075)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1075, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1075.474");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1075 = 5^{2} \cdot 43 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1075.b (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:	$8.58391821729$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 43)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			474.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	1075.474
Dual form			1075.2.b.b.474.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000i q^{2} +2.00000i q^{3} -2.00000 q^{4} +4.00000 q^{6} -1.00000 q^{9} +O(q^{10})$	\(q-2.00000i q^{2} +2.00000i q^{3} -2.00000 q^{4} +4.00000 q^{6} -1.00000 q^{9} +3.00000 q^{11} -4.00000i q^{12} +5.00000i q^{13} -4.00000 q^{16} -3.00000i q^{17} +2.00000i q^{18} +2.00000 q^{19} -6.00000i q^{22} +1.00000i q^{23} +10.0000 q^{26} +4.00000i q^{27} +6.00000 q^{29} -1.00000 q^{31} +8.00000i q^{32} +6.00000i q^{33} -6.00000 q^{34} +2.00000 q^{36} -4.00000i q^{38} -10.0000 q^{39} +5.00000 q^{41} +1.00000i q^{43} -6.00000 q^{44} +2.00000 q^{46} +4.00000i q^{47} -8.00000i q^{48} +7.00000 q^{49} +6.00000 q^{51} -10.0000i q^{52} +5.00000i q^{53} +8.00000 q^{54} +4.00000i q^{57} -12.0000i q^{58} +12.0000 q^{59} +2.00000 q^{61} +2.00000i q^{62} +8.00000 q^{64} +12.0000 q^{66} -3.00000i q^{67} +6.00000i q^{68} -2.00000 q^{69} +2.00000 q^{71} -2.00000i q^{73} -4.00000 q^{76} +20.0000i q^{78} +8.00000 q^{79} -11.0000 q^{81} -10.0000i q^{82} -15.0000i q^{83} +2.00000 q^{86} +12.0000i q^{87} +4.00000 q^{89} -2.00000i q^{92} -2.00000i q^{93} +8.00000 q^{94} -16.0000 q^{96} +7.00000i q^{97} -14.0000i q^{98} -3.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{4} + 8 q^{6} - 2 q^{9}+O(q^{10}) $ 2 * q - 4 * q^4 + 8 * q^6 - 2 * q^9	$ 2 q - 4 q^{4} + 8 q^{6} - 2 q^{9} + 6 q^{11} - 8 q^{16} + 4 q^{19} + 20 q^{26} + 12 q^{29} - 2 q^{31} - 12 q^{34} + 4 q^{36} - 20 q^{39} + 10 q^{41} - 12 q^{44} + 4 q^{46} + 14 q^{49} + 12 q^{51} + 16 q^{54} + 24 q^{59} + 4 q^{61} + 16 q^{64} + 24 q^{66} - 4 q^{69} + 4 q^{71} - 8 q^{76} + 16 q^{79} - 22 q^{81} + 4 q^{86} + 8 q^{89} + 16 q^{94} - 32 q^{96} - 6 q^{99}+O(q^{100}) $ 2 * q - 4 * q^4 + 8 * q^6 - 2 * q^9 + 6 * q^11 - 8 * q^16 + 4 * q^19 + 20 * q^26 + 12 * q^29 - 2 * q^31 - 12 * q^34 + 4 * q^36 - 20 * q^39 + 10 * q^41 - 12 * q^44 + 4 * q^46 + 14 * q^49 + 12 * q^51 + 16 * q^54 + 24 * q^59 + 4 * q^61 + 16 * q^64 + 24 * q^66 - 4 * q^69 + 4 * q^71 - 8 * q^76 + 16 * q^79 - 22 * q^81 + 4 * q^86 + 8 * q^89 + 16 * q^94 - 32 * q^96 - 6 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1075\mathbb{Z}\right)^\times$.

$n$	$302$	$476$
$\chi(n)$	$-1$	$1$

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 $

$2$	− 2.00000i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$2$	− 2.00000i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$3$	2.00000i	1.15470i	0.816497	+	0.577350i	$0.195913\pi$
$3$	2.00000i	1.15470i	−0.816497	+	0.577350i	$0.804087\pi$
$4$	−2.00000	−1.00000
$5$	0	0
$5$	0	0
$6$	4.00000	1.63299
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	3.00000	0.904534	0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000	0.904534	0.452267	+	0.891883i	$0.350615\pi$
$12$	− 4.00000i	− 1.15470i
$13$	5.00000i	1.38675i	0.720577	+	0.693375i	$0.243877\pi$
$13$	5.00000i	1.38675i	−0.720577	+	0.693375i	$0.756123\pi$
$14$	0	0
$15$	0	0
$16$	−4.00000	−1.00000
$17$	− 3.00000i	− 0.727607i	−0.931476	−	0.363803i	$-0.881478\pi$
$17$	− 3.00000i	− 0.727607i	0.931476	−	0.363803i	$-0.118522\pi$
$18$	2.00000i	0.471405i
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	0	0
$21$	0	0
$22$	− 6.00000i	− 1.27920i
$23$	1.00000i	0.208514i	0.994550	+	0.104257i	$0.0332465\pi$
$23$	1.00000i	0.208514i	−0.994550	+	0.104257i	$0.966753\pi$
$24$	0	0
$25$	0	0
$26$	10.0000	1.96116
$27$	4.00000i	0.769800i
$28$	0	0
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	−1.00000	−0.179605	−0.0898027	−	0.995960i	$-0.528624\pi$
$31$	−1.00000	−0.179605	−0.0898027	+	0.995960i	$0.528624\pi$
$32$	8.00000i	1.41421i
$33$	6.00000i	1.04447i
$34$	−6.00000	−1.02899
$35$	0	0
$36$	2.00000	0.333333
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	− 4.00000i	− 0.648886i
$39$	−10.0000	−1.60128
$40$	0	0
$41$	5.00000	0.780869	0.390434	−	0.920631i	$-0.372325\pi$
$41$	5.00000	0.780869	0.390434	+	0.920631i	$0.372325\pi$
$42$	0	0
$43$	1.00000i	0.152499i
$43$	1.00000i	0.152499i
$44$	−6.00000	−0.904534
$45$	0	0
$46$	2.00000	0.294884
$47$	4.00000i	0.583460i	0.956501	+	0.291730i	$0.0942309\pi$
$47$	4.00000i	0.583460i	−0.956501	+	0.291730i	$0.905769\pi$
$48$	− 8.00000i	− 1.15470i
$49$	7.00000	1.00000
$50$	0	0
$51$	6.00000	0.840168
$52$	− 10.0000i	− 1.38675i
$53$	5.00000i	0.686803i	0.939189	+	0.343401i	$0.111579\pi$
$53$	5.00000i	0.686803i	−0.939189	+	0.343401i	$0.888421\pi$
$54$	8.00000	1.08866
$55$	0	0
$56$	0	0
$57$	4.00000i	0.529813i
$58$	− 12.0000i	− 1.57568i
$59$	12.0000	1.56227	0.781133	−	0.624364i	$-0.214642\pi$
$59$	12.0000	1.56227	0.781133	+	0.624364i	$0.214642\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	2.00000i	0.254000i
$63$	0	0
$64$	8.00000	1.00000
$65$	0	0
$66$	12.0000	1.47710
$67$	− 3.00000i	− 0.366508i	−0.983066	−	0.183254i	$-0.941337\pi$
$67$	− 3.00000i	− 0.366508i	0.983066	−	0.183254i	$-0.0586631\pi$
$68$	6.00000i	0.727607i
$69$	−2.00000	−0.240772
$70$	0	0
$71$	2.00000	0.237356	0.118678	−	0.992933i	$-0.462134\pi$
$71$	2.00000	0.237356	0.118678	+	0.992933i	$0.462134\pi$
$72$	0	0
$73$	− 2.00000i	− 0.234082i	−0.993127	−	0.117041i	$-0.962659\pi$
$73$	− 2.00000i	− 0.234082i	0.993127	−	0.117041i	$-0.0373409\pi$
$74$	0	0
$75$	0	0
$76$	−4.00000	−0.458831
$77$	0	0
$78$	20.0000i	2.26455i
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	− 10.0000i	− 1.10432i
$83$	− 15.0000i	− 1.64646i	−0.567705	−	0.823232i	$-0.692169\pi$
$83$	− 15.0000i	− 1.64646i	0.567705	−	0.823232i	$-0.307831\pi$
$84$	0	0
$85$	0	0
$86$	2.00000	0.215666
$87$	12.0000i	1.28654i
$88$	0	0
$89$	4.00000	0.423999	0.212000	−	0.977270i	$-0.432002\pi$
$89$	4.00000	0.423999	0.212000	+	0.977270i	$0.432002\pi$
$90$	0	0
$91$	0	0
$92$	− 2.00000i	− 0.208514i
$93$	− 2.00000i	− 0.207390i
$94$	8.00000	0.825137
$95$	0	0
$96$	−16.0000	−1.63299
$97$	7.00000i	0.710742i	0.934725	+	0.355371i	$0.115646\pi$
$97$	7.00000i	0.710742i	−0.934725	+	0.355371i	$0.884354\pi$
$98$	− 14.0000i	− 1.41421i
$99$	−3.00000	−0.301511
$100$	0	0
$101$	−9.00000	−0.895533	−0.447767	−	0.894150i	$-0.647781\pi$
$101$	−9.00000	−0.895533	−0.447767	+	0.894150i	$0.647781\pi$
$102$	− 12.0000i	− 1.18818i
$103$	− 1.00000i	− 0.0985329i	−0.998786	−	0.0492665i	$-0.984312\pi$
$103$	− 1.00000i	− 0.0985329i	0.998786	−	0.0492665i	$-0.0156884\pi$
$104$	0	0
$105$	0	0
$106$	10.0000	0.971286
$107$	− 12.0000i	− 1.16008i	−0.814587	−	0.580042i	$-0.803036\pi$
$107$	− 12.0000i	− 1.16008i	0.814587	−	0.580042i	$-0.196964\pi$
$108$	− 8.00000i	− 0.769800i
$109$	−7.00000	−0.670478	−0.335239	−	0.942133i	$-0.608817\pi$
$109$	−7.00000	−0.670478	−0.335239	+	0.942133i	$0.608817\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	20.0000i	1.88144i	0.339182	+	0.940721i	$0.389850\pi$
$113$	20.0000i	1.88144i	−0.339182	+	0.940721i	$0.610150\pi$
$114$	8.00000	0.749269
$115$	0	0
$116$	−12.0000	−1.11417
$117$	− 5.00000i	− 0.462250i
$118$	− 24.0000i	− 2.20938i
$119$	0	0
$120$	0	0
$121$	−2.00000	−0.181818
$122$	− 4.00000i	− 0.362143i
$123$	10.0000i	0.901670i
$124$	2.00000	0.179605
$125$	0	0
$126$	0	0
$127$	1.00000i	0.0887357i	0.999015	+	0.0443678i	$0.0141274\pi$
$127$	1.00000i	0.0887357i	−0.999015	+	0.0443678i	$0.985873\pi$
$128$	0	0
$129$	−2.00000	−0.176090
$130$	0	0
$131$	8.00000	0.698963	0.349482	−	0.936943i	$-0.386358\pi$
$131$	8.00000	0.698963	0.349482	+	0.936943i	$0.386358\pi$
$132$	− 12.0000i	− 1.04447i
$133$	0	0
$134$	−6.00000	−0.518321
$135$	0	0
$136$	0	0
$137$	6.00000i	0.512615i	0.966595	+	0.256307i	$0.0825059\pi$
$137$	6.00000i	0.512615i	−0.966595	+	0.256307i	$0.917494\pi$
$138$	4.00000i	0.340503i
$139$	−19.0000	−1.61156	−0.805779	−	0.592216i	$-0.798253\pi$
$139$	−19.0000	−1.61156	−0.805779	+	0.592216i	$0.798253\pi$
$140$	0	0
$141$	−8.00000	−0.673722
$142$	− 4.00000i	− 0.335673i
$143$	15.0000i	1.25436i
$144$	4.00000	0.333333
$145$	0	0
$146$	−4.00000	−0.331042
$147$	14.0000i	1.15470i
$148$	0	0
$149$	−12.0000	−0.983078	−0.491539	−	0.870855i	$-0.663566\pi$
$149$	−12.0000	−0.983078	−0.491539	+	0.870855i	$0.663566\pi$
$150$	0	0
$151$	−20.0000	−1.62758	−0.813788	−	0.581161i	$-0.802599\pi$
$151$	−20.0000	−1.62758	−0.813788	+	0.581161i	$0.802599\pi$
$152$	0	0
$153$	3.00000i	0.242536i
$154$	0	0
$155$	0	0
$156$	20.0000	1.60128
$157$	− 10.0000i	− 0.798087i	−0.916932	−	0.399043i	$-0.869342\pi$
$157$	− 10.0000i	− 0.798087i	0.916932	−	0.399043i	$-0.130658\pi$
$158$	− 16.0000i	− 1.27289i
$159$	−10.0000	−0.793052
$160$	0	0
$161$	0	0
$162$	22.0000i	1.72848i
$163$	− 14.0000i	− 1.09656i	−0.836293	−	0.548282i	$-0.815282\pi$
$163$	− 14.0000i	− 1.09656i	0.836293	−	0.548282i	$-0.184718\pi$
$164$	−10.0000	−0.780869
$165$	0	0
$166$	−30.0000	−2.32845
$167$	− 9.00000i	− 0.696441i	−0.937413	−	0.348220i	$-0.886786\pi$
$167$	− 9.00000i	− 0.696441i	0.937413	−	0.348220i	$-0.113214\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	0	0
$171$	−2.00000	−0.152944
$172$	− 2.00000i	− 0.152499i
$173$	− 6.00000i	− 0.456172i	−0.973641	−	0.228086i	$-0.926753\pi$
$173$	− 6.00000i	− 0.456172i	0.973641	−	0.228086i	$-0.0732467\pi$
$174$	24.0000	1.81944
$175$	0	0
$176$	−12.0000	−0.904534
$177$	24.0000i	1.80395i
$178$	− 8.00000i	− 0.599625i
$179$	−20.0000	−1.49487	−0.747435	−	0.664335i	$-0.768715\pi$
$179$	−20.0000	−1.49487	−0.747435	+	0.664335i	$0.768715\pi$
$180$	0	0
$181$	10.0000	0.743294	0.371647	−	0.928374i	$-0.378793\pi$
$181$	10.0000	0.743294	0.371647	+	0.928374i	$0.378793\pi$
$182$	0	0
$183$	4.00000i	0.295689i
$184$	0	0
$185$	0	0
$186$	−4.00000	−0.293294
$187$	− 9.00000i	− 0.658145i
$188$	− 8.00000i	− 0.583460i
$189$	0	0
$190$	0	0
$191$	−16.0000	−1.15772	−0.578860	−	0.815427i	$-0.696502\pi$
$191$	−16.0000	−1.15772	−0.578860	+	0.815427i	$0.696502\pi$
$192$	16.0000i	1.15470i
$193$	− 3.00000i	− 0.215945i	−0.994154	−	0.107972i	$-0.965564\pi$
$193$	− 3.00000i	− 0.215945i	0.994154	−	0.107972i	$-0.0344358\pi$
$194$	14.0000	1.00514
$195$	0	0
$196$	−14.0000	−1.00000
$197$	2.00000i	0.142494i	0.997459	+	0.0712470i	$0.0226979\pi$
$197$	2.00000i	0.142494i	−0.997459	+	0.0712470i	$0.977302\pi$
$198$	6.00000i	0.426401i
$199$	−14.0000	−0.992434	−0.496217	−	0.868199i	$-0.665278\pi$
$199$	−14.0000	−0.992434	−0.496217	+	0.868199i	$0.665278\pi$
$200$	0	0
$201$	6.00000	0.423207
$202$	18.0000i	1.26648i
$203$	0	0
$204$	−12.0000	−0.840168
$205$	0	0
$206$	−2.00000	−0.139347
$207$	− 1.00000i	− 0.0695048i
$208$	− 20.0000i	− 1.38675i
$209$	6.00000	0.415029
$210$	0	0
$211$	2.00000	0.137686	0.0688428	−	0.997628i	$-0.478069\pi$
$211$	2.00000	0.137686	0.0688428	+	0.997628i	$0.478069\pi$
$212$	− 10.0000i	− 0.686803i
$213$	4.00000i	0.274075i
$214$	−24.0000	−1.64061
$215$	0	0
$216$	0	0
$217$	0	0
$218$	14.0000i	0.948200i
$219$	4.00000	0.270295
$220$	0	0
$221$	15.0000	1.00901
$222$	0	0
$223$	28.0000i	1.87502i	0.347960	+	0.937509i	$0.386874\pi$
$223$	28.0000i	1.87502i	−0.347960	+	0.937509i	$0.613126\pi$
$224$	0	0
$225$	0	0
$226$	40.0000	2.66076
$227$	− 4.00000i	− 0.265489i	−0.991150	−	0.132745i	$-0.957621\pi$
$227$	− 4.00000i	− 0.265489i	0.991150	−	0.132745i	$-0.0423790\pi$
$228$	− 8.00000i	− 0.529813i
$229$	15.0000	0.991228	0.495614	−	0.868543i	$-0.334943\pi$
$229$	15.0000	0.991228	0.495614	+	0.868543i	$0.334943\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 6.00000i	− 0.393073i	−0.980497	−	0.196537i	$-0.937031\pi$
$233$	− 6.00000i	− 0.393073i	0.980497	−	0.196537i	$-0.0629694\pi$
$234$	−10.0000	−0.653720
$235$	0	0
$236$	−24.0000	−1.56227
$237$	16.0000i	1.03931i
$238$	0	0
$239$	−16.0000	−1.03495	−0.517477	−	0.855697i	$-0.673129\pi$
$239$	−16.0000	−1.03495	−0.517477	+	0.855697i	$0.673129\pi$
$240$	0	0
$241$	−12.0000	−0.772988	−0.386494	−	0.922292i	$-0.626314\pi$
$241$	−12.0000	−0.772988	−0.386494	+	0.922292i	$0.626314\pi$
$242$	4.00000i	0.257130i
$243$	− 10.0000i	− 0.641500i
$244$	−4.00000	−0.256074
$245$	0	0
$246$	20.0000	1.27515
$247$	10.0000i	0.636285i
$248$	0	0
$249$	30.0000	1.90117
$250$	0	0
$251$	−23.0000	−1.45175	−0.725874	−	0.687828i	$-0.758564\pi$
$251$	−23.0000	−1.45175	−0.725874	+	0.687828i	$0.758564\pi$
$252$	0	0
$253$	3.00000i	0.188608i
$254$	2.00000	0.125491
$255$	0	0
$256$	16.0000	1.00000
$257$	− 24.0000i	− 1.49708i	−0.663090	−	0.748539i	$-0.730755\pi$
$257$	− 24.0000i	− 1.49708i	0.663090	−	0.748539i	$-0.269245\pi$
$258$	4.00000i	0.249029i
$259$	0	0
$260$	0	0
$261$	−6.00000	−0.371391
$262$	− 16.0000i	− 0.988483i
$263$	18.0000i	1.10993i	0.831875	+	0.554964i	$0.187268\pi$
$263$	18.0000i	1.10993i	−0.831875	+	0.554964i	$0.812732\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	8.00000i	0.489592i
$268$	6.00000i	0.366508i
$269$	25.0000	1.52428	0.762138	−	0.647414i	$-0.224150\pi$
$269$	25.0000	1.52428	0.762138	+	0.647414i	$0.224150\pi$
$270$	0	0
$271$	23.0000	1.39715	0.698575	−	0.715537i	$-0.253818\pi$
$271$	23.0000	1.39715	0.698575	+	0.715537i	$0.253818\pi$
$272$	12.0000i	0.727607i
$273$	0	0
$274$	12.0000	0.724947
$275$	0	0
$276$	4.00000	0.240772
$277$	− 32.0000i	− 1.92269i	−0.275340	−	0.961347i	$-0.588791\pi$
$277$	− 32.0000i	− 1.92269i	0.275340	−	0.961347i	$-0.411209\pi$
$278$	38.0000i	2.27909i
$279$	1.00000	0.0598684
$280$	0	0
$281$	19.0000	1.13344	0.566722	−	0.823909i	$-0.308211\pi$
$281$	19.0000	1.13344	0.566722	+	0.823909i	$0.308211\pi$
$282$	16.0000i	0.952786i
$283$	− 21.0000i	− 1.24832i	−0.781296	−	0.624160i	$-0.785441\pi$
$283$	− 21.0000i	− 1.24832i	0.781296	−	0.624160i	$-0.214559\pi$
$284$	−4.00000	−0.237356
$285$	0	0
$286$	30.0000	1.77394
$287$	0	0
$288$	− 8.00000i	− 0.471405i
$289$	8.00000	0.470588
$290$	0	0
$291$	−14.0000	−0.820695
$292$	4.00000i	0.234082i
$293$	26.0000i	1.51894i	0.650545	+	0.759468i	$0.274541\pi$
$293$	26.0000i	1.51894i	−0.650545	+	0.759468i	$0.725459\pi$
$294$	28.0000	1.63299
$295$	0	0
$296$	0	0
$297$	12.0000i	0.696311i
$298$	24.0000i	1.39028i
$299$	−5.00000	−0.289157
$300$	0	0
$301$	0	0
$302$	40.0000i	2.30174i
$303$	− 18.0000i	− 1.03407i
$304$	−8.00000	−0.458831
$305$	0	0
$306$	6.00000	0.342997
$307$	− 7.00000i	− 0.399511i	−0.979846	−	0.199756i	$-0.935985\pi$
$307$	− 7.00000i	− 0.399511i	0.979846	−	0.199756i	$-0.0640148\pi$
$308$	0	0
$309$	2.00000	0.113776
$310$	0	0
$311$	15.0000	0.850572	0.425286	−	0.905059i	$-0.360174\pi$
$311$	15.0000	0.850572	0.425286	+	0.905059i	$0.360174\pi$
$312$	0	0
$313$	− 22.0000i	− 1.24351i	−0.783210	−	0.621757i	$-0.786419\pi$
$313$	− 22.0000i	− 1.24351i	0.783210	−	0.621757i	$-0.213581\pi$
$314$	−20.0000	−1.12867
$315$	0	0
$316$	−16.0000	−0.900070
$317$	9.00000i	0.505490i	0.967533	+	0.252745i	$0.0813334\pi$
$317$	9.00000i	0.505490i	−0.967533	+	0.252745i	$0.918667\pi$
$318$	20.0000i	1.12154i
$319$	18.0000	1.00781
$320$	0	0
$321$	24.0000	1.33955
$322$	0	0
$323$	− 6.00000i	− 0.333849i
$324$	22.0000	1.22222
$325$	0	0
$326$	−28.0000	−1.55078
$327$	− 14.0000i	− 0.774202i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−26.0000	−1.42909	−0.714545	−	0.699590i	$-0.753366\pi$
$331$	−26.0000	−1.42909	−0.714545	+	0.699590i	$0.753366\pi$
$332$	30.0000i	1.64646i
$333$	0	0
$334$	−18.0000	−0.984916
$335$	0	0
$336$	0	0
$337$	− 3.00000i	− 0.163420i	−0.996656	−	0.0817102i	$-0.973962\pi$
$337$	− 3.00000i	− 0.163420i	0.996656	−	0.0817102i	$-0.0260382\pi$
$338$	24.0000i	1.30543i
$339$	−40.0000	−2.17250
$340$	0	0
$341$	−3.00000	−0.162459
$342$	4.00000i	0.216295i
$343$	0	0
$344$	0	0
$345$	0	0
$346$	−12.0000	−0.645124
$347$	28.0000i	1.50312i	0.659665	+	0.751559i	$0.270698\pi$
$347$	28.0000i	1.50312i	−0.659665	+	0.751559i	$0.729302\pi$
$348$	− 24.0000i	− 1.28654i
$349$	−14.0000	−0.749403	−0.374701	−	0.927146i	$-0.622255\pi$
$349$	−14.0000	−0.749403	−0.374701	+	0.927146i	$0.622255\pi$
$350$	0	0
$351$	−20.0000	−1.06752
$352$	24.0000i	1.27920i
$353$	31.0000i	1.64996i	0.565159	+	0.824982i	$0.308815\pi$
$353$	31.0000i	1.64996i	−0.565159	+	0.824982i	$0.691185\pi$
$354$	48.0000	2.55117
$355$	0	0
$356$	−8.00000	−0.423999
$357$	0	0
$358$	40.0000i	2.11407i
$359$	−19.0000	−1.00278	−0.501391	−	0.865221i	$-0.667178\pi$
$359$	−19.0000	−1.00278	−0.501391	+	0.865221i	$0.667178\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	− 20.0000i	− 1.05118i
$363$	− 4.00000i	− 0.209946i
$364$	0	0
$365$	0	0
$366$	8.00000	0.418167
$367$	− 32.0000i	− 1.67039i	−0.549957	−	0.835193i	$-0.685356\pi$
$367$	− 32.0000i	− 1.67039i	0.549957	−	0.835193i	$-0.314644\pi$
$368$	− 4.00000i	− 0.208514i
$369$	−5.00000	−0.260290
$370$	0	0
$371$	0	0
$372$	4.00000i	0.207390i
$373$	− 32.0000i	− 1.65690i	−0.560065	−	0.828449i	$-0.689224\pi$
$373$	− 32.0000i	− 1.65690i	0.560065	−	0.828449i	$-0.310776\pi$
$374$	−18.0000	−0.930758
$375$	0	0
$376$	0	0
$377$	30.0000i	1.54508i
$378$	0	0
$379$	−11.0000	−0.565032	−0.282516	−	0.959263i	$-0.591169\pi$
$379$	−11.0000	−0.565032	−0.282516	+	0.959263i	$0.591169\pi$
$380$	0	0
$381$	−2.00000	−0.102463
$382$	32.0000i	1.63726i
$383$	− 32.0000i	− 1.63512i	−0.575841	−	0.817562i	$-0.695325\pi$
$383$	− 32.0000i	− 1.63512i	0.575841	−	0.817562i	$-0.304675\pi$
$384$	0	0
$385$	0	0
$386$	−6.00000	−0.305392
$387$	− 1.00000i	− 0.0508329i
$388$	− 14.0000i	− 0.710742i
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	0	0
$391$	3.00000	0.151717
$392$	0	0
$393$	16.0000i	0.807093i
$394$	4.00000	0.201517
$395$	0	0
$396$	6.00000	0.301511
$397$	− 6.00000i	− 0.301131i	−0.988600	−	0.150566i	$-0.951890\pi$
$397$	− 6.00000i	− 0.301131i	0.988600	−	0.150566i	$-0.0481095\pi$
$398$	28.0000i	1.40351i
$399$	0	0
$400$	0	0
$401$	5.00000	0.249688	0.124844	−	0.992176i	$-0.460157\pi$
$401$	5.00000	0.249688	0.124844	+	0.992176i	$0.460157\pi$
$402$	− 12.0000i	− 0.598506i
$403$	− 5.00000i	− 0.249068i
$404$	18.0000	0.895533
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	24.0000	1.18672	0.593362	−	0.804936i	$-0.297800\pi$
$409$	24.0000	1.18672	0.593362	+	0.804936i	$0.297800\pi$
$410$	0	0
$411$	−12.0000	−0.591916
$412$	2.00000i	0.0985329i
$413$	0	0
$414$	−2.00000	−0.0982946
$415$	0	0
$416$	−40.0000	−1.96116
$417$	− 38.0000i	− 1.86087i
$418$	− 12.0000i	− 0.586939i
$419$	28.0000	1.36789	0.683945	−	0.729534i	$-0.260263\pi$
$419$	28.0000	1.36789	0.683945	+	0.729534i	$0.260263\pi$
$420$	0	0
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	− 4.00000i	− 0.194717i
$423$	− 4.00000i	− 0.194487i
$424$	0	0
$425$	0	0
$426$	8.00000	0.387601
$427$	0	0
$428$	24.0000i	1.16008i
$429$	−30.0000	−1.44841
$430$	0	0
$431$	−21.0000	−1.01153	−0.505767	−	0.862670i	$-0.668791\pi$
$431$	−21.0000	−1.01153	−0.505767	+	0.862670i	$0.668791\pi$
$432$	− 16.0000i	− 0.769800i
$433$	12.0000i	0.576683i	0.957528	+	0.288342i	$0.0931039\pi$
$433$	12.0000i	0.576683i	−0.957528	+	0.288342i	$0.906896\pi$
$434$	0	0
$435$	0	0
$436$	14.0000	0.670478
$437$	2.00000i	0.0956730i
$438$	− 8.00000i	− 0.382255i
$439$	−17.0000	−0.811366	−0.405683	−	0.914014i	$-0.632966\pi$
$439$	−17.0000	−0.811366	−0.405683	+	0.914014i	$0.632966\pi$
$440$	0	0
$441$	−7.00000	−0.333333
$442$	− 30.0000i	− 1.42695i
$443$	4.00000i	0.190046i	0.995475	+	0.0950229i	$0.0302924\pi$
$443$	4.00000i	0.190046i	−0.995475	+	0.0950229i	$0.969708\pi$
$444$	0	0
$445$	0	0
$446$	56.0000	2.65168
$447$	− 24.0000i	− 1.13516i
$448$	0	0
$449$	−30.0000	−1.41579	−0.707894	−	0.706319i	$-0.750354\pi$
$449$	−30.0000	−1.41579	−0.707894	+	0.706319i	$0.750354\pi$
$450$	0	0
$451$	15.0000	0.706322
$452$	− 40.0000i	− 1.88144i
$453$	− 40.0000i	− 1.87936i
$454$	−8.00000	−0.375459
$455$	0	0
$456$	0	0
$457$	− 18.0000i	− 0.842004i	−0.907060	−	0.421002i	$-0.861678\pi$
$457$	− 18.0000i	− 0.842004i	0.907060	−	0.421002i	$-0.138322\pi$
$458$	− 30.0000i	− 1.40181i
$459$	12.0000	0.560112
$460$	0	0
$461$	30.0000	1.39724	0.698620	−	0.715493i	$-0.253798\pi$
$461$	30.0000	1.39724	0.698620	+	0.715493i	$0.253798\pi$
$462$	0	0
$463$	− 4.00000i	− 0.185896i	−0.995671	−	0.0929479i	$-0.970371\pi$
$463$	− 4.00000i	− 0.185896i	0.995671	−	0.0929479i	$-0.0296290\pi$
$464$	−24.0000	−1.11417
$465$	0	0
$466$	−12.0000	−0.555889
$467$	6.00000i	0.277647i	0.990317	+	0.138823i	$0.0443321\pi$
$467$	6.00000i	0.277647i	−0.990317	+	0.138823i	$0.955668\pi$
$468$	10.0000i	0.462250i
$469$	0	0
$470$	0	0
$471$	20.0000	0.921551
$472$	0	0
$473$	3.00000i	0.137940i
$474$	32.0000	1.46981
$475$	0	0
$476$	0	0
$477$	− 5.00000i	− 0.228934i
$478$	32.0000i	1.46365i
$479$	−21.0000	−0.959514	−0.479757	−	0.877401i	$-0.659275\pi$
$479$	−21.0000	−0.959514	−0.479757	+	0.877401i	$0.659275\pi$
$480$	0	0
$481$	0	0
$482$	24.0000i	1.09317i
$483$	0	0
$484$	4.00000	0.181818
$485$	0	0
$486$	−20.0000	−0.907218
$487$	36.0000i	1.63132i	0.578535	+	0.815658i	$0.303625\pi$
$487$	36.0000i	1.63132i	−0.578535	+	0.815658i	$0.696375\pi$
$488$	0	0
$489$	28.0000	1.26620
$490$	0	0
$491$	−6.00000	−0.270776	−0.135388	−	0.990793i	$-0.543228\pi$
$491$	−6.00000	−0.270776	−0.135388	+	0.990793i	$0.543228\pi$
$492$	− 20.0000i	− 0.901670i
$493$	− 18.0000i	− 0.810679i
$494$	20.0000	0.899843
$495$	0	0
$496$	4.00000	0.179605
$497$	0	0
$498$	− 60.0000i	− 2.68866i
$499$	8.00000	0.358129	0.179065	−	0.983837i	$-0.442693\pi$
$499$	8.00000	0.358129	0.179065	+	0.983837i	$0.442693\pi$
$500$	0	0
$501$	18.0000	0.804181
$502$	46.0000i	2.05308i
$503$	− 6.00000i	− 0.267527i	−0.991013	−	0.133763i	$-0.957294\pi$
$503$	− 6.00000i	− 0.267527i	0.991013	−	0.133763i	$-0.0427062\pi$
$504$	0	0
$505$	0	0
$506$	6.00000	0.266733
$507$	− 24.0000i	− 1.06588i
$508$	− 2.00000i	− 0.0887357i
$509$	15.0000	0.664863	0.332432	−	0.943127i	$-0.392131\pi$
$509$	15.0000	0.664863	0.332432	+	0.943127i	$0.392131\pi$
$510$	0	0
$511$	0	0
$512$	− 32.0000i	− 1.41421i
$513$	8.00000i	0.353209i
$514$	−48.0000	−2.11719
$515$	0	0
$516$	4.00000	0.176090
$517$	12.0000i	0.527759i
$518$	0	0
$519$	12.0000	0.526742
$520$	0	0
$521$	14.0000	0.613351	0.306676	−	0.951814i	$-0.400783\pi$
$521$	14.0000	0.613351	0.306676	+	0.951814i	$0.400783\pi$
$522$	12.0000i	0.525226i
$523$	− 12.0000i	− 0.524723i	−0.964970	−	0.262362i	$-0.915499\pi$
$523$	− 12.0000i	− 0.524723i	0.964970	−	0.262362i	$-0.0845013\pi$
$524$	−16.0000	−0.698963
$525$	0	0
$526$	36.0000	1.56967
$527$	3.00000i	0.130682i
$528$	− 24.0000i	− 1.04447i
$529$	22.0000	0.956522
$530$	0	0
$531$	−12.0000	−0.520756
$532$	0	0
$533$	25.0000i	1.08287i
$534$	16.0000	0.692388
$535$	0	0
$536$	0	0
$537$	− 40.0000i	− 1.72613i
$538$	− 50.0000i	− 2.15565i
$539$	21.0000	0.904534
$540$	0	0
$541$	1.00000	0.0429934	0.0214967	−	0.999769i	$-0.493157\pi$
$541$	1.00000	0.0429934	0.0214967	+	0.999769i	$0.493157\pi$
$542$	− 46.0000i	− 1.97587i
$543$	20.0000i	0.858282i
$544$	24.0000	1.02899
$545$	0	0
$546$	0	0
$547$	− 29.0000i	− 1.23995i	−0.784621	−	0.619975i	$-0.787143\pi$
$547$	− 29.0000i	− 1.23995i	0.784621	−	0.619975i	$-0.212857\pi$
$548$	− 12.0000i	− 0.512615i
$549$	−2.00000	−0.0853579
$550$	0	0
$551$	12.0000	0.511217
$552$	0	0
$553$	0	0
$554$	−64.0000	−2.71910
$555$	0	0
$556$	38.0000	1.61156
$557$	− 3.00000i	− 0.127114i	−0.997978	−	0.0635570i	$-0.979756\pi$
$557$	− 3.00000i	− 0.127114i	0.997978	−	0.0635570i	$-0.0202445\pi$
$558$	− 2.00000i	− 0.0846668i
$559$	−5.00000	−0.211477
$560$	0	0
$561$	18.0000	0.759961
$562$	− 38.0000i	− 1.60293i
$563$	− 37.0000i	− 1.55936i	−0.626176	−	0.779682i	$-0.715381\pi$
$563$	− 37.0000i	− 1.55936i	0.626176	−	0.779682i	$-0.284619\pi$
$564$	16.0000	0.673722
$565$	0	0
$566$	−42.0000	−1.76539
$567$	0	0
$568$	0	0
$569$	−7.00000	−0.293455	−0.146728	−	0.989177i	$-0.546874\pi$
$569$	−7.00000	−0.293455	−0.146728	+	0.989177i	$0.546874\pi$
$570$	0	0
$571$	−14.0000	−0.585882	−0.292941	−	0.956131i	$-0.594634\pi$
$571$	−14.0000	−0.585882	−0.292941	+	0.956131i	$0.594634\pi$
$572$	− 30.0000i	− 1.25436i
$573$	− 32.0000i	− 1.33682i
$574$	0	0
$575$	0	0
$576$	−8.00000	−0.333333
$577$	− 20.0000i	− 0.832611i	−0.909225	−	0.416305i	$-0.863325\pi$
$577$	− 20.0000i	− 0.832611i	0.909225	−	0.416305i	$-0.136675\pi$
$578$	− 16.0000i	− 0.665512i
$579$	6.00000	0.249351
$580$	0	0
$581$	0	0
$582$	28.0000i	1.16064i
$583$	15.0000i	0.621237i
$584$	0	0
$585$	0	0
$586$	52.0000	2.14810
$587$	2.00000i	0.0825488i	0.999148	+	0.0412744i	$0.0131418\pi$
$587$	2.00000i	0.0825488i	−0.999148	+	0.0412744i	$0.986858\pi$
$588$	− 28.0000i	− 1.15470i
$589$	−2.00000	−0.0824086
$590$	0	0
$591$	−4.00000	−0.164538
$592$	0	0
$593$	16.0000i	0.657041i	0.944497	+	0.328521i	$0.106550\pi$
$593$	16.0000i	0.657041i	−0.944497	+	0.328521i	$0.893450\pi$
$594$	24.0000	0.984732
$595$	0	0
$596$	24.0000	0.983078
$597$	− 28.0000i	− 1.14596i
$598$	10.0000i	0.408930i
$599$	1.00000	0.0408589	0.0204294	−	0.999791i	$-0.493497\pi$
$599$	1.00000	0.0408589	0.0204294	+	0.999791i	$0.493497\pi$
$600$	0	0
$601$	−4.00000	−0.163163	−0.0815817	−	0.996667i	$-0.525997\pi$
$601$	−4.00000	−0.163163	−0.0815817	+	0.996667i	$0.525997\pi$
$602$	0	0
$603$	3.00000i	0.122169i
$604$	40.0000	1.62758
$605$	0	0
$606$	−36.0000	−1.46240
$607$	− 4.00000i	− 0.162355i	−0.996700	−	0.0811775i	$-0.974132\pi$
$607$	− 4.00000i	− 0.162355i	0.996700	−	0.0811775i	$-0.0258681\pi$
$608$	16.0000i	0.648886i
$609$	0	0
$610$	0	0
$611$	−20.0000	−0.809113
$612$	− 6.00000i	− 0.242536i
$613$	18.0000i	0.727013i	0.931592	+	0.363507i	$0.118421\pi$
$613$	18.0000i	0.727013i	−0.931592	+	0.363507i	$0.881579\pi$
$614$	−14.0000	−0.564994
$615$	0	0
$616$	0	0
$617$	− 21.0000i	− 0.845428i	−0.906263	−	0.422714i	$-0.861077\pi$
$617$	− 21.0000i	− 0.845428i	0.906263	−	0.422714i	$-0.138923\pi$
$618$	− 4.00000i	− 0.160904i
$619$	−36.0000	−1.44696	−0.723481	−	0.690344i	$-0.757459\pi$
$619$	−36.0000	−1.44696	−0.723481	+	0.690344i	$0.757459\pi$
$620$	0	0
$621$	−4.00000	−0.160514
$622$	− 30.0000i	− 1.20289i
$623$	0	0
$624$	40.0000	1.60128
$625$	0	0
$626$	−44.0000	−1.75859
$627$	12.0000i	0.479234i
$628$	20.0000i	0.798087i
$629$	0	0
$630$	0	0
$631$	−6.00000	−0.238856	−0.119428	−	0.992843i	$-0.538106\pi$
$631$	−6.00000	−0.238856	−0.119428	+	0.992843i	$0.538106\pi$
$632$	0	0
$633$	4.00000i	0.158986i
$634$	18.0000	0.714871
$635$	0	0
$636$	20.0000	0.793052
$637$	35.0000i	1.38675i
$638$	− 36.0000i	− 1.42525i
$639$	−2.00000	−0.0791188
$640$	0	0
$641$	12.0000	0.473972	0.236986	−	0.971513i	$-0.423841\pi$
$641$	12.0000	0.473972	0.236986	+	0.971513i	$0.423841\pi$
$642$	− 48.0000i	− 1.89441i
$643$	36.0000i	1.41970i	0.704352	+	0.709851i	$0.251238\pi$
$643$	36.0000i	1.41970i	−0.704352	+	0.709851i	$0.748762\pi$
$644$	0	0
$645$	0	0
$646$	−12.0000	−0.472134
$647$	− 12.0000i	− 0.471769i	−0.971781	−	0.235884i	$-0.924201\pi$
$647$	− 12.0000i	− 0.471769i	0.971781	−	0.235884i	$-0.0757987\pi$
$648$	0	0
$649$	36.0000	1.41312
$650$	0	0
$651$	0	0
$652$	28.0000i	1.09656i
$653$	14.0000i	0.547862i	0.961749	+	0.273931i	$0.0883240\pi$
$653$	14.0000i	0.547862i	−0.961749	+	0.273931i	$0.911676\pi$
$654$	−28.0000	−1.09489
$655$	0	0
$656$	−20.0000	−0.780869
$657$	2.00000i	0.0780274i
$658$	0	0
$659$	19.0000	0.740135	0.370067	−	0.929005i	$-0.379335\pi$
$659$	19.0000	0.740135	0.370067	+	0.929005i	$0.379335\pi$
$660$	0	0
$661$	31.0000	1.20576	0.602880	−	0.797832i	$-0.294020\pi$
$661$	31.0000	1.20576	0.602880	+	0.797832i	$0.294020\pi$
$662$	52.0000i	2.02104i
$663$	30.0000i	1.16510i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	6.00000i	0.232321i
$668$	18.0000i	0.696441i
$669$	−56.0000	−2.16509
$670$	0	0
$671$	6.00000	0.231627
$672$	0	0
$673$	− 14.0000i	− 0.539660i	−0.962908	−	0.269830i	$-0.913032\pi$
$673$	− 14.0000i	− 0.539660i	0.962908	−	0.269830i	$-0.0869676\pi$
$674$	−6.00000	−0.231111
$675$	0	0
$676$	24.0000	0.923077
$677$	34.0000i	1.30673i	0.757045	+	0.653363i	$0.226642\pi$
$677$	34.0000i	1.30673i	−0.757045	+	0.653363i	$0.773358\pi$
$678$	80.0000i	3.07238i
$679$	0	0
$680$	0	0
$681$	8.00000	0.306561
$682$	6.00000i	0.229752i
$683$	9.00000i	0.344375i	0.985064	+	0.172188i	$0.0550836\pi$
$683$	9.00000i	0.344375i	−0.985064	+	0.172188i	$0.944916\pi$
$684$	4.00000	0.152944
$685$	0	0
$686$	0	0
$687$	30.0000i	1.14457i
$688$	− 4.00000i	− 0.152499i
$689$	−25.0000	−0.952424
$690$	0	0
$691$	−40.0000	−1.52167	−0.760836	−	0.648944i	$-0.775211\pi$
$691$	−40.0000	−1.52167	−0.760836	+	0.648944i	$0.775211\pi$
$692$	12.0000i	0.456172i
$693$	0	0
$694$	56.0000	2.12573
$695$	0	0
$696$	0	0
$697$	− 15.0000i	− 0.568166i
$698$	28.0000i	1.05982i
$699$	12.0000	0.453882
$700$	0	0
$701$	−2.00000	−0.0755390	−0.0377695	−	0.999286i	$-0.512025\pi$
$701$	−2.00000	−0.0755390	−0.0377695	+	0.999286i	$0.512025\pi$
$702$	40.0000i	1.50970i
$703$	0	0
$704$	24.0000	0.904534
$705$	0	0
$706$	62.0000	2.33340
$707$	0	0
$708$	− 48.0000i	− 1.80395i
$709$	1.00000	0.0375558	0.0187779	−	0.999824i	$-0.494022\pi$
$709$	1.00000	0.0375558	0.0187779	+	0.999824i	$0.494022\pi$
$710$	0	0
$711$	−8.00000	−0.300023
$712$	0	0
$713$	− 1.00000i	− 0.0374503i
$714$	0	0
$715$	0	0
$716$	40.0000	1.49487
$717$	− 32.0000i	− 1.19506i
$718$	38.0000i	1.41815i
$719$	−8.00000	−0.298350	−0.149175	−	0.988811i	$-0.547662\pi$
$719$	−8.00000	−0.298350	−0.149175	+	0.988811i	$0.547662\pi$
$720$	0	0
$721$	0	0
$722$	30.0000i	1.11648i
$723$	− 24.0000i	− 0.892570i
$724$	−20.0000	−0.743294
$725$	0	0
$726$	−8.00000	−0.296908
$727$	16.0000i	0.593407i	0.954970	+	0.296704i	$0.0958873\pi$
$727$	16.0000i	0.593407i	−0.954970	+	0.296704i	$0.904113\pi$
$728$	0	0
$729$	−13.0000	−0.481481
$730$	0	0
$731$	3.00000	0.110959
$732$	− 8.00000i	− 0.295689i
$733$	− 32.0000i	− 1.18195i	−0.806691	−	0.590973i	$-0.798744\pi$
$733$	− 32.0000i	− 1.18195i	0.806691	−	0.590973i	$-0.201256\pi$
$734$	−64.0000	−2.36228
$735$	0	0
$736$	−8.00000	−0.294884
$737$	− 9.00000i	− 0.331519i
$738$	10.0000i	0.368105i
$739$	10.0000	0.367856	0.183928	−	0.982940i	$-0.441119\pi$
$739$	10.0000	0.367856	0.183928	+	0.982940i	$0.441119\pi$
$740$	0	0
$741$	−20.0000	−0.734718
$742$	0	0
$743$	24.0000i	0.880475i	0.897881	+	0.440237i	$0.145106\pi$
$743$	24.0000i	0.880475i	−0.897881	+	0.440237i	$0.854894\pi$
$744$	0	0
$745$	0	0
$746$	−64.0000	−2.34321
$747$	15.0000i	0.548821i
$748$	18.0000i	0.658145i
$749$	0	0
$750$	0	0
$751$	−6.00000	−0.218943	−0.109472	−	0.993990i	$-0.534916\pi$
$751$	−6.00000	−0.218943	−0.109472	+	0.993990i	$0.534916\pi$
$752$	− 16.0000i	− 0.583460i
$753$	− 46.0000i	− 1.67633i
$754$	60.0000	2.18507
$755$	0	0
$756$	0	0
$757$	28.0000i	1.01768i	0.860862	+	0.508839i	$0.169925\pi$
$757$	28.0000i	1.01768i	−0.860862	+	0.508839i	$0.830075\pi$
$758$	22.0000i	0.799076i
$759$	−6.00000	−0.217786
$760$	0	0
$761$	20.0000	0.724999	0.362500	−	0.931984i	$-0.381923\pi$
$761$	20.0000	0.724999	0.362500	+	0.931984i	$0.381923\pi$
$762$	4.00000i	0.144905i
$763$	0	0
$764$	32.0000	1.15772
$765$	0	0
$766$	−64.0000	−2.31241
$767$	60.0000i	2.16647i
$768$	32.0000i	1.15470i
$769$	42.0000	1.51456	0.757279	−	0.653091i	$-0.226528\pi$
$769$	42.0000	1.51456	0.757279	+	0.653091i	$0.226528\pi$
$770$	0	0
$771$	48.0000	1.72868
$772$	6.00000i	0.215945i
$773$	4.00000i	0.143870i	0.997409	+	0.0719350i	$0.0229174\pi$
$773$	4.00000i	0.143870i	−0.997409	+	0.0719350i	$0.977083\pi$
$774$	−2.00000	−0.0718885
$775$	0	0
$776$	0	0
$777$	0	0
$778$	12.0000i	0.430221i
$779$	10.0000	0.358287
$780$	0	0
$781$	6.00000	0.214697
$782$	− 6.00000i	− 0.214560i
$783$	24.0000i	0.857690i
$784$	−28.0000	−1.00000
$785$	0	0
$786$	32.0000	1.14140
$787$	4.00000i	0.142585i	0.997455	+	0.0712923i	$0.0227123\pi$
$787$	4.00000i	0.142585i	−0.997455	+	0.0712923i	$0.977288\pi$
$788$	− 4.00000i	− 0.142494i
$789$	−36.0000	−1.28163
$790$	0	0
$791$	0	0
$792$	0	0
$793$	10.0000i	0.355110i
$794$	−12.0000	−0.425864
$795$	0	0
$796$	28.0000	0.992434
$797$	− 42.0000i	− 1.48772i	−0.668338	−	0.743858i	$-0.732994\pi$
$797$	− 42.0000i	− 1.48772i	0.668338	−	0.743858i	$-0.267006\pi$
$798$	0	0
$799$	12.0000	0.424529
$800$	0	0
$801$	−4.00000	−0.141333
$802$	− 10.0000i	− 0.353112i
$803$	− 6.00000i	− 0.211735i
$804$	−12.0000	−0.423207
$805$	0	0
$806$	−10.0000	−0.352235
$807$	50.0000i	1.76008i
$808$	0	0
$809$	−26.0000	−0.914111	−0.457056	−	0.889438i	$-0.651096\pi$
$809$	−26.0000	−0.914111	−0.457056	+	0.889438i	$0.651096\pi$
$810$	0	0
$811$	−14.0000	−0.491606	−0.245803	−	0.969320i	$-0.579052\pi$
$811$	−14.0000	−0.491606	−0.245803	+	0.969320i	$0.579052\pi$
$812$	0	0
$813$	46.0000i	1.61329i
$814$	0	0
$815$	0	0
$816$	−24.0000	−0.840168
$817$	2.00000i	0.0699711i
$818$	− 48.0000i	− 1.67828i
$819$	0	0
$820$	0	0
$821$	49.0000	1.71011	0.855056	−	0.518536i	$-0.173523\pi$
$821$	49.0000	1.71011	0.855056	+	0.518536i	$0.173523\pi$
$822$	24.0000i	0.837096i
$823$	1.00000i	0.0348578i	0.999848	+	0.0174289i	$0.00554807\pi$
$823$	1.00000i	0.0348578i	−0.999848	+	0.0174289i	$0.994452\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 36.0000i	− 1.25184i	−0.779886	−	0.625921i	$-0.784723\pi$
$827$	− 36.0000i	− 1.25184i	0.779886	−	0.625921i	$-0.215277\pi$
$828$	2.00000i	0.0695048i
$829$	−44.0000	−1.52818	−0.764092	−	0.645108i	$-0.776812\pi$
$829$	−44.0000	−1.52818	−0.764092	+	0.645108i	$0.776812\pi$
$830$	0	0
$831$	64.0000	2.22014
$832$	40.0000i	1.38675i
$833$	− 21.0000i	− 0.727607i
$834$	−76.0000	−2.63166
$835$	0	0
$836$	−12.0000	−0.415029
$837$	− 4.00000i	− 0.138260i
$838$	− 56.0000i	− 1.93449i
$839$	40.0000	1.38095	0.690477	−	0.723355i	$-0.257401\pi$
$839$	40.0000	1.38095	0.690477	+	0.723355i	$0.257401\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	20.0000i	0.689246i
$843$	38.0000i	1.30879i
$844$	−4.00000	−0.137686
$845$	0	0
$846$	−8.00000	−0.275046
$847$	0	0
$848$	− 20.0000i	− 0.686803i
$849$	42.0000	1.44144
$850$	0	0
$851$	0	0
$852$	− 8.00000i	− 0.274075i
$853$	29.0000i	0.992941i	0.868054	+	0.496471i	$0.165371\pi$
$853$	29.0000i	0.992941i	−0.868054	+	0.496471i	$0.834629\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 10.0000i	− 0.341593i	−0.985306	−	0.170797i	$-0.945366\pi$
$857$	− 10.0000i	− 0.341593i	0.985306	−	0.170797i	$-0.0546341\pi$
$858$	60.0000i	2.04837i
$859$	32.0000	1.09183	0.545913	−	0.837842i	$-0.316183\pi$
$859$	32.0000	1.09183	0.545913	+	0.837842i	$0.316183\pi$
$860$	0	0
$861$	0	0
$862$	42.0000i	1.43053i
$863$	6.00000i	0.204242i	0.994772	+	0.102121i	$0.0325630\pi$
$863$	6.00000i	0.204242i	−0.994772	+	0.102121i	$0.967437\pi$
$864$	−32.0000	−1.08866
$865$	0	0
$866$	24.0000	0.815553
$867$	16.0000i	0.543388i
$868$	0	0
$869$	24.0000	0.814144
$870$	0	0
$871$	15.0000	0.508256
$872$	0	0
$873$	− 7.00000i	− 0.236914i
$874$	4.00000	0.135302
$875$	0	0
$876$	−8.00000	−0.270295
$877$	41.0000i	1.38447i	0.721671	+	0.692236i	$0.243374\pi$
$877$	41.0000i	1.38447i	−0.721671	+	0.692236i	$0.756626\pi$
$878$	34.0000i	1.14744i
$879$	−52.0000	−1.75392
$880$	0	0
$881$	37.0000	1.24656	0.623281	−	0.781998i	$-0.285799\pi$
$881$	37.0000	1.24656	0.623281	+	0.781998i	$0.285799\pi$
$882$	14.0000i	0.471405i
$883$	− 31.0000i	− 1.04323i	−0.853180	−	0.521617i	$-0.825329\pi$
$883$	− 31.0000i	− 1.04323i	0.853180	−	0.521617i	$-0.174671\pi$
$884$	−30.0000	−1.00901
$885$	0	0
$886$	8.00000	0.268765
$887$	− 22.0000i	− 0.738688i	−0.929293	−	0.369344i	$-0.879582\pi$
$887$	− 22.0000i	− 0.738688i	0.929293	−	0.369344i	$-0.120418\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−33.0000	−1.10554
$892$	− 56.0000i	− 1.87502i
$893$	8.00000i	0.267710i
$894$	−48.0000	−1.60536
$895$	0	0
$896$	0	0
$897$	− 10.0000i	− 0.333890i
$898$	60.0000i	2.00223i
$899$	−6.00000	−0.200111
$900$	0	0
$901$	15.0000	0.499722
$902$	− 30.0000i	− 0.998891i
$903$	0	0
$904$	0	0
$905$	0	0
$906$	−80.0000	−2.65782
$907$	47.0000i	1.56061i	0.625400	+	0.780305i	$0.284936\pi$
$907$	47.0000i	1.56061i	−0.625400	+	0.780305i	$0.715064\pi$
$908$	8.00000i	0.265489i
$909$	9.00000	0.298511
$910$	0	0
$911$	−22.0000	−0.728893	−0.364446	−	0.931224i	$-0.618742\pi$
$911$	−22.0000	−0.728893	−0.364446	+	0.931224i	$0.618742\pi$
$912$	− 16.0000i	− 0.529813i
$913$	− 45.0000i	− 1.48928i
$914$	−36.0000	−1.19077
$915$	0	0
$916$	−30.0000	−0.991228
$917$	0	0
$918$	− 24.0000i	− 0.792118i
$919$	49.0000	1.61636	0.808180	−	0.588935i	$-0.200453\pi$
$919$	49.0000	1.61636	0.808180	+	0.588935i	$0.200453\pi$
$920$	0	0
$921$	14.0000	0.461316
$922$	− 60.0000i	− 1.97599i
$923$	10.0000i	0.329154i
$924$	0	0
$925$	0	0
$926$	−8.00000	−0.262896
$927$	1.00000i	0.0328443i
$928$	48.0000i	1.57568i
$929$	6.00000	0.196854	0.0984268	−	0.995144i	$-0.468619\pi$
$929$	6.00000	0.196854	0.0984268	+	0.995144i	$0.468619\pi$
$930$	0	0
$931$	14.0000	0.458831
$932$	12.0000i	0.393073i
$933$	30.0000i	0.982156i
$934$	12.0000	0.392652
$935$	0	0
$936$	0	0
$937$	32.0000i	1.04539i	0.852518	+	0.522697i	$0.175074\pi$
$937$	32.0000i	1.04539i	−0.852518	+	0.522697i	$0.824926\pi$
$938$	0	0
$939$	44.0000	1.43589
$940$	0	0
$941$	33.0000	1.07577	0.537885	−	0.843018i	$-0.319224\pi$
$941$	33.0000	1.07577	0.537885	+	0.843018i	$0.319224\pi$
$942$	− 40.0000i	− 1.30327i
$943$	5.00000i	0.162822i
$944$	−48.0000	−1.56227
$945$	0	0
$946$	6.00000	0.195077
$947$	− 33.0000i	− 1.07236i	−0.844105	−	0.536178i	$-0.819868\pi$
$947$	− 33.0000i	− 1.07236i	0.844105	−	0.536178i	$-0.180132\pi$
$948$	− 32.0000i	− 1.03931i
$949$	10.0000	0.324614
$950$	0	0
$951$	−18.0000	−0.583690
$952$	0	0
$953$	− 22.0000i	− 0.712650i	−0.934362	−	0.356325i	$-0.884030\pi$
$953$	− 22.0000i	− 0.712650i	0.934362	−	0.356325i	$-0.115970\pi$
$954$	−10.0000	−0.323762
$955$	0	0
$956$	32.0000	1.03495
$957$	36.0000i	1.16371i
$958$	42.0000i	1.35696i
$959$	0	0
$960$	0	0
$961$	−30.0000	−0.967742
$962$	0	0
$963$	12.0000i	0.386695i
$964$	24.0000	0.772988
$965$	0	0
$966$	0	0
$967$	37.0000i	1.18984i	0.803785	+	0.594920i	$0.202816\pi$
$967$	37.0000i	1.18984i	−0.803785	+	0.594920i	$0.797184\pi$
$968$	0	0
$969$	12.0000	0.385496
$970$	0	0
$971$	−13.0000	−0.417190	−0.208595	−	0.978002i	$-0.566889\pi$
$971$	−13.0000	−0.417190	−0.208595	+	0.978002i	$0.566889\pi$
$972$	20.0000i	0.641500i
$973$	0	0
$974$	72.0000	2.30703
$975$	0	0
$976$	−8.00000	−0.256074
$977$	34.0000i	1.08776i	0.839164	+	0.543878i	$0.183045\pi$
$977$	34.0000i	1.08776i	−0.839164	+	0.543878i	$0.816955\pi$
$978$	− 56.0000i	− 1.79068i
$979$	12.0000	0.383522
$980$	0	0
$981$	7.00000	0.223493
$982$	12.0000i	0.382935i
$983$	24.0000i	0.765481i	0.923856	+	0.382741i	$0.125020\pi$
$983$	24.0000i	0.765481i	−0.923856	+	0.382741i	$0.874980\pi$
$984$	0	0
$985$	0	0
$986$	−36.0000	−1.14647
$987$	0	0
$988$	− 20.0000i	− 0.636285i
$989$	−1.00000	−0.0317982
$990$	0	0
$991$	2.00000	0.0635321	0.0317660	−	0.999495i	$-0.489887\pi$
$991$	2.00000	0.0635321	0.0317660	+	0.999495i	$0.489887\pi$
$992$	− 8.00000i	− 0.254000i
$993$	− 52.0000i	− 1.65017i
$994$	0	0
$995$	0	0
$996$	−60.0000	−1.90117
$997$	4.00000i	0.126681i	0.997992	+	0.0633406i	$0.0201755\pi$
$997$	4.00000i	0.126681i	−0.997992	+	0.0633406i	$0.979825\pi$
$998$	− 16.0000i	− 0.506471i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1075.2.b.b.474.1		2
5.2	odd	4		1075.2.a.h.1.1		1
5.3	odd	4		43.2.a.a.1.1	✓	1
5.4	even	2	inner	1075.2.b.b.474.2		2
15.2	even	4		9675.2.a.b.1.1		1
15.8	even	4		387.2.a.e.1.1		1
20.3	even	4		688.2.a.b.1.1		1
35.13	even	4		2107.2.a.a.1.1		1
40.3	even	4		2752.2.a.b.1.1		1
40.13	odd	4		2752.2.a.f.1.1		1
55.43	even	4		5203.2.a.a.1.1		1
60.23	odd	4		6192.2.a.ba.1.1		1
65.38	odd	4		7267.2.a.a.1.1		1
215.128	even	4		1849.2.a.d.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.2.a.a.1.1	✓	1	5.3	odd	4
387.2.a.e.1.1		1	15.8	even	4
688.2.a.b.1.1		1	20.3	even	4
1075.2.a.h.1.1		1	5.2	odd	4
1075.2.b.b.474.1		2	1.1	even	1	trivial
1075.2.b.b.474.2		2	5.4	even	2	inner
1849.2.a.d.1.1		1	215.128	even	4
2107.2.a.a.1.1		1	35.13	even	4
2752.2.a.b.1.1		1	40.3	even	4
2752.2.a.f.1.1		1	40.13	odd	4
5203.2.a.a.1.1		1	55.43	even	4
6192.2.a.ba.1.1		1	60.23	odd	4
7267.2.a.a.1.1		1	65.38	odd	4
9675.2.a.b.1.1		1	15.2	even	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:	\( N \)	\(=\)	\( 1075 = 5^{2} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1075.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.00000i q^{2} +2.00000i q^{3} -2.00000 q^{4} +4.00000 q^{6} -1.00000 q^{9} +O(q^{10})\)	\(q-2.00000i q^{2} +2.00000i q^{3} -2.00000 q^{4} +4.00000 q^{6} -1.00000 q^{9} +3.00000 q^{11} -4.00000i q^{12} +5.00000i q^{13} -4.00000 q^{16} -3.00000i q^{17} +2.00000i q^{18} +2.00000 q^{19} -6.00000i q^{22} +1.00000i q^{23} +10.0000 q^{26} +4.00000i q^{27} +6.00000 q^{29} -1.00000 q^{31} +8.00000i q^{32} +6.00000i q^{33} -6.00000 q^{34} +2.00000 q^{36} -4.00000i q^{38} -10.0000 q^{39} +5.00000 q^{41} +1.00000i q^{43} -6.00000 q^{44} +2.00000 q^{46} +4.00000i q^{47} -8.00000i q^{48} +7.00000 q^{49} +6.00000 q^{51} -10.0000i q^{52} +5.00000i q^{53} +8.00000 q^{54} +4.00000i q^{57} -12.0000i q^{58} +12.0000 q^{59} +2.00000 q^{61} +2.00000i q^{62} +8.00000 q^{64} +12.0000 q^{66} -3.00000i q^{67} +6.00000i q^{68} -2.00000 q^{69} +2.00000 q^{71} -2.00000i q^{73} -4.00000 q^{76} +20.0000i q^{78} +8.00000 q^{79} -11.0000 q^{81} -10.0000i q^{82} -15.0000i q^{83} +2.00000 q^{86} +12.0000i q^{87} +4.00000 q^{89} -2.00000i q^{92} -2.00000i q^{93} +8.00000 q^{94} -16.0000 q^{96} +7.00000i q^{97} -14.0000i q^{98} -3.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{4} + 8 q^{6} - 2 q^{9}+O(q^{10}) \) 2 * q - 4 * q^4 + 8 * q^6 - 2 * q^9	\( 2 q - 4 q^{4} + 8 q^{6} - 2 q^{9} + 6 q^{11} - 8 q^{16} + 4 q^{19} + 20 q^{26} + 12 q^{29} - 2 q^{31} - 12 q^{34} + 4 q^{36} - 20 q^{39} + 10 q^{41} - 12 q^{44} + 4 q^{46} + 14 q^{49} + 12 q^{51} + 16 q^{54} + 24 q^{59} + 4 q^{61} + 16 q^{64} + 24 q^{66} - 4 q^{69} + 4 q^{71} - 8 q^{76} + 16 q^{79} - 22 q^{81} + 4 q^{86} + 8 q^{89} + 16 q^{94} - 32 q^{96} - 6 q^{99}+O(q^{100}) \) 2 * q - 4 * q^4 + 8 * q^6 - 2 * q^9 + 6 * q^11 - 8 * q^16 + 4 * q^19 + 20 * q^26 + 12 * q^29 - 2 * q^31 - 12 * q^34 + 4 * q^36 - 20 * q^39 + 10 * q^41 - 12 * q^44 + 4 * q^46 + 14 * q^49 + 12 * q^51 + 16 * q^54 + 24 * q^59 + 4 * q^61 + 16 * q^64 + 24 * q^66 - 4 * q^69 + 4 * q^71 - 8 * q^76 + 16 * q^79 - 22 * q^81 + 4 * q^86 + 8 * q^89 + 16 * q^94 - 32 * q^96 - 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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