LMFDB - Embedded newform 2850.2.d.o.799.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2850,2,Mod(799,2850)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2850, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2850.799");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2850 = 2 \cdot 3 \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2850.d (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:	$22.7573645761$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			799.2
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	2850.799
Dual form			2850.2.d.o.799.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -4.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -4.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} -1.00000 q^{11} +1.00000i q^{12} +4.00000 q^{14} +1.00000 q^{16} -8.00000i q^{17} -1.00000i q^{18} -1.00000 q^{19} -4.00000 q^{21} -1.00000i q^{22} +3.00000i q^{23} -1.00000 q^{24} +1.00000i q^{27} +4.00000i q^{28} +1.00000 q^{29} +1.00000 q^{31} +1.00000i q^{32} +1.00000i q^{33} +8.00000 q^{34} +1.00000 q^{36} -2.00000i q^{37} -1.00000i q^{38} -10.0000 q^{41} -4.00000i q^{42} +8.00000i q^{43} +1.00000 q^{44} -3.00000 q^{46} -1.00000i q^{48} -9.00000 q^{49} -8.00000 q^{51} -3.00000i q^{53} -1.00000 q^{54} -4.00000 q^{56} +1.00000i q^{57} +1.00000i q^{58} -4.00000 q^{59} +5.00000 q^{61} +1.00000i q^{62} +4.00000i q^{63} -1.00000 q^{64} -1.00000 q^{66} -5.00000i q^{67} +8.00000i q^{68} +3.00000 q^{69} -6.00000 q^{71} +1.00000i q^{72} +13.0000i q^{73} +2.00000 q^{74} +1.00000 q^{76} +4.00000i q^{77} +5.00000 q^{79} +1.00000 q^{81} -10.0000i q^{82} -11.0000i q^{83} +4.00000 q^{84} -8.00000 q^{86} -1.00000i q^{87} +1.00000i q^{88} -3.00000 q^{89} -3.00000i q^{92} -1.00000i q^{93} +1.00000 q^{96} +10.0000i q^{97} -9.00000i q^{98} +1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9	$ 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} - 2 q^{11} + 8 q^{14} + 2 q^{16} - 2 q^{19} - 8 q^{21} - 2 q^{24} + 2 q^{29} + 2 q^{31} + 16 q^{34} + 2 q^{36} - 20 q^{41} + 2 q^{44} - 6 q^{46} - 18 q^{49} - 16 q^{51} - 2 q^{54} - 8 q^{56} - 8 q^{59} + 10 q^{61} - 2 q^{64} - 2 q^{66} + 6 q^{69} - 12 q^{71} + 4 q^{74} + 2 q^{76} + 10 q^{79} + 2 q^{81} + 8 q^{84} - 16 q^{86} - 6 q^{89} + 2 q^{96} + 2 q^{99}+O(q^{100}) $ 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 - 2 * q^11 + 8 * q^14 + 2 * q^16 - 2 * q^19 - 8 * q^21 - 2 * q^24 + 2 * q^29 + 2 * q^31 + 16 * q^34 + 2 * q^36 - 20 * q^41 + 2 * q^44 - 6 * q^46 - 18 * q^49 - 16 * q^51 - 2 * q^54 - 8 * q^56 - 8 * q^59 + 10 * q^61 - 2 * q^64 - 2 * q^66 + 6 * q^69 - 12 * q^71 + 4 * q^74 + 2 * q^76 + 10 * q^79 + 2 * q^81 + 8 * q^84 - 16 * q^86 - 6 * q^89 + 2 * q^96 + 2 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2850\mathbb{Z}\right)^\times$.

$n$	$1027$	$1351$	$1901$
$\chi(n)$	$-1$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.00000i	0.707107i
$2$	1.00000i	0.707107i
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	1.00000	0.408248
$7$	− 4.00000i	− 1.51186i	−0.654654	−	0.755929i	$-0.727186\pi$
$7$	− 4.00000i	− 1.51186i	0.654654	−	0.755929i	$-0.272814\pi$
$8$	− 1.00000i	− 0.353553i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−1.00000	−0.301511	−0.150756	−	0.988571i	$-0.548171\pi$
$11$	−1.00000	−0.301511	−0.150756	+	0.988571i	$0.548171\pi$
$12$	1.00000i	0.288675i
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	4.00000	1.06904
$15$	0	0
$16$	1.00000	0.250000
$17$	− 8.00000i	− 1.94029i	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	− 8.00000i	− 1.94029i	0.242536	−	0.970143i	$-0.422021\pi$
$18$	− 1.00000i	− 0.235702i
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−4.00000	−0.872872
$22$	− 1.00000i	− 0.213201i
$23$	3.00000i	0.625543i	0.949828	+	0.312772i	$0.101257\pi$
$23$	3.00000i	0.625543i	−0.949828	+	0.312772i	$0.898743\pi$
$24$	−1.00000	−0.204124
$25$	0	0
$26$	0	0
$27$	1.00000i	0.192450i
$28$	4.00000i	0.755929i
$29$	1.00000	0.185695	0.0928477	−	0.995680i	$-0.470403\pi$
$29$	1.00000	0.185695	0.0928477	+	0.995680i	$0.470403\pi$
$30$	0	0
$31$	1.00000	0.179605	0.0898027	−	0.995960i	$-0.471376\pi$
$31$	1.00000	0.179605	0.0898027	+	0.995960i	$0.471376\pi$
$32$	1.00000i	0.176777i
$33$	1.00000i	0.174078i
$34$	8.00000	1.37199
$35$	0	0
$36$	1.00000	0.166667
$37$	− 2.00000i	− 0.328798i	−0.986394	−	0.164399i	$-0.947432\pi$
$37$	− 2.00000i	− 0.328798i	0.986394	−	0.164399i	$-0.0525685\pi$
$38$	− 1.00000i	− 0.162221i
$39$	0	0
$40$	0	0
$41$	−10.0000	−1.56174	−0.780869	−	0.624695i	$-0.785223\pi$
$41$	−10.0000	−1.56174	−0.780869	+	0.624695i	$0.785223\pi$
$42$	− 4.00000i	− 0.617213i
$43$	8.00000i	1.21999i	0.792406	+	0.609994i	$0.208828\pi$
$43$	8.00000i	1.21999i	−0.792406	+	0.609994i	$0.791172\pi$
$44$	1.00000	0.150756
$45$	0	0
$46$	−3.00000	−0.442326
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	− 1.00000i	− 0.144338i
$49$	−9.00000	−1.28571
$50$	0	0
$51$	−8.00000	−1.12022
$52$	0	0
$53$	− 3.00000i	− 0.412082i	−0.978543	−	0.206041i	$-0.933942\pi$
$53$	− 3.00000i	− 0.412082i	0.978543	−	0.206041i	$-0.0660580\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	−4.00000	−0.534522
$57$	1.00000i	0.132453i
$58$	1.00000i	0.131306i
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	0	0
$61$	5.00000	0.640184	0.320092	−	0.947386i	$-0.396286\pi$
$61$	5.00000	0.640184	0.320092	+	0.947386i	$0.396286\pi$
$62$	1.00000i	0.127000i
$63$	4.00000i	0.503953i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	−1.00000	−0.123091
$67$	− 5.00000i	− 0.610847i	−0.952217	−	0.305424i	$-0.901202\pi$
$67$	− 5.00000i	− 0.610847i	0.952217	−	0.305424i	$-0.0987981\pi$
$68$	8.00000i	0.970143i
$69$	3.00000	0.361158
$70$	0	0
$71$	−6.00000	−0.712069	−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000	−0.712069	−0.356034	+	0.934473i	$0.615871\pi$
$72$	1.00000i	0.117851i
$73$	13.0000i	1.52153i	0.649025	+	0.760767i	$0.275177\pi$
$73$	13.0000i	1.52153i	−0.649025	+	0.760767i	$0.724823\pi$
$74$	2.00000	0.232495
$75$	0	0
$76$	1.00000	0.114708
$77$	4.00000i	0.455842i
$78$	0	0
$79$	5.00000	0.562544	0.281272	−	0.959628i	$-0.409244\pi$
$79$	5.00000	0.562544	0.281272	+	0.959628i	$0.409244\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	− 10.0000i	− 1.10432i
$83$	− 11.0000i	− 1.20741i	−0.797209	−	0.603703i	$-0.793691\pi$
$83$	− 11.0000i	− 1.20741i	0.797209	−	0.603703i	$-0.206309\pi$
$84$	4.00000	0.436436
$85$	0	0
$86$	−8.00000	−0.862662
$87$	− 1.00000i	− 0.107211i
$88$	1.00000i	0.106600i
$89$	−3.00000	−0.317999	−0.159000	−	0.987279i	$-0.550827\pi$
$89$	−3.00000	−0.317999	−0.159000	+	0.987279i	$0.550827\pi$
$90$	0	0
$91$	0	0
$92$	− 3.00000i	− 0.312772i
$93$	− 1.00000i	− 0.103695i
$94$	0	0
$95$	0	0
$96$	1.00000	0.102062
$97$	10.0000i	1.01535i	0.861550	+	0.507673i	$0.169494\pi$
$97$	10.0000i	1.01535i	−0.861550	+	0.507673i	$0.830506\pi$
$98$	− 9.00000i	− 0.909137i
$99$	1.00000	0.100504
$100$	0	0
$101$	−12.0000	−1.19404	−0.597022	−	0.802225i	$-0.703650\pi$
$101$	−12.0000	−1.19404	−0.597022	+	0.802225i	$0.703650\pi$
$102$	− 8.00000i	− 0.792118i
$103$	− 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
$106$	3.00000	0.291386
$107$	14.0000i	1.35343i	0.736245	+	0.676716i	$0.236597\pi$
$107$	14.0000i	1.35343i	−0.736245	+	0.676716i	$0.763403\pi$
$108$	− 1.00000i	− 0.0962250i
$109$	−6.00000	−0.574696	−0.287348	−	0.957826i	$-0.592774\pi$
$109$	−6.00000	−0.574696	−0.287348	+	0.957826i	$0.592774\pi$
$110$	0	0
$111$	−2.00000	−0.189832
$112$	− 4.00000i	− 0.377964i
$113$	1.00000i	0.0940721i	0.998893	+	0.0470360i	$0.0149776\pi$
$113$	1.00000i	0.0940721i	−0.998893	+	0.0470360i	$0.985022\pi$
$114$	−1.00000	−0.0936586
$115$	0	0
$116$	−1.00000	−0.0928477
$117$	0	0
$118$	− 4.00000i	− 0.368230i
$119$	−32.0000	−2.93344
$120$	0	0
$121$	−10.0000	−0.909091
$122$	5.00000i	0.452679i
$123$	10.0000i	0.901670i
$124$	−1.00000	−0.0898027
$125$	0	0
$126$	−4.00000	−0.356348
$127$	− 17.0000i	− 1.50851i	−0.656584	−	0.754253i	$-0.727999\pi$
$127$	− 17.0000i	− 1.50851i	0.656584	−	0.754253i	$-0.272001\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	8.00000	0.704361
$130$	0	0
$131$	−15.0000	−1.31056	−0.655278	−	0.755388i	$-0.727449\pi$
$131$	−15.0000	−1.31056	−0.655278	+	0.755388i	$0.727449\pi$
$132$	− 1.00000i	− 0.0870388i
$133$	4.00000i	0.346844i
$134$	5.00000	0.431934
$135$	0	0
$136$	−8.00000	−0.685994
$137$	4.00000i	0.341743i	0.985293	+	0.170872i	$0.0546583\pi$
$137$	4.00000i	0.341743i	−0.985293	+	0.170872i	$0.945342\pi$
$138$	3.00000i	0.255377i
$139$	−6.00000	−0.508913	−0.254457	−	0.967084i	$-0.581897\pi$
$139$	−6.00000	−0.508913	−0.254457	+	0.967084i	$0.581897\pi$
$140$	0	0
$141$	0	0
$142$	− 6.00000i	− 0.503509i
$143$	0	0
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	−13.0000	−1.07589
$147$	9.00000i	0.742307i
$148$	2.00000i	0.164399i
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	1.00000i	0.0811107i
$153$	8.00000i	0.646762i
$154$	−4.00000	−0.322329
$155$	0	0
$156$	0	0
$157$	2.00000i	0.159617i	0.996810	+	0.0798087i	$0.0254309\pi$
$157$	2.00000i	0.159617i	−0.996810	+	0.0798087i	$0.974569\pi$
$158$	5.00000i	0.397779i
$159$	−3.00000	−0.237915
$160$	0	0
$161$	12.0000	0.945732
$162$	1.00000i	0.0785674i
$163$	− 10.0000i	− 0.783260i	−0.920123	−	0.391630i	$-0.871911\pi$
$163$	− 10.0000i	− 0.783260i	0.920123	−	0.391630i	$-0.128089\pi$
$164$	10.0000	0.780869
$165$	0	0
$166$	11.0000	0.853766
$167$	− 22.0000i	− 1.70241i	−0.524832	−	0.851206i	$-0.675872\pi$
$167$	− 22.0000i	− 1.70241i	0.524832	−	0.851206i	$-0.324128\pi$
$168$	4.00000i	0.308607i
$169$	13.0000	1.00000
$170$	0	0
$171$	1.00000	0.0764719
$172$	− 8.00000i	− 0.609994i
$173$	11.0000i	0.836315i	0.908375	+	0.418157i	$0.137324\pi$
$173$	11.0000i	0.836315i	−0.908375	+	0.418157i	$0.862676\pi$
$174$	1.00000	0.0758098
$175$	0	0
$176$	−1.00000	−0.0753778
$177$	4.00000i	0.300658i
$178$	− 3.00000i	− 0.224860i
$179$	−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
$181$	12.0000	0.891953	0.445976	−	0.895045i	$-0.352856\pi$
$181$	12.0000	0.891953	0.445976	+	0.895045i	$0.352856\pi$
$182$	0	0
$183$	− 5.00000i	− 0.369611i
$184$	3.00000	0.221163
$185$	0	0
$186$	1.00000	0.0733236
$187$	8.00000i	0.585018i
$188$	0	0
$189$	4.00000	0.290957
$190$	0	0
$191$	13.0000	0.940647	0.470323	−	0.882494i	$-0.344137\pi$
$191$	13.0000	0.940647	0.470323	+	0.882494i	$0.344137\pi$
$192$	1.00000i	0.0721688i
$193$	14.0000i	1.00774i	0.863779	+	0.503871i	$0.168091\pi$
$193$	14.0000i	1.00774i	−0.863779	+	0.503871i	$0.831909\pi$
$194$	−10.0000	−0.717958
$195$	0	0
$196$	9.00000	0.642857
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	1.00000i	0.0710669i
$199$	−26.0000	−1.84309	−0.921546	−	0.388270i	$-0.873073\pi$
$199$	−26.0000	−1.84309	−0.921546	+	0.388270i	$0.873073\pi$
$200$	0	0
$201$	−5.00000	−0.352673
$202$	− 12.0000i	− 0.844317i
$203$	− 4.00000i	− 0.280745i
$204$	8.00000	0.560112
$205$	0	0
$206$	1.00000	0.0696733
$207$	− 3.00000i	− 0.208514i
$208$	0	0
$209$	1.00000	0.0691714
$210$	0	0
$211$	25.0000	1.72107	0.860535	−	0.509390i	$-0.170129\pi$
$211$	25.0000	1.72107	0.860535	+	0.509390i	$0.170129\pi$
$212$	3.00000i	0.206041i
$213$	6.00000i	0.411113i
$214$	−14.0000	−0.957020
$215$	0	0
$216$	1.00000	0.0680414
$217$	− 4.00000i	− 0.271538i
$218$	− 6.00000i	− 0.406371i
$219$	13.0000	0.878459
$220$	0	0
$221$	0	0
$222$	− 2.00000i	− 0.134231i
$223$	− 29.0000i	− 1.94198i	−0.239113	−	0.970992i	$-0.576857\pi$
$223$	− 29.0000i	− 1.94198i	0.239113	−	0.970992i	$-0.423143\pi$
$224$	4.00000	0.267261
$225$	0	0
$226$	−1.00000	−0.0665190
$227$	− 30.0000i	− 1.99117i	−0.0938647	−	0.995585i	$-0.529922\pi$
$227$	− 30.0000i	− 1.99117i	0.0938647	−	0.995585i	$-0.470078\pi$
$228$	− 1.00000i	− 0.0662266i
$229$	−13.0000	−0.859064	−0.429532	−	0.903052i	$-0.641321\pi$
$229$	−13.0000	−0.859064	−0.429532	+	0.903052i	$0.641321\pi$
$230$	0	0
$231$	4.00000	0.263181
$232$	− 1.00000i	− 0.0656532i
$233$	4.00000i	0.262049i	0.991379	+	0.131024i	$0.0418266\pi$
$233$	4.00000i	0.262049i	−0.991379	+	0.131024i	$0.958173\pi$
$234$	0	0
$235$	0	0
$236$	4.00000	0.260378
$237$	− 5.00000i	− 0.324785i
$238$	− 32.0000i	− 2.07425i
$239$	−24.0000	−1.55243	−0.776215	−	0.630468i	$-0.782863\pi$
$239$	−24.0000	−1.55243	−0.776215	+	0.630468i	$0.782863\pi$
$240$	0	0
$241$	8.00000	0.515325	0.257663	−	0.966235i	$-0.417048\pi$
$241$	8.00000	0.515325	0.257663	+	0.966235i	$0.417048\pi$
$242$	− 10.0000i	− 0.642824i
$243$	− 1.00000i	− 0.0641500i
$244$	−5.00000	−0.320092
$245$	0	0
$246$	−10.0000	−0.637577
$247$	0	0
$248$	− 1.00000i	− 0.0635001i
$249$	−11.0000	−0.697097
$250$	0	0
$251$	16.0000	1.00991	0.504956	−	0.863145i	$-0.331509\pi$
$251$	16.0000	1.00991	0.504956	+	0.863145i	$0.331509\pi$
$252$	− 4.00000i	− 0.251976i
$253$	− 3.00000i	− 0.188608i
$254$	17.0000	1.06667
$255$	0	0
$256$	1.00000	0.0625000
$257$	− 27.0000i	− 1.68421i	−0.539311	−	0.842107i	$-0.681315\pi$
$257$	− 27.0000i	− 1.68421i	0.539311	−	0.842107i	$-0.318685\pi$
$258$	8.00000i	0.498058i
$259$	−8.00000	−0.497096
$260$	0	0
$261$	−1.00000	−0.0618984
$262$	− 15.0000i	− 0.926703i
$263$	19.0000i	1.17159i	0.810459	+	0.585795i	$0.199218\pi$
$263$	19.0000i	1.17159i	−0.810459	+	0.585795i	$0.800782\pi$
$264$	1.00000	0.0615457
$265$	0	0
$266$	−4.00000	−0.245256
$267$	3.00000i	0.183597i
$268$	5.00000i	0.305424i
$269$	26.0000	1.58525	0.792624	−	0.609711i	$-0.208714\pi$
$269$	26.0000	1.58525	0.792624	+	0.609711i	$0.208714\pi$
$270$	0	0
$271$	−14.0000	−0.850439	−0.425220	−	0.905090i	$-0.639803\pi$
$271$	−14.0000	−0.850439	−0.425220	+	0.905090i	$0.639803\pi$
$272$	− 8.00000i	− 0.485071i
$273$	0	0
$274$	−4.00000	−0.241649
$275$	0	0
$276$	−3.00000	−0.180579
$277$	17.0000i	1.02143i	0.859750	+	0.510716i	$0.170619\pi$
$277$	17.0000i	1.02143i	−0.859750	+	0.510716i	$0.829381\pi$
$278$	− 6.00000i	− 0.359856i
$279$	−1.00000	−0.0598684
$280$	0	0
$281$	−15.0000	−0.894825	−0.447412	−	0.894328i	$-0.647654\pi$
$281$	−15.0000	−0.894825	−0.447412	+	0.894328i	$0.647654\pi$
$282$	0	0
$283$	− 32.0000i	− 1.90220i	−0.308879	−	0.951101i	$-0.599954\pi$
$283$	− 32.0000i	− 1.90220i	0.308879	−	0.951101i	$-0.400046\pi$
$284$	6.00000	0.356034
$285$	0	0
$286$	0	0
$287$	40.0000i	2.36113i
$288$	− 1.00000i	− 0.0589256i
$289$	−47.0000	−2.76471
$290$	0	0
$291$	10.0000	0.586210
$292$	− 13.0000i	− 0.760767i
$293$	− 1.00000i	− 0.0584206i	−0.999573	−	0.0292103i	$-0.990701\pi$
$293$	− 1.00000i	− 0.0584206i	0.999573	−	0.0292103i	$-0.00929925\pi$
$294$	−9.00000	−0.524891
$295$	0	0
$296$	−2.00000	−0.116248
$297$	− 1.00000i	− 0.0580259i
$298$	0	0
$299$	0	0
$300$	0	0
$301$	32.0000	1.84445
$302$	0	0
$303$	12.0000i	0.689382i
$304$	−1.00000	−0.0573539
$305$	0	0
$306$	−8.00000	−0.457330
$307$	− 5.00000i	− 0.285365i	−0.989769	−	0.142683i	$-0.954427\pi$
$307$	− 5.00000i	− 0.285365i	0.989769	−	0.142683i	$-0.0455728\pi$
$308$	− 4.00000i	− 0.227921i
$309$	−1.00000	−0.0568880
$310$	0	0
$311$	−16.0000	−0.907277	−0.453638	−	0.891186i	$-0.649874\pi$
$311$	−16.0000	−0.907277	−0.453638	+	0.891186i	$0.649874\pi$
$312$	0	0
$313$	21.0000i	1.18699i	0.804838	+	0.593495i	$0.202252\pi$
$313$	21.0000i	1.18699i	−0.804838	+	0.593495i	$0.797748\pi$
$314$	−2.00000	−0.112867
$315$	0	0
$316$	−5.00000	−0.281272
$317$	9.00000i	0.505490i	0.967533	+	0.252745i	$0.0813334\pi$
$317$	9.00000i	0.505490i	−0.967533	+	0.252745i	$0.918667\pi$
$318$	− 3.00000i	− 0.168232i
$319$	−1.00000	−0.0559893
$320$	0	0
$321$	14.0000	0.781404
$322$	12.0000i	0.668734i
$323$	8.00000i	0.445132i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	10.0000	0.553849
$327$	6.00000i	0.331801i
$328$	10.0000i	0.552158i
$329$	0	0
$330$	0	0
$331$	−1.00000	−0.0549650	−0.0274825	−	0.999622i	$-0.508749\pi$
$331$	−1.00000	−0.0549650	−0.0274825	+	0.999622i	$0.508749\pi$
$332$	11.0000i	0.603703i
$333$	2.00000i	0.109599i
$334$	22.0000	1.20379
$335$	0	0
$336$	−4.00000	−0.218218
$337$	− 16.0000i	− 0.871576i	−0.900049	−	0.435788i	$-0.856470\pi$
$337$	− 16.0000i	− 0.871576i	0.900049	−	0.435788i	$-0.143530\pi$
$338$	13.0000i	0.707107i
$339$	1.00000	0.0543125
$340$	0	0
$341$	−1.00000	−0.0541530
$342$	1.00000i	0.0540738i
$343$	8.00000i	0.431959i
$344$	8.00000	0.431331
$345$	0	0
$346$	−11.0000	−0.591364
$347$	12.0000i	0.644194i	0.946707	+	0.322097i	$0.104388\pi$
$347$	12.0000i	0.644194i	−0.946707	+	0.322097i	$0.895612\pi$
$348$	1.00000i	0.0536056i
$349$	25.0000	1.33822	0.669110	−	0.743164i	$-0.266676\pi$
$349$	25.0000	1.33822	0.669110	+	0.743164i	$0.266676\pi$
$350$	0	0
$351$	0	0
$352$	− 1.00000i	− 0.0533002i
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	−4.00000	−0.212598
$355$	0	0
$356$	3.00000	0.159000
$357$	32.0000i	1.69362i
$358$	− 20.0000i	− 1.05703i
$359$	−16.0000	−0.844448	−0.422224	−	0.906492i	$-0.638750\pi$
$359$	−16.0000	−0.844448	−0.422224	+	0.906492i	$0.638750\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	12.0000i	0.630706i
$363$	10.0000i	0.524864i
$364$	0	0
$365$	0	0
$366$	5.00000	0.261354
$367$	− 22.0000i	− 1.14839i	−0.818718	−	0.574195i	$-0.805315\pi$
$367$	− 22.0000i	− 1.14839i	0.818718	−	0.574195i	$-0.194685\pi$
$368$	3.00000i	0.156386i
$369$	10.0000	0.520579
$370$	0	0
$371$	−12.0000	−0.623009
$372$	1.00000i	0.0518476i
$373$	− 32.0000i	− 1.65690i	−0.560065	−	0.828449i	$-0.689224\pi$
$373$	− 32.0000i	− 1.65690i	0.560065	−	0.828449i	$-0.310776\pi$
$374$	−8.00000	−0.413670
$375$	0	0
$376$	0	0
$377$	0	0
$378$	4.00000i	0.205738i
$379$	32.0000	1.64373	0.821865	−	0.569683i	$-0.192934\pi$
$379$	32.0000	1.64373	0.821865	+	0.569683i	$0.192934\pi$
$380$	0	0
$381$	−17.0000	−0.870936
$382$	13.0000i	0.665138i
$383$	18.0000i	0.919757i	0.887982	+	0.459879i	$0.152107\pi$
$383$	18.0000i	0.919757i	−0.887982	+	0.459879i	$0.847893\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	−14.0000	−0.712581
$387$	− 8.00000i	− 0.406663i
$388$	− 10.0000i	− 0.507673i
$389$	22.0000	1.11544	0.557722	−	0.830028i	$-0.311675\pi$
$389$	22.0000	1.11544	0.557722	+	0.830028i	$0.311675\pi$
$390$	0	0
$391$	24.0000	1.21373
$392$	9.00000i	0.454569i
$393$	15.0000i	0.756650i
$394$	0	0
$395$	0	0
$396$	−1.00000	−0.0502519
$397$	35.0000i	1.75660i	0.478110	+	0.878300i	$0.341322\pi$
$397$	35.0000i	1.75660i	−0.478110	+	0.878300i	$0.658678\pi$
$398$	− 26.0000i	− 1.30326i
$399$	4.00000	0.200250
$400$	0	0
$401$	−5.00000	−0.249688	−0.124844	−	0.992176i	$-0.539843\pi$
$401$	−5.00000	−0.249688	−0.124844	+	0.992176i	$0.539843\pi$
$402$	− 5.00000i	− 0.249377i
$403$	0	0
$404$	12.0000	0.597022
$405$	0	0
$406$	4.00000	0.198517
$407$	2.00000i	0.0991363i
$408$	8.00000i	0.396059i
$409$	18.0000	0.890043	0.445021	−	0.895520i	$-0.353196\pi$
$409$	18.0000	0.890043	0.445021	+	0.895520i	$0.353196\pi$
$410$	0	0
$411$	4.00000	0.197305
$412$	1.00000i	0.0492665i
$413$	16.0000i	0.787309i
$414$	3.00000	0.147442
$415$	0	0
$416$	0	0
$417$	6.00000i	0.293821i
$418$	1.00000i	0.0489116i
$419$	12.0000	0.586238	0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000	0.586238	0.293119	+	0.956076i	$0.405307\pi$
$420$	0	0
$421$	22.0000	1.07221	0.536107	−	0.844150i	$-0.319894\pi$
$421$	22.0000	1.07221	0.536107	+	0.844150i	$0.319894\pi$
$422$	25.0000i	1.21698i
$423$	0	0
$424$	−3.00000	−0.145693
$425$	0	0
$426$	−6.00000	−0.290701
$427$	− 20.0000i	− 0.967868i
$428$	− 14.0000i	− 0.676716i
$429$	0	0
$430$	0	0
$431$	−14.0000	−0.674356	−0.337178	−	0.941441i	$-0.609472\pi$
$431$	−14.0000	−0.674356	−0.337178	+	0.941441i	$0.609472\pi$
$432$	1.00000i	0.0481125i
$433$	22.0000i	1.05725i	0.848855	+	0.528626i	$0.177293\pi$
$433$	22.0000i	1.05725i	−0.848855	+	0.528626i	$0.822707\pi$
$434$	4.00000	0.192006
$435$	0	0
$436$	6.00000	0.287348
$437$	− 3.00000i	− 0.143509i
$438$	13.0000i	0.621164i
$439$	−7.00000	−0.334092	−0.167046	−	0.985949i	$-0.553423\pi$
$439$	−7.00000	−0.334092	−0.167046	+	0.985949i	$0.553423\pi$
$440$	0	0
$441$	9.00000	0.428571
$442$	0	0
$443$	− 3.00000i	− 0.142534i	−0.997457	−	0.0712672i	$-0.977296\pi$
$443$	− 3.00000i	− 0.142534i	0.997457	−	0.0712672i	$-0.0227043\pi$
$444$	2.00000	0.0949158
$445$	0	0
$446$	29.0000	1.37319
$447$	0	0
$448$	4.00000i	0.188982i
$449$	23.0000	1.08544	0.542719	−	0.839915i	$-0.317395\pi$
$449$	23.0000	1.08544	0.542719	+	0.839915i	$0.317395\pi$
$450$	0	0
$451$	10.0000	0.470882
$452$	− 1.00000i	− 0.0470360i
$453$	0	0
$454$	30.0000	1.40797
$455$	0	0
$456$	1.00000	0.0468293
$457$	− 22.0000i	− 1.02912i	−0.857455	−	0.514558i	$-0.827956\pi$
$457$	− 22.0000i	− 1.02912i	0.857455	−	0.514558i	$-0.172044\pi$
$458$	− 13.0000i	− 0.607450i
$459$	8.00000	0.373408
$460$	0	0
$461$	−12.0000	−0.558896	−0.279448	−	0.960161i	$-0.590151\pi$
$461$	−12.0000	−0.558896	−0.279448	+	0.960161i	$0.590151\pi$
$462$	4.00000i	0.186097i
$463$	− 4.00000i	− 0.185896i	−0.995671	−	0.0929479i	$-0.970371\pi$
$463$	− 4.00000i	− 0.185896i	0.995671	−	0.0929479i	$-0.0296290\pi$
$464$	1.00000	0.0464238
$465$	0	0
$466$	−4.00000	−0.185296
$467$	− 3.00000i	− 0.138823i	−0.997588	−	0.0694117i	$-0.977888\pi$
$467$	− 3.00000i	− 0.138823i	0.997588	−	0.0694117i	$-0.0221122\pi$
$468$	0	0
$469$	−20.0000	−0.923514
$470$	0	0
$471$	2.00000	0.0921551
$472$	4.00000i	0.184115i
$473$	− 8.00000i	− 0.367840i
$474$	5.00000	0.229658
$475$	0	0
$476$	32.0000	1.46672
$477$	3.00000i	0.137361i
$478$	− 24.0000i	− 1.09773i
$479$	−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
$482$	8.00000i	0.364390i
$483$	− 12.0000i	− 0.546019i
$484$	10.0000	0.454545
$485$	0	0
$486$	1.00000	0.0453609
$487$	− 32.0000i	− 1.45006i	−0.688718	−	0.725029i	$-0.741826\pi$
$487$	− 32.0000i	− 1.45006i	0.688718	−	0.725029i	$-0.258174\pi$
$488$	− 5.00000i	− 0.226339i
$489$	−10.0000	−0.452216
$490$	0	0
$491$	4.00000	0.180517	0.0902587	−	0.995918i	$-0.471231\pi$
$491$	4.00000	0.180517	0.0902587	+	0.995918i	$0.471231\pi$
$492$	− 10.0000i	− 0.450835i
$493$	− 8.00000i	− 0.360302i
$494$	0	0
$495$	0	0
$496$	1.00000	0.0449013
$497$	24.0000i	1.07655i
$498$	− 11.0000i	− 0.492922i
$499$	4.00000	0.179065	0.0895323	−	0.995984i	$-0.471463\pi$
$499$	4.00000	0.179065	0.0895323	+	0.995984i	$0.471463\pi$
$500$	0	0
$501$	−22.0000	−0.982888
$502$	16.0000i	0.714115i
$503$	4.00000i	0.178351i	0.996016	+	0.0891756i	$0.0284232\pi$
$503$	4.00000i	0.178351i	−0.996016	+	0.0891756i	$0.971577\pi$
$504$	4.00000	0.178174
$505$	0	0
$506$	3.00000	0.133366
$507$	− 13.0000i	− 0.577350i
$508$	17.0000i	0.754253i
$509$	−25.0000	−1.10811	−0.554053	−	0.832482i	$-0.686919\pi$
$509$	−25.0000	−1.10811	−0.554053	+	0.832482i	$0.686919\pi$
$510$	0	0
$511$	52.0000	2.30034
$512$	1.00000i	0.0441942i
$513$	− 1.00000i	− 0.0441511i
$514$	27.0000	1.19092
$515$	0	0
$516$	−8.00000	−0.352180
$517$	0	0
$518$	− 8.00000i	− 0.351500i
$519$	11.0000	0.482846
$520$	0	0
$521$	−1.00000	−0.0438108	−0.0219054	−	0.999760i	$-0.506973\pi$
$521$	−1.00000	−0.0438108	−0.0219054	+	0.999760i	$0.506973\pi$
$522$	− 1.00000i	− 0.0437688i
$523$	− 4.00000i	− 0.174908i	−0.996169	−	0.0874539i	$-0.972127\pi$
$523$	− 4.00000i	− 0.174908i	0.996169	−	0.0874539i	$-0.0278730\pi$
$524$	15.0000	0.655278
$525$	0	0
$526$	−19.0000	−0.828439
$527$	− 8.00000i	− 0.348485i
$528$	1.00000i	0.0435194i
$529$	14.0000	0.608696
$530$	0	0
$531$	4.00000	0.173585
$532$	− 4.00000i	− 0.173422i
$533$	0	0
$534$	−3.00000	−0.129823
$535$	0	0
$536$	−5.00000	−0.215967
$537$	20.0000i	0.863064i
$538$	26.0000i	1.12094i
$539$	9.00000	0.387657
$540$	0	0
$541$	27.0000	1.16082	0.580410	−	0.814324i	$-0.302892\pi$
$541$	27.0000	1.16082	0.580410	+	0.814324i	$0.302892\pi$
$542$	− 14.0000i	− 0.601351i
$543$	− 12.0000i	− 0.514969i
$544$	8.00000	0.342997
$545$	0	0
$546$	0	0
$547$	21.0000i	0.897895i	0.893558	+	0.448948i	$0.148201\pi$
$547$	21.0000i	0.897895i	−0.893558	+	0.448948i	$0.851799\pi$
$548$	− 4.00000i	− 0.170872i
$549$	−5.00000	−0.213395
$550$	0	0
$551$	−1.00000	−0.0426014
$552$	− 3.00000i	− 0.127688i
$553$	− 20.0000i	− 0.850487i
$554$	−17.0000	−0.722261
$555$	0	0
$556$	6.00000	0.254457
$557$	36.0000i	1.52537i	0.646771	+	0.762684i	$0.276119\pi$
$557$	36.0000i	1.52537i	−0.646771	+	0.762684i	$0.723881\pi$
$558$	− 1.00000i	− 0.0423334i
$559$	0	0
$560$	0	0
$561$	8.00000	0.337760
$562$	− 15.0000i	− 0.632737i
$563$	− 22.0000i	− 0.927189i	−0.886047	−	0.463595i	$-0.846559\pi$
$563$	− 22.0000i	− 0.927189i	0.886047	−	0.463595i	$-0.153441\pi$
$564$	0	0
$565$	0	0
$566$	32.0000	1.34506
$567$	− 4.00000i	− 0.167984i
$568$	6.00000i	0.251754i
$569$	−6.00000	−0.251533	−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000	−0.251533	−0.125767	+	0.992060i	$0.540139\pi$
$570$	0	0
$571$	6.00000	0.251092	0.125546	−	0.992088i	$-0.459932\pi$
$571$	6.00000	0.251092	0.125546	+	0.992088i	$0.459932\pi$
$572$	0	0
$573$	− 13.0000i	− 0.543083i
$574$	−40.0000	−1.66957
$575$	0	0
$576$	1.00000	0.0416667
$577$	− 21.0000i	− 0.874241i	−0.899403	−	0.437121i	$-0.855998\pi$
$577$	− 21.0000i	− 0.874241i	0.899403	−	0.437121i	$-0.144002\pi$
$578$	− 47.0000i	− 1.95494i
$579$	14.0000	0.581820
$580$	0	0
$581$	−44.0000	−1.82543
$582$	10.0000i	0.414513i
$583$	3.00000i	0.124247i
$584$	13.0000	0.537944
$585$	0	0
$586$	1.00000	0.0413096
$587$	− 33.0000i	− 1.36206i	−0.732257	−	0.681028i	$-0.761533\pi$
$587$	− 33.0000i	− 1.36206i	0.732257	−	0.681028i	$-0.238467\pi$
$588$	− 9.00000i	− 0.371154i
$589$	−1.00000	−0.0412043
$590$	0	0
$591$	0	0
$592$	− 2.00000i	− 0.0821995i
$593$	20.0000i	0.821302i	0.911793	+	0.410651i	$0.134698\pi$
$593$	20.0000i	0.821302i	−0.911793	+	0.410651i	$0.865302\pi$
$594$	1.00000	0.0410305
$595$	0	0
$596$	0	0
$597$	26.0000i	1.06411i
$598$	0	0
$599$	24.0000	0.980613	0.490307	−	0.871550i	$-0.336885\pi$
$599$	24.0000	0.980613	0.490307	+	0.871550i	$0.336885\pi$
$600$	0	0
$601$	−38.0000	−1.55005	−0.775026	−	0.631929i	$-0.782263\pi$
$601$	−38.0000	−1.55005	−0.775026	+	0.631929i	$0.782263\pi$
$602$	32.0000i	1.30422i
$603$	5.00000i	0.203616i
$604$	0	0
$605$	0	0
$606$	−12.0000	−0.487467
$607$	27.0000i	1.09590i	0.836512	+	0.547948i	$0.184591\pi$
$607$	27.0000i	1.09590i	−0.836512	+	0.547948i	$0.815409\pi$
$608$	− 1.00000i	− 0.0405554i
$609$	−4.00000	−0.162088
$610$	0	0
$611$	0	0
$612$	− 8.00000i	− 0.323381i
$613$	− 38.0000i	− 1.53481i	−0.641165	−	0.767403i	$-0.721549\pi$
$613$	− 38.0000i	− 1.53481i	0.641165	−	0.767403i	$-0.278451\pi$
$614$	5.00000	0.201784
$615$	0	0
$616$	4.00000	0.161165
$617$	6.00000i	0.241551i	0.992680	+	0.120775i	$0.0385381\pi$
$617$	6.00000i	0.241551i	−0.992680	+	0.120775i	$0.961462\pi$
$618$	− 1.00000i	− 0.0402259i
$619$	20.0000	0.803868	0.401934	−	0.915669i	$-0.368338\pi$
$619$	20.0000	0.803868	0.401934	+	0.915669i	$0.368338\pi$
$620$	0	0
$621$	−3.00000	−0.120386
$622$	− 16.0000i	− 0.641542i
$623$	12.0000i	0.480770i
$624$	0	0
$625$	0	0
$626$	−21.0000	−0.839329
$627$	− 1.00000i	− 0.0399362i
$628$	− 2.00000i	− 0.0798087i
$629$	−16.0000	−0.637962
$630$	0	0
$631$	44.0000	1.75161	0.875806	−	0.482663i	$-0.160330\pi$
$631$	44.0000	1.75161	0.875806	+	0.482663i	$0.160330\pi$
$632$	− 5.00000i	− 0.198889i
$633$	− 25.0000i	− 0.993661i
$634$	−9.00000	−0.357436
$635$	0	0
$636$	3.00000	0.118958
$637$	0	0
$638$	− 1.00000i	− 0.0395904i
$639$	6.00000	0.237356
$640$	0	0
$641$	6.00000	0.236986	0.118493	−	0.992955i	$-0.462194\pi$
$641$	6.00000	0.236986	0.118493	+	0.992955i	$0.462194\pi$
$642$	14.0000i	0.552536i
$643$	− 2.00000i	− 0.0788723i	−0.999222	−	0.0394362i	$-0.987444\pi$
$643$	− 2.00000i	− 0.0788723i	0.999222	−	0.0394362i	$-0.0125562\pi$
$644$	−12.0000	−0.472866
$645$	0	0
$646$	−8.00000	−0.314756
$647$	43.0000i	1.69050i	0.534368	+	0.845252i	$0.320550\pi$
$647$	43.0000i	1.69050i	−0.534368	+	0.845252i	$0.679450\pi$
$648$	− 1.00000i	− 0.0392837i
$649$	4.00000	0.157014
$650$	0	0
$651$	−4.00000	−0.156772
$652$	10.0000i	0.391630i
$653$	24.0000i	0.939193i	0.882881	+	0.469596i	$0.155601\pi$
$653$	24.0000i	0.939193i	−0.882881	+	0.469596i	$0.844399\pi$
$654$	−6.00000	−0.234619
$655$	0	0
$656$	−10.0000	−0.390434
$657$	− 13.0000i	− 0.507178i
$658$	0	0
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	0	0
$661$	−8.00000	−0.311164	−0.155582	−	0.987823i	$-0.549725\pi$
$661$	−8.00000	−0.311164	−0.155582	+	0.987823i	$0.549725\pi$
$662$	− 1.00000i	− 0.0388661i
$663$	0	0
$664$	−11.0000	−0.426883
$665$	0	0
$666$	−2.00000	−0.0774984
$667$	3.00000i	0.116160i
$668$	22.0000i	0.851206i
$669$	−29.0000	−1.12120
$670$	0	0
$671$	−5.00000	−0.193023
$672$	− 4.00000i	− 0.154303i
$673$	− 16.0000i	− 0.616755i	−0.951264	−	0.308377i	$-0.900214\pi$
$673$	− 16.0000i	− 0.616755i	0.951264	−	0.308377i	$-0.0997859\pi$
$674$	16.0000	0.616297
$675$	0	0
$676$	−13.0000	−0.500000
$677$	3.00000i	0.115299i	0.998337	+	0.0576497i	$0.0183606\pi$
$677$	3.00000i	0.115299i	−0.998337	+	0.0576497i	$0.981639\pi$
$678$	1.00000i	0.0384048i
$679$	40.0000	1.53506
$680$	0	0
$681$	−30.0000	−1.14960
$682$	− 1.00000i	− 0.0382920i
$683$	6.00000i	0.229584i	0.993390	+	0.114792i	$0.0366201\pi$
$683$	6.00000i	0.229584i	−0.993390	+	0.114792i	$0.963380\pi$
$684$	−1.00000	−0.0382360
$685$	0	0
$686$	−8.00000	−0.305441
$687$	13.0000i	0.495981i
$688$	8.00000i	0.304997i
$689$	0	0
$690$	0	0
$691$	10.0000	0.380418	0.190209	−	0.981744i	$-0.439083\pi$
$691$	10.0000	0.380418	0.190209	+	0.981744i	$0.439083\pi$
$692$	− 11.0000i	− 0.418157i
$693$	− 4.00000i	− 0.151947i
$694$	−12.0000	−0.455514
$695$	0	0
$696$	−1.00000	−0.0379049
$697$	80.0000i	3.03022i
$698$	25.0000i	0.946264i
$699$	4.00000	0.151294
$700$	0	0
$701$	−8.00000	−0.302156	−0.151078	−	0.988522i	$-0.548274\pi$
$701$	−8.00000	−0.302156	−0.151078	+	0.988522i	$0.548274\pi$
$702$	0	0
$703$	2.00000i	0.0754314i
$704$	1.00000	0.0376889
$705$	0	0
$706$	0	0
$707$	48.0000i	1.80523i
$708$	− 4.00000i	− 0.150329i
$709$	35.0000	1.31445	0.657226	−	0.753693i	$-0.271730\pi$
$709$	35.0000	1.31445	0.657226	+	0.753693i	$0.271730\pi$
$710$	0	0
$711$	−5.00000	−0.187515
$712$	3.00000i	0.112430i
$713$	3.00000i	0.112351i
$714$	−32.0000	−1.19757
$715$	0	0
$716$	20.0000	0.747435
$717$	24.0000i	0.896296i
$718$	− 16.0000i	− 0.597115i
$719$	−29.0000	−1.08152	−0.540759	−	0.841178i	$-0.681863\pi$
$719$	−29.0000	−1.08152	−0.540759	+	0.841178i	$0.681863\pi$
$720$	0	0
$721$	−4.00000	−0.148968
$722$	1.00000i	0.0372161i
$723$	− 8.00000i	− 0.297523i
$724$	−12.0000	−0.445976
$725$	0	0
$726$	−10.0000	−0.371135
$727$	− 18.0000i	− 0.667583i	−0.942647	−	0.333792i	$-0.891672\pi$
$727$	− 18.0000i	− 0.667583i	0.942647	−	0.333792i	$-0.108328\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	64.0000	2.36713
$732$	5.00000i	0.184805i
$733$	− 23.0000i	− 0.849524i	−0.905305	−	0.424762i	$-0.860358\pi$
$733$	− 23.0000i	− 0.849524i	0.905305	−	0.424762i	$-0.139642\pi$
$734$	22.0000	0.812035
$735$	0	0
$736$	−3.00000	−0.110581
$737$	5.00000i	0.184177i
$738$	10.0000i	0.368105i
$739$	−4.00000	−0.147142	−0.0735712	−	0.997290i	$-0.523440\pi$
$739$	−4.00000	−0.147142	−0.0735712	+	0.997290i	$0.523440\pi$
$740$	0	0
$741$	0	0
$742$	− 12.0000i	− 0.440534i
$743$	24.0000i	0.880475i	0.897881	+	0.440237i	$0.145106\pi$
$743$	24.0000i	0.880475i	−0.897881	+	0.440237i	$0.854894\pi$
$744$	−1.00000	−0.0366618
$745$	0	0
$746$	32.0000	1.17160
$747$	11.0000i	0.402469i
$748$	− 8.00000i	− 0.292509i
$749$	56.0000	2.04620
$750$	0	0
$751$	40.0000	1.45962	0.729810	−	0.683650i	$-0.239608\pi$
$751$	40.0000	1.45962	0.729810	+	0.683650i	$0.239608\pi$
$752$	0	0
$753$	− 16.0000i	− 0.583072i
$754$	0	0
$755$	0	0
$756$	−4.00000	−0.145479
$757$	23.0000i	0.835949i	0.908459	+	0.417975i	$0.137260\pi$
$757$	23.0000i	0.835949i	−0.908459	+	0.417975i	$0.862740\pi$
$758$	32.0000i	1.16229i
$759$	−3.00000	−0.108893
$760$	0	0
$761$	−26.0000	−0.942499	−0.471250	−	0.882000i	$-0.656197\pi$
$761$	−26.0000	−0.942499	−0.471250	+	0.882000i	$0.656197\pi$
$762$	− 17.0000i	− 0.615845i
$763$	24.0000i	0.868858i
$764$	−13.0000	−0.470323
$765$	0	0
$766$	−18.0000	−0.650366
$767$	0	0
$768$	− 1.00000i	− 0.0360844i
$769$	45.0000	1.62274	0.811371	−	0.584532i	$-0.198722\pi$
$769$	45.0000	1.62274	0.811371	+	0.584532i	$0.198722\pi$
$770$	0	0
$771$	−27.0000	−0.972381
$772$	− 14.0000i	− 0.503871i
$773$	− 54.0000i	− 1.94225i	−0.238581	−	0.971123i	$-0.576682\pi$
$773$	− 54.0000i	− 1.94225i	0.238581	−	0.971123i	$-0.423318\pi$
$774$	8.00000	0.287554
$775$	0	0
$776$	10.0000	0.358979
$777$	8.00000i	0.286998i
$778$	22.0000i	0.788738i
$779$	10.0000	0.358287
$780$	0	0
$781$	6.00000	0.214697
$782$	24.0000i	0.858238i
$783$	1.00000i	0.0357371i
$784$	−9.00000	−0.321429
$785$	0	0
$786$	−15.0000	−0.535032
$787$	1.00000i	0.0356462i	0.999841	+	0.0178231i	$0.00567356\pi$
$787$	1.00000i	0.0356462i	−0.999841	+	0.0178231i	$0.994326\pi$
$788$	0	0
$789$	19.0000	0.676418
$790$	0	0
$791$	4.00000	0.142224
$792$	− 1.00000i	− 0.0355335i
$793$	0	0
$794$	−35.0000	−1.24210
$795$	0	0
$796$	26.0000	0.921546
$797$	− 46.0000i	− 1.62940i	−0.579880	−	0.814702i	$-0.696901\pi$
$797$	− 46.0000i	− 1.62940i	0.579880	−	0.814702i	$-0.303099\pi$
$798$	4.00000i	0.141598i
$799$	0	0
$800$	0	0
$801$	3.00000	0.106000
$802$	− 5.00000i	− 0.176556i
$803$	− 13.0000i	− 0.458760i
$804$	5.00000	0.176336
$805$	0	0
$806$	0	0
$807$	− 26.0000i	− 0.915243i
$808$	12.0000i	0.422159i
$809$	16.0000	0.562530	0.281265	−	0.959630i	$-0.409246\pi$
$809$	16.0000	0.562530	0.281265	+	0.959630i	$0.409246\pi$
$810$	0	0
$811$	−49.0000	−1.72062	−0.860311	−	0.509769i	$-0.829731\pi$
$811$	−49.0000	−1.72062	−0.860311	+	0.509769i	$0.829731\pi$
$812$	4.00000i	0.140372i
$813$	14.0000i	0.491001i
$814$	−2.00000	−0.0701000
$815$	0	0
$816$	−8.00000	−0.280056
$817$	− 8.00000i	− 0.279885i
$818$	18.0000i	0.629355i
$819$	0	0
$820$	0	0
$821$	−10.0000	−0.349002	−0.174501	−	0.984657i	$-0.555831\pi$
$821$	−10.0000	−0.349002	−0.174501	+	0.984657i	$0.555831\pi$
$822$	4.00000i	0.139516i
$823$	26.0000i	0.906303i	0.891434	+	0.453152i	$0.149700\pi$
$823$	26.0000i	0.906303i	−0.891434	+	0.453152i	$0.850300\pi$
$824$	−1.00000	−0.0348367
$825$	0	0
$826$	−16.0000	−0.556711
$827$	− 28.0000i	− 0.973655i	−0.873498	−	0.486828i	$-0.838154\pi$
$827$	− 28.0000i	− 0.973655i	0.873498	−	0.486828i	$-0.161846\pi$
$828$	3.00000i	0.104257i
$829$	−26.0000	−0.903017	−0.451509	−	0.892267i	$-0.649114\pi$
$829$	−26.0000	−0.903017	−0.451509	+	0.892267i	$0.649114\pi$
$830$	0	0
$831$	17.0000	0.589723
$832$	0	0
$833$	72.0000i	2.49465i
$834$	−6.00000	−0.207763
$835$	0	0
$836$	−1.00000	−0.0345857
$837$	1.00000i	0.0345651i
$838$	12.0000i	0.414533i
$839$	48.0000	1.65714	0.828572	−	0.559883i	$-0.189154\pi$
$839$	48.0000	1.65714	0.828572	+	0.559883i	$0.189154\pi$
$840$	0	0
$841$	−28.0000	−0.965517
$842$	22.0000i	0.758170i
$843$	15.0000i	0.516627i
$844$	−25.0000	−0.860535
$845$	0	0
$846$	0	0
$847$	40.0000i	1.37442i
$848$	− 3.00000i	− 0.103020i
$849$	−32.0000	−1.09824
$850$	0	0
$851$	6.00000	0.205677
$852$	− 6.00000i	− 0.205557i
$853$	− 46.0000i	− 1.57501i	−0.616308	−	0.787505i	$-0.711372\pi$
$853$	− 46.0000i	− 1.57501i	0.616308	−	0.787505i	$-0.288628\pi$
$854$	20.0000	0.684386
$855$	0	0
$856$	14.0000	0.478510
$857$	− 18.0000i	− 0.614868i	−0.951569	−	0.307434i	$-0.900530\pi$
$857$	− 18.0000i	− 0.614868i	0.951569	−	0.307434i	$-0.0994704\pi$
$858$	0	0
$859$	−34.0000	−1.16007	−0.580033	−	0.814593i	$-0.696960\pi$
$859$	−34.0000	−1.16007	−0.580033	+	0.814593i	$0.696960\pi$
$860$	0	0
$861$	40.0000	1.36320
$862$	− 14.0000i	− 0.476842i
$863$	− 28.0000i	− 0.953131i	−0.879139	−	0.476566i	$-0.841881\pi$
$863$	− 28.0000i	− 0.953131i	0.879139	−	0.476566i	$-0.158119\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	−22.0000	−0.747590
$867$	47.0000i	1.59620i
$868$	4.00000i	0.135769i
$869$	−5.00000	−0.169613
$870$	0	0
$871$	0	0
$872$	6.00000i	0.203186i
$873$	− 10.0000i	− 0.338449i
$874$	3.00000	0.101477
$875$	0	0
$876$	−13.0000	−0.439229
$877$	42.0000i	1.41824i	0.705088	+	0.709120i	$0.250907\pi$
$877$	42.0000i	1.41824i	−0.705088	+	0.709120i	$0.749093\pi$
$878$	− 7.00000i	− 0.236239i
$879$	−1.00000	−0.0337292
$880$	0	0
$881$	−8.00000	−0.269527	−0.134763	−	0.990878i	$-0.543027\pi$
$881$	−8.00000	−0.269527	−0.134763	+	0.990878i	$0.543027\pi$
$882$	9.00000i	0.303046i
$883$	26.0000i	0.874970i	0.899226	+	0.437485i	$0.144131\pi$
$883$	26.0000i	0.874970i	−0.899226	+	0.437485i	$0.855869\pi$
$884$	0	0
$885$	0	0
$886$	3.00000	0.100787
$887$	− 48.0000i	− 1.61168i	−0.592132	−	0.805841i	$-0.701714\pi$
$887$	− 48.0000i	− 1.61168i	0.592132	−	0.805841i	$-0.298286\pi$
$888$	2.00000i	0.0671156i
$889$	−68.0000	−2.28065
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	29.0000i	0.970992i
$893$	0	0
$894$	0	0
$895$	0	0
$896$	−4.00000	−0.133631
$897$	0	0
$898$	23.0000i	0.767520i
$899$	1.00000	0.0333519
$900$	0	0
$901$	−24.0000	−0.799556
$902$	10.0000i	0.332964i
$903$	− 32.0000i	− 1.06489i
$904$	1.00000	0.0332595
$905$	0	0
$906$	0	0
$907$	− 48.0000i	− 1.59381i	−0.604102	−	0.796907i	$-0.706468\pi$
$907$	− 48.0000i	− 1.59381i	0.604102	−	0.796907i	$-0.293532\pi$
$908$	30.0000i	0.995585i
$909$	12.0000	0.398015
$910$	0	0
$911$	18.0000	0.596367	0.298183	−	0.954509i	$-0.403619\pi$
$911$	18.0000	0.596367	0.298183	+	0.954509i	$0.403619\pi$
$912$	1.00000i	0.0331133i
$913$	11.0000i	0.364047i
$914$	22.0000	0.727695
$915$	0	0
$916$	13.0000	0.429532
$917$	60.0000i	1.98137i
$918$	8.00000i	0.264039i
$919$	2.00000	0.0659739	0.0329870	−	0.999456i	$-0.489498\pi$
$919$	2.00000	0.0659739	0.0329870	+	0.999456i	$0.489498\pi$
$920$	0	0
$921$	−5.00000	−0.164756
$922$	− 12.0000i	− 0.395199i
$923$	0	0
$924$	−4.00000	−0.131590
$925$	0	0
$926$	4.00000	0.131448
$927$	1.00000i	0.0328443i
$928$	1.00000i	0.0328266i
$929$	12.0000	0.393707	0.196854	−	0.980433i	$-0.436928\pi$
$929$	12.0000	0.393707	0.196854	+	0.980433i	$0.436928\pi$
$930$	0	0
$931$	9.00000	0.294963
$932$	− 4.00000i	− 0.131024i
$933$	16.0000i	0.523816i
$934$	3.00000	0.0981630
$935$	0	0
$936$	0	0
$937$	42.0000i	1.37208i	0.727564	+	0.686040i	$0.240653\pi$
$937$	42.0000i	1.37208i	−0.727564	+	0.686040i	$0.759347\pi$
$938$	− 20.0000i	− 0.653023i
$939$	21.0000	0.685309
$940$	0	0
$941$	−19.0000	−0.619382	−0.309691	−	0.950837i	$-0.600226\pi$
$941$	−19.0000	−0.619382	−0.309691	+	0.950837i	$0.600226\pi$
$942$	2.00000i	0.0651635i
$943$	− 30.0000i	− 0.976934i
$944$	−4.00000	−0.130189
$945$	0	0
$946$	8.00000	0.260102
$947$	− 52.0000i	− 1.68977i	−0.534946	−	0.844886i	$-0.679668\pi$
$947$	− 52.0000i	− 1.68977i	0.534946	−	0.844886i	$-0.320332\pi$
$948$	5.00000i	0.162392i
$949$	0	0
$950$	0	0
$951$	9.00000	0.291845
$952$	32.0000i	1.03713i
$953$	15.0000i	0.485898i	0.970039	+	0.242949i	$0.0781147\pi$
$953$	15.0000i	0.485898i	−0.970039	+	0.242949i	$0.921885\pi$
$954$	−3.00000	−0.0971286
$955$	0	0
$956$	24.0000	0.776215
$957$	1.00000i	0.0323254i
$958$	− 21.0000i	− 0.678479i
$959$	16.0000	0.516667
$960$	0	0
$961$	−30.0000	−0.967742
$962$	0	0
$963$	− 14.0000i	− 0.451144i
$964$	−8.00000	−0.257663
$965$	0	0
$966$	12.0000	0.386094
$967$	38.0000i	1.22200i	0.791632	+	0.610999i	$0.209232\pi$
$967$	38.0000i	1.22200i	−0.791632	+	0.610999i	$0.790768\pi$
$968$	10.0000i	0.321412i
$969$	8.00000	0.256997
$970$	0	0
$971$	−8.00000	−0.256732	−0.128366	−	0.991727i	$-0.540973\pi$
$971$	−8.00000	−0.256732	−0.128366	+	0.991727i	$0.540973\pi$
$972$	1.00000i	0.0320750i
$973$	24.0000i	0.769405i
$974$	32.0000	1.02535
$975$	0	0
$976$	5.00000	0.160046
$977$	2.00000i	0.0639857i	0.999488	+	0.0319928i	$0.0101854\pi$
$977$	2.00000i	0.0639857i	−0.999488	+	0.0319928i	$0.989815\pi$
$978$	− 10.0000i	− 0.319765i
$979$	3.00000	0.0958804
$980$	0	0
$981$	6.00000	0.191565
$982$	4.00000i	0.127645i
$983$	− 30.0000i	− 0.956851i	−0.878128	−	0.478426i	$-0.841208\pi$
$983$	− 30.0000i	− 0.956851i	0.878128	−	0.478426i	$-0.158792\pi$
$984$	10.0000	0.318788
$985$	0	0
$986$	8.00000	0.254772
$987$	0	0
$988$	0	0
$989$	−24.0000	−0.763156
$990$	0	0
$991$	−25.0000	−0.794151	−0.397076	−	0.917786i	$-0.629975\pi$
$991$	−25.0000	−0.794151	−0.397076	+	0.917786i	$0.629975\pi$
$992$	1.00000i	0.0317500i
$993$	1.00000i	0.0317340i
$994$	−24.0000	−0.761234
$995$	0	0
$996$	11.0000	0.348548
$997$	− 13.0000i	− 0.411714i	−0.978582	−	0.205857i	$-0.934002\pi$
$997$	− 13.0000i	− 0.411714i	0.978582	−	0.205857i	$-0.0659982\pi$
$998$	4.00000i	0.126618i
$999$	2.00000	0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2850.2.d.o.799.2		2
5.2	odd	4		2850.2.a.f.1.1	✓	1
5.3	odd	4		2850.2.a.w.1.1	yes	1
5.4	even	2	inner	2850.2.d.o.799.1		2
15.2	even	4		8550.2.a.bk.1.1		1
15.8	even	4		8550.2.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2850.2.a.f.1.1	✓	1	5.2	odd	4
2850.2.a.w.1.1	yes	1	5.3	odd	4
2850.2.d.o.799.1		2	5.4	even	2	inner
2850.2.d.o.799.2		2	1.1	even	1	trivial
8550.2.a.c.1.1		1	15.8	even	4
8550.2.a.bk.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:	\( N \)	\(=\)	\( 2850 = 2 \cdot 3 \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2850.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -4.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -4.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} -1.00000 q^{11} +1.00000i q^{12} +4.00000 q^{14} +1.00000 q^{16} -8.00000i q^{17} -1.00000i q^{18} -1.00000 q^{19} -4.00000 q^{21} -1.00000i q^{22} +3.00000i q^{23} -1.00000 q^{24} +1.00000i q^{27} +4.00000i q^{28} +1.00000 q^{29} +1.00000 q^{31} +1.00000i q^{32} +1.00000i q^{33} +8.00000 q^{34} +1.00000 q^{36} -2.00000i q^{37} -1.00000i q^{38} -10.0000 q^{41} -4.00000i q^{42} +8.00000i q^{43} +1.00000 q^{44} -3.00000 q^{46} -1.00000i q^{48} -9.00000 q^{49} -8.00000 q^{51} -3.00000i q^{53} -1.00000 q^{54} -4.00000 q^{56} +1.00000i q^{57} +1.00000i q^{58} -4.00000 q^{59} +5.00000 q^{61} +1.00000i q^{62} +4.00000i q^{63} -1.00000 q^{64} -1.00000 q^{66} -5.00000i q^{67} +8.00000i q^{68} +3.00000 q^{69} -6.00000 q^{71} +1.00000i q^{72} +13.0000i q^{73} +2.00000 q^{74} +1.00000 q^{76} +4.00000i q^{77} +5.00000 q^{79} +1.00000 q^{81} -10.0000i q^{82} -11.0000i q^{83} +4.00000 q^{84} -8.00000 q^{86} -1.00000i q^{87} +1.00000i q^{88} -3.00000 q^{89} -3.00000i q^{92} -1.00000i q^{93} +1.00000 q^{96} +10.0000i q^{97} -9.00000i q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9	\( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} - 2 q^{11} + 8 q^{14} + 2 q^{16} - 2 q^{19} - 8 q^{21} - 2 q^{24} + 2 q^{29} + 2 q^{31} + 16 q^{34} + 2 q^{36} - 20 q^{41} + 2 q^{44} - 6 q^{46} - 18 q^{49} - 16 q^{51} - 2 q^{54} - 8 q^{56} - 8 q^{59} + 10 q^{61} - 2 q^{64} - 2 q^{66} + 6 q^{69} - 12 q^{71} + 4 q^{74} + 2 q^{76} + 10 q^{79} + 2 q^{81} + 8 q^{84} - 16 q^{86} - 6 q^{89} + 2 q^{96} + 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 - 2 * q^11 + 8 * q^14 + 2 * q^16 - 2 * q^19 - 8 * q^21 - 2 * q^24 + 2 * q^29 + 2 * q^31 + 16 * q^34 + 2 * q^36 - 20 * q^41 + 2 * q^44 - 6 * q^46 - 18 * q^49 - 16 * q^51 - 2 * q^54 - 8 * q^56 - 8 * q^59 + 10 * q^61 - 2 * q^64 - 2 * q^66 + 6 * q^69 - 12 * q^71 + 4 * q^74 + 2 * q^76 + 10 * q^79 + 2 * q^81 + 8 * q^84 - 16 * q^86 - 6 * q^89 + 2 * q^96 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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