LMFDB - Embedded newform 2850.2.d.g.799.1

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

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} +1.00000 q^{11} +1.00000i q^{12} -4.00000i q^{13} +1.00000 q^{16} +4.00000i q^{17} +1.00000i q^{18} +1.00000 q^{19} -1.00000i q^{22} -5.00000i q^{23} +1.00000 q^{24} -4.00000 q^{26} +1.00000i q^{27} -3.00000 q^{29} -5.00000 q^{31} -1.00000i q^{32} -1.00000i q^{33} +4.00000 q^{34} +1.00000 q^{36} -6.00000i q^{37} -1.00000i q^{38} -4.00000 q^{39} -2.00000 q^{41} -4.00000i q^{43} -1.00000 q^{44} -5.00000 q^{46} -1.00000i q^{48} +7.00000 q^{49} +4.00000 q^{51} +4.00000i q^{52} -9.00000i q^{53} +1.00000 q^{54} -1.00000i q^{57} +3.00000i q^{58} -11.0000 q^{61} +5.00000i q^{62} -1.00000 q^{64} -1.00000 q^{66} -1.00000i q^{67} -4.00000i q^{68} -5.00000 q^{69} +2.00000 q^{71} -1.00000i q^{72} +3.00000i q^{73} -6.00000 q^{74} -1.00000 q^{76} +4.00000i q^{78} -17.0000 q^{79} +1.00000 q^{81} +2.00000i q^{82} -3.00000i q^{83} -4.00000 q^{86} +3.00000i q^{87} +1.00000i q^{88} -7.00000 q^{89} +5.00000i q^{92} +5.00000i q^{93} -1.00000 q^{96} +10.0000i q^{97} -7.00000i q^{98} -1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 2 q^{16} + 2 q^{19} + 2 q^{24} - 8 q^{26} - 6 q^{29} - 10 q^{31} + 8 q^{34} + 2 q^{36} - 8 q^{39} - 4 q^{41} - 2 q^{44} - 10 q^{46} + 14 q^{49} + 8 q^{51} + 2 q^{54} - 22 q^{61} - 2 q^{64} - 2 q^{66} - 10 q^{69} + 4 q^{71} - 12 q^{74} - 2 q^{76} - 34 q^{79} + 2 q^{81} - 8 q^{86} - 14 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 + 2 * q^16 + 2 * q^19 + 2 * q^24 - 8 * q^26 - 6 * q^29 - 10 * q^31 + 8 * q^34 + 2 * q^36 - 8 * q^39 - 4 * q^41 - 2 * q^44 - 10 * q^46 + 14 * q^49 + 8 * q^51 + 2 * q^54 - 22 * q^61 - 2 * q^64 - 2 * q^66 - 10 * q^69 + 4 * q^71 - 12 * q^74 - 2 * q^76 - 34 * q^79 + 2 * q^81 - 8 * q^86 - 14 * 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
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	1.00000i	0.353553i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	1.00000	0.301511	0.150756	−	0.988571i	$-0.451829\pi$
$11$	1.00000	0.301511	0.150756	+	0.988571i	$0.451829\pi$
$12$	1.00000i	0.288675i
$13$	− 4.00000i	− 1.10940i	−0.832050	−	0.554700i	$-0.812833\pi$
$13$	− 4.00000i	− 1.10940i	0.832050	−	0.554700i	$-0.187167\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	4.00000i	0.970143i	0.874475	+	0.485071i	$0.161206\pi$
$17$	4.00000i	0.970143i	−0.874475	+	0.485071i	$0.838794\pi$
$18$	1.00000i	0.235702i
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	− 1.00000i	− 0.213201i
$23$	− 5.00000i	− 1.04257i	−0.853382	−	0.521286i	$-0.825452\pi$
$23$	− 5.00000i	− 1.04257i	0.853382	−	0.521286i	$-0.174548\pi$
$24$	1.00000	0.204124
$25$	0	0
$26$	−4.00000	−0.784465
$27$	1.00000i	0.192450i
$28$	0	0
$29$	−3.00000	−0.557086	−0.278543	−	0.960424i	$-0.589851\pi$
$29$	−3.00000	−0.557086	−0.278543	+	0.960424i	$0.589851\pi$
$30$	0	0
$31$	−5.00000	−0.898027	−0.449013	−	0.893525i	$-0.648224\pi$
$31$	−5.00000	−0.898027	−0.449013	+	0.893525i	$0.648224\pi$
$32$	− 1.00000i	− 0.176777i
$33$	− 1.00000i	− 0.174078i
$34$	4.00000	0.685994
$35$	0	0
$36$	1.00000	0.166667
$37$	− 6.00000i	− 0.986394i	−0.869918	−	0.493197i	$-0.835828\pi$
$37$	− 6.00000i	− 0.986394i	0.869918	−	0.493197i	$-0.164172\pi$
$38$	− 1.00000i	− 0.162221i
$39$	−4.00000	−0.640513
$40$	0	0
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	0	0
$43$	− 4.00000i	− 0.609994i	−0.952353	−	0.304997i	$-0.901344\pi$
$43$	− 4.00000i	− 0.609994i	0.952353	−	0.304997i	$-0.0986555\pi$
$44$	−1.00000	−0.150756
$45$	0	0
$46$	−5.00000	−0.737210
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	− 1.00000i	− 0.144338i
$49$	7.00000	1.00000
$50$	0	0
$51$	4.00000	0.560112
$52$	4.00000i	0.554700i
$53$	− 9.00000i	− 1.23625i	−0.786082	−	0.618123i	$-0.787894\pi$
$53$	− 9.00000i	− 1.23625i	0.786082	−	0.618123i	$-0.212106\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	0	0
$57$	− 1.00000i	− 0.132453i
$58$	3.00000i	0.393919i
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	−11.0000	−1.40841	−0.704203	−	0.709999i	$-0.748695\pi$
$61$	−11.0000	−1.40841	−0.704203	+	0.709999i	$0.748695\pi$
$62$	5.00000i	0.635001i
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	−1.00000	−0.123091
$67$	− 1.00000i	− 0.122169i	−0.998133	−	0.0610847i	$-0.980544\pi$
$67$	− 1.00000i	− 0.122169i	0.998133	−	0.0610847i	$-0.0194560\pi$
$68$	− 4.00000i	− 0.485071i
$69$	−5.00000	−0.601929
$70$	0	0
$71$	2.00000	0.237356	0.118678	−	0.992933i	$-0.462134\pi$
$71$	2.00000	0.237356	0.118678	+	0.992933i	$0.462134\pi$
$72$	− 1.00000i	− 0.117851i
$73$	3.00000i	0.351123i	0.984468	+	0.175562i	$0.0561742\pi$
$73$	3.00000i	0.351123i	−0.984468	+	0.175562i	$0.943826\pi$
$74$	−6.00000	−0.697486
$75$	0	0
$76$	−1.00000	−0.114708
$77$	0	0
$78$	4.00000i	0.452911i
$79$	−17.0000	−1.91265	−0.956325	−	0.292306i	$-0.905577\pi$
$79$	−17.0000	−1.91265	−0.956325	+	0.292306i	$0.905577\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	2.00000i	0.220863i
$83$	− 3.00000i	− 0.329293i	−0.986353	−	0.164646i	$-0.947352\pi$
$83$	− 3.00000i	− 0.329293i	0.986353	−	0.164646i	$-0.0526483\pi$
$84$	0	0
$85$	0	0
$86$	−4.00000	−0.431331
$87$	3.00000i	0.321634i
$88$	1.00000i	0.106600i
$89$	−7.00000	−0.741999	−0.370999	−	0.928633i	$-0.620985\pi$
$89$	−7.00000	−0.741999	−0.370999	+	0.928633i	$0.620985\pi$
$90$	0	0
$91$	0	0
$92$	5.00000i	0.521286i
$93$	5.00000i	0.518476i
$94$	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	$0.830506\pi$
$98$	− 7.00000i	− 0.707107i
$99$	−1.00000	−0.100504
$100$	0	0
$101$	−8.00000	−0.796030	−0.398015	−	0.917379i	$-0.630301\pi$
$101$	−8.00000	−0.796030	−0.398015	+	0.917379i	$0.630301\pi$
$102$	− 4.00000i	− 0.396059i
$103$	− 13.0000i	− 1.28093i	−0.767988	−	0.640464i	$-0.778742\pi$
$103$	− 13.0000i	− 1.28093i	0.767988	−	0.640464i	$-0.221258\pi$
$104$	4.00000	0.392232
$105$	0	0
$106$	−9.00000	−0.874157
$107$	10.0000i	0.966736i	0.875417	+	0.483368i	$0.160587\pi$
$107$	10.0000i	0.966736i	−0.875417	+	0.483368i	$0.839413\pi$
$108$	− 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
$111$	−6.00000	−0.569495
$112$	0	0
$113$	− 5.00000i	− 0.470360i	−0.971952	−	0.235180i	$-0.924432\pi$
$113$	− 5.00000i	− 0.470360i	0.971952	−	0.235180i	$-0.0755680\pi$
$114$	−1.00000	−0.0936586
$115$	0	0
$116$	3.00000	0.278543
$117$	4.00000i	0.369800i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−10.0000	−0.909091
$122$	11.0000i	0.995893i
$123$	2.00000i	0.180334i
$124$	5.00000	0.449013
$125$	0	0
$126$	0	0
$127$	− 5.00000i	− 0.443678i	−0.975083	−	0.221839i	$-0.928794\pi$
$127$	− 5.00000i	− 0.443678i	0.975083	−	0.221839i	$-0.0712060\pi$
$128$	1.00000i	0.0883883i
$129$	−4.00000	−0.352180
$130$	0	0
$131$	15.0000	1.31056	0.655278	−	0.755388i	$-0.272551\pi$
$131$	15.0000	1.31056	0.655278	+	0.755388i	$0.272551\pi$
$132$	1.00000i	0.0870388i
$133$	0	0
$134$	−1.00000	−0.0863868
$135$	0	0
$136$	−4.00000	−0.342997
$137$	− 4.00000i	− 0.341743i	−0.985293	−	0.170872i	$-0.945342\pi$
$137$	− 4.00000i	− 0.341743i	0.985293	−	0.170872i	$-0.0546583\pi$
$138$	5.00000i	0.425628i
$139$	2.00000	0.169638	0.0848189	−	0.996396i	$-0.472969\pi$
$139$	2.00000	0.169638	0.0848189	+	0.996396i	$0.472969\pi$
$140$	0	0
$141$	0	0
$142$	− 2.00000i	− 0.167836i
$143$	− 4.00000i	− 0.334497i
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	3.00000	0.248282
$147$	− 7.00000i	− 0.577350i
$148$	6.00000i	0.493197i
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	−24.0000	−1.95309	−0.976546	−	0.215308i	$-0.930924\pi$
$151$	−24.0000	−1.95309	−0.976546	+	0.215308i	$0.930924\pi$
$152$	1.00000i	0.0811107i
$153$	− 4.00000i	− 0.323381i
$154$	0	0
$155$	0	0
$156$	4.00000	0.320256
$157$	14.0000i	1.11732i	0.829396	+	0.558661i	$0.188685\pi$
$157$	14.0000i	1.11732i	−0.829396	+	0.558661i	$0.811315\pi$
$158$	17.0000i	1.35245i
$159$	−9.00000	−0.713746
$160$	0	0
$161$	0	0
$162$	− 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	$-0.128089\pi$
$164$	2.00000	0.156174
$165$	0	0
$166$	−3.00000	−0.232845
$167$	18.0000i	1.39288i	0.717614	+	0.696441i	$0.245234\pi$
$167$	18.0000i	1.39288i	−0.717614	+	0.696441i	$0.754766\pi$
$168$	0	0
$169$	−3.00000	−0.230769
$170$	0	0
$171$	−1.00000	−0.0764719
$172$	4.00000i	0.304997i
$173$	− 7.00000i	− 0.532200i	−0.963945	−	0.266100i	$-0.914265\pi$
$173$	− 7.00000i	− 0.532200i	0.963945	−	0.266100i	$-0.0857352\pi$
$174$	3.00000	0.227429
$175$	0	0
$176$	1.00000	0.0753778
$177$	0	0
$178$	7.00000i	0.524672i
$179$	4.00000	0.298974	0.149487	−	0.988764i	$-0.452238\pi$
$179$	4.00000	0.298974	0.149487	+	0.988764i	$0.452238\pi$
$180$	0	0
$181$	0	0		−	1.00000i	$-0.5\pi$
$181$	0	0			1.00000i	$0.5\pi$
$182$	0	0
$183$	11.0000i	0.813143i
$184$	5.00000	0.368605
$185$	0	0
$186$	5.00000	0.366618
$187$	4.00000i	0.292509i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	11.0000	0.795932	0.397966	−	0.917400i	$-0.369716\pi$
$191$	11.0000	0.795932	0.397966	+	0.917400i	$0.369716\pi$
$192$	1.00000i	0.0721688i
$193$	− 10.0000i	− 0.719816i	−0.932988	−	0.359908i	$-0.882808\pi$
$193$	− 10.0000i	− 0.719816i	0.932988	−	0.359908i	$-0.117192\pi$
$194$	10.0000	0.717958
$195$	0	0
$196$	−7.00000	−0.500000
$197$	− 20.0000i	− 1.42494i	−0.701702	−	0.712470i	$-0.747576\pi$
$197$	− 20.0000i	− 1.42494i	0.701702	−	0.712470i	$-0.252424\pi$
$198$	1.00000i	0.0710669i
$199$	−6.00000	−0.425329	−0.212664	−	0.977125i	$-0.568214\pi$
$199$	−6.00000	−0.425329	−0.212664	+	0.977125i	$0.568214\pi$
$200$	0	0
$201$	−1.00000	−0.0705346
$202$	8.00000i	0.562878i
$203$	0	0
$204$	−4.00000	−0.280056
$205$	0	0
$206$	−13.0000	−0.905753
$207$	5.00000i	0.347524i
$208$	− 4.00000i	− 0.277350i
$209$	1.00000	0.0691714
$210$	0	0
$211$	−13.0000	−0.894957	−0.447478	−	0.894295i	$-0.647678\pi$
$211$	−13.0000	−0.894957	−0.447478	+	0.894295i	$0.647678\pi$
$212$	9.00000i	0.618123i
$213$	− 2.00000i	− 0.137038i
$214$	10.0000	0.683586
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	0	0
$218$	6.00000i	0.406371i
$219$	3.00000	0.202721
$220$	0	0
$221$	16.0000	1.07628
$222$	6.00000i	0.402694i
$223$	15.0000i	1.00447i	0.864730	+	0.502237i	$0.167490\pi$
$223$	15.0000i	1.00447i	−0.864730	+	0.502237i	$0.832510\pi$
$224$	0	0
$225$	0	0
$226$	−5.00000	−0.332595
$227$	2.00000i	0.132745i	0.997795	+	0.0663723i	$0.0211425\pi$
$227$	2.00000i	0.132745i	−0.997795	+	0.0663723i	$0.978857\pi$
$228$	1.00000i	0.0662266i
$229$	−5.00000	−0.330409	−0.165205	−	0.986259i	$-0.552828\pi$
$229$	−5.00000	−0.330409	−0.165205	+	0.986259i	$0.552828\pi$
$230$	0	0
$231$	0	0
$232$	− 3.00000i	− 0.196960i
$233$	24.0000i	1.57229i	0.618041	+	0.786146i	$0.287927\pi$
$233$	24.0000i	1.57229i	−0.618041	+	0.786146i	$0.712073\pi$
$234$	4.00000	0.261488
$235$	0	0
$236$	0	0
$237$	17.0000i	1.10427i
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	−4.00000	−0.257663	−0.128831	−	0.991667i	$-0.541123\pi$
$241$	−4.00000	−0.257663	−0.128831	+	0.991667i	$0.541123\pi$
$242$	10.0000i	0.642824i
$243$	− 1.00000i	− 0.0641500i
$244$	11.0000	0.704203
$245$	0	0
$246$	2.00000	0.127515
$247$	− 4.00000i	− 0.254514i
$248$	− 5.00000i	− 0.317500i
$249$	−3.00000	−0.190117
$250$	0	0
$251$	16.0000	1.00991	0.504956	−	0.863145i	$-0.331509\pi$
$251$	16.0000	1.00991	0.504956	+	0.863145i	$0.331509\pi$
$252$	0	0
$253$	− 5.00000i	− 0.314347i
$254$	−5.00000	−0.313728
$255$	0	0
$256$	1.00000	0.0625000
$257$	7.00000i	0.436648i	0.975876	+	0.218324i	$0.0700590\pi$
$257$	7.00000i	0.436648i	−0.975876	+	0.218324i	$0.929941\pi$
$258$	4.00000i	0.249029i
$259$	0	0
$260$	0	0
$261$	3.00000	0.185695
$262$	− 15.0000i	− 0.926703i
$263$	11.0000i	0.678289i	0.940734	+	0.339145i	$0.110138\pi$
$263$	11.0000i	0.678289i	−0.940734	+	0.339145i	$0.889862\pi$
$264$	1.00000	0.0615457
$265$	0	0
$266$	0	0
$267$	7.00000i	0.428393i
$268$	1.00000i	0.0610847i
$269$	2.00000	0.121942	0.0609711	−	0.998140i	$-0.480580\pi$
$269$	2.00000	0.121942	0.0609711	+	0.998140i	$0.480580\pi$
$270$	0	0
$271$	2.00000	0.121491	0.0607457	−	0.998153i	$-0.480652\pi$
$271$	2.00000	0.121491	0.0607457	+	0.998153i	$0.480652\pi$
$272$	4.00000i	0.242536i
$273$	0	0
$274$	−4.00000	−0.241649
$275$	0	0
$276$	5.00000	0.300965
$277$	− 1.00000i	− 0.0600842i	−0.999549	−	0.0300421i	$-0.990436\pi$
$277$	− 1.00000i	− 0.0600842i	0.999549	−	0.0300421i	$-0.00956413\pi$
$278$	− 2.00000i	− 0.119952i
$279$	5.00000	0.299342
$280$	0	0
$281$	−3.00000	−0.178965	−0.0894825	−	0.995988i	$-0.528521\pi$
$281$	−3.00000	−0.178965	−0.0894825	+	0.995988i	$0.528521\pi$
$282$	0	0
$283$	8.00000i	0.475551i	0.971320	+	0.237775i	$0.0764182\pi$
$283$	8.00000i	0.475551i	−0.971320	+	0.237775i	$0.923582\pi$
$284$	−2.00000	−0.118678
$285$	0	0
$286$	−4.00000	−0.236525
$287$	0	0
$288$	1.00000i	0.0589256i
$289$	1.00000	0.0588235
$290$	0	0
$291$	10.0000	0.586210
$292$	− 3.00000i	− 0.175562i
$293$	− 11.0000i	− 0.642627i	−0.946973	−	0.321313i	$-0.895876\pi$
$293$	− 11.0000i	− 0.642627i	0.946973	−	0.321313i	$-0.104124\pi$
$294$	−7.00000	−0.408248
$295$	0	0
$296$	6.00000	0.348743
$297$	1.00000i	0.0580259i
$298$	0	0
$299$	−20.0000	−1.15663
$300$	0	0
$301$	0	0
$302$	24.0000i	1.38104i
$303$	8.00000i	0.459588i
$304$	1.00000	0.0573539
$305$	0	0
$306$	−4.00000	−0.228665
$307$	− 25.0000i	− 1.42683i	−0.700744	−	0.713413i	$-0.747149\pi$
$307$	− 25.0000i	− 1.42683i	0.700744	−	0.713413i	$-0.252851\pi$
$308$	0	0
$309$	−13.0000	−0.739544
$310$	0	0
$311$	32.0000	1.81455	0.907277	−	0.420534i	$-0.138157\pi$
$311$	32.0000	1.81455	0.907277	+	0.420534i	$0.138157\pi$
$312$	− 4.00000i	− 0.226455i
$313$	− 21.0000i	− 1.18699i	−0.804838	−	0.593495i	$-0.797748\pi$
$313$	− 21.0000i	− 1.18699i	0.804838	−	0.593495i	$-0.202252\pi$
$314$	14.0000	0.790066
$315$	0	0
$316$	17.0000	0.956325
$317$	− 13.0000i	− 0.730153i	−0.930978	−	0.365076i	$-0.881043\pi$
$317$	− 13.0000i	− 0.730153i	0.930978	−	0.365076i	$-0.118957\pi$
$318$	9.00000i	0.504695i
$319$	−3.00000	−0.167968
$320$	0	0
$321$	10.0000	0.558146
$322$	0	0
$323$	4.00000i	0.222566i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	−10.0000	−0.553849
$327$	6.00000i	0.331801i
$328$	− 2.00000i	− 0.110432i
$329$	0	0
$330$	0	0
$331$	13.0000	0.714545	0.357272	−	0.934000i	$-0.383707\pi$
$331$	13.0000	0.714545	0.357272	+	0.934000i	$0.383707\pi$
$332$	3.00000i	0.164646i
$333$	6.00000i	0.328798i
$334$	18.0000	0.984916
$335$	0	0
$336$	0	0
$337$	− 8.00000i	− 0.435788i	−0.975972	−	0.217894i	$-0.930081\pi$
$337$	− 8.00000i	− 0.435788i	0.975972	−	0.217894i	$-0.0699187\pi$
$338$	3.00000i	0.163178i
$339$	−5.00000	−0.271563
$340$	0	0
$341$	−5.00000	−0.270765
$342$	1.00000i	0.0540738i
$343$	0	0
$344$	4.00000	0.215666
$345$	0	0
$346$	−7.00000	−0.376322
$347$	− 12.0000i	− 0.644194i	−0.946707	−	0.322097i	$-0.895612\pi$
$347$	− 12.0000i	− 0.644194i	0.946707	−	0.322097i	$-0.104388\pi$
$348$	− 3.00000i	− 0.160817i
$349$	1.00000	0.0535288	0.0267644	−	0.999642i	$-0.491480\pi$
$349$	1.00000	0.0535288	0.0267644	+	0.999642i	$0.491480\pi$
$350$	0	0
$351$	4.00000	0.213504
$352$	− 1.00000i	− 0.0533002i
$353$	− 28.0000i	− 1.49029i	−0.666903	−	0.745145i	$-0.732380\pi$
$353$	− 28.0000i	− 1.49029i	0.666903	−	0.745145i	$-0.267620\pi$
$354$	0	0
$355$	0	0
$356$	7.00000	0.370999
$357$	0	0
$358$	− 4.00000i	− 0.211407i
$359$	−24.0000	−1.26667	−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000	−1.26667	−0.633336	+	0.773877i	$0.718315\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	10.0000i	0.524864i
$364$	0	0
$365$	0	0
$366$	11.0000	0.574979
$367$	− 18.0000i	− 0.939592i	−0.882775	−	0.469796i	$-0.844327\pi$
$367$	− 18.0000i	− 0.939592i	0.882775	−	0.469796i	$-0.155673\pi$
$368$	− 5.00000i	− 0.260643i
$369$	2.00000	0.104116
$370$	0	0
$371$	0	0
$372$	− 5.00000i	− 0.259238i
$373$	− 28.0000i	− 1.44979i	−0.688862	−	0.724893i	$-0.741889\pi$
$373$	− 28.0000i	− 1.44979i	0.688862	−	0.724893i	$-0.258111\pi$
$374$	4.00000	0.206835
$375$	0	0
$376$	0	0
$377$	12.0000i	0.618031i
$378$	0	0
$379$	8.00000	0.410932	0.205466	−	0.978664i	$-0.434129\pi$
$379$	8.00000	0.410932	0.205466	+	0.978664i	$0.434129\pi$
$380$	0	0
$381$	−5.00000	−0.256158
$382$	− 11.0000i	− 0.562809i
$383$	6.00000i	0.306586i	0.988181	+	0.153293i	$0.0489878\pi$
$383$	6.00000i	0.306586i	−0.988181	+	0.153293i	$0.951012\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	−10.0000	−0.508987
$387$	4.00000i	0.203331i
$388$	− 10.0000i	− 0.507673i
$389$	−26.0000	−1.31825	−0.659126	−	0.752032i	$-0.729074\pi$
$389$	−26.0000	−1.31825	−0.659126	+	0.752032i	$0.729074\pi$
$390$	0	0
$391$	20.0000	1.01144
$392$	7.00000i	0.353553i
$393$	− 15.0000i	− 0.756650i
$394$	−20.0000	−1.00759
$395$	0	0
$396$	1.00000	0.0502519
$397$	− 3.00000i	− 0.150566i	−0.997162	−	0.0752828i	$-0.976014\pi$
$397$	− 3.00000i	− 0.150566i	0.997162	−	0.0752828i	$-0.0239860\pi$
$398$	6.00000i	0.300753i
$399$	0	0
$400$	0	0
$401$	−17.0000	−0.848939	−0.424470	−	0.905442i	$-0.639539\pi$
$401$	−17.0000	−0.848939	−0.424470	+	0.905442i	$0.639539\pi$
$402$	1.00000i	0.0498755i
$403$	20.0000i	0.996271i
$404$	8.00000	0.398015
$405$	0	0
$406$	0	0
$407$	− 6.00000i	− 0.297409i
$408$	4.00000i	0.198030i
$409$	−6.00000	−0.296681	−0.148340	−	0.988936i	$-0.547393\pi$
$409$	−6.00000	−0.296681	−0.148340	+	0.988936i	$0.547393\pi$
$410$	0	0
$411$	−4.00000	−0.197305
$412$	13.0000i	0.640464i
$413$	0	0
$414$	5.00000	0.245737
$415$	0	0
$416$	−4.00000	−0.196116
$417$	− 2.00000i	− 0.0979404i
$418$	− 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
$421$	−34.0000	−1.65706	−0.828529	−	0.559946i	$-0.810822\pi$
$421$	−34.0000	−1.65706	−0.828529	+	0.559946i	$0.810822\pi$
$422$	13.0000i	0.632830i
$423$	0	0
$424$	9.00000	0.437079
$425$	0	0
$426$	−2.00000	−0.0969003
$427$	0	0
$428$	− 10.0000i	− 0.483368i
$429$	−4.00000	−0.193122
$430$	0	0
$431$	38.0000	1.83040	0.915198	−	0.403005i	$-0.132034\pi$
$431$	38.0000	1.83040	0.915198	+	0.403005i	$0.132034\pi$
$432$	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	$0.822707\pi$
$434$	0	0
$435$	0	0
$436$	6.00000	0.287348
$437$	− 5.00000i	− 0.239182i
$438$	− 3.00000i	− 0.143346i
$439$	−21.0000	−1.00228	−0.501138	−	0.865368i	$-0.667085\pi$
$439$	−21.0000	−1.00228	−0.501138	+	0.865368i	$0.667085\pi$
$440$	0	0
$441$	−7.00000	−0.333333
$442$	− 16.0000i	− 0.761042i
$443$	13.0000i	0.617649i	0.951119	+	0.308824i	$0.0999355\pi$
$443$	13.0000i	0.617649i	−0.951119	+	0.308824i	$0.900064\pi$
$444$	6.00000	0.284747
$445$	0	0
$446$	15.0000	0.710271
$447$	0	0
$448$	0	0
$449$	27.0000	1.27421	0.637104	−	0.770778i	$-0.280132\pi$
$449$	27.0000	1.27421	0.637104	+	0.770778i	$0.280132\pi$
$450$	0	0
$451$	−2.00000	−0.0941763
$452$	5.00000i	0.235180i
$453$	24.0000i	1.12762i
$454$	2.00000	0.0938647
$455$	0	0
$456$	1.00000	0.0468293
$457$	− 18.0000i	− 0.842004i	−0.907060	−	0.421002i	$-0.861678\pi$
$457$	− 18.0000i	− 0.842004i	0.907060	−	0.421002i	$-0.138322\pi$
$458$	5.00000i	0.233635i
$459$	−4.00000	−0.186704
$460$	0	0
$461$	4.00000	0.186299	0.0931493	−	0.995652i	$-0.470307\pi$
$461$	4.00000	0.186299	0.0931493	+	0.995652i	$0.470307\pi$
$462$	0	0
$463$	− 4.00000i	− 0.185896i	−0.995671	−	0.0929479i	$-0.970371\pi$
$463$	− 4.00000i	− 0.185896i	0.995671	−	0.0929479i	$-0.0296290\pi$
$464$	−3.00000	−0.139272
$465$	0	0
$466$	24.0000	1.11178
$467$	− 11.0000i	− 0.509019i	−0.967070	−	0.254510i	$-0.918086\pi$
$467$	− 11.0000i	− 0.509019i	0.967070	−	0.254510i	$-0.0819141\pi$
$468$	− 4.00000i	− 0.184900i
$469$	0	0
$470$	0	0
$471$	14.0000	0.645086
$472$	0	0
$473$	− 4.00000i	− 0.183920i
$474$	17.0000	0.780836
$475$	0	0
$476$	0	0
$477$	9.00000i	0.412082i
$478$	0	0
$479$	37.0000	1.69057	0.845287	−	0.534313i	$-0.179430\pi$
$479$	37.0000	1.69057	0.845287	+	0.534313i	$0.179430\pi$
$480$	0	0
$481$	−24.0000	−1.09431
$482$	4.00000i	0.182195i
$483$	0	0
$484$	10.0000	0.454545
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	8.00000i	0.362515i	0.983436	+	0.181257i	$0.0580167\pi$
$487$	8.00000i	0.362515i	−0.983436	+	0.181257i	$0.941983\pi$
$488$	− 11.0000i	− 0.497947i
$489$	−10.0000	−0.452216
$490$	0	0
$491$	36.0000	1.62466	0.812329	−	0.583200i	$-0.198200\pi$
$491$	36.0000	1.62466	0.812329	+	0.583200i	$0.198200\pi$
$492$	− 2.00000i	− 0.0901670i
$493$	− 12.0000i	− 0.540453i
$494$	−4.00000	−0.179969
$495$	0	0
$496$	−5.00000	−0.224507
$497$	0	0
$498$	3.00000i	0.134433i
$499$	24.0000	1.07439	0.537194	−	0.843459i	$-0.319484\pi$
$499$	24.0000	1.07439	0.537194	+	0.843459i	$0.319484\pi$
$500$	0	0
$501$	18.0000	0.804181
$502$	− 16.0000i	− 0.714115i
$503$	36.0000i	1.60516i	0.596544	+	0.802580i	$0.296540\pi$
$503$	36.0000i	1.60516i	−0.596544	+	0.802580i	$0.703460\pi$
$504$	0	0
$505$	0	0
$506$	−5.00000	−0.222277
$507$	3.00000i	0.133235i
$508$	5.00000i	0.221839i
$509$	27.0000	1.19675	0.598377	−	0.801215i	$-0.295813\pi$
$509$	27.0000	1.19675	0.598377	+	0.801215i	$0.295813\pi$
$510$	0	0
$511$	0	0
$512$	− 1.00000i	− 0.0441942i
$513$	1.00000i	0.0441511i
$514$	7.00000	0.308757
$515$	0	0
$516$	4.00000	0.176090
$517$	0	0
$518$	0	0
$519$	−7.00000	−0.307266
$520$	0	0
$521$	3.00000	0.131432	0.0657162	−	0.997838i	$-0.479067\pi$
$521$	3.00000	0.131432	0.0657162	+	0.997838i	$0.479067\pi$
$522$	− 3.00000i	− 0.131306i
$523$	− 20.0000i	− 0.874539i	−0.899331	−	0.437269i	$-0.855946\pi$
$523$	− 20.0000i	− 0.874539i	0.899331	−	0.437269i	$-0.144054\pi$
$524$	−15.0000	−0.655278
$525$	0	0
$526$	11.0000	0.479623
$527$	− 20.0000i	− 0.871214i
$528$	− 1.00000i	− 0.0435194i
$529$	−2.00000	−0.0869565
$530$	0	0
$531$	0	0
$532$	0	0
$533$	8.00000i	0.346518i
$534$	7.00000	0.302920
$535$	0	0
$536$	1.00000	0.0431934
$537$	− 4.00000i	− 0.172613i
$538$	− 2.00000i	− 0.0862261i
$539$	7.00000	0.301511
$540$	0	0
$541$	−13.0000	−0.558914	−0.279457	−	0.960158i	$-0.590154\pi$
$541$	−13.0000	−0.558914	−0.279457	+	0.960158i	$0.590154\pi$
$542$	− 2.00000i	− 0.0859074i
$543$	0	0
$544$	4.00000	0.171499
$545$	0	0
$546$	0	0
$547$	1.00000i	0.0427569i	0.999771	+	0.0213785i	$0.00680549\pi$
$547$	1.00000i	0.0427569i	−0.999771	+	0.0213785i	$0.993195\pi$
$548$	4.00000i	0.170872i
$549$	11.0000	0.469469
$550$	0	0
$551$	−3.00000	−0.127804
$552$	− 5.00000i	− 0.212814i
$553$	0	0
$554$	−1.00000	−0.0424859
$555$	0	0
$556$	−2.00000	−0.0848189
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	− 5.00000i	− 0.211667i
$559$	−16.0000	−0.676728
$560$	0	0
$561$	4.00000	0.168880
$562$	3.00000i	0.126547i
$563$	− 26.0000i	− 1.09577i	−0.836554	−	0.547885i	$-0.815433\pi$
$563$	− 26.0000i	− 1.09577i	0.836554	−	0.547885i	$-0.184567\pi$
$564$	0	0
$565$	0	0
$566$	8.00000	0.336265
$567$	0	0
$568$	2.00000i	0.0839181i
$569$	−22.0000	−0.922288	−0.461144	−	0.887325i	$-0.652561\pi$
$569$	−22.0000	−0.922288	−0.461144	+	0.887325i	$0.652561\pi$
$570$	0	0
$571$	−26.0000	−1.08807	−0.544033	−	0.839064i	$-0.683103\pi$
$571$	−26.0000	−1.08807	−0.544033	+	0.839064i	$0.683103\pi$
$572$	4.00000i	0.167248i
$573$	− 11.0000i	− 0.459532i
$574$	0	0
$575$	0	0
$576$	1.00000	0.0416667
$577$	− 27.0000i	− 1.12402i	−0.827129	−	0.562012i	$-0.810027\pi$
$577$	− 27.0000i	− 1.12402i	0.827129	−	0.562012i	$-0.189973\pi$
$578$	− 1.00000i	− 0.0415945i
$579$	−10.0000	−0.415586
$580$	0	0
$581$	0	0
$582$	− 10.0000i	− 0.414513i
$583$	− 9.00000i	− 0.372742i
$584$	−3.00000	−0.124141
$585$	0	0
$586$	−11.0000	−0.454406
$587$	− 17.0000i	− 0.701665i	−0.936438	−	0.350833i	$-0.885899\pi$
$587$	− 17.0000i	− 0.701665i	0.936438	−	0.350833i	$-0.114101\pi$
$588$	7.00000i	0.288675i
$589$	−5.00000	−0.206021
$590$	0	0
$591$	−20.0000	−0.822690
$592$	− 6.00000i	− 0.246598i
$593$	28.0000i	1.14982i	0.818216	+	0.574911i	$0.194963\pi$
$593$	28.0000i	1.14982i	−0.818216	+	0.574911i	$0.805037\pi$
$594$	1.00000	0.0410305
$595$	0	0
$596$	0	0
$597$	6.00000i	0.245564i
$598$	20.0000i	0.817861i
$599$	−28.0000	−1.14405	−0.572024	−	0.820237i	$-0.693842\pi$
$599$	−28.0000	−1.14405	−0.572024	+	0.820237i	$0.693842\pi$
$600$	0	0
$601$	−2.00000	−0.0815817	−0.0407909	−	0.999168i	$-0.512988\pi$
$601$	−2.00000	−0.0815817	−0.0407909	+	0.999168i	$0.512988\pi$
$602$	0	0
$603$	1.00000i	0.0407231i
$604$	24.0000	0.976546
$605$	0	0
$606$	8.00000	0.324978
$607$	− 25.0000i	− 1.01472i	−0.861735	−	0.507359i	$-0.830622\pi$
$607$	− 25.0000i	− 1.01472i	0.861735	−	0.507359i	$-0.169378\pi$
$608$	− 1.00000i	− 0.0405554i
$609$	0	0
$610$	0	0
$611$	0	0
$612$	4.00000i	0.161690i
$613$	− 26.0000i	− 1.05013i	−0.851062	−	0.525065i	$-0.824041\pi$
$613$	− 26.0000i	− 1.05013i	0.851062	−	0.525065i	$-0.175959\pi$
$614$	−25.0000	−1.00892
$615$	0	0
$616$	0	0
$617$	− 46.0000i	− 1.85189i	−0.377658	−	0.925945i	$-0.623271\pi$
$617$	− 46.0000i	− 1.85189i	0.377658	−	0.925945i	$-0.376729\pi$
$618$	13.0000i	0.522937i
$619$	8.00000	0.321547	0.160774	−	0.986991i	$-0.448601\pi$
$619$	8.00000	0.321547	0.160774	+	0.986991i	$0.448601\pi$
$620$	0	0
$621$	5.00000	0.200643
$622$	− 32.0000i	− 1.28308i
$623$	0	0
$624$	−4.00000	−0.160128
$625$	0	0
$626$	−21.0000	−0.839329
$627$	− 1.00000i	− 0.0399362i
$628$	− 14.0000i	− 0.558661i
$629$	24.0000	0.956943
$630$	0	0
$631$	28.0000	1.11466	0.557331	−	0.830290i	$-0.311825\pi$
$631$	28.0000	1.11466	0.557331	+	0.830290i	$0.311825\pi$
$632$	− 17.0000i	− 0.676224i
$633$	13.0000i	0.516704i
$634$	−13.0000	−0.516296
$635$	0	0
$636$	9.00000	0.356873
$637$	− 28.0000i	− 1.10940i
$638$	3.00000i	0.118771i
$639$	−2.00000	−0.0791188
$640$	0	0
$641$	6.00000	0.236986	0.118493	−	0.992955i	$-0.462194\pi$
$641$	6.00000	0.236986	0.118493	+	0.992955i	$0.462194\pi$
$642$	− 10.0000i	− 0.394669i
$643$	14.0000i	0.552106i	0.961142	+	0.276053i	$0.0890266\pi$
$643$	14.0000i	0.552106i	−0.961142	+	0.276053i	$0.910973\pi$
$644$	0	0
$645$	0	0
$646$	4.00000	0.157378
$647$	− 21.0000i	− 0.825595i	−0.910823	−	0.412798i	$-0.864552\pi$
$647$	− 21.0000i	− 0.825595i	0.910823	−	0.412798i	$-0.135448\pi$
$648$	1.00000i	0.0392837i
$649$	0	0
$650$	0	0
$651$	0	0
$652$	10.0000i	0.391630i
$653$	48.0000i	1.87839i	0.343391	+	0.939193i	$0.388424\pi$
$653$	48.0000i	1.87839i	−0.343391	+	0.939193i	$0.611576\pi$
$654$	6.00000	0.234619
$655$	0	0
$656$	−2.00000	−0.0780869
$657$	− 3.00000i	− 0.117041i
$658$	0	0
$659$	36.0000	1.40236	0.701180	−	0.712984i	$-0.252657\pi$
$659$	36.0000	1.40236	0.701180	+	0.712984i	$0.252657\pi$
$660$	0	0
$661$	−32.0000	−1.24466	−0.622328	−	0.782757i	$-0.713813\pi$
$661$	−32.0000	−1.24466	−0.622328	+	0.782757i	$0.713813\pi$
$662$	− 13.0000i	− 0.505259i
$663$	− 16.0000i	− 0.621389i
$664$	3.00000	0.116423
$665$	0	0
$666$	6.00000	0.232495
$667$	15.0000i	0.580802i
$668$	− 18.0000i	− 0.696441i
$669$	15.0000	0.579934
$670$	0	0
$671$	−11.0000	−0.424650
$672$	0	0
$673$	− 44.0000i	− 1.69608i	−0.529936	−	0.848038i	$-0.677784\pi$
$673$	− 44.0000i	− 1.69608i	0.529936	−	0.848038i	$-0.322216\pi$
$674$	−8.00000	−0.308148
$675$	0	0
$676$	3.00000	0.115385
$677$	− 15.0000i	− 0.576497i	−0.957556	−	0.288248i	$-0.906927\pi$
$677$	− 15.0000i	− 0.576497i	0.957556	−	0.288248i	$-0.0930729\pi$
$678$	5.00000i	0.192024i
$679$	0	0
$680$	0	0
$681$	2.00000	0.0766402
$682$	5.00000i	0.191460i
$683$	18.0000i	0.688751i	0.938832	+	0.344375i	$0.111909\pi$
$683$	18.0000i	0.688751i	−0.938832	+	0.344375i	$0.888091\pi$
$684$	1.00000	0.0382360
$685$	0	0
$686$	0	0
$687$	5.00000i	0.190762i
$688$	− 4.00000i	− 0.152499i
$689$	−36.0000	−1.37149
$690$	0	0
$691$	26.0000	0.989087	0.494543	−	0.869153i	$-0.335335\pi$
$691$	26.0000	0.989087	0.494543	+	0.869153i	$0.335335\pi$
$692$	7.00000i	0.266100i
$693$	0	0
$694$	−12.0000	−0.455514
$695$	0	0
$696$	−3.00000	−0.113715
$697$	− 8.00000i	− 0.303022i
$698$	− 1.00000i	− 0.0378506i
$699$	24.0000	0.907763
$700$	0	0
$701$	0	0		−	1.00000i	$-0.5\pi$
$701$	0	0			1.00000i	$0.5\pi$
$702$	− 4.00000i	− 0.150970i
$703$	− 6.00000i	− 0.226294i
$704$	−1.00000	−0.0376889
$705$	0	0
$706$	−28.0000	−1.05379
$707$	0	0
$708$	0	0
$709$	43.0000	1.61490	0.807449	−	0.589937i	$-0.200847\pi$
$709$	43.0000	1.61490	0.807449	+	0.589937i	$0.200847\pi$
$710$	0	0
$711$	17.0000	0.637550
$712$	− 7.00000i	− 0.262336i
$713$	25.0000i	0.936257i
$714$	0	0
$715$	0	0
$716$	−4.00000	−0.149487
$717$	0	0
$718$	24.0000i	0.895672i
$719$	29.0000	1.08152	0.540759	−	0.841178i	$-0.318137\pi$
$719$	29.0000	1.08152	0.540759	+	0.841178i	$0.318137\pi$
$720$	0	0
$721$	0	0
$722$	− 1.00000i	− 0.0372161i
$723$	4.00000i	0.148762i
$724$	0	0
$725$	0	0
$726$	10.0000	0.371135
$727$	− 10.0000i	− 0.370879i	−0.982656	−	0.185440i	$-0.940629\pi$
$727$	− 10.0000i	− 0.370879i	0.982656	−	0.185440i	$-0.0593710\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	16.0000	0.591781
$732$	− 11.0000i	− 0.406572i
$733$	− 41.0000i	− 1.51437i	−0.653201	−	0.757185i	$-0.726574\pi$
$733$	− 41.0000i	− 1.51437i	0.653201	−	0.757185i	$-0.273426\pi$
$734$	−18.0000	−0.664392
$735$	0	0
$736$	−5.00000	−0.184302
$737$	− 1.00000i	− 0.0368355i
$738$	− 2.00000i	− 0.0736210i
$739$	20.0000	0.735712	0.367856	−	0.929883i	$-0.380092\pi$
$739$	20.0000	0.735712	0.367856	+	0.929883i	$0.380092\pi$
$740$	0	0
$741$	−4.00000	−0.146944
$742$	0	0
$743$	16.0000i	0.586983i	0.955962	+	0.293492i	$0.0948173\pi$
$743$	16.0000i	0.586983i	−0.955962	+	0.293492i	$0.905183\pi$
$744$	−5.00000	−0.183309
$745$	0	0
$746$	−28.0000	−1.02515
$747$	3.00000i	0.109764i
$748$	− 4.00000i	− 0.146254i
$749$	0	0
$750$	0	0
$751$	32.0000	1.16770	0.583848	−	0.811863i	$-0.301546\pi$
$751$	32.0000	1.16770	0.583848	+	0.811863i	$0.301546\pi$
$752$	0	0
$753$	− 16.0000i	− 0.583072i
$754$	12.0000	0.437014
$755$	0	0
$756$	0	0
$757$	− 23.0000i	− 0.835949i	−0.908459	−	0.417975i	$-0.862740\pi$
$757$	− 23.0000i	− 0.835949i	0.908459	−	0.417975i	$-0.137260\pi$
$758$	− 8.00000i	− 0.290573i
$759$	−5.00000	−0.181489
$760$	0	0
$761$	−30.0000	−1.08750	−0.543750	−	0.839248i	$-0.682996\pi$
$761$	−30.0000	−1.08750	−0.543750	+	0.839248i	$0.682996\pi$
$762$	5.00000i	0.181131i
$763$	0	0
$764$	−11.0000	−0.397966
$765$	0	0
$766$	6.00000	0.216789
$767$	0	0
$768$	− 1.00000i	− 0.0360844i
$769$	−35.0000	−1.26213	−0.631066	−	0.775729i	$-0.717382\pi$
$769$	−35.0000	−1.26213	−0.631066	+	0.775729i	$0.717382\pi$
$770$	0	0
$771$	7.00000	0.252099
$772$	10.0000i	0.359908i
$773$	− 18.0000i	− 0.647415i	−0.946157	−	0.323708i	$-0.895071\pi$
$773$	− 18.0000i	− 0.647415i	0.946157	−	0.323708i	$-0.104929\pi$
$774$	4.00000	0.143777
$775$	0	0
$776$	−10.0000	−0.358979
$777$	0	0
$778$	26.0000i	0.932145i
$779$	−2.00000	−0.0716574
$780$	0	0
$781$	2.00000	0.0715656
$782$	− 20.0000i	− 0.715199i
$783$	− 3.00000i	− 0.107211i
$784$	7.00000	0.250000
$785$	0	0
$786$	−15.0000	−0.535032
$787$	37.0000i	1.31891i	0.751745	+	0.659454i	$0.229212\pi$
$787$	37.0000i	1.31891i	−0.751745	+	0.659454i	$0.770788\pi$
$788$	20.0000i	0.712470i
$789$	11.0000	0.391610
$790$	0	0
$791$	0	0
$792$	− 1.00000i	− 0.0355335i
$793$	44.0000i	1.56249i
$794$	−3.00000	−0.106466
$795$	0	0
$796$	6.00000	0.212664
$797$	− 18.0000i	− 0.637593i	−0.947823	−	0.318796i	$-0.896721\pi$
$797$	− 18.0000i	− 0.637593i	0.947823	−	0.318796i	$-0.103279\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	7.00000	0.247333
$802$	17.0000i	0.600291i
$803$	3.00000i	0.105868i
$804$	1.00000	0.0352673
$805$	0	0
$806$	20.0000	0.704470
$807$	− 2.00000i	− 0.0704033i
$808$	− 8.00000i	− 0.281439i
$809$	44.0000	1.54696	0.773479	−	0.633822i	$-0.218515\pi$
$809$	44.0000	1.54696	0.773479	+	0.633822i	$0.218515\pi$
$810$	0	0
$811$	−35.0000	−1.22902	−0.614508	−	0.788911i	$-0.710645\pi$
$811$	−35.0000	−1.22902	−0.614508	+	0.788911i	$0.710645\pi$
$812$	0	0
$813$	− 2.00000i	− 0.0701431i
$814$	−6.00000	−0.210300
$815$	0	0
$816$	4.00000	0.140028
$817$	− 4.00000i	− 0.139942i
$818$	6.00000i	0.209785i
$819$	0	0
$820$	0	0
$821$	26.0000	0.907406	0.453703	−	0.891153i	$-0.350103\pi$
$821$	26.0000	0.907406	0.453703	+	0.891153i	$0.350103\pi$
$822$	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	$0.850300\pi$
$824$	13.0000	0.452876
$825$	0	0
$826$	0	0
$827$	48.0000i	1.66912i	0.550914	+	0.834562i	$0.314279\pi$
$827$	48.0000i	1.66912i	−0.550914	+	0.834562i	$0.685721\pi$
$828$	− 5.00000i	− 0.173762i
$829$	50.0000	1.73657	0.868286	−	0.496064i	$-0.165222\pi$
$829$	50.0000	1.73657	0.868286	+	0.496064i	$0.165222\pi$
$830$	0	0
$831$	−1.00000	−0.0346896
$832$	4.00000i	0.138675i
$833$	28.0000i	0.970143i
$834$	−2.00000	−0.0692543
$835$	0	0
$836$	−1.00000	−0.0345857
$837$	− 5.00000i	− 0.172825i
$838$	− 12.0000i	− 0.414533i
$839$	12.0000	0.414286	0.207143	−	0.978311i	$-0.433583\pi$
$839$	12.0000	0.414286	0.207143	+	0.978311i	$0.433583\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	34.0000i	1.17172i
$843$	3.00000i	0.103325i
$844$	13.0000	0.447478
$845$	0	0
$846$	0	0
$847$	0	0
$848$	− 9.00000i	− 0.309061i
$849$	8.00000	0.274559
$850$	0	0
$851$	−30.0000	−1.02839
$852$	2.00000i	0.0685189i
$853$	− 10.0000i	− 0.342393i	−0.985237	−	0.171197i	$-0.945237\pi$
$853$	− 10.0000i	− 0.342393i	0.985237	−	0.171197i	$-0.0547634\pi$
$854$	0	0
$855$	0	0
$856$	−10.0000	−0.341793
$857$	− 38.0000i	− 1.29806i	−0.760765	−	0.649028i	$-0.775176\pi$
$857$	− 38.0000i	− 1.29806i	0.760765	−	0.649028i	$-0.224824\pi$
$858$	4.00000i	0.136558i
$859$	18.0000	0.614152	0.307076	−	0.951685i	$-0.400649\pi$
$859$	18.0000	0.614152	0.307076	+	0.951685i	$0.400649\pi$
$860$	0	0
$861$	0	0
$862$	− 38.0000i	− 1.29429i
$863$	− 4.00000i	− 0.136162i	−0.997680	−	0.0680808i	$-0.978312\pi$
$863$	− 4.00000i	− 0.136162i	0.997680	−	0.0680808i	$-0.0216876\pi$
$864$	1.00000	0.0340207
$865$	0	0
$866$	22.0000	0.747590
$867$	− 1.00000i	− 0.0339618i
$868$	0	0
$869$	−17.0000	−0.576686
$870$	0	0
$871$	−4.00000	−0.135535
$872$	− 6.00000i	− 0.203186i
$873$	− 10.0000i	− 0.338449i
$874$	−5.00000	−0.169128
$875$	0	0
$876$	−3.00000	−0.101361
$877$	− 2.00000i	− 0.0675352i	−0.999430	−	0.0337676i	$-0.989249\pi$
$877$	− 2.00000i	− 0.0675352i	0.999430	−	0.0337676i	$-0.0107506\pi$
$878$	21.0000i	0.708716i
$879$	−11.0000	−0.371021
$880$	0	0
$881$	−48.0000	−1.61716	−0.808581	−	0.588386i	$-0.799764\pi$
$881$	−48.0000	−1.61716	−0.808581	+	0.588386i	$0.799764\pi$
$882$	7.00000i	0.235702i
$883$	38.0000i	1.27880i	0.768874	+	0.639401i	$0.220818\pi$
$883$	38.0000i	1.27880i	−0.768874	+	0.639401i	$0.779182\pi$
$884$	−16.0000	−0.538138
$885$	0	0
$886$	13.0000	0.436744
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	− 6.00000i	− 0.201347i
$889$	0	0
$890$	0	0
$891$	1.00000	0.0335013
$892$	− 15.0000i	− 0.502237i
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	20.0000i	0.667781i
$898$	− 27.0000i	− 0.901002i
$899$	15.0000	0.500278
$900$	0	0
$901$	36.0000	1.19933
$902$	2.00000i	0.0665927i
$903$	0	0
$904$	5.00000	0.166298
$905$	0	0
$906$	24.0000	0.797347
$907$	8.00000i	0.265636i	0.991140	+	0.132818i	$0.0424025\pi$
$907$	8.00000i	0.265636i	−0.991140	+	0.132818i	$0.957597\pi$
$908$	− 2.00000i	− 0.0663723i
$909$	8.00000	0.265343
$910$	0	0
$911$	−18.0000	−0.596367	−0.298183	−	0.954509i	$-0.596381\pi$
$911$	−18.0000	−0.596367	−0.298183	+	0.954509i	$0.596381\pi$
$912$	− 1.00000i	− 0.0331133i
$913$	− 3.00000i	− 0.0992855i
$914$	−18.0000	−0.595387
$915$	0	0
$916$	5.00000	0.165205
$917$	0	0
$918$	4.00000i	0.132020i
$919$	−14.0000	−0.461817	−0.230909	−	0.972975i	$-0.574170\pi$
$919$	−14.0000	−0.461817	−0.230909	+	0.972975i	$0.574170\pi$
$920$	0	0
$921$	−25.0000	−0.823778
$922$	− 4.00000i	− 0.131733i
$923$	− 8.00000i	− 0.263323i
$924$	0	0
$925$	0	0
$926$	−4.00000	−0.131448
$927$	13.0000i	0.426976i
$928$	3.00000i	0.0984798i
$929$	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
$931$	7.00000	0.229416
$932$	− 24.0000i	− 0.786146i
$933$	− 32.0000i	− 1.04763i
$934$	−11.0000	−0.359931
$935$	0	0
$936$	−4.00000	−0.130744
$937$	− 10.0000i	− 0.326686i	−0.986569	−	0.163343i	$-0.947772\pi$
$937$	− 10.0000i	− 0.326686i	0.986569	−	0.163343i	$-0.0522277\pi$
$938$	0	0
$939$	−21.0000	−0.685309
$940$	0	0
$941$	33.0000	1.07577	0.537885	−	0.843018i	$-0.319224\pi$
$941$	33.0000	1.07577	0.537885	+	0.843018i	$0.319224\pi$
$942$	− 14.0000i	− 0.456145i
$943$	10.0000i	0.325645i
$944$	0	0
$945$	0	0
$946$	−4.00000	−0.130051
$947$	20.0000i	0.649913i	0.945729	+	0.324956i	$0.105350\pi$
$947$	20.0000i	0.649913i	−0.945729	+	0.324956i	$0.894650\pi$
$948$	− 17.0000i	− 0.552134i
$949$	12.0000	0.389536
$950$	0	0
$951$	−13.0000	−0.421554
$952$	0	0
$953$	− 3.00000i	− 0.0971795i	−0.998819	−	0.0485898i	$-0.984527\pi$
$953$	− 3.00000i	− 0.0971795i	0.998819	−	0.0485898i	$-0.0154727\pi$
$954$	9.00000	0.291386
$955$	0	0
$956$	0	0
$957$	3.00000i	0.0969762i
$958$	− 37.0000i	− 1.19542i
$959$	0	0
$960$	0	0
$961$	−6.00000	−0.193548
$962$	24.0000i	0.773791i
$963$	− 10.0000i	− 0.322245i
$964$	4.00000	0.128831
$965$	0	0
$966$	0	0
$967$	14.0000i	0.450210i	0.974335	+	0.225105i	$0.0722725\pi$
$967$	14.0000i	0.450210i	−0.974335	+	0.225105i	$0.927728\pi$
$968$	− 10.0000i	− 0.321412i
$969$	4.00000	0.128499
$970$	0	0
$971$	−12.0000	−0.385098	−0.192549	−	0.981287i	$-0.561675\pi$
$971$	−12.0000	−0.385098	−0.192549	+	0.981287i	$0.561675\pi$
$972$	1.00000i	0.0320750i
$973$	0	0
$974$	8.00000	0.256337
$975$	0	0
$976$	−11.0000	−0.352101
$977$	46.0000i	1.47167i	0.677161	+	0.735835i	$0.263210\pi$
$977$	46.0000i	1.47167i	−0.677161	+	0.735835i	$0.736790\pi$
$978$	10.0000i	0.319765i
$979$	−7.00000	−0.223721
$980$	0	0
$981$	6.00000	0.191565
$982$	− 36.0000i	− 1.14881i
$983$	− 34.0000i	− 1.08443i	−0.840239	−	0.542216i	$-0.817586\pi$
$983$	− 34.0000i	− 1.08443i	0.840239	−	0.542216i	$-0.182414\pi$
$984$	−2.00000	−0.0637577
$985$	0	0
$986$	−12.0000	−0.382158
$987$	0	0
$988$	4.00000i	0.127257i
$989$	−20.0000	−0.635963
$990$	0	0
$991$	13.0000	0.412959	0.206479	−	0.978451i	$-0.433799\pi$
$991$	13.0000	0.412959	0.206479	+	0.978451i	$0.433799\pi$
$992$	5.00000i	0.158750i
$993$	− 13.0000i	− 0.412543i
$994$	0	0
$995$	0	0
$996$	3.00000	0.0950586
$997$	37.0000i	1.17180i	0.810383	+	0.585901i	$0.199259\pi$
$997$	37.0000i	1.17180i	−0.810383	+	0.585901i	$0.800741\pi$
$998$	− 24.0000i	− 0.759707i
$999$	6.00000	0.189832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2850.2.d.g.799.1		2
5.2	odd	4		2850.2.a.u.1.1	yes	1
5.3	odd	4		2850.2.a.l.1.1	✓	1
5.4	even	2	inner	2850.2.d.g.799.2		2
15.2	even	4		8550.2.a.h.1.1		1
15.8	even	4		8550.2.a.z.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2850.2.a.l.1.1	✓	1	5.3	odd	4
2850.2.a.u.1.1	yes	1	5.2	odd	4
2850.2.d.g.799.1		2	1.1	even	1	trivial
2850.2.d.g.799.2		2	5.4	even	2	inner
8550.2.a.h.1.1		1	15.2	even	4
8550.2.a.z.1.1		1	15.8	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)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} +1.00000 q^{11} +1.00000i q^{12} -4.00000i q^{13} +1.00000 q^{16} +4.00000i q^{17} +1.00000i q^{18} +1.00000 q^{19} -1.00000i q^{22} -5.00000i q^{23} +1.00000 q^{24} -4.00000 q^{26} +1.00000i q^{27} -3.00000 q^{29} -5.00000 q^{31} -1.00000i q^{32} -1.00000i q^{33} +4.00000 q^{34} +1.00000 q^{36} -6.00000i q^{37} -1.00000i q^{38} -4.00000 q^{39} -2.00000 q^{41} -4.00000i q^{43} -1.00000 q^{44} -5.00000 q^{46} -1.00000i q^{48} +7.00000 q^{49} +4.00000 q^{51} +4.00000i q^{52} -9.00000i q^{53} +1.00000 q^{54} -1.00000i q^{57} +3.00000i q^{58} -11.0000 q^{61} +5.00000i q^{62} -1.00000 q^{64} -1.00000 q^{66} -1.00000i q^{67} -4.00000i q^{68} -5.00000 q^{69} +2.00000 q^{71} -1.00000i q^{72} +3.00000i q^{73} -6.00000 q^{74} -1.00000 q^{76} +4.00000i q^{78} -17.0000 q^{79} +1.00000 q^{81} +2.00000i q^{82} -3.00000i q^{83} -4.00000 q^{86} +3.00000i q^{87} +1.00000i q^{88} -7.00000 q^{89} +5.00000i q^{92} +5.00000i q^{93} -1.00000 q^{96} +10.0000i q^{97} -7.00000i q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9	\( 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9} + 2 q^{11} + 2 q^{16} + 2 q^{19} + 2 q^{24} - 8 q^{26} - 6 q^{29} - 10 q^{31} + 8 q^{34} + 2 q^{36} - 8 q^{39} - 4 q^{41} - 2 q^{44} - 10 q^{46} + 14 q^{49} + 8 q^{51} + 2 q^{54} - 22 q^{61} - 2 q^{64} - 2 q^{66} - 10 q^{69} + 4 q^{71} - 12 q^{74} - 2 q^{76} - 34 q^{79} + 2 q^{81} - 8 q^{86} - 14 q^{89} - 2 q^{96} - 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9 + 2 * q^11 + 2 * q^16 + 2 * q^19 + 2 * q^24 - 8 * q^26 - 6 * q^29 - 10 * q^31 + 8 * q^34 + 2 * q^36 - 8 * q^39 - 4 * q^41 - 2 * q^44 - 10 * q^46 + 14 * q^49 + 8 * q^51 + 2 * q^54 - 22 * q^61 - 2 * q^64 - 2 * q^66 - 10 * q^69 + 4 * q^71 - 12 * q^74 - 2 * q^76 - 34 * q^79 + 2 * q^81 - 8 * q^86 - 14 * q^89 - 2 * q^96 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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