LMFDB - Embedded newform 2450.2.c.g.99.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2450,2,Mod(99,2450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2450, 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("2450.99");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2450 = 2 \cdot 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2450.c (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:	$19.5633484952$
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:	no (minimal twist has level 70)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			99.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	2450.99
Dual form			2450.2.c.g.99.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +1.00000i q^{8} +2.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +1.00000i q^{8} +2.00000 q^{9} -6.00000 q^{11} +1.00000i q^{12} +4.00000i q^{13} +1.00000 q^{16} -2.00000i q^{18} -2.00000 q^{19} +6.00000i q^{22} +3.00000i q^{23} +1.00000 q^{24} +4.00000 q^{26} -5.00000i q^{27} +3.00000 q^{29} +8.00000 q^{31} -1.00000i q^{32} +6.00000i q^{33} -2.00000 q^{36} -4.00000i q^{37} +2.00000i q^{38} +4.00000 q^{39} +9.00000 q^{41} +7.00000i q^{43} +6.00000 q^{44} +3.00000 q^{46} -1.00000i q^{48} -4.00000i q^{52} +6.00000i q^{53} -5.00000 q^{54} +2.00000i q^{57} -3.00000i q^{58} +6.00000 q^{59} +5.00000 q^{61} -8.00000i q^{62} -1.00000 q^{64} +6.00000 q^{66} +5.00000i q^{67} +3.00000 q^{69} -6.00000 q^{71} +2.00000i q^{72} +16.0000i q^{73} -4.00000 q^{74} +2.00000 q^{76} -4.00000i q^{78} -2.00000 q^{79} +1.00000 q^{81} -9.00000i q^{82} -3.00000i q^{83} +7.00000 q^{86} -3.00000i q^{87} -6.00000i q^{88} +15.0000 q^{89} -3.00000i q^{92} -8.00000i q^{93} -1.00000 q^{96} +14.0000i q^{97} -12.0000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} - 2 q^{6} + 4 q^{9}+O(q^{10}) $ 2 * q - 2 * q^4 - 2 * q^6 + 4 * q^9	$ 2 q - 2 q^{4} - 2 q^{6} + 4 q^{9} - 12 q^{11} + 2 q^{16} - 4 q^{19} + 2 q^{24} + 8 q^{26} + 6 q^{29} + 16 q^{31} - 4 q^{36} + 8 q^{39} + 18 q^{41} + 12 q^{44} + 6 q^{46} - 10 q^{54} + 12 q^{59} + 10 q^{61} - 2 q^{64} + 12 q^{66} + 6 q^{69} - 12 q^{71} - 8 q^{74} + 4 q^{76} - 4 q^{79} + 2 q^{81} + 14 q^{86} + 30 q^{89} - 2 q^{96} - 24 q^{99}+O(q^{100}) $ 2 * q - 2 * q^4 - 2 * q^6 + 4 * q^9 - 12 * q^11 + 2 * q^16 - 4 * q^19 + 2 * q^24 + 8 * q^26 + 6 * q^29 + 16 * q^31 - 4 * q^36 + 8 * q^39 + 18 * q^41 + 12 * q^44 + 6 * q^46 - 10 * q^54 + 12 * q^59 + 10 * q^61 - 2 * q^64 + 12 * q^66 + 6 * q^69 - 12 * q^71 - 8 * q^74 + 4 * q^76 - 4 * q^79 + 2 * q^81 + 14 * q^86 + 30 * q^89 - 2 * q^96 - 24 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2450\mathbb{Z}\right)^\times$.

$n$	$101$	$1177$
$\chi(n)$	$1$	$-1$

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 $

$2$	− 1.00000i	− 0.707107i
$2$	− 1.00000i	− 0.707107i
$3$	− 1.00000i	− 0.577350i	−0.957427	−	0.288675i	$-0.906785\pi$
$3$	− 1.00000i	− 0.577350i	0.957427	−	0.288675i	$-0.0932147\pi$
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	−1.00000	−0.408248
$7$	0	0
$7$	0	0
$8$	1.00000i	0.353553i
$9$	2.00000	0.666667
$10$	0	0
$11$	−6.00000	−1.80907	−0.904534	−	0.426401i	$-0.859781\pi$
$11$	−6.00000	−1.80907	−0.904534	+	0.426401i	$0.859781\pi$
$12$	1.00000i	0.288675i
$13$	4.00000i	1.10940i	0.832050	+	0.554700i	$0.187167\pi$
$13$	4.00000i	1.10940i	−0.832050	+	0.554700i	$0.812833\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	− 2.00000i	− 0.471405i
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	0	0
$21$	0	0
$22$	6.00000i	1.27920i
$23$	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
$26$	4.00000	0.784465
$27$	− 5.00000i	− 0.962250i
$28$	0	0
$29$	3.00000	0.557086	0.278543	−	0.960424i	$-0.410149\pi$
$29$	3.00000	0.557086	0.278543	+	0.960424i	$0.410149\pi$
$30$	0	0
$31$	8.00000	1.43684	0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000	1.43684	0.718421	+	0.695608i	$0.244865\pi$
$32$	− 1.00000i	− 0.176777i
$33$	6.00000i	1.04447i
$34$	0	0
$35$	0	0
$36$	−2.00000	−0.333333
$37$	− 4.00000i	− 0.657596i	−0.944400	−	0.328798i	$-0.893356\pi$
$37$	− 4.00000i	− 0.657596i	0.944400	−	0.328798i	$-0.106644\pi$
$38$	2.00000i	0.324443i
$39$	4.00000	0.640513
$40$	0	0
$41$	9.00000	1.40556	0.702782	−	0.711405i	$-0.251941\pi$
$41$	9.00000	1.40556	0.702782	+	0.711405i	$0.251941\pi$
$42$	0	0
$43$	7.00000i	1.06749i	0.845645	+	0.533745i	$0.179216\pi$
$43$	7.00000i	1.06749i	−0.845645	+	0.533745i	$0.820784\pi$
$44$	6.00000	0.904534
$45$	0	0
$46$	3.00000	0.442326
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	− 1.00000i	− 0.144338i
$49$	0	0
$50$	0	0
$51$	0	0
$52$	− 4.00000i	− 0.554700i
$53$	6.00000i	0.824163i	0.911147	+	0.412082i	$0.135198\pi$
$53$	6.00000i	0.824163i	−0.911147	+	0.412082i	$0.864802\pi$
$54$	−5.00000	−0.680414
$55$	0	0
$56$	0	0
$57$	2.00000i	0.264906i
$58$	− 3.00000i	− 0.393919i
$59$	6.00000	0.781133	0.390567	−	0.920575i	$-0.372279\pi$
$59$	6.00000	0.781133	0.390567	+	0.920575i	$0.372279\pi$
$60$	0	0
$61$	5.00000	0.640184	0.320092	−	0.947386i	$-0.396286\pi$
$61$	5.00000	0.640184	0.320092	+	0.947386i	$0.396286\pi$
$62$	− 8.00000i	− 1.01600i
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	6.00000	0.738549
$67$	5.00000i	0.610847i	0.952217	+	0.305424i	$0.0987981\pi$
$67$	5.00000i	0.610847i	−0.952217	+	0.305424i	$0.901202\pi$
$68$	0	0
$69$	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	$0.615871\pi$
$72$	2.00000i	0.235702i
$73$	16.0000i	1.87266i	0.351123	+	0.936329i	$0.385800\pi$
$73$	16.0000i	1.87266i	−0.351123	+	0.936329i	$0.614200\pi$
$74$	−4.00000	−0.464991
$75$	0	0
$76$	2.00000	0.229416
$77$	0	0
$78$	− 4.00000i	− 0.452911i
$79$	−2.00000	−0.225018	−0.112509	−	0.993651i	$-0.535889\pi$
$79$	−2.00000	−0.225018	−0.112509	+	0.993651i	$0.535889\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	− 9.00000i	− 0.993884i
$83$	− 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
$86$	7.00000	0.754829
$87$	− 3.00000i	− 0.321634i
$88$	− 6.00000i	− 0.639602i
$89$	15.0000	1.59000	0.794998	−	0.606612i	$-0.207472\pi$
$89$	15.0000	1.59000	0.794998	+	0.606612i	$0.207472\pi$
$90$	0	0
$91$	0	0
$92$	− 3.00000i	− 0.312772i
$93$	− 8.00000i	− 0.829561i
$94$	0	0
$95$	0	0
$96$	−1.00000	−0.102062
$97$	14.0000i	1.42148i	0.703452	+	0.710742i	$0.251641\pi$
$97$	14.0000i	1.42148i	−0.703452	+	0.710742i	$0.748359\pi$
$98$	0	0
$99$	−12.0000	−1.20605
$100$	0	0
$101$	15.0000	1.49256	0.746278	−	0.665635i	$-0.231839\pi$
$101$	15.0000	1.49256	0.746278	+	0.665635i	$0.231839\pi$
$102$	0	0
$103$	1.00000i	0.0985329i	0.998786	+	0.0492665i	$0.0156884\pi$
$103$	1.00000i	0.0985329i	−0.998786	+	0.0492665i	$0.984312\pi$
$104$	−4.00000	−0.392232
$105$	0	0
$106$	6.00000	0.582772
$107$	− 15.0000i	− 1.45010i	−0.688694	−	0.725052i	$-0.741816\pi$
$107$	− 15.0000i	− 1.45010i	0.688694	−	0.725052i	$-0.258184\pi$
$108$	5.00000i	0.481125i
$109$	−11.0000	−1.05361	−0.526804	−	0.849987i	$-0.676610\pi$
$109$	−11.0000	−1.05361	−0.526804	+	0.849987i	$0.676610\pi$
$110$	0	0
$111$	−4.00000	−0.379663
$112$	0	0
$113$	− 6.00000i	− 0.564433i	−0.959351	−	0.282216i	$-0.908930\pi$
$113$	− 6.00000i	− 0.564433i	0.959351	−	0.282216i	$-0.0910696\pi$
$114$	2.00000	0.187317
$115$	0	0
$116$	−3.00000	−0.278543
$117$	8.00000i	0.739600i
$118$	− 6.00000i	− 0.552345i
$119$	0	0
$120$	0	0
$121$	25.0000	2.27273
$122$	− 5.00000i	− 0.452679i
$123$	− 9.00000i	− 0.811503i
$124$	−8.00000	−0.718421
$125$	0	0
$126$	0	0
$127$	8.00000i	0.709885i	0.934888	+	0.354943i	$0.115500\pi$
$127$	8.00000i	0.709885i	−0.934888	+	0.354943i	$0.884500\pi$
$128$	1.00000i	0.0883883i
$129$	7.00000	0.616316
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	− 6.00000i	− 0.522233i
$133$	0	0
$134$	5.00000	0.431934
$135$	0	0
$136$	0	0
$137$	12.0000i	1.02523i	0.858619	+	0.512615i	$0.171323\pi$
$137$	12.0000i	1.02523i	−0.858619	+	0.512615i	$0.828677\pi$
$138$	− 3.00000i	− 0.255377i
$139$	10.0000	0.848189	0.424094	−	0.905618i	$-0.360592\pi$
$139$	10.0000	0.848189	0.424094	+	0.905618i	$0.360592\pi$
$140$	0	0
$141$	0	0
$142$	6.00000i	0.503509i
$143$	− 24.0000i	− 2.00698i
$144$	2.00000	0.166667
$145$	0	0
$146$	16.0000	1.32417
$147$	0	0
$148$	4.00000i	0.328798i
$149$	−15.0000	−1.22885	−0.614424	−	0.788976i	$-0.710612\pi$
$149$	−15.0000	−1.22885	−0.614424	+	0.788976i	$0.710612\pi$
$150$	0	0
$151$	−4.00000	−0.325515	−0.162758	−	0.986666i	$-0.552039\pi$
$151$	−4.00000	−0.325515	−0.162758	+	0.986666i	$0.552039\pi$
$152$	− 2.00000i	− 0.162221i
$153$	0	0
$154$	0	0
$155$	0	0
$156$	−4.00000	−0.320256
$157$	− 22.0000i	− 1.75579i	−0.478852	−	0.877896i	$-0.658947\pi$
$157$	− 22.0000i	− 1.75579i	0.478852	−	0.877896i	$-0.341053\pi$
$158$	2.00000i	0.159111i
$159$	6.00000	0.475831
$160$	0	0
$161$	0	0
$162$	− 1.00000i	− 0.0785674i
$163$	4.00000i	0.313304i	0.987654	+	0.156652i	$0.0500701\pi$
$163$	4.00000i	0.313304i	−0.987654	+	0.156652i	$0.949930\pi$
$164$	−9.00000	−0.702782
$165$	0	0
$166$	−3.00000	−0.232845
$167$	− 3.00000i	− 0.232147i	−0.993241	−	0.116073i	$-0.962969\pi$
$167$	− 3.00000i	− 0.232147i	0.993241	−	0.116073i	$-0.0370308\pi$
$168$	0	0
$169$	−3.00000	−0.230769
$170$	0	0
$171$	−4.00000	−0.305888
$172$	− 7.00000i	− 0.533745i
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	−3.00000	−0.227429
$175$	0	0
$176$	−6.00000	−0.452267
$177$	− 6.00000i	− 0.450988i
$178$	− 15.0000i	− 1.12430i
$179$	24.0000	1.79384	0.896922	−	0.442189i	$-0.145798\pi$
$179$	24.0000	1.79384	0.896922	+	0.442189i	$0.145798\pi$
$180$	0	0
$181$	11.0000	0.817624	0.408812	−	0.912619i	$-0.365943\pi$
$181$	11.0000	0.817624	0.408812	+	0.912619i	$0.365943\pi$
$182$	0	0
$183$	− 5.00000i	− 0.369611i
$184$	−3.00000	−0.221163
$185$	0	0
$186$	−8.00000	−0.586588
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−6.00000	−0.434145	−0.217072	−	0.976156i	$-0.569651\pi$
$191$	−6.00000	−0.434145	−0.217072	+	0.976156i	$0.569651\pi$
$192$	1.00000i	0.0721688i
$193$	− 2.00000i	− 0.143963i	−0.997406	−	0.0719816i	$-0.977068\pi$
$193$	− 2.00000i	− 0.143963i	0.997406	−	0.0719816i	$-0.0229323\pi$
$194$	14.0000	1.00514
$195$	0	0
$196$	0	0
$197$	− 6.00000i	− 0.427482i	−0.976890	−	0.213741i	$-0.931435\pi$
$197$	− 6.00000i	− 0.427482i	0.976890	−	0.213741i	$-0.0685649\pi$
$198$	12.0000i	0.852803i
$199$	4.00000	0.283552	0.141776	−	0.989899i	$-0.454719\pi$
$199$	4.00000	0.283552	0.141776	+	0.989899i	$0.454719\pi$
$200$	0	0
$201$	5.00000	0.352673
$202$	− 15.0000i	− 1.05540i
$203$	0	0
$204$	0	0
$205$	0	0
$206$	1.00000	0.0696733
$207$	6.00000i	0.417029i
$208$	4.00000i	0.277350i
$209$	12.0000	0.830057
$210$	0	0
$211$	−10.0000	−0.688428	−0.344214	−	0.938891i	$-0.611855\pi$
$211$	−10.0000	−0.688428	−0.344214	+	0.938891i	$0.611855\pi$
$212$	− 6.00000i	− 0.412082i
$213$	6.00000i	0.411113i
$214$	−15.0000	−1.02538
$215$	0	0
$216$	5.00000	0.340207
$217$	0	0
$218$	11.0000i	0.745014i
$219$	16.0000	1.08118
$220$	0	0
$221$	0	0
$222$	4.00000i	0.268462i
$223$	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
$226$	−6.00000	−0.399114
$227$	− 12.0000i	− 0.796468i	−0.917284	−	0.398234i	$-0.869623\pi$
$227$	− 12.0000i	− 0.796468i	0.917284	−	0.398234i	$-0.130377\pi$
$228$	− 2.00000i	− 0.132453i
$229$	−14.0000	−0.925146	−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000	−0.925146	−0.462573	+	0.886581i	$0.653074\pi$
$230$	0	0
$231$	0	0
$232$	3.00000i	0.196960i
$233$	12.0000i	0.786146i	0.919507	+	0.393073i	$0.128588\pi$
$233$	12.0000i	0.786146i	−0.919507	+	0.393073i	$0.871412\pi$
$234$	8.00000	0.522976
$235$	0	0
$236$	−6.00000	−0.390567
$237$	2.00000i	0.129914i
$238$	0	0
$239$	−12.0000	−0.776215	−0.388108	−	0.921614i	$-0.626871\pi$
$239$	−12.0000	−0.776215	−0.388108	+	0.921614i	$0.626871\pi$
$240$	0	0
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	− 25.0000i	− 1.60706i
$243$	− 16.0000i	− 1.02640i
$244$	−5.00000	−0.320092
$245$	0	0
$246$	−9.00000	−0.573819
$247$	− 8.00000i	− 0.509028i
$248$	8.00000i	0.508001i
$249$	−3.00000	−0.190117
$250$	0	0
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	0	0
$253$	− 18.0000i	− 1.13165i
$254$	8.00000	0.501965
$255$	0	0
$256$	1.00000	0.0625000
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	− 7.00000i	− 0.435801i
$259$	0	0
$260$	0	0
$261$	6.00000	0.371391
$262$	0	0
$263$	21.0000i	1.29492i	0.762101	+	0.647458i	$0.224168\pi$
$263$	21.0000i	1.29492i	−0.762101	+	0.647458i	$0.775832\pi$
$264$	−6.00000	−0.369274
$265$	0	0
$266$	0	0
$267$	− 15.0000i	− 0.917985i
$268$	− 5.00000i	− 0.305424i
$269$	−15.0000	−0.914566	−0.457283	−	0.889321i	$-0.651177\pi$
$269$	−15.0000	−0.914566	−0.457283	+	0.889321i	$0.651177\pi$
$270$	0	0
$271$	2.00000	0.121491	0.0607457	−	0.998153i	$-0.480652\pi$
$271$	2.00000	0.121491	0.0607457	+	0.998153i	$0.480652\pi$
$272$	0	0
$273$	0	0
$274$	12.0000	0.724947
$275$	0	0
$276$	−3.00000	−0.180579
$277$	8.00000i	0.480673i	0.970690	+	0.240337i	$0.0772579\pi$
$277$	8.00000i	0.480673i	−0.970690	+	0.240337i	$0.922742\pi$
$278$	− 10.0000i	− 0.599760i
$279$	16.0000	0.957895
$280$	0	0
$281$	−6.00000	−0.357930	−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000	−0.357930	−0.178965	+	0.983855i	$0.557275\pi$
$282$	0	0
$283$	4.00000i	0.237775i	0.992908	+	0.118888i	$0.0379328\pi$
$283$	4.00000i	0.237775i	−0.992908	+	0.118888i	$0.962067\pi$
$284$	6.00000	0.356034
$285$	0	0
$286$	−24.0000	−1.41915
$287$	0	0
$288$	− 2.00000i	− 0.117851i
$289$	17.0000	1.00000
$290$	0	0
$291$	14.0000	0.820695
$292$	− 16.0000i	− 0.936329i
$293$	− 12.0000i	− 0.701047i	−0.936554	−	0.350524i	$-0.886004\pi$
$293$	− 12.0000i	− 0.701047i	0.936554	−	0.350524i	$-0.113996\pi$
$294$	0	0
$295$	0	0
$296$	4.00000	0.232495
$297$	30.0000i	1.74078i
$298$	15.0000i	0.868927i
$299$	−12.0000	−0.693978
$300$	0	0
$301$	0	0
$302$	4.00000i	0.230174i
$303$	− 15.0000i	− 0.861727i
$304$	−2.00000	−0.114708
$305$	0	0
$306$	0	0
$307$	5.00000i	0.285365i	0.989769	+	0.142683i	$0.0455728\pi$
$307$	5.00000i	0.285365i	−0.989769	+	0.142683i	$0.954427\pi$
$308$	0	0
$309$	1.00000	0.0568880
$310$	0	0
$311$	−18.0000	−1.02069	−0.510343	−	0.859971i	$-0.670482\pi$
$311$	−18.0000	−1.02069	−0.510343	+	0.859971i	$0.670482\pi$
$312$	4.00000i	0.226455i
$313$	− 8.00000i	− 0.452187i	−0.974106	−	0.226093i	$-0.927405\pi$
$313$	− 8.00000i	− 0.452187i	0.974106	−	0.226093i	$-0.0725954\pi$
$314$	−22.0000	−1.24153
$315$	0	0
$316$	2.00000	0.112509
$317$	12.0000i	0.673987i	0.941507	+	0.336994i	$0.109410\pi$
$317$	12.0000i	0.673987i	−0.941507	+	0.336994i	$0.890590\pi$
$318$	− 6.00000i	− 0.336463i
$319$	−18.0000	−1.00781
$320$	0	0
$321$	−15.0000	−0.837218
$322$	0	0
$323$	0	0
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	4.00000	0.221540
$327$	11.0000i	0.608301i
$328$	9.00000i	0.496942i
$329$	0	0
$330$	0	0
$331$	−28.0000	−1.53902	−0.769510	−	0.638635i	$-0.779499\pi$
$331$	−28.0000	−1.53902	−0.769510	+	0.638635i	$0.779499\pi$
$332$	3.00000i	0.164646i
$333$	− 8.00000i	− 0.438397i
$334$	−3.00000	−0.164153
$335$	0	0
$336$	0	0
$337$	− 22.0000i	− 1.19842i	−0.800593	−	0.599208i	$-0.795482\pi$
$337$	− 22.0000i	− 1.19842i	0.800593	−	0.599208i	$-0.204518\pi$
$338$	3.00000i	0.163178i
$339$	−6.00000	−0.325875
$340$	0	0
$341$	−48.0000	−2.59935
$342$	4.00000i	0.216295i
$343$	0	0
$344$	−7.00000	−0.377415
$345$	0	0
$346$	0	0
$347$	9.00000i	0.483145i	0.970383	+	0.241573i	$0.0776632\pi$
$347$	9.00000i	0.483145i	−0.970383	+	0.241573i	$0.922337\pi$
$348$	3.00000i	0.160817i
$349$	−17.0000	−0.909989	−0.454995	−	0.890494i	$-0.650359\pi$
$349$	−17.0000	−0.909989	−0.454995	+	0.890494i	$0.650359\pi$
$350$	0	0
$351$	20.0000	1.06752
$352$	6.00000i	0.319801i
$353$	6.00000i	0.319348i	0.987170	+	0.159674i	$0.0510443\pi$
$353$	6.00000i	0.319348i	−0.987170	+	0.159674i	$0.948956\pi$
$354$	−6.00000	−0.318896
$355$	0	0
$356$	−15.0000	−0.794998
$357$	0	0
$358$	− 24.0000i	− 1.26844i
$359$	−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
$361$	−15.0000	−0.789474
$362$	− 11.0000i	− 0.578147i
$363$	− 25.0000i	− 1.31216i
$364$	0	0
$365$	0	0
$366$	−5.00000	−0.261354
$367$	35.0000i	1.82699i	0.406855	+	0.913493i	$0.366625\pi$
$367$	35.0000i	1.82699i	−0.406855	+	0.913493i	$0.633375\pi$
$368$	3.00000i	0.156386i
$369$	18.0000	0.937043
$370$	0	0
$371$	0	0
$372$	8.00000i	0.414781i
$373$	4.00000i	0.207112i	0.994624	+	0.103556i	$0.0330221\pi$
$373$	4.00000i	0.207112i	−0.994624	+	0.103556i	$0.966978\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	12.0000i	0.618031i
$378$	0	0
$379$	34.0000	1.74646	0.873231	−	0.487306i	$-0.162020\pi$
$379$	34.0000	1.74646	0.873231	+	0.487306i	$0.162020\pi$
$380$	0	0
$381$	8.00000	0.409852
$382$	6.00000i	0.306987i
$383$	15.0000i	0.766464i	0.923652	+	0.383232i	$0.125189\pi$
$383$	15.0000i	0.766464i	−0.923652	+	0.383232i	$0.874811\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	−2.00000	−0.101797
$387$	14.0000i	0.711660i
$388$	− 14.0000i	− 0.710742i
$389$	30.0000	1.52106	0.760530	−	0.649303i	$-0.224939\pi$
$389$	30.0000	1.52106	0.760530	+	0.649303i	$0.224939\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	−6.00000	−0.302276
$395$	0	0
$396$	12.0000	0.603023
$397$	14.0000i	0.702640i	0.936255	+	0.351320i	$0.114267\pi$
$397$	14.0000i	0.702640i	−0.936255	+	0.351320i	$0.885733\pi$
$398$	− 4.00000i	− 0.200502i
$399$	0	0
$400$	0	0
$401$	15.0000	0.749064	0.374532	−	0.927214i	$-0.377803\pi$
$401$	15.0000	0.749064	0.374532	+	0.927214i	$0.377803\pi$
$402$	− 5.00000i	− 0.249377i
$403$	32.0000i	1.59403i
$404$	−15.0000	−0.746278
$405$	0	0
$406$	0	0
$407$	24.0000i	1.18964i
$408$	0	0
$409$	13.0000	0.642809	0.321404	−	0.946942i	$-0.395845\pi$
$409$	13.0000	0.642809	0.321404	+	0.946942i	$0.395845\pi$
$410$	0	0
$411$	12.0000	0.591916
$412$	− 1.00000i	− 0.0492665i
$413$	0	0
$414$	6.00000	0.294884
$415$	0	0
$416$	4.00000	0.196116
$417$	− 10.0000i	− 0.489702i
$418$	− 12.0000i	− 0.586939i
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	17.0000	0.828529	0.414265	−	0.910156i	$-0.364039\pi$
$421$	17.0000	0.828529	0.414265	+	0.910156i	$0.364039\pi$
$422$	10.0000i	0.486792i
$423$	0	0
$424$	−6.00000	−0.291386
$425$	0	0
$426$	6.00000	0.290701
$427$	0	0
$428$	15.0000i	0.725052i
$429$	−24.0000	−1.15873
$430$	0	0
$431$	30.0000	1.44505	0.722525	−	0.691345i	$-0.242982\pi$
$431$	30.0000	1.44505	0.722525	+	0.691345i	$0.242982\pi$
$432$	− 5.00000i	− 0.240563i
$433$	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
$436$	11.0000	0.526804
$437$	− 6.00000i	− 0.287019i
$438$	− 16.0000i	− 0.764510i
$439$	28.0000	1.33637	0.668184	−	0.743996i	$-0.267072\pi$
$439$	28.0000	1.33637	0.668184	+	0.743996i	$0.267072\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 21.0000i	− 0.997740i	−0.866677	−	0.498870i	$-0.833748\pi$
$443$	− 21.0000i	− 0.997740i	0.866677	−	0.498870i	$-0.166252\pi$
$444$	4.00000	0.189832
$445$	0	0
$446$	28.0000	1.32584
$447$	15.0000i	0.709476i
$448$	0	0
$449$	9.00000	0.424736	0.212368	−	0.977190i	$-0.431882\pi$
$449$	9.00000	0.424736	0.212368	+	0.977190i	$0.431882\pi$
$450$	0	0
$451$	−54.0000	−2.54276
$452$	6.00000i	0.282216i
$453$	4.00000i	0.187936i
$454$	−12.0000	−0.563188
$455$	0	0
$456$	−2.00000	−0.0936586
$457$	32.0000i	1.49690i	0.663193	+	0.748448i	$0.269201\pi$
$457$	32.0000i	1.49690i	−0.663193	+	0.748448i	$0.730799\pi$
$458$	14.0000i	0.654177i
$459$	0	0
$460$	0	0
$461$	18.0000	0.838344	0.419172	−	0.907907i	$-0.362320\pi$
$461$	18.0000	0.838344	0.419172	+	0.907907i	$0.362320\pi$
$462$	0	0
$463$	13.0000i	0.604161i	0.953282	+	0.302081i	$0.0976812\pi$
$463$	13.0000i	0.604161i	−0.953282	+	0.302081i	$0.902319\pi$
$464$	3.00000	0.139272
$465$	0	0
$466$	12.0000	0.555889
$467$	− 15.0000i	− 0.694117i	−0.937843	−	0.347059i	$-0.887180\pi$
$467$	− 15.0000i	− 0.694117i	0.937843	−	0.347059i	$-0.112820\pi$
$468$	− 8.00000i	− 0.369800i
$469$	0	0
$470$	0	0
$471$	−22.0000	−1.01371
$472$	6.00000i	0.276172i
$473$	− 42.0000i	− 1.93116i
$474$	2.00000	0.0918630
$475$	0	0
$476$	0	0
$477$	12.0000i	0.549442i
$478$	12.0000i	0.548867i
$479$	−12.0000	−0.548294	−0.274147	−	0.961688i	$-0.588395\pi$
$479$	−12.0000	−0.548294	−0.274147	+	0.961688i	$0.588395\pi$
$480$	0	0
$481$	16.0000	0.729537
$482$	− 2.00000i	− 0.0910975i
$483$	0	0
$484$	−25.0000	−1.13636
$485$	0	0
$486$	−16.0000	−0.725775
$487$	− 16.0000i	− 0.725029i	−0.931978	−	0.362515i	$-0.881918\pi$
$487$	− 16.0000i	− 0.725029i	0.931978	−	0.362515i	$-0.118082\pi$
$488$	5.00000i	0.226339i
$489$	4.00000	0.180886
$490$	0	0
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	9.00000i	0.405751i
$493$	0	0
$494$	−8.00000	−0.359937
$495$	0	0
$496$	8.00000	0.359211
$497$	0	0
$498$	3.00000i	0.134433i
$499$	22.0000	0.984855	0.492428	−	0.870353i	$-0.336110\pi$
$499$	22.0000	0.984855	0.492428	+	0.870353i	$0.336110\pi$
$500$	0	0
$501$	−3.00000	−0.134030
$502$	12.0000i	0.535586i
$503$	− 21.0000i	− 0.936344i	−0.883637	−	0.468172i	$-0.844913\pi$
$503$	− 21.0000i	− 0.936344i	0.883637	−	0.468172i	$-0.155087\pi$
$504$	0	0
$505$	0	0
$506$	−18.0000	−0.800198
$507$	3.00000i	0.133235i
$508$	− 8.00000i	− 0.354943i
$509$	−21.0000	−0.930809	−0.465404	−	0.885098i	$-0.654091\pi$
$509$	−21.0000	−0.930809	−0.465404	+	0.885098i	$0.654091\pi$
$510$	0	0
$511$	0	0
$512$	− 1.00000i	− 0.0441942i
$513$	10.0000i	0.441511i
$514$	0	0
$515$	0	0
$516$	−7.00000	−0.308158
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−18.0000	−0.788594	−0.394297	−	0.918983i	$-0.629012\pi$
$521$	−18.0000	−0.788594	−0.394297	+	0.918983i	$0.629012\pi$
$522$	− 6.00000i	− 0.262613i
$523$	28.0000i	1.22435i	0.790721	+	0.612177i	$0.209706\pi$
$523$	28.0000i	1.22435i	−0.790721	+	0.612177i	$0.790294\pi$
$524$	0	0
$525$	0	0
$526$	21.0000	0.915644
$527$	0	0
$528$	6.00000i	0.261116i
$529$	14.0000	0.608696
$530$	0	0
$531$	12.0000	0.520756
$532$	0	0
$533$	36.0000i	1.55933i
$534$	−15.0000	−0.649113
$535$	0	0
$536$	−5.00000	−0.215967
$537$	− 24.0000i	− 1.03568i
$538$	15.0000i	0.646696i
$539$	0	0
$540$	0	0
$541$	−25.0000	−1.07483	−0.537417	−	0.843317i	$-0.680600\pi$
$541$	−25.0000	−1.07483	−0.537417	+	0.843317i	$0.680600\pi$
$542$	− 2.00000i	− 0.0859074i
$543$	− 11.0000i	− 0.472055i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 19.0000i	− 0.812381i	−0.913788	−	0.406191i	$-0.866857\pi$
$547$	− 19.0000i	− 0.812381i	0.913788	−	0.406191i	$-0.133143\pi$
$548$	− 12.0000i	− 0.512615i
$549$	10.0000	0.426790
$550$	0	0
$551$	−6.00000	−0.255609
$552$	3.00000i	0.127688i
$553$	0	0
$554$	8.00000	0.339887
$555$	0	0
$556$	−10.0000	−0.424094
$557$	− 18.0000i	− 0.762684i	−0.924434	−	0.381342i	$-0.875462\pi$
$557$	− 18.0000i	− 0.762684i	0.924434	−	0.381342i	$-0.124538\pi$
$558$	− 16.0000i	− 0.677334i
$559$	−28.0000	−1.18427
$560$	0	0
$561$	0	0
$562$	6.00000i	0.253095i
$563$	− 27.0000i	− 1.13791i	−0.822367	−	0.568957i	$-0.807347\pi$
$563$	− 27.0000i	− 1.13791i	0.822367	−	0.568957i	$-0.192653\pi$
$564$	0	0
$565$	0	0
$566$	4.00000	0.168133
$567$	0	0
$568$	− 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
$571$	−22.0000	−0.920671	−0.460336	−	0.887745i	$-0.652271\pi$
$571$	−22.0000	−0.920671	−0.460336	+	0.887745i	$0.652271\pi$
$572$	24.0000i	1.00349i
$573$	6.00000i	0.250654i
$574$	0	0
$575$	0	0
$576$	−2.00000	−0.0833333
$577$	26.0000i	1.08239i	0.840896	+	0.541197i	$0.182029\pi$
$577$	26.0000i	1.08239i	−0.840896	+	0.541197i	$0.817971\pi$
$578$	− 17.0000i	− 0.707107i
$579$	−2.00000	−0.0831172
$580$	0	0
$581$	0	0
$582$	− 14.0000i	− 0.580319i
$583$	− 36.0000i	− 1.49097i
$584$	−16.0000	−0.662085
$585$	0	0
$586$	−12.0000	−0.495715
$587$	12.0000i	0.495293i	0.968850	+	0.247647i	$0.0796572\pi$
$587$	12.0000i	0.495293i	−0.968850	+	0.247647i	$0.920343\pi$
$588$	0	0
$589$	−16.0000	−0.659269
$590$	0	0
$591$	−6.00000	−0.246807
$592$	− 4.00000i	− 0.164399i
$593$	− 6.00000i	− 0.246390i	−0.992382	−	0.123195i	$-0.960686\pi$
$593$	− 6.00000i	− 0.246390i	0.992382	−	0.123195i	$-0.0393141\pi$
$594$	30.0000	1.23091
$595$	0	0
$596$	15.0000	0.614424
$597$	− 4.00000i	− 0.163709i
$598$	12.0000i	0.490716i
$599$	−12.0000	−0.490307	−0.245153	−	0.969484i	$-0.578838\pi$
$599$	−12.0000	−0.490307	−0.245153	+	0.969484i	$0.578838\pi$
$600$	0	0
$601$	−46.0000	−1.87638	−0.938190	−	0.346122i	$-0.887498\pi$
$601$	−46.0000	−1.87638	−0.938190	+	0.346122i	$0.887498\pi$
$602$	0	0
$603$	10.0000i	0.407231i
$604$	4.00000	0.162758
$605$	0	0
$606$	−15.0000	−0.609333
$607$	23.0000i	0.933541i	0.884378	+	0.466771i	$0.154583\pi$
$607$	23.0000i	0.933541i	−0.884378	+	0.466771i	$0.845417\pi$
$608$	2.00000i	0.0811107i
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	16.0000i	0.646234i	0.946359	+	0.323117i	$0.104731\pi$
$613$	16.0000i	0.646234i	−0.946359	+	0.323117i	$0.895269\pi$
$614$	5.00000	0.201784
$615$	0	0
$616$	0	0
$617$	− 12.0000i	− 0.483102i	−0.970388	−	0.241551i	$-0.922344\pi$
$617$	− 12.0000i	− 0.483102i	0.970388	−	0.241551i	$-0.0776561\pi$
$618$	− 1.00000i	− 0.0402259i
$619$	−14.0000	−0.562708	−0.281354	−	0.959604i	$-0.590783\pi$
$619$	−14.0000	−0.562708	−0.281354	+	0.959604i	$0.590783\pi$
$620$	0	0
$621$	15.0000	0.601929
$622$	18.0000i	0.721734i
$623$	0	0
$624$	4.00000	0.160128
$625$	0	0
$626$	−8.00000	−0.319744
$627$	− 12.0000i	− 0.479234i
$628$	22.0000i	0.877896i
$629$	0	0
$630$	0	0
$631$	14.0000	0.557331	0.278666	−	0.960388i	$-0.410108\pi$
$631$	14.0000	0.557331	0.278666	+	0.960388i	$0.410108\pi$
$632$	− 2.00000i	− 0.0795557i
$633$	10.0000i	0.397464i
$634$	12.0000	0.476581
$635$	0	0
$636$	−6.00000	−0.237915
$637$	0	0
$638$	18.0000i	0.712627i
$639$	−12.0000	−0.474713
$640$	0	0
$641$	−3.00000	−0.118493	−0.0592464	−	0.998243i	$-0.518870\pi$
$641$	−3.00000	−0.118493	−0.0592464	+	0.998243i	$0.518870\pi$
$642$	15.0000i	0.592003i
$643$	− 20.0000i	− 0.788723i	−0.918955	−	0.394362i	$-0.870966\pi$
$643$	− 20.0000i	− 0.788723i	0.918955	−	0.394362i	$-0.129034\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	3.00000i	0.117942i	0.998260	+	0.0589711i	$0.0187820\pi$
$647$	3.00000i	0.117942i	−0.998260	+	0.0589711i	$0.981218\pi$
$648$	1.00000i	0.0392837i
$649$	−36.0000	−1.41312
$650$	0	0
$651$	0	0
$652$	− 4.00000i	− 0.156652i
$653$	48.0000i	1.87839i	0.343391	+	0.939193i	$0.388424\pi$
$653$	48.0000i	1.87839i	−0.343391	+	0.939193i	$0.611576\pi$
$654$	11.0000	0.430134
$655$	0	0
$656$	9.00000	0.351391
$657$	32.0000i	1.24844i
$658$	0	0
$659$	−6.00000	−0.233727	−0.116863	−	0.993148i	$-0.537284\pi$
$659$	−6.00000	−0.233727	−0.116863	+	0.993148i	$0.537284\pi$
$660$	0	0
$661$	41.0000	1.59472	0.797358	−	0.603507i	$-0.206231\pi$
$661$	41.0000	1.59472	0.797358	+	0.603507i	$0.206231\pi$
$662$	28.0000i	1.08825i
$663$	0	0
$664$	3.00000	0.116423
$665$	0	0
$666$	−8.00000	−0.309994
$667$	9.00000i	0.348481i
$668$	3.00000i	0.116073i
$669$	28.0000	1.08254
$670$	0	0
$671$	−30.0000	−1.15814
$672$	0	0
$673$	− 8.00000i	− 0.308377i	−0.988041	−	0.154189i	$-0.950724\pi$
$673$	− 8.00000i	− 0.308377i	0.988041	−	0.154189i	$-0.0492764\pi$
$674$	−22.0000	−0.847408
$675$	0	0
$676$	3.00000	0.115385
$677$	12.0000i	0.461197i	0.973049	+	0.230599i	$0.0740685\pi$
$677$	12.0000i	0.461197i	−0.973049	+	0.230599i	$0.925932\pi$
$678$	6.00000i	0.230429i
$679$	0	0
$680$	0	0
$681$	−12.0000	−0.459841
$682$	48.0000i	1.83801i
$683$	− 9.00000i	− 0.344375i	−0.985064	−	0.172188i	$-0.944916\pi$
$683$	− 9.00000i	− 0.344375i	0.985064	−	0.172188i	$-0.0550836\pi$
$684$	4.00000	0.152944
$685$	0	0
$686$	0	0
$687$	14.0000i	0.534133i
$688$	7.00000i	0.266872i
$689$	−24.0000	−0.914327
$690$	0	0
$691$	−22.0000	−0.836919	−0.418460	−	0.908235i	$-0.637430\pi$
$691$	−22.0000	−0.836919	−0.418460	+	0.908235i	$0.637430\pi$
$692$	0	0
$693$	0	0
$694$	9.00000	0.341635
$695$	0	0
$696$	3.00000	0.113715
$697$	0	0
$698$	17.0000i	0.643459i
$699$	12.0000	0.453882
$700$	0	0
$701$	−3.00000	−0.113308	−0.0566542	−	0.998394i	$-0.518043\pi$
$701$	−3.00000	−0.113308	−0.0566542	+	0.998394i	$0.518043\pi$
$702$	− 20.0000i	− 0.754851i
$703$	8.00000i	0.301726i
$704$	6.00000	0.226134
$705$	0	0
$706$	6.00000	0.225813
$707$	0	0
$708$	6.00000i	0.225494i
$709$	31.0000	1.16423	0.582115	−	0.813107i	$-0.302225\pi$
$709$	31.0000	1.16423	0.582115	+	0.813107i	$0.302225\pi$
$710$	0	0
$711$	−4.00000	−0.150012
$712$	15.0000i	0.562149i
$713$	24.0000i	0.898807i
$714$	0	0
$715$	0	0
$716$	−24.0000	−0.896922
$717$	12.0000i	0.448148i
$718$	24.0000i	0.895672i
$719$	18.0000	0.671287	0.335643	−	0.941989i	$-0.391046\pi$
$719$	18.0000	0.671287	0.335643	+	0.941989i	$0.391046\pi$
$720$	0	0
$721$	0	0
$722$	15.0000i	0.558242i
$723$	− 2.00000i	− 0.0743808i
$724$	−11.0000	−0.408812
$725$	0	0
$726$	−25.0000	−0.927837
$727$	− 19.0000i	− 0.704671i	−0.935874	−	0.352335i	$-0.885388\pi$
$727$	− 19.0000i	− 0.704671i	0.935874	−	0.352335i	$-0.114612\pi$
$728$	0	0
$729$	−13.0000	−0.481481
$730$	0	0
$731$	0	0
$732$	5.00000i	0.184805i
$733$	34.0000i	1.25582i	0.778287	+	0.627909i	$0.216089\pi$
$733$	34.0000i	1.25582i	−0.778287	+	0.627909i	$0.783911\pi$
$734$	35.0000	1.29187
$735$	0	0
$736$	3.00000	0.110581
$737$	− 30.0000i	− 1.10506i
$738$	− 18.0000i	− 0.662589i
$739$	−26.0000	−0.956425	−0.478213	−	0.878244i	$-0.658715\pi$
$739$	−26.0000	−0.956425	−0.478213	+	0.878244i	$0.658715\pi$
$740$	0	0
$741$	−8.00000	−0.293887
$742$	0	0
$743$	− 39.0000i	− 1.43077i	−0.698730	−	0.715386i	$-0.746251\pi$
$743$	− 39.0000i	− 1.43077i	0.698730	−	0.715386i	$-0.253749\pi$
$744$	8.00000	0.293294
$745$	0	0
$746$	4.00000	0.146450
$747$	− 6.00000i	− 0.219529i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−4.00000	−0.145962	−0.0729810	−	0.997333i	$-0.523251\pi$
$751$	−4.00000	−0.145962	−0.0729810	+	0.997333i	$0.523251\pi$
$752$	0	0
$753$	12.0000i	0.437304i
$754$	12.0000	0.437014
$755$	0	0
$756$	0	0
$757$	− 28.0000i	− 1.01768i	−0.860862	−	0.508839i	$-0.830075\pi$
$757$	− 28.0000i	− 1.01768i	0.860862	−	0.508839i	$-0.169925\pi$
$758$	− 34.0000i	− 1.23494i
$759$	−18.0000	−0.653359
$760$	0	0
$761$	42.0000	1.52250	0.761249	−	0.648459i	$-0.224586\pi$
$761$	42.0000	1.52250	0.761249	+	0.648459i	$0.224586\pi$
$762$	− 8.00000i	− 0.289809i
$763$	0	0
$764$	6.00000	0.217072
$765$	0	0
$766$	15.0000	0.541972
$767$	24.0000i	0.866590i
$768$	− 1.00000i	− 0.0360844i
$769$	−50.0000	−1.80305	−0.901523	−	0.432731i	$-0.857550\pi$
$769$	−50.0000	−1.80305	−0.901523	+	0.432731i	$0.857550\pi$
$770$	0	0
$771$	0	0
$772$	2.00000i	0.0719816i
$773$	12.0000i	0.431610i	0.976436	+	0.215805i	$0.0692376\pi$
$773$	12.0000i	0.431610i	−0.976436	+	0.215805i	$0.930762\pi$
$774$	14.0000	0.503220
$775$	0	0
$776$	−14.0000	−0.502571
$777$	0	0
$778$	− 30.0000i	− 1.07555i
$779$	−18.0000	−0.644917
$780$	0	0
$781$	36.0000	1.28818
$782$	0	0
$783$	− 15.0000i	− 0.536056i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 43.0000i	− 1.53278i	−0.642373	−	0.766392i	$-0.722050\pi$
$787$	− 43.0000i	− 1.53278i	0.642373	−	0.766392i	$-0.277950\pi$
$788$	6.00000i	0.213741i
$789$	21.0000	0.747620
$790$	0	0
$791$	0	0
$792$	− 12.0000i	− 0.426401i
$793$	20.0000i	0.710221i
$794$	14.0000	0.496841
$795$	0	0
$796$	−4.00000	−0.141776
$797$	− 48.0000i	− 1.70025i	−0.526583	−	0.850124i	$-0.676527\pi$
$797$	− 48.0000i	− 1.70025i	0.526583	−	0.850124i	$-0.323473\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	30.0000	1.06000
$802$	− 15.0000i	− 0.529668i
$803$	− 96.0000i	− 3.38777i
$804$	−5.00000	−0.176336
$805$	0	0
$806$	32.0000	1.12715
$807$	15.0000i	0.528025i
$808$	15.0000i	0.527698i
$809$	−21.0000	−0.738321	−0.369160	−	0.929366i	$-0.620355\pi$
$809$	−21.0000	−0.738321	−0.369160	+	0.929366i	$0.620355\pi$
$810$	0	0
$811$	−16.0000	−0.561836	−0.280918	−	0.959732i	$-0.590639\pi$
$811$	−16.0000	−0.561836	−0.280918	+	0.959732i	$0.590639\pi$
$812$	0	0
$813$	− 2.00000i	− 0.0701431i
$814$	24.0000	0.841200
$815$	0	0
$816$	0	0
$817$	− 14.0000i	− 0.489798i
$818$	− 13.0000i	− 0.454534i
$819$	0	0
$820$	0	0
$821$	−18.0000	−0.628204	−0.314102	−	0.949389i	$-0.601703\pi$
$821$	−18.0000	−0.628204	−0.314102	+	0.949389i	$0.601703\pi$
$822$	− 12.0000i	− 0.418548i
$823$	19.0000i	0.662298i	0.943578	+	0.331149i	$0.107436\pi$
$823$	19.0000i	0.662298i	−0.943578	+	0.331149i	$0.892564\pi$
$824$	−1.00000	−0.0348367
$825$	0	0
$826$	0	0
$827$	15.0000i	0.521601i	0.965393	+	0.260801i	$0.0839865\pi$
$827$	15.0000i	0.521601i	−0.965393	+	0.260801i	$0.916014\pi$
$828$	− 6.00000i	− 0.208514i
$829$	−2.00000	−0.0694629	−0.0347314	−	0.999397i	$-0.511058\pi$
$829$	−2.00000	−0.0694629	−0.0347314	+	0.999397i	$0.511058\pi$
$830$	0	0
$831$	8.00000	0.277517
$832$	− 4.00000i	− 0.138675i
$833$	0	0
$834$	−10.0000	−0.346272
$835$	0	0
$836$	−12.0000	−0.415029
$837$	− 40.0000i	− 1.38260i
$838$	0	0
$839$	30.0000	1.03572	0.517858	−	0.855467i	$-0.326730\pi$
$839$	30.0000	1.03572	0.517858	+	0.855467i	$0.326730\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	− 17.0000i	− 0.585859i
$843$	6.00000i	0.206651i
$844$	10.0000	0.344214
$845$	0	0
$846$	0	0
$847$	0	0
$848$	6.00000i	0.206041i
$849$	4.00000	0.137280
$850$	0	0
$851$	12.0000	0.411355
$852$	− 6.00000i	− 0.205557i
$853$	46.0000i	1.57501i	0.616308	+	0.787505i	$0.288628\pi$
$853$	46.0000i	1.57501i	−0.616308	+	0.787505i	$0.711372\pi$
$854$	0	0
$855$	0	0
$856$	15.0000	0.512689
$857$	6.00000i	0.204956i	0.994735	+	0.102478i	$0.0326771\pi$
$857$	6.00000i	0.204956i	−0.994735	+	0.102478i	$0.967323\pi$
$858$	24.0000i	0.819346i
$859$	−32.0000	−1.09183	−0.545913	−	0.837842i	$-0.683817\pi$
$859$	−32.0000	−1.09183	−0.545913	+	0.837842i	$0.683817\pi$
$860$	0	0
$861$	0	0
$862$	− 30.0000i	− 1.02180i
$863$	− 27.0000i	− 0.919091i	−0.888154	−	0.459545i	$-0.848012\pi$
$863$	− 27.0000i	− 0.919091i	0.888154	−	0.459545i	$-0.151988\pi$
$864$	−5.00000	−0.170103
$865$	0	0
$866$	22.0000	0.747590
$867$	− 17.0000i	− 0.577350i
$868$	0	0
$869$	12.0000	0.407072
$870$	0	0
$871$	−20.0000	−0.677674
$872$	− 11.0000i	− 0.372507i
$873$	28.0000i	0.947656i
$874$	−6.00000	−0.202953
$875$	0	0
$876$	−16.0000	−0.540590
$877$	2.00000i	0.0675352i	0.999430	+	0.0337676i	$0.0107506\pi$
$877$	2.00000i	0.0675352i	−0.999430	+	0.0337676i	$0.989249\pi$
$878$	− 28.0000i	− 0.944954i
$879$	−12.0000	−0.404750
$880$	0	0
$881$	57.0000	1.92038	0.960189	−	0.279350i	$-0.0901189\pi$
$881$	57.0000	1.92038	0.960189	+	0.279350i	$0.0901189\pi$
$882$	0	0
$883$	52.0000i	1.74994i	0.484178	+	0.874970i	$0.339119\pi$
$883$	52.0000i	1.74994i	−0.484178	+	0.874970i	$0.660881\pi$
$884$	0	0
$885$	0	0
$886$	−21.0000	−0.705509
$887$	− 21.0000i	− 0.705111i	−0.935791	−	0.352555i	$-0.885313\pi$
$887$	− 21.0000i	− 0.705111i	0.935791	−	0.352555i	$-0.114687\pi$
$888$	− 4.00000i	− 0.134231i
$889$	0	0
$890$	0	0
$891$	−6.00000	−0.201008
$892$	− 28.0000i	− 0.937509i
$893$	0	0
$894$	15.0000	0.501675
$895$	0	0
$896$	0	0
$897$	12.0000i	0.400668i
$898$	− 9.00000i	− 0.300334i
$899$	24.0000	0.800445
$900$	0	0
$901$	0	0
$902$	54.0000i	1.79800i
$903$	0	0
$904$	6.00000	0.199557
$905$	0	0
$906$	4.00000	0.132891
$907$	− 25.0000i	− 0.830111i	−0.909796	−	0.415056i	$-0.863762\pi$
$907$	− 25.0000i	− 0.830111i	0.909796	−	0.415056i	$-0.136238\pi$
$908$	12.0000i	0.398234i
$909$	30.0000	0.995037
$910$	0	0
$911$	18.0000	0.596367	0.298183	−	0.954509i	$-0.403619\pi$
$911$	18.0000	0.596367	0.298183	+	0.954509i	$0.403619\pi$
$912$	2.00000i	0.0662266i
$913$	18.0000i	0.595713i
$914$	32.0000	1.05847
$915$	0	0
$916$	14.0000	0.462573
$917$	0	0
$918$	0	0
$919$	−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
$921$	5.00000	0.164756
$922$	− 18.0000i	− 0.592798i
$923$	− 24.0000i	− 0.789970i
$924$	0	0
$925$	0	0
$926$	13.0000	0.427207
$927$	2.00000i	0.0656886i
$928$	− 3.00000i	− 0.0984798i
$929$	21.0000	0.688988	0.344494	−	0.938789i	$-0.388051\pi$
$929$	21.0000	0.688988	0.344494	+	0.938789i	$0.388051\pi$
$930$	0	0
$931$	0	0
$932$	− 12.0000i	− 0.393073i
$933$	18.0000i	0.589294i
$934$	−15.0000	−0.490815
$935$	0	0
$936$	−8.00000	−0.261488
$937$	− 28.0000i	− 0.914720i	−0.889282	−	0.457360i	$-0.848795\pi$
$937$	− 28.0000i	− 0.914720i	0.889282	−	0.457360i	$-0.151205\pi$
$938$	0	0
$939$	−8.00000	−0.261070
$940$	0	0
$941$	−6.00000	−0.195594	−0.0977972	−	0.995206i	$-0.531180\pi$
$941$	−6.00000	−0.195594	−0.0977972	+	0.995206i	$0.531180\pi$
$942$	22.0000i	0.716799i
$943$	27.0000i	0.879241i
$944$	6.00000	0.195283
$945$	0	0
$946$	−42.0000	−1.36554
$947$	− 3.00000i	− 0.0974869i	−0.998811	−	0.0487435i	$-0.984478\pi$
$947$	− 3.00000i	− 0.0974869i	0.998811	−	0.0487435i	$-0.0155217\pi$
$948$	− 2.00000i	− 0.0649570i
$949$	−64.0000	−2.07753
$950$	0	0
$951$	12.0000	0.389127
$952$	0	0
$953$	− 60.0000i	− 1.94359i	−0.235826	−	0.971795i	$-0.575780\pi$
$953$	− 60.0000i	− 1.94359i	0.235826	−	0.971795i	$-0.424220\pi$
$954$	12.0000	0.388514
$955$	0	0
$956$	12.0000	0.388108
$957$	18.0000i	0.581857i
$958$	12.0000i	0.387702i
$959$	0	0
$960$	0	0
$961$	33.0000	1.06452
$962$	− 16.0000i	− 0.515861i
$963$	− 30.0000i	− 0.966736i
$964$	−2.00000	−0.0644157
$965$	0	0
$966$	0	0
$967$	35.0000i	1.12552i	0.826619	+	0.562762i	$0.190261\pi$
$967$	35.0000i	1.12552i	−0.826619	+	0.562762i	$0.809739\pi$
$968$	25.0000i	0.803530i
$969$	0	0
$970$	0	0
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	16.0000i	0.513200i
$973$	0	0
$974$	−16.0000	−0.512673
$975$	0	0
$976$	5.00000	0.160046
$977$	− 6.00000i	− 0.191957i	−0.995383	−	0.0959785i	$-0.969402\pi$
$977$	− 6.00000i	− 0.191957i	0.995383	−	0.0959785i	$-0.0305980\pi$
$978$	− 4.00000i	− 0.127906i
$979$	−90.0000	−2.87641
$980$	0	0
$981$	−22.0000	−0.702406
$982$	0	0
$983$	− 39.0000i	− 1.24391i	−0.783054	−	0.621953i	$-0.786339\pi$
$983$	− 39.0000i	− 1.24391i	0.783054	−	0.621953i	$-0.213661\pi$
$984$	9.00000	0.286910
$985$	0	0
$986$	0	0
$987$	0	0
$988$	8.00000i	0.254514i
$989$	−21.0000	−0.667761
$990$	0	0
$991$	−28.0000	−0.889449	−0.444725	−	0.895667i	$-0.646698\pi$
$991$	−28.0000	−0.889449	−0.444725	+	0.895667i	$0.646698\pi$
$992$	− 8.00000i	− 0.254000i
$993$	28.0000i	0.888553i
$994$	0	0
$995$	0	0
$996$	3.00000	0.0950586
$997$	14.0000i	0.443384i	0.975117	+	0.221692i	$0.0711580\pi$
$997$	14.0000i	0.443384i	−0.975117	+	0.221692i	$0.928842\pi$
$998$	− 22.0000i	− 0.696398i
$999$	−20.0000	−0.632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2450.2.c.g.99.1		2
5.2	odd	4		2450.2.a.w.1.1		1
5.3	odd	4		490.2.a.c.1.1		1
5.4	even	2	inner	2450.2.c.g.99.2		2
7.2	even	3		350.2.j.b.249.1		4
7.4	even	3		350.2.j.b.149.2		4
7.6	odd	2		2450.2.c.l.99.1		2
15.8	even	4		4410.2.a.bm.1.1		1
20.3	even	4		3920.2.a.p.1.1		1
35.2	odd	12		350.2.e.e.151.1		2
35.3	even	12		490.2.e.h.471.1		2
35.4	even	6		350.2.j.b.149.1		4
35.9	even	6		350.2.j.b.249.2		4
35.13	even	4		490.2.a.b.1.1		1
35.18	odd	12		70.2.e.c.51.1	yes	2
35.23	odd	12		70.2.e.c.11.1	✓	2
35.27	even	4		2450.2.a.bc.1.1		1
35.32	odd	12		350.2.e.e.51.1		2
35.33	even	12		490.2.e.h.361.1		2
35.34	odd	2		2450.2.c.l.99.2		2
105.23	even	12		630.2.k.b.361.1		2
105.53	even	12		630.2.k.b.541.1		2
105.83	odd	4		4410.2.a.bd.1.1		1
140.23	even	12		560.2.q.g.81.1		2
140.83	odd	4		3920.2.a.bc.1.1		1
140.123	even	12		560.2.q.g.401.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
70.2.e.c.11.1	✓	2	35.23	odd	12
70.2.e.c.51.1	yes	2	35.18	odd	12
350.2.e.e.51.1		2	35.32	odd	12
350.2.e.e.151.1		2	35.2	odd	12
350.2.j.b.149.1		4	35.4	even	6
350.2.j.b.149.2		4	7.4	even	3
350.2.j.b.249.1		4	7.2	even	3
350.2.j.b.249.2		4	35.9	even	6
490.2.a.b.1.1		1	35.13	even	4
490.2.a.c.1.1		1	5.3	odd	4
490.2.e.h.361.1		2	35.33	even	12
490.2.e.h.471.1		2	35.3	even	12
560.2.q.g.81.1		2	140.23	even	12
560.2.q.g.401.1		2	140.123	even	12
630.2.k.b.361.1		2	105.23	even	12
630.2.k.b.541.1		2	105.53	even	12
2450.2.a.w.1.1		1	5.2	odd	4
2450.2.a.bc.1.1		1	35.27	even	4
2450.2.c.g.99.1		2	1.1	even	1	trivial
2450.2.c.g.99.2		2	5.4	even	2	inner
2450.2.c.l.99.1		2	7.6	odd	2
2450.2.c.l.99.2		2	35.34	odd	2
3920.2.a.p.1.1		1	20.3	even	4
3920.2.a.bc.1.1		1	140.83	odd	4
4410.2.a.bd.1.1		1	105.83	odd	4
4410.2.a.bm.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:	\( N \)	\(=\)	\( 2450 = 2 \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2450.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +1.00000i q^{8} +2.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +1.00000i q^{8} +2.00000 q^{9} -6.00000 q^{11} +1.00000i q^{12} +4.00000i q^{13} +1.00000 q^{16} -2.00000i q^{18} -2.00000 q^{19} +6.00000i q^{22} +3.00000i q^{23} +1.00000 q^{24} +4.00000 q^{26} -5.00000i q^{27} +3.00000 q^{29} +8.00000 q^{31} -1.00000i q^{32} +6.00000i q^{33} -2.00000 q^{36} -4.00000i q^{37} +2.00000i q^{38} +4.00000 q^{39} +9.00000 q^{41} +7.00000i q^{43} +6.00000 q^{44} +3.00000 q^{46} -1.00000i q^{48} -4.00000i q^{52} +6.00000i q^{53} -5.00000 q^{54} +2.00000i q^{57} -3.00000i q^{58} +6.00000 q^{59} +5.00000 q^{61} -8.00000i q^{62} -1.00000 q^{64} +6.00000 q^{66} +5.00000i q^{67} +3.00000 q^{69} -6.00000 q^{71} +2.00000i q^{72} +16.0000i q^{73} -4.00000 q^{74} +2.00000 q^{76} -4.00000i q^{78} -2.00000 q^{79} +1.00000 q^{81} -9.00000i q^{82} -3.00000i q^{83} +7.00000 q^{86} -3.00000i q^{87} -6.00000i q^{88} +15.0000 q^{89} -3.00000i q^{92} -8.00000i q^{93} -1.00000 q^{96} +14.0000i q^{97} -12.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} - 2 q^{6} + 4 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 - 2 * q^6 + 4 * q^9	\( 2 q - 2 q^{4} - 2 q^{6} + 4 q^{9} - 12 q^{11} + 2 q^{16} - 4 q^{19} + 2 q^{24} + 8 q^{26} + 6 q^{29} + 16 q^{31} - 4 q^{36} + 8 q^{39} + 18 q^{41} + 12 q^{44} + 6 q^{46} - 10 q^{54} + 12 q^{59} + 10 q^{61} - 2 q^{64} + 12 q^{66} + 6 q^{69} - 12 q^{71} - 8 q^{74} + 4 q^{76} - 4 q^{79} + 2 q^{81} + 14 q^{86} + 30 q^{89} - 2 q^{96} - 24 q^{99}+O(q^{100}) \) 2 * q - 2 * q^4 - 2 * q^6 + 4 * q^9 - 12 * q^11 + 2 * q^16 - 4 * q^19 + 2 * q^24 + 8 * q^26 + 6 * q^29 + 16 * q^31 - 4 * q^36 + 8 * q^39 + 18 * q^41 + 12 * q^44 + 6 * q^46 - 10 * q^54 + 12 * q^59 + 10 * q^61 - 2 * q^64 + 12 * q^66 + 6 * q^69 - 12 * q^71 - 8 * q^74 + 4 * q^76 - 4 * q^79 + 2 * q^81 + 14 * q^86 + 30 * q^89 - 2 * q^96 - 24 * q^99

Introduction

L-functions

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Database

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