LMFDB - Embedded newform 3450.2.d.f.2899.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3450,2,Mod(2899,3450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3450, 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("3450.2899");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3450.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:	$27.5483886973$
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 138)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2899.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	3450.2899
Dual form			3450.2.d.f.2899.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} +1.00000i q^{12} -2.00000i q^{13} +1.00000 q^{16} -2.00000i q^{17} +1.00000i q^{18} +8.00000 q^{19} -1.00000i q^{23} +1.00000 q^{24} -2.00000 q^{26} +1.00000i q^{27} +2.00000 q^{29} -8.00000 q^{31} -1.00000i q^{32} -2.00000 q^{34} +1.00000 q^{36} -2.00000i q^{37} -8.00000i q^{38} -2.00000 q^{39} +10.0000 q^{41} +8.00000i q^{43} -1.00000 q^{46} -8.00000i q^{47} -1.00000i q^{48} +7.00000 q^{49} -2.00000 q^{51} +2.00000i q^{52} +2.00000i q^{53} +1.00000 q^{54} -8.00000i q^{57} -2.00000i q^{58} +4.00000 q^{59} +2.00000 q^{61} +8.00000i q^{62} -1.00000 q^{64} -8.00000i q^{67} +2.00000i q^{68} -1.00000 q^{69} -1.00000i q^{72} -6.00000i q^{73} -2.00000 q^{74} -8.00000 q^{76} +2.00000i q^{78} -8.00000 q^{79} +1.00000 q^{81} -10.0000i q^{82} -16.0000i q^{83} +8.00000 q^{86} -2.00000i q^{87} -18.0000 q^{89} +1.00000i q^{92} +8.00000i q^{93} -8.00000 q^{94} -1.00000 q^{96} -10.0000i q^{97} -7.00000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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^{16} + 16 q^{19} + 2 q^{24} - 4 q^{26} + 4 q^{29} - 16 q^{31} - 4 q^{34} + 2 q^{36} - 4 q^{39} + 20 q^{41} - 2 q^{46} + 14 q^{49} - 4 q^{51} + 2 q^{54} + 8 q^{59} + 4 q^{61} - 2 q^{64} - 2 q^{69} - 4 q^{74} - 16 q^{76} - 16 q^{79} + 2 q^{81} + 16 q^{86} - 36 q^{89} - 16 q^{94} - 2 q^{96}+O(q^{100}) $ 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9 + 2 * q^16 + 16 * q^19 + 2 * q^24 - 4 * q^26 + 4 * q^29 - 16 * q^31 - 4 * q^34 + 2 * q^36 - 4 * q^39 + 20 * q^41 - 2 * q^46 + 14 * q^49 - 4 * q^51 + 2 * q^54 + 8 * q^59 + 4 * q^61 - 2 * q^64 - 2 * q^69 - 4 * q^74 - 16 * q^76 - 16 * q^79 + 2 * q^81 + 16 * q^86 - 36 * q^89 - 16 * q^94 - 2 * q^96

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3450\mathbb{Z}\right)^\times$.

$n$	$277$	$1151$	$1201$
$\chi(n)$	$-1$	$1$	$1$

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
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	1.00000i	0.288675i
$13$	− 2.00000i	− 0.554700i	−0.960769	−	0.277350i	$-0.910544\pi$
$13$	− 2.00000i	− 0.554700i	0.960769	−	0.277350i	$-0.0894562\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	− 2.00000i	− 0.485071i	−0.970143	−	0.242536i	$-0.922021\pi$
$17$	− 2.00000i	− 0.485071i	0.970143	−	0.242536i	$-0.0779791\pi$
$18$	1.00000i	0.235702i
$19$	8.00000	1.83533	0.917663	−	0.397360i	$-0.130073\pi$
$19$	8.00000	1.83533	0.917663	+	0.397360i	$0.130073\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 1.00000i	− 0.208514i
$23$	− 1.00000i	− 0.208514i
$24$	1.00000	0.204124
$25$	0	0
$26$	−2.00000	−0.392232
$27$	1.00000i	0.192450i
$28$	0	0
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	−8.00000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	− 1.00000i	− 0.176777i
$33$	0	0
$34$	−2.00000	−0.342997
$35$	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	$-0.0525685\pi$
$38$	− 8.00000i	− 1.29777i
$39$	−2.00000	−0.320256
$40$	0	0
$41$	10.0000	1.56174	0.780869	−	0.624695i	$-0.214777\pi$
$41$	10.0000	1.56174	0.780869	+	0.624695i	$0.214777\pi$
$42$	0	0
$43$	8.00000i	1.21999i	0.792406	+	0.609994i	$0.208828\pi$
$43$	8.00000i	1.21999i	−0.792406	+	0.609994i	$0.791172\pi$
$44$	0	0
$45$	0	0
$46$	−1.00000	−0.147442
$47$	− 8.00000i	− 1.16692i	−0.812142	−	0.583460i	$-0.801699\pi$
$47$	− 8.00000i	− 1.16692i	0.812142	−	0.583460i	$-0.198301\pi$
$48$	− 1.00000i	− 0.144338i
$49$	7.00000	1.00000
$50$	0	0
$51$	−2.00000	−0.280056
$52$	2.00000i	0.277350i
$53$	2.00000i	0.274721i	0.990521	+	0.137361i	$0.0438619\pi$
$53$	2.00000i	0.274721i	−0.990521	+	0.137361i	$0.956138\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	0	0
$57$	− 8.00000i	− 1.05963i
$58$	− 2.00000i	− 0.262613i
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	8.00000i	1.01600i
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	− 8.00000i	− 0.977356i	−0.872464	−	0.488678i	$-0.837479\pi$
$67$	− 8.00000i	− 0.977356i	0.872464	−	0.488678i	$-0.162521\pi$
$68$	2.00000i	0.242536i
$69$	−1.00000	−0.120386
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	− 1.00000i	− 0.117851i
$73$	− 6.00000i	− 0.702247i	−0.936329	−	0.351123i	$-0.885800\pi$
$73$	− 6.00000i	− 0.702247i	0.936329	−	0.351123i	$-0.114200\pi$
$74$	−2.00000	−0.232495
$75$	0	0
$76$	−8.00000	−0.917663
$77$	0	0
$78$	2.00000i	0.226455i
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	− 10.0000i	− 1.10432i
$83$	− 16.0000i	− 1.75623i	−0.478451	−	0.878114i	$-0.658802\pi$
$83$	− 16.0000i	− 1.75623i	0.478451	−	0.878114i	$-0.341198\pi$
$84$	0	0
$85$	0	0
$86$	8.00000	0.862662
$87$	− 2.00000i	− 0.214423i
$88$	0	0
$89$	−18.0000	−1.90800	−0.953998	−	0.299813i	$-0.903076\pi$
$89$	−18.0000	−1.90800	−0.953998	+	0.299813i	$0.903076\pi$
$90$	0	0
$91$	0	0
$92$	1.00000i	0.104257i
$93$	8.00000i	0.829561i
$94$	−8.00000	−0.825137
$95$	0	0
$96$	−1.00000	−0.102062
$97$	− 10.0000i	− 1.01535i	−0.861550	−	0.507673i	$-0.830506\pi$
$97$	− 10.0000i	− 1.01535i	0.861550	−	0.507673i	$-0.169494\pi$
$98$	− 7.00000i	− 0.707107i
$99$	0	0
$100$	0	0
$101$	−18.0000	−1.79107	−0.895533	−	0.444994i	$-0.853206\pi$
$101$	−18.0000	−1.79107	−0.895533	+	0.444994i	$0.853206\pi$
$102$	2.00000i	0.198030i
$103$	8.00000i	0.788263i	0.919054	+	0.394132i	$0.128955\pi$
$103$	8.00000i	0.788263i	−0.919054	+	0.394132i	$0.871045\pi$
$104$	2.00000	0.196116
$105$	0	0
$106$	2.00000	0.194257
$107$	− 8.00000i	− 0.773389i	−0.922208	−	0.386695i	$-0.873617\pi$
$107$	− 8.00000i	− 0.773389i	0.922208	−	0.386695i	$-0.126383\pi$
$108$	− 1.00000i	− 0.0962250i
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	0	0
$111$	−2.00000	−0.189832
$112$	0	0
$113$	− 6.00000i	− 0.564433i	−0.959351	−	0.282216i	$-0.908930\pi$
$113$	− 6.00000i	− 0.564433i	0.959351	−	0.282216i	$-0.0910696\pi$
$114$	−8.00000	−0.749269
$115$	0	0
$116$	−2.00000	−0.185695
$117$	2.00000i	0.184900i
$118$	− 4.00000i	− 0.368230i
$119$	0	0
$120$	0	0
$121$	−11.0000	−1.00000
$122$	− 2.00000i	− 0.181071i
$123$	− 10.0000i	− 0.901670i
$124$	8.00000	0.718421
$125$	0	0
$126$	0	0
$127$	− 16.0000i	− 1.41977i	−0.704317	−	0.709885i	$-0.748747\pi$
$127$	− 16.0000i	− 1.41977i	0.704317	−	0.709885i	$-0.251253\pi$
$128$	1.00000i	0.0883883i
$129$	8.00000	0.704361
$130$	0	0
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	0	0
$133$	0	0
$134$	−8.00000	−0.691095
$135$	0	0
$136$	2.00000	0.171499
$137$	− 10.0000i	− 0.854358i	−0.904167	−	0.427179i	$-0.859507\pi$
$137$	− 10.0000i	− 0.854358i	0.904167	−	0.427179i	$-0.140493\pi$
$138$	1.00000i	0.0851257i
$139$	20.0000	1.69638	0.848189	−	0.529694i	$-0.177693\pi$
$139$	20.0000	1.69638	0.848189	+	0.529694i	$0.177693\pi$
$140$	0	0
$141$	−8.00000	−0.673722
$142$	0	0
$143$	0	0
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	−6.00000	−0.496564
$147$	− 7.00000i	− 0.577350i
$148$	2.00000i	0.164399i
$149$	−2.00000	−0.163846	−0.0819232	−	0.996639i	$-0.526106\pi$
$149$	−2.00000	−0.163846	−0.0819232	+	0.996639i	$0.526106\pi$
$150$	0	0
$151$	16.0000	1.30206	0.651031	−	0.759051i	$-0.274337\pi$
$151$	16.0000	1.30206	0.651031	+	0.759051i	$0.274337\pi$
$152$	8.00000i	0.648886i
$153$	2.00000i	0.161690i
$154$	0	0
$155$	0	0
$156$	2.00000	0.160128
$157$	22.0000i	1.75579i	0.478852	+	0.877896i	$0.341053\pi$
$157$	22.0000i	1.75579i	−0.478852	+	0.877896i	$0.658947\pi$
$158$	8.00000i	0.636446i
$159$	2.00000	0.158610
$160$	0	0
$161$	0	0
$162$	− 1.00000i	− 0.0785674i
$163$	− 4.00000i	− 0.313304i	−0.987654	−	0.156652i	$-0.949930\pi$
$163$	− 4.00000i	− 0.313304i	0.987654	−	0.156652i	$-0.0500701\pi$
$164$	−10.0000	−0.780869
$165$	0	0
$166$	−16.0000	−1.24184
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	0	0
$169$	9.00000	0.692308
$170$	0	0
$171$	−8.00000	−0.611775
$172$	− 8.00000i	− 0.609994i
$173$	14.0000i	1.06440i	0.846619	+	0.532200i	$0.178635\pi$
$173$	14.0000i	1.06440i	−0.846619	+	0.532200i	$0.821365\pi$
$174$	−2.00000	−0.151620
$175$	0	0
$176$	0	0
$177$	− 4.00000i	− 0.300658i
$178$	18.0000i	1.34916i
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	0	0
$181$	10.0000	0.743294	0.371647	−	0.928374i	$-0.378793\pi$
$181$	10.0000	0.743294	0.371647	+	0.928374i	$0.378793\pi$
$182$	0	0
$183$	− 2.00000i	− 0.147844i
$184$	1.00000	0.0737210
$185$	0	0
$186$	8.00000	0.586588
$187$	0	0
$188$	8.00000i	0.583460i
$189$	0	0
$190$	0	0
$191$	24.0000	1.73658	0.868290	−	0.496058i	$-0.165220\pi$
$191$	24.0000	1.73658	0.868290	+	0.496058i	$0.165220\pi$
$192$	1.00000i	0.0721688i
$193$	− 14.0000i	− 1.00774i	−0.863779	−	0.503871i	$-0.831909\pi$
$193$	− 14.0000i	− 1.00774i	0.863779	−	0.503871i	$-0.168091\pi$
$194$	−10.0000	−0.717958
$195$	0	0
$196$	−7.00000	−0.500000
$197$	− 22.0000i	− 1.56744i	−0.621117	−	0.783718i	$-0.713321\pi$
$197$	− 22.0000i	− 1.56744i	0.621117	−	0.783718i	$-0.286679\pi$
$198$	0	0
$199$	−16.0000	−1.13421	−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000	−1.13421	−0.567105	+	0.823646i	$0.691937\pi$
$200$	0	0
$201$	−8.00000	−0.564276
$202$	18.0000i	1.26648i
$203$	0	0
$204$	2.00000	0.140028
$205$	0	0
$206$	8.00000	0.557386
$207$	1.00000i	0.0695048i
$208$	− 2.00000i	− 0.138675i
$209$	0	0
$210$	0	0
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	− 2.00000i	− 0.137361i
$213$	0	0
$214$	−8.00000	−0.546869
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	0	0
$218$	2.00000i	0.135457i
$219$	−6.00000	−0.405442
$220$	0	0
$221$	−4.00000	−0.269069
$222$	2.00000i	0.134231i
$223$	24.0000i	1.60716i	0.595198	+	0.803579i	$0.297074\pi$
$223$	24.0000i	1.60716i	−0.595198	+	0.803579i	$0.702926\pi$
$224$	0	0
$225$	0	0
$226$	−6.00000	−0.399114
$227$	8.00000i	0.530979i	0.964114	+	0.265489i	$0.0855335\pi$
$227$	8.00000i	0.530979i	−0.964114	+	0.265489i	$0.914466\pi$
$228$	8.00000i	0.529813i
$229$	6.00000	0.396491	0.198246	−	0.980152i	$-0.436476\pi$
$229$	6.00000	0.396491	0.198246	+	0.980152i	$0.436476\pi$
$230$	0	0
$231$	0	0
$232$	2.00000i	0.131306i
$233$	− 22.0000i	− 1.44127i	−0.693316	−	0.720634i	$-0.743851\pi$
$233$	− 22.0000i	− 1.44127i	0.693316	−	0.720634i	$-0.256149\pi$
$234$	2.00000	0.130744
$235$	0	0
$236$	−4.00000	−0.260378
$237$	8.00000i	0.519656i
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	−22.0000	−1.41714	−0.708572	−	0.705638i	$-0.750660\pi$
$241$	−22.0000	−1.41714	−0.708572	+	0.705638i	$0.750660\pi$
$242$	11.0000i	0.707107i
$243$	− 1.00000i	− 0.0641500i
$244$	−2.00000	−0.128037
$245$	0	0
$246$	−10.0000	−0.637577
$247$	− 16.0000i	− 1.01806i
$248$	− 8.00000i	− 0.508001i
$249$	−16.0000	−1.01396
$250$	0	0
$251$	8.00000	0.504956	0.252478	−	0.967603i	$-0.418755\pi$
$251$	8.00000	0.504956	0.252478	+	0.967603i	$0.418755\pi$
$252$	0	0
$253$	0	0
$254$	−16.0000	−1.00393
$255$	0	0
$256$	1.00000	0.0625000
$257$	− 18.0000i	− 1.12281i	−0.827541	−	0.561405i	$-0.810261\pi$
$257$	− 18.0000i	− 1.12281i	0.827541	−	0.561405i	$-0.189739\pi$
$258$	− 8.00000i	− 0.498058i
$259$	0	0
$260$	0	0
$261$	−2.00000	−0.123797
$262$	12.0000i	0.741362i
$263$	8.00000i	0.493301i	0.969104	+	0.246651i	$0.0793300\pi$
$263$	8.00000i	0.493301i	−0.969104	+	0.246651i	$0.920670\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	18.0000i	1.10158i
$268$	8.00000i	0.488678i
$269$	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
$271$	−16.0000	−0.971931	−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000	−0.971931	−0.485965	+	0.873978i	$0.661532\pi$
$272$	− 2.00000i	− 0.121268i
$273$	0	0
$274$	−10.0000	−0.604122
$275$	0	0
$276$	1.00000	0.0601929
$277$	2.00000i	0.120168i	0.998193	+	0.0600842i	$0.0191369\pi$
$277$	2.00000i	0.120168i	−0.998193	+	0.0600842i	$0.980863\pi$
$278$	− 20.0000i	− 1.19952i
$279$	8.00000	0.478947
$280$	0	0
$281$	2.00000	0.119310	0.0596550	−	0.998219i	$-0.481000\pi$
$281$	2.00000	0.119310	0.0596550	+	0.998219i	$0.481000\pi$
$282$	8.00000i	0.476393i
$283$	24.0000i	1.42665i	0.700832	+	0.713326i	$0.252812\pi$
$283$	24.0000i	1.42665i	−0.700832	+	0.713326i	$0.747188\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	1.00000i	0.0589256i
$289$	13.0000	0.764706
$290$	0	0
$291$	−10.0000	−0.586210
$292$	6.00000i	0.351123i
$293$	− 30.0000i	− 1.75262i	−0.481749	−	0.876309i	$-0.659998\pi$
$293$	− 30.0000i	− 1.75262i	0.481749	−	0.876309i	$-0.340002\pi$
$294$	−7.00000	−0.408248
$295$	0	0
$296$	2.00000	0.116248
$297$	0	0
$298$	2.00000i	0.115857i
$299$	−2.00000	−0.115663
$300$	0	0
$301$	0	0
$302$	− 16.0000i	− 0.920697i
$303$	18.0000i	1.03407i
$304$	8.00000	0.458831
$305$	0	0
$306$	2.00000	0.114332
$307$	4.00000i	0.228292i	0.993464	+	0.114146i	$0.0364132\pi$
$307$	4.00000i	0.228292i	−0.993464	+	0.114146i	$0.963587\pi$
$308$	0	0
$309$	8.00000	0.455104
$310$	0	0
$311$	−8.00000	−0.453638	−0.226819	−	0.973937i	$-0.572833\pi$
$311$	−8.00000	−0.453638	−0.226819	+	0.973937i	$0.572833\pi$
$312$	− 2.00000i	− 0.113228i
$313$	2.00000i	0.113047i	0.998401	+	0.0565233i	$0.0180015\pi$
$313$	2.00000i	0.113047i	−0.998401	+	0.0565233i	$0.981998\pi$
$314$	22.0000	1.24153
$315$	0	0
$316$	8.00000	0.450035
$317$	− 14.0000i	− 0.786318i	−0.919470	−	0.393159i	$-0.871382\pi$
$317$	− 14.0000i	− 0.786318i	0.919470	−	0.393159i	$-0.128618\pi$
$318$	− 2.00000i	− 0.112154i
$319$	0	0
$320$	0	0
$321$	−8.00000	−0.446516
$322$	0	0
$323$	− 16.0000i	− 0.890264i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	−4.00000	−0.221540
$327$	2.00000i	0.110600i
$328$	10.0000i	0.552158i
$329$	0	0
$330$	0	0
$331$	−4.00000	−0.219860	−0.109930	−	0.993939i	$-0.535063\pi$
$331$	−4.00000	−0.219860	−0.109930	+	0.993939i	$0.535063\pi$
$332$	16.0000i	0.878114i
$333$	2.00000i	0.109599i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	30.0000i	1.63420i	0.576493	+	0.817102i	$0.304421\pi$
$337$	30.0000i	1.63420i	−0.576493	+	0.817102i	$0.695579\pi$
$338$	− 9.00000i	− 0.489535i
$339$	−6.00000	−0.325875
$340$	0	0
$341$	0	0
$342$	8.00000i	0.432590i
$343$	0	0
$344$	−8.00000	−0.431331
$345$	0	0
$346$	14.0000	0.752645
$347$	12.0000i	0.644194i	0.946707	+	0.322097i	$0.104388\pi$
$347$	12.0000i	0.644194i	−0.946707	+	0.322097i	$0.895612\pi$
$348$	2.00000i	0.107211i
$349$	−6.00000	−0.321173	−0.160586	−	0.987022i	$-0.551338\pi$
$349$	−6.00000	−0.321173	−0.160586	+	0.987022i	$0.551338\pi$
$350$	0	0
$351$	2.00000	0.106752
$352$	0	0
$353$	− 14.0000i	− 0.745145i	−0.928003	−	0.372572i	$-0.878476\pi$
$353$	− 14.0000i	− 0.745145i	0.928003	−	0.372572i	$-0.121524\pi$
$354$	−4.00000	−0.212598
$355$	0	0
$356$	18.0000	0.953998
$357$	0	0
$358$	12.0000i	0.634220i
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	45.0000	2.36842
$362$	− 10.0000i	− 0.525588i
$363$	11.0000i	0.577350i
$364$	0	0
$365$	0	0
$366$	−2.00000	−0.104542
$367$	24.0000i	1.25279i	0.779506	+	0.626395i	$0.215470\pi$
$367$	24.0000i	1.25279i	−0.779506	+	0.626395i	$0.784530\pi$
$368$	− 1.00000i	− 0.0521286i
$369$	−10.0000	−0.520579
$370$	0	0
$371$	0	0
$372$	− 8.00000i	− 0.414781i
$373$	18.0000i	0.932005i	0.884783	+	0.466002i	$0.154306\pi$
$373$	18.0000i	0.932005i	−0.884783	+	0.466002i	$0.845694\pi$
$374$	0	0
$375$	0	0
$376$	8.00000	0.412568
$377$	− 4.00000i	− 0.206010i
$378$	0	0
$379$	16.0000	0.821865	0.410932	−	0.911666i	$-0.365203\pi$
$379$	16.0000	0.821865	0.410932	+	0.911666i	$0.365203\pi$
$380$	0	0
$381$	−16.0000	−0.819705
$382$	− 24.0000i	− 1.22795i
$383$	16.0000i	0.817562i	0.912633	+	0.408781i	$0.134046\pi$
$383$	16.0000i	0.817562i	−0.912633	+	0.408781i	$0.865954\pi$
$384$	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
$391$	−2.00000	−0.101144
$392$	7.00000i	0.353553i
$393$	12.0000i	0.605320i
$394$	−22.0000	−1.10834
$395$	0	0
$396$	0	0
$397$	− 30.0000i	− 1.50566i	−0.658217	−	0.752828i	$-0.728689\pi$
$397$	− 30.0000i	− 1.50566i	0.658217	−	0.752828i	$-0.271311\pi$
$398$	16.0000i	0.802008i
$399$	0	0
$400$	0	0
$401$	−14.0000	−0.699127	−0.349563	−	0.936913i	$-0.613670\pi$
$401$	−14.0000	−0.699127	−0.349563	+	0.936913i	$0.613670\pi$
$402$	8.00000i	0.399004i
$403$	16.0000i	0.797017i
$404$	18.0000	0.895533
$405$	0	0
$406$	0	0
$407$	0	0
$408$	− 2.00000i	− 0.0990148i
$409$	22.0000	1.08783	0.543915	−	0.839140i	$-0.316941\pi$
$409$	22.0000	1.08783	0.543915	+	0.839140i	$0.316941\pi$
$410$	0	0
$411$	−10.0000	−0.493264
$412$	− 8.00000i	− 0.394132i
$413$	0	0
$414$	1.00000	0.0491473
$415$	0	0
$416$	−2.00000	−0.0980581
$417$	− 20.0000i	− 0.979404i
$418$	0	0
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	−30.0000	−1.46211	−0.731055	−	0.682318i	$-0.760972\pi$
$421$	−30.0000	−1.46211	−0.731055	+	0.682318i	$0.760972\pi$
$422$	12.0000i	0.584151i
$423$	8.00000i	0.388973i
$424$	−2.00000	−0.0971286
$425$	0	0
$426$	0	0
$427$	0	0
$428$	8.00000i	0.386695i
$429$	0	0
$430$	0	0
$431$	−24.0000	−1.15604	−0.578020	−	0.816023i	$-0.696174\pi$
$431$	−24.0000	−1.15604	−0.578020	+	0.816023i	$0.696174\pi$
$432$	1.00000i	0.0481125i
$433$	− 22.0000i	− 1.05725i	−0.848855	−	0.528626i	$-0.822707\pi$
$433$	− 22.0000i	− 1.05725i	0.848855	−	0.528626i	$-0.177293\pi$
$434$	0	0
$435$	0	0
$436$	2.00000	0.0957826
$437$	− 8.00000i	− 0.382692i
$438$	6.00000i	0.286691i
$439$	8.00000	0.381819	0.190910	−	0.981608i	$-0.438856\pi$
$439$	8.00000	0.381819	0.190910	+	0.981608i	$0.438856\pi$
$440$	0	0
$441$	−7.00000	−0.333333
$442$	4.00000i	0.190261i
$443$	4.00000i	0.190046i	0.995475	+	0.0950229i	$0.0302924\pi$
$443$	4.00000i	0.190046i	−0.995475	+	0.0950229i	$0.969708\pi$
$444$	2.00000	0.0949158
$445$	0	0
$446$	24.0000	1.13643
$447$	2.00000i	0.0945968i
$448$	0	0
$449$	30.0000	1.41579	0.707894	−	0.706319i	$-0.249646\pi$
$449$	30.0000	1.41579	0.707894	+	0.706319i	$0.249646\pi$
$450$	0	0
$451$	0	0
$452$	6.00000i	0.282216i
$453$	− 16.0000i	− 0.751746i
$454$	8.00000	0.375459
$455$	0	0
$456$	8.00000	0.374634
$457$	− 18.0000i	− 0.842004i	−0.907060	−	0.421002i	$-0.861678\pi$
$457$	− 18.0000i	− 0.842004i	0.907060	−	0.421002i	$-0.138322\pi$
$458$	− 6.00000i	− 0.280362i
$459$	2.00000	0.0933520
$460$	0	0
$461$	30.0000	1.39724	0.698620	−	0.715493i	$-0.253798\pi$
$461$	30.0000	1.39724	0.698620	+	0.715493i	$0.253798\pi$
$462$	0	0
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	2.00000	0.0928477
$465$	0	0
$466$	−22.0000	−1.01913
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	− 2.00000i	− 0.0924500i
$469$	0	0
$470$	0	0
$471$	22.0000	1.01371
$472$	4.00000i	0.184115i
$473$	0	0
$474$	8.00000	0.367452
$475$	0	0
$476$	0	0
$477$	− 2.00000i	− 0.0915737i
$478$	0	0
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	−4.00000	−0.182384
$482$	22.0000i	1.00207i
$483$	0	0
$484$	11.0000	0.500000
$485$	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	$0.941983\pi$
$488$	2.00000i	0.0905357i
$489$	−4.00000	−0.180886
$490$	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
$493$	− 4.00000i	− 0.180151i
$494$	−16.0000	−0.719874
$495$	0	0
$496$	−8.00000	−0.359211
$497$	0	0
$498$	16.0000i	0.716977i
$499$	20.0000	0.895323	0.447661	−	0.894203i	$-0.352257\pi$
$499$	20.0000	0.895323	0.447661	+	0.894203i	$0.352257\pi$
$500$	0	0
$501$	0	0
$502$	− 8.00000i	− 0.357057i
$503$	− 32.0000i	− 1.42681i	−0.700752	−	0.713405i	$-0.747152\pi$
$503$	− 32.0000i	− 1.42681i	0.700752	−	0.713405i	$-0.252848\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 9.00000i	− 0.399704i
$508$	16.0000i	0.709885i
$509$	−30.0000	−1.32973	−0.664863	−	0.746965i	$-0.731510\pi$
$509$	−30.0000	−1.32973	−0.664863	+	0.746965i	$0.731510\pi$
$510$	0	0
$511$	0	0
$512$	− 1.00000i	− 0.0441942i
$513$	8.00000i	0.353209i
$514$	−18.0000	−0.793946
$515$	0	0
$516$	−8.00000	−0.352180
$517$	0	0
$518$	0	0
$519$	14.0000	0.614532
$520$	0	0
$521$	26.0000	1.13908	0.569540	−	0.821963i	$-0.307121\pi$
$521$	26.0000	1.13908	0.569540	+	0.821963i	$0.307121\pi$
$522$	2.00000i	0.0875376i
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	12.0000	0.524222
$525$	0	0
$526$	8.00000	0.348817
$527$	16.0000i	0.696971i
$528$	0	0
$529$	−1.00000	−0.0434783
$530$	0	0
$531$	−4.00000	−0.173585
$532$	0	0
$533$	− 20.0000i	− 0.866296i
$534$	18.0000	0.778936
$535$	0	0
$536$	8.00000	0.345547
$537$	12.0000i	0.517838i
$538$	− 2.00000i	− 0.0862261i
$539$	0	0
$540$	0	0
$541$	−10.0000	−0.429934	−0.214967	−	0.976621i	$-0.568964\pi$
$541$	−10.0000	−0.429934	−0.214967	+	0.976621i	$0.568964\pi$
$542$	16.0000i	0.687259i
$543$	− 10.0000i	− 0.429141i
$544$	−2.00000	−0.0857493
$545$	0	0
$546$	0	0
$547$	44.0000i	1.88130i	0.339372	+	0.940652i	$0.389785\pi$
$547$	44.0000i	1.88130i	−0.339372	+	0.940652i	$0.610215\pi$
$548$	10.0000i	0.427179i
$549$	−2.00000	−0.0853579
$550$	0	0
$551$	16.0000	0.681623
$552$	− 1.00000i	− 0.0425628i
$553$	0	0
$554$	2.00000	0.0849719
$555$	0	0
$556$	−20.0000	−0.848189
$557$	− 2.00000i	− 0.0847427i	−0.999102	−	0.0423714i	$-0.986509\pi$
$557$	− 2.00000i	− 0.0847427i	0.999102	−	0.0423714i	$-0.0134913\pi$
$558$	− 8.00000i	− 0.338667i
$559$	16.0000	0.676728
$560$	0	0
$561$	0	0
$562$	− 2.00000i	− 0.0843649i
$563$	− 24.0000i	− 1.01148i	−0.862686	−	0.505740i	$-0.831220\pi$
$563$	− 24.0000i	− 1.01148i	0.862686	−	0.505740i	$-0.168780\pi$
$564$	8.00000	0.336861
$565$	0	0
$566$	24.0000	1.00880
$567$	0	0
$568$	0	0
$569$	−10.0000	−0.419222	−0.209611	−	0.977785i	$-0.567220\pi$
$569$	−10.0000	−0.419222	−0.209611	+	0.977785i	$0.567220\pi$
$570$	0	0
$571$	16.0000	0.669579	0.334790	−	0.942293i	$-0.391335\pi$
$571$	16.0000	0.669579	0.334790	+	0.942293i	$0.391335\pi$
$572$	0	0
$573$	− 24.0000i	− 1.00261i
$574$	0	0
$575$	0	0
$576$	1.00000	0.0416667
$577$	− 18.0000i	− 0.749350i	−0.927156	−	0.374675i	$-0.877754\pi$
$577$	− 18.0000i	− 0.749350i	0.927156	−	0.374675i	$-0.122246\pi$
$578$	− 13.0000i	− 0.540729i
$579$	−14.0000	−0.581820
$580$	0	0
$581$	0	0
$582$	10.0000i	0.414513i
$583$	0	0
$584$	6.00000	0.248282
$585$	0	0
$586$	−30.0000	−1.23929
$587$	− 12.0000i	− 0.495293i	−0.968850	−	0.247647i	$-0.920343\pi$
$587$	− 12.0000i	− 0.495293i	0.968850	−	0.247647i	$-0.0796572\pi$
$588$	7.00000i	0.288675i
$589$	−64.0000	−2.63707
$590$	0	0
$591$	−22.0000	−0.904959
$592$	− 2.00000i	− 0.0821995i
$593$	34.0000i	1.39621i	0.715994	+	0.698106i	$0.245974\pi$
$593$	34.0000i	1.39621i	−0.715994	+	0.698106i	$0.754026\pi$
$594$	0	0
$595$	0	0
$596$	2.00000	0.0819232
$597$	16.0000i	0.654836i
$598$	2.00000i	0.0817861i
$599$	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
$601$	42.0000	1.71322	0.856608	−	0.515968i	$-0.172568\pi$
$601$	42.0000	1.71322	0.856608	+	0.515968i	$0.172568\pi$
$602$	0	0
$603$	8.00000i	0.325785i
$604$	−16.0000	−0.651031
$605$	0	0
$606$	18.0000	0.731200
$607$	24.0000i	0.974130i	0.873366	+	0.487065i	$0.161933\pi$
$607$	24.0000i	0.974130i	−0.873366	+	0.487065i	$0.838067\pi$
$608$	− 8.00000i	− 0.324443i
$609$	0	0
$610$	0	0
$611$	−16.0000	−0.647291
$612$	− 2.00000i	− 0.0808452i
$613$	10.0000i	0.403896i	0.979396	+	0.201948i	$0.0647272\pi$
$613$	10.0000i	0.403896i	−0.979396	+	0.201948i	$0.935273\pi$
$614$	4.00000	0.161427
$615$	0	0
$616$	0	0
$617$	− 18.0000i	− 0.724653i	−0.932051	−	0.362326i	$-0.881983\pi$
$617$	− 18.0000i	− 0.724653i	0.932051	−	0.362326i	$-0.118017\pi$
$618$	− 8.00000i	− 0.321807i
$619$	40.0000	1.60774	0.803868	−	0.594808i	$-0.202772\pi$
$619$	40.0000	1.60774	0.803868	+	0.594808i	$0.202772\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	8.00000i	0.320771i
$623$	0	0
$624$	−2.00000	−0.0800641
$625$	0	0
$626$	2.00000	0.0799361
$627$	0	0
$628$	− 22.0000i	− 0.877896i
$629$	−4.00000	−0.159490
$630$	0	0
$631$	24.0000	0.955425	0.477712	−	0.878516i	$-0.341466\pi$
$631$	24.0000	0.955425	0.477712	+	0.878516i	$0.341466\pi$
$632$	− 8.00000i	− 0.318223i
$633$	12.0000i	0.476957i
$634$	−14.0000	−0.556011
$635$	0	0
$636$	−2.00000	−0.0793052
$637$	− 14.0000i	− 0.554700i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	18.0000	0.710957	0.355479	−	0.934684i	$-0.384318\pi$
$641$	18.0000	0.710957	0.355479	+	0.934684i	$0.384318\pi$
$642$	8.00000i	0.315735i
$643$	16.0000i	0.630978i	0.948929	+	0.315489i	$0.102169\pi$
$643$	16.0000i	0.630978i	−0.948929	+	0.315489i	$0.897831\pi$
$644$	0	0
$645$	0	0
$646$	−16.0000	−0.629512
$647$	− 24.0000i	− 0.943537i	−0.881722	−	0.471769i	$-0.843616\pi$
$647$	− 24.0000i	− 0.943537i	0.881722	−	0.471769i	$-0.156384\pi$
$648$	1.00000i	0.0392837i
$649$	0	0
$650$	0	0
$651$	0	0
$652$	4.00000i	0.156652i
$653$	− 10.0000i	− 0.391330i	−0.980671	−	0.195665i	$-0.937313\pi$
$653$	− 10.0000i	− 0.391330i	0.980671	−	0.195665i	$-0.0626866\pi$
$654$	2.00000	0.0782062
$655$	0	0
$656$	10.0000	0.390434
$657$	6.00000i	0.234082i
$658$	0	0
$659$	−32.0000	−1.24654	−0.623272	−	0.782006i	$-0.714197\pi$
$659$	−32.0000	−1.24654	−0.623272	+	0.782006i	$0.714197\pi$
$660$	0	0
$661$	42.0000	1.63361	0.816805	−	0.576913i	$-0.195743\pi$
$661$	42.0000	1.63361	0.816805	+	0.576913i	$0.195743\pi$
$662$	4.00000i	0.155464i
$663$	4.00000i	0.155347i
$664$	16.0000	0.620920
$665$	0	0
$666$	2.00000	0.0774984
$667$	− 2.00000i	− 0.0774403i
$668$	0	0
$669$	24.0000	0.927894
$670$	0	0
$671$	0	0
$672$	0	0
$673$	18.0000i	0.693849i	0.937893	+	0.346925i	$0.112774\pi$
$673$	18.0000i	0.693849i	−0.937893	+	0.346925i	$0.887226\pi$
$674$	30.0000	1.15556
$675$	0	0
$676$	−9.00000	−0.346154
$677$	− 2.00000i	− 0.0768662i	−0.999261	−	0.0384331i	$-0.987763\pi$
$677$	− 2.00000i	− 0.0768662i	0.999261	−	0.0384331i	$-0.0122367\pi$
$678$	6.00000i	0.230429i
$679$	0	0
$680$	0	0
$681$	8.00000	0.306561
$682$	0	0
$683$	− 4.00000i	− 0.153056i	−0.997067	−	0.0765279i	$-0.975617\pi$
$683$	− 4.00000i	− 0.153056i	0.997067	−	0.0765279i	$-0.0243834\pi$
$684$	8.00000	0.305888
$685$	0	0
$686$	0	0
$687$	− 6.00000i	− 0.228914i
$688$	8.00000i	0.304997i
$689$	4.00000	0.152388
$690$	0	0
$691$	20.0000	0.760836	0.380418	−	0.924815i	$-0.375780\pi$
$691$	20.0000	0.760836	0.380418	+	0.924815i	$0.375780\pi$
$692$	− 14.0000i	− 0.532200i
$693$	0	0
$694$	12.0000	0.455514
$695$	0	0
$696$	2.00000	0.0758098
$697$	− 20.0000i	− 0.757554i
$698$	6.00000i	0.227103i
$699$	−22.0000	−0.832116
$700$	0	0
$701$	18.0000	0.679851	0.339925	−	0.940452i	$-0.389598\pi$
$701$	18.0000	0.679851	0.339925	+	0.940452i	$0.389598\pi$
$702$	− 2.00000i	− 0.0754851i
$703$	− 16.0000i	− 0.603451i
$704$	0	0
$705$	0	0
$706$	−14.0000	−0.526897
$707$	0	0
$708$	4.00000i	0.150329i
$709$	14.0000	0.525781	0.262891	−	0.964826i	$-0.415324\pi$
$709$	14.0000	0.525781	0.262891	+	0.964826i	$0.415324\pi$
$710$	0	0
$711$	8.00000	0.300023
$712$	− 18.0000i	− 0.674579i
$713$	8.00000i	0.299602i
$714$	0	0
$715$	0	0
$716$	12.0000	0.448461
$717$	0	0
$718$	0	0
$719$	−48.0000	−1.79010	−0.895049	−	0.445968i	$-0.852860\pi$
$719$	−48.0000	−1.79010	−0.895049	+	0.445968i	$0.852860\pi$
$720$	0	0
$721$	0	0
$722$	− 45.0000i	− 1.67473i
$723$	22.0000i	0.818189i
$724$	−10.0000	−0.371647
$725$	0	0
$726$	11.0000	0.408248
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	16.0000	0.591781
$732$	2.00000i	0.0739221i
$733$	− 6.00000i	− 0.221615i	−0.993842	−	0.110808i	$-0.964656\pi$
$733$	− 6.00000i	− 0.221615i	0.993842	−	0.110808i	$-0.0353437\pi$
$734$	24.0000	0.885856
$735$	0	0
$736$	−1.00000	−0.0368605
$737$	0	0
$738$	10.0000i	0.368105i
$739$	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
$741$	−16.0000	−0.587775
$742$	0	0
$743$	− 24.0000i	− 0.880475i	−0.897881	−	0.440237i	$-0.854894\pi$
$743$	− 24.0000i	− 0.880475i	0.897881	−	0.440237i	$-0.145106\pi$
$744$	−8.00000	−0.293294
$745$	0	0
$746$	18.0000	0.659027
$747$	16.0000i	0.585409i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	16.0000	0.583848	0.291924	−	0.956441i	$-0.405705\pi$
$751$	16.0000	0.583848	0.291924	+	0.956441i	$0.405705\pi$
$752$	− 8.00000i	− 0.291730i
$753$	− 8.00000i	− 0.291536i
$754$	−4.00000	−0.145671
$755$	0	0
$756$	0	0
$757$	38.0000i	1.38113i	0.723269	+	0.690567i	$0.242639\pi$
$757$	38.0000i	1.38113i	−0.723269	+	0.690567i	$0.757361\pi$
$758$	− 16.0000i	− 0.581146i
$759$	0	0
$760$	0	0
$761$	42.0000	1.52250	0.761249	−	0.648459i	$-0.224586\pi$
$761$	42.0000	1.52250	0.761249	+	0.648459i	$0.224586\pi$
$762$	16.0000i	0.579619i
$763$	0	0
$764$	−24.0000	−0.868290
$765$	0	0
$766$	16.0000	0.578103
$767$	− 8.00000i	− 0.288863i
$768$	− 1.00000i	− 0.0360844i
$769$	−26.0000	−0.937584	−0.468792	−	0.883309i	$-0.655311\pi$
$769$	−26.0000	−0.937584	−0.468792	+	0.883309i	$0.655311\pi$
$770$	0	0
$771$	−18.0000	−0.648254
$772$	14.0000i	0.503871i
$773$	− 38.0000i	− 1.36677i	−0.730061	−	0.683383i	$-0.760508\pi$
$773$	− 38.0000i	− 1.36677i	0.730061	−	0.683383i	$-0.239492\pi$
$774$	−8.00000	−0.287554
$775$	0	0
$776$	10.0000	0.358979
$777$	0	0
$778$	− 22.0000i	− 0.788738i
$779$	80.0000	2.86630
$780$	0	0
$781$	0	0
$782$	2.00000i	0.0715199i
$783$	2.00000i	0.0714742i
$784$	7.00000	0.250000
$785$	0	0
$786$	12.0000	0.428026
$787$	32.0000i	1.14068i	0.821410	+	0.570338i	$0.193188\pi$
$787$	32.0000i	1.14068i	−0.821410	+	0.570338i	$0.806812\pi$
$788$	22.0000i	0.783718i
$789$	8.00000	0.284808
$790$	0	0
$791$	0	0
$792$	0	0
$793$	− 4.00000i	− 0.142044i
$794$	−30.0000	−1.06466
$795$	0	0
$796$	16.0000	0.567105
$797$	− 2.00000i	− 0.0708436i	−0.999372	−	0.0354218i	$-0.988723\pi$
$797$	− 2.00000i	− 0.0708436i	0.999372	−	0.0354218i	$-0.0112775\pi$
$798$	0	0
$799$	−16.0000	−0.566039
$800$	0	0
$801$	18.0000	0.635999
$802$	14.0000i	0.494357i
$803$	0	0
$804$	8.00000	0.282138
$805$	0	0
$806$	16.0000	0.563576
$807$	− 2.00000i	− 0.0704033i
$808$	− 18.0000i	− 0.633238i
$809$	38.0000	1.33601	0.668004	−	0.744157i	$-0.267149\pi$
$809$	38.0000	1.33601	0.668004	+	0.744157i	$0.267149\pi$
$810$	0	0
$811$	−4.00000	−0.140459	−0.0702295	−	0.997531i	$-0.522373\pi$
$811$	−4.00000	−0.140459	−0.0702295	+	0.997531i	$0.522373\pi$
$812$	0	0
$813$	16.0000i	0.561144i
$814$	0	0
$815$	0	0
$816$	−2.00000	−0.0700140
$817$	64.0000i	2.23908i
$818$	− 22.0000i	− 0.769212i
$819$	0	0
$820$	0	0
$821$	−50.0000	−1.74501	−0.872506	−	0.488603i	$-0.837507\pi$
$821$	−50.0000	−1.74501	−0.872506	+	0.488603i	$0.837507\pi$
$822$	10.0000i	0.348790i
$823$	− 32.0000i	− 1.11545i	−0.830026	−	0.557725i	$-0.811674\pi$
$823$	− 32.0000i	− 1.11545i	0.830026	−	0.557725i	$-0.188326\pi$
$824$	−8.00000	−0.278693
$825$	0	0
$826$	0	0
$827$	− 24.0000i	− 0.834562i	−0.908778	−	0.417281i	$-0.862983\pi$
$827$	− 24.0000i	− 0.834562i	0.908778	−	0.417281i	$-0.137017\pi$
$828$	− 1.00000i	− 0.0347524i
$829$	−22.0000	−0.764092	−0.382046	−	0.924143i	$-0.624780\pi$
$829$	−22.0000	−0.764092	−0.382046	+	0.924143i	$0.624780\pi$
$830$	0	0
$831$	2.00000	0.0693792
$832$	2.00000i	0.0693375i
$833$	− 14.0000i	− 0.485071i
$834$	−20.0000	−0.692543
$835$	0	0
$836$	0	0
$837$	− 8.00000i	− 0.276520i
$838$	0	0
$839$	56.0000	1.93333	0.966667	−	0.256036i	$-0.0824164\pi$
$839$	56.0000	1.93333	0.966667	+	0.256036i	$0.0824164\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	30.0000i	1.03387i
$843$	− 2.00000i	− 0.0688837i
$844$	12.0000	0.413057
$845$	0	0
$846$	8.00000	0.275046
$847$	0	0
$848$	2.00000i	0.0686803i
$849$	24.0000	0.823678
$850$	0	0
$851$	−2.00000	−0.0685591
$852$	0	0
$853$	− 26.0000i	− 0.890223i	−0.895475	−	0.445112i	$-0.853164\pi$
$853$	− 26.0000i	− 0.890223i	0.895475	−	0.445112i	$-0.146836\pi$
$854$	0	0
$855$	0	0
$856$	8.00000	0.273434
$857$	54.0000i	1.84460i	0.386469	+	0.922302i	$0.373695\pi$
$857$	54.0000i	1.84460i	−0.386469	+	0.922302i	$0.626305\pi$
$858$	0	0
$859$	−36.0000	−1.22830	−0.614152	−	0.789188i	$-0.710502\pi$
$859$	−36.0000	−1.22830	−0.614152	+	0.789188i	$0.710502\pi$
$860$	0	0
$861$	0	0
$862$	24.0000i	0.817443i
$863$	32.0000i	1.08929i	0.838666	+	0.544646i	$0.183336\pi$
$863$	32.0000i	1.08929i	−0.838666	+	0.544646i	$0.816664\pi$
$864$	1.00000	0.0340207
$865$	0	0
$866$	−22.0000	−0.747590
$867$	− 13.0000i	− 0.441503i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−16.0000	−0.542139
$872$	− 2.00000i	− 0.0677285i
$873$	10.0000i	0.338449i
$874$	−8.00000	−0.270604
$875$	0	0
$876$	6.00000	0.202721
$877$	− 22.0000i	− 0.742887i	−0.928456	−	0.371444i	$-0.878863\pi$
$877$	− 22.0000i	− 0.742887i	0.928456	−	0.371444i	$-0.121137\pi$
$878$	− 8.00000i	− 0.269987i
$879$	−30.0000	−1.01187
$880$	0	0
$881$	−22.0000	−0.741199	−0.370599	−	0.928793i	$-0.620848\pi$
$881$	−22.0000	−0.741199	−0.370599	+	0.928793i	$0.620848\pi$
$882$	7.00000i	0.235702i
$883$	4.00000i	0.134611i	0.997732	+	0.0673054i	$0.0214402\pi$
$883$	4.00000i	0.134611i	−0.997732	+	0.0673054i	$0.978560\pi$
$884$	4.00000	0.134535
$885$	0	0
$886$	4.00000	0.134383
$887$	48.0000i	1.61168i	0.592132	+	0.805841i	$0.298286\pi$
$887$	48.0000i	1.61168i	−0.592132	+	0.805841i	$0.701714\pi$
$888$	− 2.00000i	− 0.0671156i
$889$	0	0
$890$	0	0
$891$	0	0
$892$	− 24.0000i	− 0.803579i
$893$	− 64.0000i	− 2.14168i
$894$	2.00000	0.0668900
$895$	0	0
$896$	0	0
$897$	2.00000i	0.0667781i
$898$	− 30.0000i	− 1.00111i
$899$	−16.0000	−0.533630
$900$	0	0
$901$	4.00000	0.133259
$902$	0	0
$903$	0	0
$904$	6.00000	0.199557
$905$	0	0
$906$	−16.0000	−0.531564
$907$	− 24.0000i	− 0.796907i	−0.917189	−	0.398453i	$-0.869547\pi$
$907$	− 24.0000i	− 0.796907i	0.917189	−	0.398453i	$-0.130453\pi$
$908$	− 8.00000i	− 0.265489i
$909$	18.0000	0.597022
$910$	0	0
$911$	40.0000	1.32526	0.662630	−	0.748947i	$-0.269440\pi$
$911$	40.0000	1.32526	0.662630	+	0.748947i	$0.269440\pi$
$912$	− 8.00000i	− 0.264906i
$913$	0	0
$914$	−18.0000	−0.595387
$915$	0	0
$916$	−6.00000	−0.198246
$917$	0	0
$918$	− 2.00000i	− 0.0660098i
$919$	−16.0000	−0.527791	−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000	−0.527791	−0.263896	+	0.964551i	$0.585007\pi$
$920$	0	0
$921$	4.00000	0.131804
$922$	− 30.0000i	− 0.987997i
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 8.00000i	− 0.262754i
$928$	− 2.00000i	− 0.0656532i
$929$	−34.0000	−1.11550	−0.557752	−	0.830008i	$-0.688336\pi$
$929$	−34.0000	−1.11550	−0.557752	+	0.830008i	$0.688336\pi$
$930$	0	0
$931$	56.0000	1.83533
$932$	22.0000i	0.720634i
$933$	8.00000i	0.261908i
$934$	0	0
$935$	0	0
$936$	−2.00000	−0.0653720
$937$	22.0000i	0.718709i	0.933201	+	0.359354i	$0.117003\pi$
$937$	22.0000i	0.718709i	−0.933201	+	0.359354i	$0.882997\pi$
$938$	0	0
$939$	2.00000	0.0652675
$940$	0	0
$941$	58.0000	1.89075	0.945373	−	0.325991i	$-0.105698\pi$
$941$	58.0000	1.89075	0.945373	+	0.325991i	$0.105698\pi$
$942$	− 22.0000i	− 0.716799i
$943$	− 10.0000i	− 0.325645i
$944$	4.00000	0.130189
$945$	0	0
$946$	0	0
$947$	− 44.0000i	− 1.42981i	−0.699223	−	0.714904i	$-0.746470\pi$
$947$	− 44.0000i	− 1.42981i	0.699223	−	0.714904i	$-0.253530\pi$
$948$	− 8.00000i	− 0.259828i
$949$	−12.0000	−0.389536
$950$	0	0
$951$	−14.0000	−0.453981
$952$	0	0
$953$	− 22.0000i	− 0.712650i	−0.934362	−	0.356325i	$-0.884030\pi$
$953$	− 22.0000i	− 0.712650i	0.934362	−	0.356325i	$-0.115970\pi$
$954$	−2.00000	−0.0647524
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	33.0000	1.06452
$962$	4.00000i	0.128965i
$963$	8.00000i	0.257796i
$964$	22.0000	0.708572
$965$	0	0
$966$	0	0
$967$	32.0000i	1.02905i	0.857475	+	0.514525i	$0.172032\pi$
$967$	32.0000i	1.02905i	−0.857475	+	0.514525i	$0.827968\pi$
$968$	− 11.0000i	− 0.353553i
$969$	−16.0000	−0.513994
$970$	0	0
$971$	−48.0000	−1.54039	−0.770197	−	0.637806i	$-0.779842\pi$
$971$	−48.0000	−1.54039	−0.770197	+	0.637806i	$0.779842\pi$
$972$	1.00000i	0.0320750i
$973$	0	0
$974$	8.00000	0.256337
$975$	0	0
$976$	2.00000	0.0640184
$977$	14.0000i	0.447900i	0.974601	+	0.223950i	$0.0718952\pi$
$977$	14.0000i	0.447900i	−0.974601	+	0.223950i	$0.928105\pi$
$978$	4.00000i	0.127906i
$979$	0	0
$980$	0	0
$981$	2.00000	0.0638551
$982$	− 4.00000i	− 0.127645i
$983$	48.0000i	1.53096i	0.643458	+	0.765481i	$0.277499\pi$
$983$	48.0000i	1.53096i	−0.643458	+	0.765481i	$0.722501\pi$
$984$	10.0000	0.318788
$985$	0	0
$986$	−4.00000	−0.127386
$987$	0	0
$988$	16.0000i	0.509028i
$989$	8.00000	0.254385
$990$	0	0
$991$	−56.0000	−1.77890	−0.889449	−	0.457034i	$-0.848912\pi$
$991$	−56.0000	−1.77890	−0.889449	+	0.457034i	$0.848912\pi$
$992$	8.00000i	0.254000i
$993$	4.00000i	0.126936i
$994$	0	0
$995$	0	0
$996$	16.0000	0.506979
$997$	− 14.0000i	− 0.443384i	−0.975117	−	0.221692i	$-0.928842\pi$
$997$	− 14.0000i	− 0.443384i	0.975117	−	0.221692i	$-0.0711580\pi$
$998$	− 20.0000i	− 0.633089i
$999$	2.00000	0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3450.2.d.f.2899.1		2
5.2	odd	4		138.2.a.c.1.1	✓	1
5.3	odd	4		3450.2.a.k.1.1		1
5.4	even	2	inner	3450.2.d.f.2899.2		2
15.2	even	4		414.2.a.a.1.1		1
20.7	even	4		1104.2.a.g.1.1		1
35.27	even	4		6762.2.a.bg.1.1		1
40.27	even	4		4416.2.a.c.1.1		1
40.37	odd	4		4416.2.a.s.1.1		1
60.47	odd	4		3312.2.a.d.1.1		1
115.22	even	4		3174.2.a.e.1.1		1
345.137	odd	4		9522.2.a.d.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.2.a.c.1.1	✓	1	5.2	odd	4
414.2.a.a.1.1		1	15.2	even	4
1104.2.a.g.1.1		1	20.7	even	4
3174.2.a.e.1.1		1	115.22	even	4
3312.2.a.d.1.1		1	60.47	odd	4
3450.2.a.k.1.1		1	5.3	odd	4
3450.2.d.f.2899.1		2	1.1	even	1	trivial
3450.2.d.f.2899.2		2	5.4	even	2	inner
4416.2.a.c.1.1		1	40.27	even	4
4416.2.a.s.1.1		1	40.37	odd	4
6762.2.a.bg.1.1		1	35.27	even	4
9522.2.a.d.1.1		1	345.137	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3450.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} +1.00000i q^{12} -2.00000i q^{13} +1.00000 q^{16} -2.00000i q^{17} +1.00000i q^{18} +8.00000 q^{19} -1.00000i q^{23} +1.00000 q^{24} -2.00000 q^{26} +1.00000i q^{27} +2.00000 q^{29} -8.00000 q^{31} -1.00000i q^{32} -2.00000 q^{34} +1.00000 q^{36} -2.00000i q^{37} -8.00000i q^{38} -2.00000 q^{39} +10.0000 q^{41} +8.00000i q^{43} -1.00000 q^{46} -8.00000i q^{47} -1.00000i q^{48} +7.00000 q^{49} -2.00000 q^{51} +2.00000i q^{52} +2.00000i q^{53} +1.00000 q^{54} -8.00000i q^{57} -2.00000i q^{58} +4.00000 q^{59} +2.00000 q^{61} +8.00000i q^{62} -1.00000 q^{64} -8.00000i q^{67} +2.00000i q^{68} -1.00000 q^{69} -1.00000i q^{72} -6.00000i q^{73} -2.00000 q^{74} -8.00000 q^{76} +2.00000i q^{78} -8.00000 q^{79} +1.00000 q^{81} -10.0000i q^{82} -16.0000i q^{83} +8.00000 q^{86} -2.00000i q^{87} -18.0000 q^{89} +1.00000i q^{92} +8.00000i q^{93} -8.00000 q^{94} -1.00000 q^{96} -10.0000i q^{97} -7.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9	\( 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9} + 2 q^{16} + 16 q^{19} + 2 q^{24} - 4 q^{26} + 4 q^{29} - 16 q^{31} - 4 q^{34} + 2 q^{36} - 4 q^{39} + 20 q^{41} - 2 q^{46} + 14 q^{49} - 4 q^{51} + 2 q^{54} + 8 q^{59} + 4 q^{61} - 2 q^{64} - 2 q^{69} - 4 q^{74} - 16 q^{76} - 16 q^{79} + 2 q^{81} + 16 q^{86} - 36 q^{89} - 16 q^{94} - 2 q^{96}+O(q^{100}) \) 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9 + 2 * q^16 + 16 * q^19 + 2 * q^24 - 4 * q^26 + 4 * q^29 - 16 * q^31 - 4 * q^34 + 2 * q^36 - 4 * q^39 + 20 * q^41 - 2 * q^46 + 14 * q^49 - 4 * q^51 + 2 * q^54 + 8 * q^59 + 4 * q^61 - 2 * q^64 - 2 * q^69 - 4 * q^74 - 16 * q^76 - 16 * q^79 + 2 * q^81 + 16 * q^86 - 36 * q^89 - 16 * q^94 - 2 * q^96

Introduction

L-functions

Modular forms

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Database

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