LMFDB - Embedded newform 4950.2.c.m.199.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4950,2,Mod(199,4950)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4950.199"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4950, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4950 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4950.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,-2,0,0,0,0,0,0,-2,0,0,6,0,2,0,0,-10,0,0,0,0,0,0,12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(26)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$39.5259490005$
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 110)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.2
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	4950.199
Dual form			4950.2.c.m.199.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{4} -3.00000i q^{7} -1.00000i q^{8} -1.00000 q^{11} -6.00000i q^{13} +3.00000 q^{14} +1.00000 q^{16} -7.00000i q^{17} -5.00000 q^{19} -1.00000i q^{22} +6.00000i q^{23} +6.00000 q^{26} +3.00000i q^{28} +5.00000 q^{29} -3.00000 q^{31} +1.00000i q^{32} +7.00000 q^{34} -3.00000i q^{37} -5.00000i q^{38} -2.00000 q^{41} +4.00000i q^{43} +1.00000 q^{44} -6.00000 q^{46} -2.00000i q^{47} -2.00000 q^{49} +6.00000i q^{52} +1.00000i q^{53} -3.00000 q^{56} +5.00000i q^{58} -10.0000 q^{59} +7.00000 q^{61} -3.00000i q^{62} -1.00000 q^{64} -8.00000i q^{67} +7.00000i q^{68} -7.00000 q^{71} +14.0000i q^{73} +3.00000 q^{74} +5.00000 q^{76} +3.00000i q^{77} -10.0000 q^{79} -2.00000i q^{82} +6.00000i q^{83} -4.00000 q^{86} +1.00000i q^{88} -15.0000 q^{89} -18.0000 q^{91} -6.00000i q^{92} +2.00000 q^{94} +12.0000i q^{97} -2.00000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} - 2 q^{11} + 6 q^{14} + 2 q^{16} - 10 q^{19} + 12 q^{26} + 10 q^{29} - 6 q^{31} + 14 q^{34} - 4 q^{41} + 2 q^{44} - 12 q^{46} - 4 q^{49} - 6 q^{56} - 20 q^{59} + 14 q^{61} - 2 q^{64} - 14 q^{71}+ \cdots + 4 q^{94}+O(q^{100}) $ 2 * q - 2 * q^4 - 2 * q^11 + 6 * q^14 + 2 * q^16 - 10 * q^19 + 12 * q^26 + 10 * q^29 - 6 * q^31 + 14 * q^34 - 4 * q^41 + 2 * q^44 - 12 * q^46 - 4 * q^49 - 6 * q^56 - 20 * q^59 + 14 * q^61 - 2 * q^64 - 14 * q^71 + 6 * q^74 + 10 * q^76 - 20 * q^79 - 8 * q^86 - 30 * q^89 - 36 * q^91 + 4 * q^94

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4950\mathbb{Z}\right)^\times$.

$n$	$551$	$2377$	$4501$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	0	0
$3$	0	0
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 3.00000i	− 1.13389i	−0.823754	−	0.566947i	$-0.808125\pi$
$7$	− 3.00000i	− 1.13389i	0.823754	−	0.566947i	$-0.191875\pi$
$8$	− 1.00000i	− 0.353553i
$9$	0	0
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	− 6.00000i	− 1.66410i	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	− 6.00000i	− 1.66410i	0.554700	−	0.832050i	$-0.312833\pi$
$14$	3.00000	0.801784
$15$	0	0
$16$	1.00000	0.250000
$17$	− 7.00000i	− 1.69775i	−0.528594	−	0.848875i	$-0.677281\pi$
$17$	− 7.00000i	− 1.69775i	0.528594	−	0.848875i	$-0.322719\pi$
$18$	0	0
$19$	−5.00000	−1.14708	−0.573539	−	0.819178i	$-0.694430\pi$
$19$	−5.00000	−1.14708	−0.573539	+	0.819178i	$0.694430\pi$
$20$	0	0
$21$	0	0
$22$	− 1.00000i	− 0.213201i
$23$	6.00000i	1.25109i	0.780189	+	0.625543i	$0.215123\pi$
$23$	6.00000i	1.25109i	−0.780189	+	0.625543i	$0.784877\pi$
$24$	0	0
$25$	0	0
$26$	6.00000	1.17670
$27$	0	0
$28$	3.00000i	0.566947i
$29$	5.00000	0.928477	0.464238	−	0.885710i	$-0.346328\pi$
$29$	5.00000	0.928477	0.464238	+	0.885710i	$0.346328\pi$
$30$	0	0
$31$	−3.00000	−0.538816	−0.269408	−	0.963026i	$-0.586828\pi$
$31$	−3.00000	−0.538816	−0.269408	+	0.963026i	$0.586828\pi$
$32$	1.00000i	0.176777i
$33$	0	0
$34$	7.00000	1.20049
$35$	0	0
$36$	0	0
$37$	− 3.00000i	− 0.493197i	−0.969118	−	0.246598i	$-0.920687\pi$
$37$	− 3.00000i	− 0.493197i	0.969118	−	0.246598i	$-0.0793129\pi$
$38$	− 5.00000i	− 0.811107i
$39$	0	0
$40$	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	$0.0986555\pi$
$43$	4.00000i	0.609994i	−0.952353	+	0.304997i	$0.901344\pi$
$44$	1.00000	0.150756
$45$	0	0
$46$	−6.00000	−0.884652
$47$	− 2.00000i	− 0.291730i	−0.989305	−	0.145865i	$-0.953403\pi$
$47$	− 2.00000i	− 0.291730i	0.989305	−	0.145865i	$-0.0465965\pi$
$48$	0	0
$49$	−2.00000	−0.285714
$50$	0	0
$51$	0	0
$52$	6.00000i	0.832050i
$53$	1.00000i	0.137361i	0.997639	+	0.0686803i	$0.0218788\pi$
$53$	1.00000i	0.137361i	−0.997639	+	0.0686803i	$0.978121\pi$
$54$	0	0
$55$	0	0
$56$	−3.00000	−0.400892
$57$	0	0
$58$	5.00000i	0.656532i
$59$	−10.0000	−1.30189	−0.650945	−	0.759125i	$-0.725627\pi$
$59$	−10.0000	−1.30189	−0.650945	+	0.759125i	$0.725627\pi$
$60$	0	0
$61$	7.00000	0.896258	0.448129	−	0.893969i	$-0.352090\pi$
$61$	7.00000	0.896258	0.448129	+	0.893969i	$0.352090\pi$
$62$	− 3.00000i	− 0.381000i
$63$	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	$-0.162521\pi$
$68$	7.00000i	0.848875i
$69$	0	0
$70$	0	0
$71$	−7.00000	−0.830747	−0.415374	−	0.909651i	$-0.636349\pi$
$71$	−7.00000	−0.830747	−0.415374	+	0.909651i	$0.636349\pi$
$72$	0	0
$73$	14.0000i	1.63858i	0.573382	+	0.819288i	$0.305631\pi$
$73$	14.0000i	1.63858i	−0.573382	+	0.819288i	$0.694369\pi$
$74$	3.00000	0.348743
$75$	0	0
$76$	5.00000	0.573539
$77$	3.00000i	0.341882i
$78$	0	0
$79$	−10.0000	−1.12509	−0.562544	−	0.826767i	$-0.690177\pi$
$79$	−10.0000	−1.12509	−0.562544	+	0.826767i	$0.690177\pi$
$80$	0	0
$81$	0	0
$82$	− 2.00000i	− 0.220863i
$83$	6.00000i	0.658586i	0.944228	+	0.329293i	$0.106810\pi$
$83$	6.00000i	0.658586i	−0.944228	+	0.329293i	$0.893190\pi$
$84$	0	0
$85$	0	0
$86$	−4.00000	−0.431331
$87$	0	0
$88$	1.00000i	0.106600i
$89$	−15.0000	−1.59000	−0.794998	−	0.606612i	$-0.792528\pi$
$89$	−15.0000	−1.59000	−0.794998	+	0.606612i	$0.792528\pi$
$90$	0	0
$91$	−18.0000	−1.88691
$92$	− 6.00000i	− 0.625543i
$93$	0	0
$94$	2.00000	0.206284
$95$	0	0
$96$	0	0
$97$	12.0000i	1.21842i	0.793011	+	0.609208i	$0.208512\pi$
$97$	12.0000i	1.21842i	−0.793011	+	0.609208i	$0.791488\pi$
$98$	− 2.00000i	− 0.202031i
$99$	0	0
$100$	0	0
$101$	−2.00000	−0.199007	−0.0995037	−	0.995037i	$-0.531726\pi$
$101$	−2.00000	−0.199007	−0.0995037	+	0.995037i	$0.531726\pi$
$102$	0	0
$103$	4.00000i	0.394132i	0.980390	+	0.197066i	$0.0631413\pi$
$103$	4.00000i	0.394132i	−0.980390	+	0.197066i	$0.936859\pi$
$104$	−6.00000	−0.588348
$105$	0	0
$106$	−1.00000	−0.0971286
$107$	8.00000i	0.773389i	0.922208	+	0.386695i	$0.126383\pi$
$107$	8.00000i	0.773389i	−0.922208	+	0.386695i	$0.873617\pi$
$108$	0	0
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	0	0
$111$	0	0
$112$	− 3.00000i	− 0.283473i
$113$	16.0000i	1.50515i	0.658505	+	0.752577i	$0.271189\pi$
$113$	16.0000i	1.50515i	−0.658505	+	0.752577i	$0.728811\pi$
$114$	0	0
$115$	0	0
$116$	−5.00000	−0.464238
$117$	0	0
$118$	− 10.0000i	− 0.920575i
$119$	−21.0000	−1.92507
$120$	0	0
$121$	1.00000	0.0909091
$122$	7.00000i	0.633750i
$123$	0	0
$124$	3.00000	0.269408
$125$	0	0
$126$	0	0
$127$	− 8.00000i	− 0.709885i	−0.934888	−	0.354943i	$-0.884500\pi$
$127$	− 8.00000i	− 0.709885i	0.934888	−	0.354943i	$-0.115500\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	0	0
$130$	0	0
$131$	−17.0000	−1.48530	−0.742648	−	0.669681i	$-0.766431\pi$
$131$	−17.0000	−1.48530	−0.742648	+	0.669681i	$0.766431\pi$
$132$	0	0
$133$	15.0000i	1.30066i
$134$	8.00000	0.691095
$135$	0	0
$136$	−7.00000	−0.600245
$137$	− 12.0000i	− 1.02523i	−0.858619	−	0.512615i	$-0.828677\pi$
$137$	− 12.0000i	− 1.02523i	0.858619	−	0.512615i	$-0.171323\pi$
$138$	0	0
$139$	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
$141$	0	0
$142$	− 7.00000i	− 0.587427i
$143$	6.00000i	0.501745i
$144$	0	0
$145$	0	0
$146$	−14.0000	−1.15865
$147$	0	0
$148$	3.00000i	0.246598i
$149$	15.0000	1.22885	0.614424	−	0.788976i	$-0.289388\pi$
$149$	15.0000	1.22885	0.614424	+	0.788976i	$0.289388\pi$
$150$	0	0
$151$	2.00000	0.162758	0.0813788	−	0.996683i	$-0.474068\pi$
$151$	2.00000	0.162758	0.0813788	+	0.996683i	$0.474068\pi$
$152$	5.00000i	0.405554i
$153$	0	0
$154$	−3.00000	−0.241747
$155$	0	0
$156$	0	0
$157$	− 3.00000i	− 0.239426i	−0.992809	−	0.119713i	$-0.961803\pi$
$157$	− 3.00000i	− 0.239426i	0.992809	−	0.119713i	$-0.0381975\pi$
$158$	− 10.0000i	− 0.795557i
$159$	0	0
$160$	0	0
$161$	18.0000	1.41860
$162$	0	0
$163$	19.0000i	1.48819i	0.668071	+	0.744097i	$0.267120\pi$
$163$	19.0000i	1.48819i	−0.668071	+	0.744097i	$0.732880\pi$
$164$	2.00000	0.156174
$165$	0	0
$166$	−6.00000	−0.465690
$167$	3.00000i	0.232147i	0.993241	+	0.116073i	$0.0370308\pi$
$167$	3.00000i	0.232147i	−0.993241	+	0.116073i	$0.962969\pi$
$168$	0	0
$169$	−23.0000	−1.76923
$170$	0	0
$171$	0	0
$172$	− 4.00000i	− 0.304997i
$173$	− 14.0000i	− 1.06440i	−0.846619	−	0.532200i	$-0.821365\pi$
$173$	− 14.0000i	− 1.06440i	0.846619	−	0.532200i	$-0.178635\pi$
$174$	0	0
$175$	0	0
$176$	−1.00000	−0.0753778
$177$	0	0
$178$	− 15.0000i	− 1.12430i
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	− 18.0000i	− 1.33425i
$183$	0	0
$184$	6.00000	0.442326
$185$	0	0
$186$	0	0
$187$	7.00000i	0.511891i
$188$	2.00000i	0.145865i
$189$	0	0
$190$	0	0
$191$	−12.0000	−0.868290	−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000	−0.868290	−0.434145	+	0.900843i	$0.642949\pi$
$192$	0	0
$193$	− 11.0000i	− 0.791797i	−0.918294	−	0.395899i	$-0.870433\pi$
$193$	− 11.0000i	− 0.791797i	0.918294	−	0.395899i	$-0.129567\pi$
$194$	−12.0000	−0.861550
$195$	0	0
$196$	2.00000	0.142857
$197$	− 12.0000i	− 0.854965i	−0.904024	−	0.427482i	$-0.859401\pi$
$197$	− 12.0000i	− 0.854965i	0.904024	−	0.427482i	$-0.140599\pi$
$198$	0	0
$199$	25.0000	1.77220	0.886102	−	0.463491i	$-0.153403\pi$
$199$	25.0000	1.77220	0.886102	+	0.463491i	$0.153403\pi$
$200$	0	0
$201$	0	0
$202$	− 2.00000i	− 0.140720i
$203$	− 15.0000i	− 1.05279i
$204$	0	0
$205$	0	0
$206$	−4.00000	−0.278693
$207$	0	0
$208$	− 6.00000i	− 0.416025i
$209$	5.00000	0.345857
$210$	0	0
$211$	−23.0000	−1.58339	−0.791693	−	0.610920i	$-0.790800\pi$
$211$	−23.0000	−1.58339	−0.791693	+	0.610920i	$0.790800\pi$
$212$	− 1.00000i	− 0.0686803i
$213$	0	0
$214$	−8.00000	−0.546869
$215$	0	0
$216$	0	0
$217$	9.00000i	0.610960i
$218$	10.0000i	0.677285i
$219$	0	0
$220$	0	0
$221$	−42.0000	−2.82523
$222$	0	0
$223$	− 6.00000i	− 0.401790i	−0.979613	−	0.200895i	$-0.935615\pi$
$223$	− 6.00000i	− 0.401790i	0.979613	−	0.200895i	$-0.0643850\pi$
$224$	3.00000	0.200446
$225$	0	0
$226$	−16.0000	−1.06430
$227$	− 2.00000i	− 0.132745i	−0.997795	−	0.0663723i	$-0.978857\pi$
$227$	− 2.00000i	− 0.132745i	0.997795	−	0.0663723i	$-0.0211425\pi$
$228$	0	0
$229$	−10.0000	−0.660819	−0.330409	−	0.943838i	$-0.607187\pi$
$229$	−10.0000	−0.660819	−0.330409	+	0.943838i	$0.607187\pi$
$230$	0	0
$231$	0	0
$232$	− 5.00000i	− 0.328266i
$233$	− 9.00000i	− 0.589610i	−0.955557	−	0.294805i	$-0.904745\pi$
$233$	− 9.00000i	− 0.589610i	0.955557	−	0.294805i	$-0.0952546\pi$
$234$	0	0
$235$	0	0
$236$	10.0000	0.650945
$237$	0	0
$238$	− 21.0000i	− 1.36123i
$239$	10.0000	0.646846	0.323423	−	0.946254i	$-0.395166\pi$
$239$	10.0000	0.646846	0.323423	+	0.946254i	$0.395166\pi$
$240$	0	0
$241$	−18.0000	−1.15948	−0.579741	−	0.814801i	$-0.696846\pi$
$241$	−18.0000	−1.15948	−0.579741	+	0.814801i	$0.696846\pi$
$242$	1.00000i	0.0642824i
$243$	0	0
$244$	−7.00000	−0.448129
$245$	0	0
$246$	0	0
$247$	30.0000i	1.90885i
$248$	3.00000i	0.190500i
$249$	0	0
$250$	0	0
$251$	−2.00000	−0.126239	−0.0631194	−	0.998006i	$-0.520105\pi$
$251$	−2.00000	−0.126239	−0.0631194	+	0.998006i	$0.520105\pi$
$252$	0	0
$253$	− 6.00000i	− 0.377217i
$254$	8.00000	0.501965
$255$	0	0
$256$	1.00000	0.0625000
$257$	− 2.00000i	− 0.124757i	−0.998053	−	0.0623783i	$-0.980131\pi$
$257$	− 2.00000i	− 0.124757i	0.998053	−	0.0623783i	$-0.0198685\pi$
$258$	0	0
$259$	−9.00000	−0.559233
$260$	0	0
$261$	0	0
$262$	− 17.0000i	− 1.05026i
$263$	− 9.00000i	− 0.554964i	−0.960731	−	0.277482i	$-0.910500\pi$
$263$	− 9.00000i	− 0.554964i	0.960731	−	0.277482i	$-0.0894999\pi$
$264$	0	0
$265$	0	0
$266$	−15.0000	−0.919709
$267$	0	0
$268$	8.00000i	0.488678i
$269$	−20.0000	−1.21942	−0.609711	−	0.792624i	$-0.708714\pi$
$269$	−20.0000	−1.21942	−0.609711	+	0.792624i	$0.708714\pi$
$270$	0	0
$271$	−8.00000	−0.485965	−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000	−0.485965	−0.242983	+	0.970031i	$0.578126\pi$
$272$	− 7.00000i	− 0.424437i
$273$	0	0
$274$	12.0000	0.724947
$275$	0	0
$276$	0	0
$277$	12.0000i	0.721010i	0.932757	+	0.360505i	$0.117396\pi$
$277$	12.0000i	0.721010i	−0.932757	+	0.360505i	$0.882604\pi$
$278$	20.0000i	1.19952i
$279$	0	0
$280$	0	0
$281$	18.0000	1.07379	0.536895	−	0.843649i	$-0.319597\pi$
$281$	18.0000	1.07379	0.536895	+	0.843649i	$0.319597\pi$
$282$	0	0
$283$	− 6.00000i	− 0.356663i	−0.983970	−	0.178331i	$-0.942930\pi$
$283$	− 6.00000i	− 0.356663i	0.983970	−	0.178331i	$-0.0570699\pi$
$284$	7.00000	0.415374
$285$	0	0
$286$	−6.00000	−0.354787
$287$	6.00000i	0.354169i
$288$	0	0
$289$	−32.0000	−1.88235
$290$	0	0
$291$	0	0
$292$	− 14.0000i	− 0.819288i
$293$	6.00000i	0.350524i	0.984522	+	0.175262i	$0.0560772\pi$
$293$	6.00000i	0.350524i	−0.984522	+	0.175262i	$0.943923\pi$
$294$	0	0
$295$	0	0
$296$	−3.00000	−0.174371
$297$	0	0
$298$	15.0000i	0.868927i
$299$	36.0000	2.08193
$300$	0	0
$301$	12.0000	0.691669
$302$	2.00000i	0.115087i
$303$	0	0
$304$	−5.00000	−0.286770
$305$	0	0
$306$	0	0
$307$	2.00000i	0.114146i	0.998370	+	0.0570730i	$0.0181768\pi$
$307$	2.00000i	0.114146i	−0.998370	+	0.0570730i	$0.981823\pi$
$308$	− 3.00000i	− 0.170941i
$309$	0	0
$310$	0	0
$311$	3.00000	0.170114	0.0850572	−	0.996376i	$-0.472893\pi$
$311$	3.00000	0.170114	0.0850572	+	0.996376i	$0.472893\pi$
$312$	0	0
$313$	− 6.00000i	− 0.339140i	−0.985518	−	0.169570i	$-0.945762\pi$
$313$	− 6.00000i	− 0.339140i	0.985518	−	0.169570i	$-0.0542379\pi$
$314$	3.00000	0.169300
$315$	0	0
$316$	10.0000	0.562544
$317$	− 7.00000i	− 0.393159i	−0.980488	−	0.196580i	$-0.937017\pi$
$317$	− 7.00000i	− 0.393159i	0.980488	−	0.196580i	$-0.0629834\pi$
$318$	0	0
$319$	−5.00000	−0.279946
$320$	0	0
$321$	0	0
$322$	18.0000i	1.00310i
$323$	35.0000i	1.94745i
$324$	0	0
$325$	0	0
$326$	−19.0000	−1.05231
$327$	0	0
$328$	2.00000i	0.110432i
$329$	−6.00000	−0.330791
$330$	0	0
$331$	−28.0000	−1.53902	−0.769510	−	0.638635i	$-0.779499\pi$
$331$	−28.0000	−1.53902	−0.769510	+	0.638635i	$0.779499\pi$
$332$	− 6.00000i	− 0.329293i
$333$	0	0
$334$	−3.00000	−0.164153
$335$	0	0
$336$	0	0
$337$	17.0000i	0.926049i	0.886345	+	0.463025i	$0.153236\pi$
$337$	17.0000i	0.926049i	−0.886345	+	0.463025i	$0.846764\pi$
$338$	− 23.0000i	− 1.25104i
$339$	0	0
$340$	0	0
$341$	3.00000	0.162459
$342$	0	0
$343$	− 15.0000i	− 0.809924i
$344$	4.00000	0.215666
$345$	0	0
$346$	14.0000	0.752645
$347$	18.0000i	0.966291i	0.875540	+	0.483145i	$0.160506\pi$
$347$	18.0000i	0.966291i	−0.875540	+	0.483145i	$0.839494\pi$
$348$	0	0
$349$	−30.0000	−1.60586	−0.802932	−	0.596071i	$-0.796728\pi$
$349$	−30.0000	−1.60586	−0.802932	+	0.596071i	$0.796728\pi$
$350$	0	0
$351$	0	0
$352$	− 1.00000i	− 0.0533002i
$353$	− 34.0000i	− 1.80964i	−0.425797	−	0.904819i	$-0.640006\pi$
$353$	− 34.0000i	− 1.80964i	0.425797	−	0.904819i	$-0.359994\pi$
$354$	0	0
$355$	0	0
$356$	15.0000	0.794998
$357$	0	0
$358$	0	0
$359$	−20.0000	−1.05556	−0.527780	−	0.849381i	$-0.676975\pi$
$359$	−20.0000	−1.05556	−0.527780	+	0.849381i	$0.676975\pi$
$360$	0	0
$361$	6.00000	0.315789
$362$	2.00000i	0.105118i
$363$	0	0
$364$	18.0000	0.943456
$365$	0	0
$366$	0	0
$367$	− 28.0000i	− 1.46159i	−0.682598	−	0.730794i	$-0.739150\pi$
$367$	− 28.0000i	− 1.46159i	0.682598	−	0.730794i	$-0.260850\pi$
$368$	6.00000i	0.312772i
$369$	0	0
$370$	0	0
$371$	3.00000	0.155752
$372$	0	0
$373$	− 6.00000i	− 0.310668i	−0.987862	−	0.155334i	$-0.950355\pi$
$373$	− 6.00000i	− 0.310668i	0.987862	−	0.155334i	$-0.0496454\pi$
$374$	−7.00000	−0.361961
$375$	0	0
$376$	−2.00000	−0.103142
$377$	− 30.0000i	− 1.54508i
$378$	0	0
$379$	30.0000	1.54100	0.770498	−	0.637442i	$-0.220007\pi$
$379$	30.0000	1.54100	0.770498	+	0.637442i	$0.220007\pi$
$380$	0	0
$381$	0	0
$382$	− 12.0000i	− 0.613973i
$383$	− 34.0000i	− 1.73732i	−0.495410	−	0.868659i	$-0.664982\pi$
$383$	− 34.0000i	− 1.73732i	0.495410	−	0.868659i	$-0.335018\pi$
$384$	0	0
$385$	0	0
$386$	11.0000	0.559885
$387$	0	0
$388$	− 12.0000i	− 0.609208i
$389$	−30.0000	−1.52106	−0.760530	−	0.649303i	$-0.775061\pi$
$389$	−30.0000	−1.52106	−0.760530	+	0.649303i	$0.775061\pi$
$390$	0	0
$391$	42.0000	2.12403
$392$	2.00000i	0.101015i
$393$	0	0
$394$	12.0000	0.604551
$395$	0	0
$396$	0	0
$397$	2.00000i	0.100377i	0.998740	+	0.0501886i	$0.0159822\pi$
$397$	2.00000i	0.100377i	−0.998740	+	0.0501886i	$0.984018\pi$
$398$	25.0000i	1.25314i
$399$	0	0
$400$	0	0
$401$	13.0000	0.649189	0.324595	−	0.945853i	$-0.394772\pi$
$401$	13.0000	0.649189	0.324595	+	0.945853i	$0.394772\pi$
$402$	0	0
$403$	18.0000i	0.896644i
$404$	2.00000	0.0995037
$405$	0	0
$406$	15.0000	0.744438
$407$	3.00000i	0.148704i
$408$	0	0
$409$	20.0000	0.988936	0.494468	−	0.869196i	$-0.335363\pi$
$409$	20.0000	0.988936	0.494468	+	0.869196i	$0.335363\pi$
$410$	0	0
$411$	0	0
$412$	− 4.00000i	− 0.197066i
$413$	30.0000i	1.47620i
$414$	0	0
$415$	0	0
$416$	6.00000	0.294174
$417$	0	0
$418$	5.00000i	0.244558i
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	32.0000	1.55958	0.779792	−	0.626038i	$-0.215325\pi$
$421$	32.0000	1.55958	0.779792	+	0.626038i	$0.215325\pi$
$422$	− 23.0000i	− 1.11962i
$423$	0	0
$424$	1.00000	0.0485643
$425$	0	0
$426$	0	0
$427$	− 21.0000i	− 1.01626i
$428$	− 8.00000i	− 0.386695i
$429$	0	0
$430$	0	0
$431$	8.00000	0.385346	0.192673	−	0.981263i	$-0.438284\pi$
$431$	8.00000	0.385346	0.192673	+	0.981263i	$0.438284\pi$
$432$	0	0
$433$	− 16.0000i	− 0.768911i	−0.923144	−	0.384455i	$-0.874389\pi$
$433$	− 16.0000i	− 0.768911i	0.923144	−	0.384455i	$-0.125611\pi$
$434$	−9.00000	−0.432014
$435$	0	0
$436$	−10.0000	−0.478913
$437$	− 30.0000i	− 1.43509i
$438$	0	0
$439$	−20.0000	−0.954548	−0.477274	−	0.878755i	$-0.658375\pi$
$439$	−20.0000	−0.954548	−0.477274	+	0.878755i	$0.658375\pi$
$440$	0	0
$441$	0	0
$442$	− 42.0000i	− 1.99774i
$443$	− 4.00000i	− 0.190046i	−0.995475	−	0.0950229i	$-0.969708\pi$
$443$	− 4.00000i	− 0.190046i	0.995475	−	0.0950229i	$-0.0302924\pi$
$444$	0	0
$445$	0	0
$446$	6.00000	0.284108
$447$	0	0
$448$	3.00000i	0.141737i
$449$	−30.0000	−1.41579	−0.707894	−	0.706319i	$-0.750354\pi$
$449$	−30.0000	−1.41579	−0.707894	+	0.706319i	$0.750354\pi$
$450$	0	0
$451$	2.00000	0.0941763
$452$	− 16.0000i	− 0.752577i
$453$	0	0
$454$	2.00000	0.0938647
$455$	0	0
$456$	0	0
$457$	− 3.00000i	− 0.140334i	−0.997535	−	0.0701670i	$-0.977647\pi$
$457$	− 3.00000i	− 0.140334i	0.997535	−	0.0701670i	$-0.0223532\pi$
$458$	− 10.0000i	− 0.467269i
$459$	0	0
$460$	0	0
$461$	−27.0000	−1.25752	−0.628758	−	0.777601i	$-0.716436\pi$
$461$	−27.0000	−1.25752	−0.628758	+	0.777601i	$0.716436\pi$
$462$	0	0
$463$	34.0000i	1.58011i	0.613033	+	0.790057i	$0.289949\pi$
$463$	34.0000i	1.58011i	−0.613033	+	0.790057i	$0.710051\pi$
$464$	5.00000	0.232119
$465$	0	0
$466$	9.00000	0.416917
$467$	23.0000i	1.06431i	0.846646	+	0.532157i	$0.178618\pi$
$467$	23.0000i	1.06431i	−0.846646	+	0.532157i	$0.821382\pi$
$468$	0	0
$469$	−24.0000	−1.10822
$470$	0	0
$471$	0	0
$472$	10.0000i	0.460287i
$473$	− 4.00000i	− 0.183920i
$474$	0	0
$475$	0	0
$476$	21.0000	0.962533
$477$	0	0
$478$	10.0000i	0.457389i
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	−18.0000	−0.820729
$482$	− 18.0000i	− 0.819878i
$483$	0	0
$484$	−1.00000	−0.0454545
$485$	0	0
$486$	0	0
$487$	12.0000i	0.543772i	0.962329	+	0.271886i	$0.0876473\pi$
$487$	12.0000i	0.543772i	−0.962329	+	0.271886i	$0.912353\pi$
$488$	− 7.00000i	− 0.316875i
$489$	0	0
$490$	0	0
$491$	3.00000	0.135388	0.0676941	−	0.997706i	$-0.478436\pi$
$491$	3.00000	0.135388	0.0676941	+	0.997706i	$0.478436\pi$
$492$	0	0
$493$	− 35.0000i	− 1.57632i
$494$	−30.0000	−1.34976
$495$	0	0
$496$	−3.00000	−0.134704
$497$	21.0000i	0.941979i
$498$	0	0
$499$	−20.0000	−0.895323	−0.447661	−	0.894203i	$-0.647743\pi$
$499$	−20.0000	−0.895323	−0.447661	+	0.894203i	$0.647743\pi$
$500$	0	0
$501$	0	0
$502$	− 2.00000i	− 0.0892644i
$503$	− 24.0000i	− 1.07011i	−0.844818	−	0.535054i	$-0.820291\pi$
$503$	− 24.0000i	− 1.07011i	0.844818	−	0.535054i	$-0.179709\pi$
$504$	0	0
$505$	0	0
$506$	6.00000	0.266733
$507$	0	0
$508$	8.00000i	0.354943i
$509$	20.0000	0.886484	0.443242	−	0.896402i	$-0.353828\pi$
$509$	20.0000	0.886484	0.443242	+	0.896402i	$0.353828\pi$
$510$	0	0
$511$	42.0000	1.85797
$512$	1.00000i	0.0441942i
$513$	0	0
$514$	2.00000	0.0882162
$515$	0	0
$516$	0	0
$517$	2.00000i	0.0879599i
$518$	− 9.00000i	− 0.395437i
$519$	0	0
$520$	0	0
$521$	−22.0000	−0.963837	−0.481919	−	0.876216i	$-0.660060\pi$
$521$	−22.0000	−0.963837	−0.481919	+	0.876216i	$0.660060\pi$
$522$	0	0
$523$	− 16.0000i	− 0.699631i	−0.936819	−	0.349816i	$-0.886244\pi$
$523$	− 16.0000i	− 0.699631i	0.936819	−	0.349816i	$-0.113756\pi$
$524$	17.0000	0.742648
$525$	0	0
$526$	9.00000	0.392419
$527$	21.0000i	0.914774i
$528$	0	0
$529$	−13.0000	−0.565217
$530$	0	0
$531$	0	0
$532$	− 15.0000i	− 0.650332i
$533$	12.0000i	0.519778i
$534$	0	0
$535$	0	0
$536$	−8.00000	−0.345547
$537$	0	0
$538$	− 20.0000i	− 0.862261i
$539$	2.00000	0.0861461
$540$	0	0
$541$	−23.0000	−0.988847	−0.494424	−	0.869221i	$-0.664621\pi$
$541$	−23.0000	−0.988847	−0.494424	+	0.869221i	$0.664621\pi$
$542$	− 8.00000i	− 0.343629i
$543$	0	0
$544$	7.00000	0.300123
$545$	0	0
$546$	0	0
$547$	− 8.00000i	− 0.342055i	−0.985266	−	0.171028i	$-0.945291\pi$
$547$	− 8.00000i	− 0.342055i	0.985266	−	0.171028i	$-0.0547087\pi$
$548$	12.0000i	0.512615i
$549$	0	0
$550$	0	0
$551$	−25.0000	−1.06504
$552$	0	0
$553$	30.0000i	1.27573i
$554$	−12.0000	−0.509831
$555$	0	0
$556$	−20.0000	−0.848189
$557$	18.0000i	0.762684i	0.924434	+	0.381342i	$0.124538\pi$
$557$	18.0000i	0.762684i	−0.924434	+	0.381342i	$0.875462\pi$
$558$	0	0
$559$	24.0000	1.01509
$560$	0	0
$561$	0	0
$562$	18.0000i	0.759284i
$563$	6.00000i	0.252870i	0.991975	+	0.126435i	$0.0403535\pi$
$563$	6.00000i	0.252870i	−0.991975	+	0.126435i	$0.959647\pi$
$564$	0	0
$565$	0	0
$566$	6.00000	0.252199
$567$	0	0
$568$	7.00000i	0.293713i
$569$	0	0		−	1.00000i	$-0.5\pi$
$569$	0	0			1.00000i	$0.5\pi$
$570$	0	0
$571$	27.0000	1.12991	0.564957	−	0.825120i	$-0.308893\pi$
$571$	27.0000	1.12991	0.564957	+	0.825120i	$0.308893\pi$
$572$	− 6.00000i	− 0.250873i
$573$	0	0
$574$	−6.00000	−0.250435
$575$	0	0
$576$	0	0
$577$	− 38.0000i	− 1.58196i	−0.611842	−	0.790980i	$-0.709571\pi$
$577$	− 38.0000i	− 1.58196i	0.611842	−	0.790980i	$-0.290429\pi$
$578$	− 32.0000i	− 1.33102i
$579$	0	0
$580$	0	0
$581$	18.0000	0.746766
$582$	0	0
$583$	− 1.00000i	− 0.0414158i
$584$	14.0000	0.579324
$585$	0	0
$586$	−6.00000	−0.247858
$587$	− 27.0000i	− 1.11441i	−0.830375	−	0.557205i	$-0.811874\pi$
$587$	− 27.0000i	− 1.11441i	0.830375	−	0.557205i	$-0.188126\pi$
$588$	0	0
$589$	15.0000	0.618064
$590$	0	0
$591$	0	0
$592$	− 3.00000i	− 0.123299i
$593$	− 14.0000i	− 0.574911i	−0.957794	−	0.287456i	$-0.907191\pi$
$593$	− 14.0000i	− 0.574911i	0.957794	−	0.287456i	$-0.0928094\pi$
$594$	0	0
$595$	0	0
$596$	−15.0000	−0.614424
$597$	0	0
$598$	36.0000i	1.47215i
$599$	45.0000	1.83865	0.919325	−	0.393499i	$-0.128735\pi$
$599$	45.0000	1.83865	0.919325	+	0.393499i	$0.128735\pi$
$600$	0	0
$601$	42.0000	1.71322	0.856608	−	0.515968i	$-0.172568\pi$
$601$	42.0000	1.71322	0.856608	+	0.515968i	$0.172568\pi$
$602$	12.0000i	0.489083i
$603$	0	0
$604$	−2.00000	−0.0813788
$605$	0	0
$606$	0	0
$607$	47.0000i	1.90767i	0.300329	+	0.953836i	$0.402903\pi$
$607$	47.0000i	1.90767i	−0.300329	+	0.953836i	$0.597097\pi$
$608$	− 5.00000i	− 0.202777i
$609$	0	0
$610$	0	0
$611$	−12.0000	−0.485468
$612$	0	0
$613$	− 26.0000i	− 1.05013i	−0.851062	−	0.525065i	$-0.824041\pi$
$613$	− 26.0000i	− 1.05013i	0.851062	−	0.525065i	$-0.175959\pi$
$614$	−2.00000	−0.0807134
$615$	0	0
$616$	3.00000	0.120873
$617$	8.00000i	0.322068i	0.986949	+	0.161034i	$0.0514829\pi$
$617$	8.00000i	0.322068i	−0.986949	+	0.161034i	$0.948517\pi$
$618$	0	0
$619$	−20.0000	−0.803868	−0.401934	−	0.915669i	$-0.631662\pi$
$619$	−20.0000	−0.803868	−0.401934	+	0.915669i	$0.631662\pi$
$620$	0	0
$621$	0	0
$622$	3.00000i	0.120289i
$623$	45.0000i	1.80289i
$624$	0	0
$625$	0	0
$626$	6.00000	0.239808
$627$	0	0
$628$	3.00000i	0.119713i
$629$	−21.0000	−0.837325
$630$	0	0
$631$	−33.0000	−1.31371	−0.656855	−	0.754017i	$-0.728113\pi$
$631$	−33.0000	−1.31371	−0.656855	+	0.754017i	$0.728113\pi$
$632$	10.0000i	0.397779i
$633$	0	0
$634$	7.00000	0.278006
$635$	0	0
$636$	0	0
$637$	12.0000i	0.475457i
$638$	− 5.00000i	− 0.197952i
$639$	0	0
$640$	0	0
$641$	33.0000	1.30342	0.651711	−	0.758468i	$-0.274052\pi$
$641$	33.0000	1.30342	0.651711	+	0.758468i	$0.274052\pi$
$642$	0	0
$643$	19.0000i	0.749287i	0.927169	+	0.374643i	$0.122235\pi$
$643$	19.0000i	0.749287i	−0.927169	+	0.374643i	$0.877765\pi$
$644$	−18.0000	−0.709299
$645$	0	0
$646$	−35.0000	−1.37706
$647$	− 42.0000i	− 1.65119i	−0.564263	−	0.825595i	$-0.690840\pi$
$647$	− 42.0000i	− 1.65119i	0.564263	−	0.825595i	$-0.309160\pi$
$648$	0	0
$649$	10.0000	0.392534
$650$	0	0
$651$	0	0
$652$	− 19.0000i	− 0.744097i
$653$	31.0000i	1.21312i	0.795036	+	0.606562i	$0.207452\pi$
$653$	31.0000i	1.21312i	−0.795036	+	0.606562i	$0.792548\pi$
$654$	0	0
$655$	0	0
$656$	−2.00000	−0.0780869
$657$	0	0
$658$	− 6.00000i	− 0.233904i
$659$	15.0000	0.584317	0.292159	−	0.956370i	$-0.405627\pi$
$659$	15.0000	0.584317	0.292159	+	0.956370i	$0.405627\pi$
$660$	0	0
$661$	2.00000	0.0777910	0.0388955	−	0.999243i	$-0.487616\pi$
$661$	2.00000	0.0777910	0.0388955	+	0.999243i	$0.487616\pi$
$662$	− 28.0000i	− 1.08825i
$663$	0	0
$664$	6.00000	0.232845
$665$	0	0
$666$	0	0
$667$	30.0000i	1.16160i
$668$	− 3.00000i	− 0.116073i
$669$	0	0
$670$	0	0
$671$	−7.00000	−0.270232
$672$	0	0
$673$	29.0000i	1.11787i	0.829212	+	0.558934i	$0.188789\pi$
$673$	29.0000i	1.11787i	−0.829212	+	0.558934i	$0.811211\pi$
$674$	−17.0000	−0.654816
$675$	0	0
$676$	23.0000	0.884615
$677$	28.0000i	1.07613i	0.842904	+	0.538064i	$0.180844\pi$
$677$	28.0000i	1.07613i	−0.842904	+	0.538064i	$0.819156\pi$
$678$	0	0
$679$	36.0000	1.38155
$680$	0	0
$681$	0	0
$682$	3.00000i	0.114876i
$683$	31.0000i	1.18618i	0.805135	+	0.593091i	$0.202093\pi$
$683$	31.0000i	1.18618i	−0.805135	+	0.593091i	$0.797907\pi$
$684$	0	0
$685$	0	0
$686$	15.0000	0.572703
$687$	0	0
$688$	4.00000i	0.152499i
$689$	6.00000	0.228582
$690$	0	0
$691$	−38.0000	−1.44559	−0.722794	−	0.691063i	$-0.757142\pi$
$691$	−38.0000	−1.44559	−0.722794	+	0.691063i	$0.757142\pi$
$692$	14.0000i	0.532200i
$693$	0	0
$694$	−18.0000	−0.683271
$695$	0	0
$696$	0	0
$697$	14.0000i	0.530288i
$698$	− 30.0000i	− 1.13552i
$699$	0	0
$700$	0	0
$701$	−7.00000	−0.264386	−0.132193	−	0.991224i	$-0.542202\pi$
$701$	−7.00000	−0.264386	−0.132193	+	0.991224i	$0.542202\pi$
$702$	0	0
$703$	15.0000i	0.565736i
$704$	1.00000	0.0376889
$705$	0	0
$706$	34.0000	1.27961
$707$	6.00000i	0.225653i
$708$	0	0
$709$	10.0000	0.375558	0.187779	−	0.982211i	$-0.439871\pi$
$709$	10.0000	0.375558	0.187779	+	0.982211i	$0.439871\pi$
$710$	0	0
$711$	0	0
$712$	15.0000i	0.562149i
$713$	− 18.0000i	− 0.674105i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	− 20.0000i	− 0.746393i
$719$	25.0000	0.932343	0.466171	−	0.884694i	$-0.345633\pi$
$719$	25.0000	0.932343	0.466171	+	0.884694i	$0.345633\pi$
$720$	0	0
$721$	12.0000	0.446903
$722$	6.00000i	0.223297i
$723$	0	0
$724$	−2.00000	−0.0743294
$725$	0	0
$726$	0	0
$727$	22.0000i	0.815935i	0.912996	+	0.407967i	$0.133762\pi$
$727$	22.0000i	0.815935i	−0.912996	+	0.407967i	$0.866238\pi$
$728$	18.0000i	0.667124i
$729$	0	0
$730$	0	0
$731$	28.0000	1.03562
$732$	0	0
$733$	24.0000i	0.886460i	0.896408	+	0.443230i	$0.146168\pi$
$733$	24.0000i	0.886460i	−0.896408	+	0.443230i	$0.853832\pi$
$734$	28.0000	1.03350
$735$	0	0
$736$	−6.00000	−0.221163
$737$	8.00000i	0.294684i
$738$	0	0
$739$	−40.0000	−1.47142	−0.735712	−	0.677295i	$-0.763152\pi$
$739$	−40.0000	−1.47142	−0.735712	+	0.677295i	$0.763152\pi$
$740$	0	0
$741$	0	0
$742$	3.00000i	0.110133i
$743$	21.0000i	0.770415i	0.922830	+	0.385208i	$0.125870\pi$
$743$	21.0000i	0.770415i	−0.922830	+	0.385208i	$0.874130\pi$
$744$	0	0
$745$	0	0
$746$	6.00000	0.219676
$747$	0	0
$748$	− 7.00000i	− 0.255945i
$749$	24.0000	0.876941
$750$	0	0
$751$	17.0000	0.620339	0.310169	−	0.950681i	$-0.399614\pi$
$751$	17.0000	0.620339	0.310169	+	0.950681i	$0.399614\pi$
$752$	− 2.00000i	− 0.0729325i
$753$	0	0
$754$	30.0000	1.09254
$755$	0	0
$756$	0	0
$757$	− 38.0000i	− 1.38113i	−0.723269	−	0.690567i	$-0.757361\pi$
$757$	− 38.0000i	− 1.38113i	0.723269	−	0.690567i	$-0.242639\pi$
$758$	30.0000i	1.08965i
$759$	0	0
$760$	0	0
$761$	18.0000	0.652499	0.326250	−	0.945284i	$-0.394215\pi$
$761$	18.0000	0.652499	0.326250	+	0.945284i	$0.394215\pi$
$762$	0	0
$763$	− 30.0000i	− 1.08607i
$764$	12.0000	0.434145
$765$	0	0
$766$	34.0000	1.22847
$767$	60.0000i	2.16647i
$768$	0	0
$769$	10.0000	0.360609	0.180305	−	0.983611i	$-0.442292\pi$
$769$	10.0000	0.360609	0.180305	+	0.983611i	$0.442292\pi$
$770$	0	0
$771$	0	0
$772$	11.0000i	0.395899i
$773$	− 19.0000i	− 0.683383i	−0.939812	−	0.341691i	$-0.889000\pi$
$773$	− 19.0000i	− 0.683383i	0.939812	−	0.341691i	$-0.111000\pi$
$774$	0	0
$775$	0	0
$776$	12.0000	0.430775
$777$	0	0
$778$	− 30.0000i	− 1.07555i
$779$	10.0000	0.358287
$780$	0	0
$781$	7.00000	0.250480
$782$	42.0000i	1.50192i
$783$	0	0
$784$	−2.00000	−0.0714286
$785$	0	0
$786$	0	0
$787$	− 28.0000i	− 0.998092i	−0.866575	−	0.499046i	$-0.833684\pi$
$787$	− 28.0000i	− 0.998092i	0.866575	−	0.499046i	$-0.166316\pi$
$788$	12.0000i	0.427482i
$789$	0	0
$790$	0	0
$791$	48.0000	1.70668
$792$	0	0
$793$	− 42.0000i	− 1.49146i
$794$	−2.00000	−0.0709773
$795$	0	0
$796$	−25.0000	−0.886102
$797$	18.0000i	0.637593i	0.947823	+	0.318796i	$0.103279\pi$
$797$	18.0000i	0.637593i	−0.947823	+	0.318796i	$0.896721\pi$
$798$	0	0
$799$	−14.0000	−0.495284
$800$	0	0
$801$	0	0
$802$	13.0000i	0.459046i
$803$	− 14.0000i	− 0.494049i
$804$	0	0
$805$	0	0
$806$	−18.0000	−0.634023
$807$	0	0
$808$	2.00000i	0.0703598i
$809$	−30.0000	−1.05474	−0.527372	−	0.849635i	$-0.676823\pi$
$809$	−30.0000	−1.05474	−0.527372	+	0.849635i	$0.676823\pi$
$810$	0	0
$811$	7.00000	0.245803	0.122902	−	0.992419i	$-0.460780\pi$
$811$	7.00000	0.245803	0.122902	+	0.992419i	$0.460780\pi$
$812$	15.0000i	0.526397i
$813$	0	0
$814$	−3.00000	−0.105150
$815$	0	0
$816$	0	0
$817$	− 20.0000i	− 0.699711i
$818$	20.0000i	0.699284i
$819$	0	0
$820$	0	0
$821$	18.0000	0.628204	0.314102	−	0.949389i	$-0.398297\pi$
$821$	18.0000	0.628204	0.314102	+	0.949389i	$0.398297\pi$
$822$	0	0
$823$	− 16.0000i	− 0.557725i	−0.960331	−	0.278862i	$-0.910043\pi$
$823$	− 16.0000i	− 0.557725i	0.960331	−	0.278862i	$-0.0899574\pi$
$824$	4.00000	0.139347
$825$	0	0
$826$	−30.0000	−1.04383
$827$	8.00000i	0.278187i	0.990279	+	0.139094i	$0.0444189\pi$
$827$	8.00000i	0.278187i	−0.990279	+	0.139094i	$0.955581\pi$
$828$	0	0
$829$	20.0000	0.694629	0.347314	−	0.937749i	$-0.387094\pi$
$829$	20.0000	0.694629	0.347314	+	0.937749i	$0.387094\pi$
$830$	0	0
$831$	0	0
$832$	6.00000i	0.208013i
$833$	14.0000i	0.485071i
$834$	0	0
$835$	0	0
$836$	−5.00000	−0.172929
$837$	0	0
$838$	0	0
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	−4.00000	−0.137931
$842$	32.0000i	1.10279i
$843$	0	0
$844$	23.0000	0.791693
$845$	0	0
$846$	0	0
$847$	− 3.00000i	− 0.103081i
$848$	1.00000i	0.0343401i
$849$	0	0
$850$	0	0
$851$	18.0000	0.617032
$852$	0	0
$853$	− 26.0000i	− 0.890223i	−0.895475	−	0.445112i	$-0.853164\pi$
$853$	− 26.0000i	− 0.890223i	0.895475	−	0.445112i	$-0.146836\pi$
$854$	21.0000	0.718605
$855$	0	0
$856$	8.00000	0.273434
$857$	− 7.00000i	− 0.239115i	−0.992827	−	0.119558i	$-0.961852\pi$
$857$	− 7.00000i	− 0.239115i	0.992827	−	0.119558i	$-0.0381477\pi$
$858$	0	0
$859$	−30.0000	−1.02359	−0.511793	−	0.859109i	$-0.671019\pi$
$859$	−30.0000	−1.02359	−0.511793	+	0.859109i	$0.671019\pi$
$860$	0	0
$861$	0	0
$862$	8.00000i	0.272481i
$863$	6.00000i	0.204242i	0.994772	+	0.102121i	$0.0325630\pi$
$863$	6.00000i	0.204242i	−0.994772	+	0.102121i	$0.967437\pi$
$864$	0	0
$865$	0	0
$866$	16.0000	0.543702
$867$	0	0
$868$	− 9.00000i	− 0.305480i
$869$	10.0000	0.339227
$870$	0	0
$871$	−48.0000	−1.62642
$872$	− 10.0000i	− 0.338643i
$873$	0	0
$874$	30.0000	1.01477
$875$	0	0
$876$	0	0
$877$	− 38.0000i	− 1.28317i	−0.767052	−	0.641584i	$-0.778277\pi$
$877$	− 38.0000i	− 1.28317i	0.767052	−	0.641584i	$-0.221723\pi$
$878$	− 20.0000i	− 0.674967i
$879$	0	0
$880$	0	0
$881$	−2.00000	−0.0673817	−0.0336909	−	0.999432i	$-0.510726\pi$
$881$	−2.00000	−0.0673817	−0.0336909	+	0.999432i	$0.510726\pi$
$882$	0	0
$883$	9.00000i	0.302874i	0.988467	+	0.151437i	$0.0483901\pi$
$883$	9.00000i	0.302874i	−0.988467	+	0.151437i	$0.951610\pi$
$884$	42.0000	1.41261
$885$	0	0
$886$	4.00000	0.134383
$887$	8.00000i	0.268614i	0.990940	+	0.134307i	$0.0428808\pi$
$887$	8.00000i	0.268614i	−0.990940	+	0.134307i	$0.957119\pi$
$888$	0	0
$889$	−24.0000	−0.804934
$890$	0	0
$891$	0	0
$892$	6.00000i	0.200895i
$893$	10.0000i	0.334637i
$894$	0	0
$895$	0	0
$896$	−3.00000	−0.100223
$897$	0	0
$898$	− 30.0000i	− 1.00111i
$899$	−15.0000	−0.500278
$900$	0	0
$901$	7.00000	0.233204
$902$	2.00000i	0.0665927i
$903$	0	0
$904$	16.0000	0.532152
$905$	0	0
$906$	0	0
$907$	57.0000i	1.89265i	0.323211	+	0.946327i	$0.395238\pi$
$907$	57.0000i	1.89265i	−0.323211	+	0.946327i	$0.604762\pi$
$908$	2.00000i	0.0663723i
$909$	0	0
$910$	0	0
$911$	−27.0000	−0.894550	−0.447275	−	0.894397i	$-0.647605\pi$
$911$	−27.0000	−0.894550	−0.447275	+	0.894397i	$0.647605\pi$
$912$	0	0
$913$	− 6.00000i	− 0.198571i
$914$	3.00000	0.0992312
$915$	0	0
$916$	10.0000	0.330409
$917$	51.0000i	1.68417i
$918$	0	0
$919$	−40.0000	−1.31948	−0.659739	−	0.751495i	$-0.729333\pi$
$919$	−40.0000	−1.31948	−0.659739	+	0.751495i	$0.729333\pi$
$920$	0	0
$921$	0	0
$922$	− 27.0000i	− 0.889198i
$923$	42.0000i	1.38245i
$924$	0	0
$925$	0	0
$926$	−34.0000	−1.11731
$927$	0	0
$928$	5.00000i	0.164133i
$929$	35.0000	1.14831	0.574156	−	0.818746i	$-0.305330\pi$
$929$	35.0000	1.14831	0.574156	+	0.818746i	$0.305330\pi$
$930$	0	0
$931$	10.0000	0.327737
$932$	9.00000i	0.294805i
$933$	0	0
$934$	−23.0000	−0.752583
$935$	0	0
$936$	0	0
$937$	− 38.0000i	− 1.24141i	−0.784046	−	0.620703i	$-0.786847\pi$
$937$	− 38.0000i	− 1.24141i	0.784046	−	0.620703i	$-0.213153\pi$
$938$	− 24.0000i	− 0.783628i
$939$	0	0
$940$	0	0
$941$	−17.0000	−0.554184	−0.277092	−	0.960843i	$-0.589371\pi$
$941$	−17.0000	−0.554184	−0.277092	+	0.960843i	$0.589371\pi$
$942$	0	0
$943$	− 12.0000i	− 0.390774i
$944$	−10.0000	−0.325472
$945$	0	0
$946$	4.00000	0.130051
$947$	− 27.0000i	− 0.877382i	−0.898638	−	0.438691i	$-0.855442\pi$
$947$	− 27.0000i	− 0.877382i	0.898638	−	0.438691i	$-0.144558\pi$
$948$	0	0
$949$	84.0000	2.72676
$950$	0	0
$951$	0	0
$952$	21.0000i	0.680614i
$953$	− 39.0000i	− 1.26333i	−0.775240	−	0.631667i	$-0.782371\pi$
$953$	− 39.0000i	− 1.26333i	0.775240	−	0.631667i	$-0.217629\pi$
$954$	0	0
$955$	0	0
$956$	−10.0000	−0.323423
$957$	0	0
$958$	0	0
$959$	−36.0000	−1.16250
$960$	0	0
$961$	−22.0000	−0.709677
$962$	− 18.0000i	− 0.580343i
$963$	0	0
$964$	18.0000	0.579741
$965$	0	0
$966$	0	0
$967$	27.0000i	0.868261i	0.900850	+	0.434131i	$0.142944\pi$
$967$	27.0000i	0.868261i	−0.900850	+	0.434131i	$0.857056\pi$
$968$	− 1.00000i	− 0.0321412i
$969$	0	0
$970$	0	0
$971$	48.0000	1.54039	0.770197	−	0.637806i	$-0.220158\pi$
$971$	48.0000	1.54039	0.770197	+	0.637806i	$0.220158\pi$
$972$	0	0
$973$	− 60.0000i	− 1.92351i
$974$	−12.0000	−0.384505
$975$	0	0
$976$	7.00000	0.224065
$977$	− 12.0000i	− 0.383914i	−0.981403	−	0.191957i	$-0.938517\pi$
$977$	− 12.0000i	− 0.383914i	0.981403	−	0.191957i	$-0.0614834\pi$
$978$	0	0
$979$	15.0000	0.479402
$980$	0	0
$981$	0	0
$982$	3.00000i	0.0957338i
$983$	− 54.0000i	− 1.72233i	−0.508323	−	0.861166i	$-0.669735\pi$
$983$	− 54.0000i	− 1.72233i	0.508323	−	0.861166i	$-0.330265\pi$
$984$	0	0
$985$	0	0
$986$	35.0000	1.11463
$987$	0	0
$988$	− 30.0000i	− 0.954427i
$989$	−24.0000	−0.763156
$990$	0	0
$991$	−8.00000	−0.254128	−0.127064	−	0.991894i	$-0.540555\pi$
$991$	−8.00000	−0.254128	−0.127064	+	0.991894i	$0.540555\pi$
$992$	− 3.00000i	− 0.0952501i
$993$	0	0
$994$	−21.0000	−0.666080
$995$	0	0
$996$	0	0
$997$	32.0000i	1.01345i	0.862108	+	0.506725i	$0.169144\pi$
$997$	32.0000i	1.01345i	−0.862108	+	0.506725i	$0.830856\pi$
$998$	− 20.0000i	− 0.633089i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4950.2.c.m.199.2		2
3.2	odd	2		550.2.b.a.199.1		2
5.2	odd	4		990.2.a.d.1.1		1
5.3	odd	4		4950.2.a.bc.1.1		1
5.4	even	2	inner	4950.2.c.m.199.1		2
12.11	even	2		4400.2.b.i.4049.2		2
15.2	even	4		110.2.a.b.1.1	✓	1
15.8	even	4		550.2.a.f.1.1		1
15.14	odd	2		550.2.b.a.199.2		2
20.7	even	4		7920.2.a.d.1.1		1
60.23	odd	4		4400.2.a.l.1.1		1
60.47	odd	4		880.2.a.i.1.1		1
60.59	even	2		4400.2.b.i.4049.1		2
105.62	odd	4		5390.2.a.bf.1.1		1
120.77	even	4		3520.2.a.y.1.1		1
120.107	odd	4		3520.2.a.h.1.1		1
165.32	odd	4		1210.2.a.b.1.1		1
165.98	odd	4		6050.2.a.bj.1.1		1
660.527	even	4		9680.2.a.x.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
110.2.a.b.1.1	✓	1	15.2	even	4
550.2.a.f.1.1		1	15.8	even	4
550.2.b.a.199.1		2	3.2	odd	2
550.2.b.a.199.2		2	15.14	odd	2
880.2.a.i.1.1		1	60.47	odd	4
990.2.a.d.1.1		1	5.2	odd	4
1210.2.a.b.1.1		1	165.32	odd	4
3520.2.a.h.1.1		1	120.107	odd	4
3520.2.a.y.1.1		1	120.77	even	4
4400.2.a.l.1.1		1	60.23	odd	4
4400.2.b.i.4049.1		2	60.59	even	2
4400.2.b.i.4049.2		2	12.11	even	2
4950.2.a.bc.1.1		1	5.3	odd	4
4950.2.c.m.199.1		2	5.4	even	2	inner
4950.2.c.m.199.2		2	1.1	even	1	trivial
5390.2.a.bf.1.1		1	105.62	odd	4
6050.2.a.bj.1.1		1	165.98	odd	4
7920.2.a.d.1.1		1	20.7	even	4
9680.2.a.x.1.1		1	660.527	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 4950 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4950.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{4} -3.00000i q^{7} -1.00000i q^{8} -1.00000 q^{11} -6.00000i q^{13} +3.00000 q^{14} +1.00000 q^{16} -7.00000i q^{17} -5.00000 q^{19} -1.00000i q^{22} +6.00000i q^{23} +6.00000 q^{26} +3.00000i q^{28} +5.00000 q^{29} -3.00000 q^{31} +1.00000i q^{32} +7.00000 q^{34} -3.00000i q^{37} -5.00000i q^{38} -2.00000 q^{41} +4.00000i q^{43} +1.00000 q^{44} -6.00000 q^{46} -2.00000i q^{47} -2.00000 q^{49} +6.00000i q^{52} +1.00000i q^{53} -3.00000 q^{56} +5.00000i q^{58} -10.0000 q^{59} +7.00000 q^{61} -3.00000i q^{62} -1.00000 q^{64} -8.00000i q^{67} +7.00000i q^{68} -7.00000 q^{71} +14.0000i q^{73} +3.00000 q^{74} +5.00000 q^{76} +3.00000i q^{77} -10.0000 q^{79} -2.00000i q^{82} +6.00000i q^{83} -4.00000 q^{86} +1.00000i q^{88} -15.0000 q^{89} -18.0000 q^{91} -6.00000i q^{92} +2.00000 q^{94} +12.0000i q^{97} -2.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} - 2 q^{11} + 6 q^{14} + 2 q^{16} - 10 q^{19} + 12 q^{26} + 10 q^{29} - 6 q^{31} + 14 q^{34} - 4 q^{41} + 2 q^{44} - 12 q^{46} - 4 q^{49} - 6 q^{56} - 20 q^{59} + 14 q^{61} - 2 q^{64} - 14 q^{71}+ \cdots + 4 q^{94}+O(q^{100}) \) 2 * q - 2 * q^4 - 2 * q^11 + 6 * q^14 + 2 * q^16 - 10 * q^19 + 12 * q^26 + 10 * q^29 - 6 * q^31 + 14 * q^34 - 4 * q^41 + 2 * q^44 - 12 * q^46 - 4 * q^49 - 6 * q^56 - 20 * q^59 + 14 * q^61 - 2 * q^64 - 14 * q^71 + 6 * q^74 + 10 * q^76 - 20 * q^79 - 8 * q^86 - 30 * q^89 - 36 * q^91 + 4 * q^94

Introduction

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