LMFDB - Embedded newform 3150.2.g.d.2899.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3150,2,Mod(2899,3150)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3150, 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("3150.2899");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3150.g (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:	$25.1528766367$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2899.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	3150.2899
Dual form			3150.2.g.d.2899.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{4} -1.00000i q^{7} +1.00000i q^{8} +O(q^{10})$	$q-1.00000i q^{2} -1.00000 q^{4} -1.00000i q^{7} +1.00000i q^{8} -4.00000 q^{11} +3.00000i q^{13} -1.00000 q^{14} +1.00000 q^{16} +7.00000i q^{17} +6.00000 q^{19} +4.00000i q^{22} -9.00000i q^{23} +3.00000 q^{26} +1.00000i q^{28} +3.00000 q^{29} -7.00000 q^{31} -1.00000i q^{32} +7.00000 q^{34} -10.0000i q^{37} -6.00000i q^{38} +1.00000 q^{41} -13.0000i q^{43} +4.00000 q^{44} -9.00000 q^{46} -2.00000i q^{47} -1.00000 q^{49} -3.00000i q^{52} +1.00000i q^{53} +1.00000 q^{56} -3.00000i q^{58} -11.0000 q^{59} +13.0000 q^{61} +7.00000i q^{62} -1.00000 q^{64} -7.00000i q^{68} -8.00000 q^{71} -8.00000i q^{73} -10.0000 q^{74} -6.00000 q^{76} +4.00000i q^{77} -4.00000 q^{79} -1.00000i q^{82} -7.00000i q^{83} -13.0000 q^{86} -4.00000i q^{88} -14.0000 q^{89} +3.00000 q^{91} +9.00000i q^{92} -2.00000 q^{94} +8.00000i q^{97} +1.00000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4}+O(q^{10}) $ 2 * q - 2 * q^4	$ 2 q - 2 q^{4} - 8 q^{11} - 2 q^{14} + 2 q^{16} + 12 q^{19} + 6 q^{26} + 6 q^{29} - 14 q^{31} + 14 q^{34} + 2 q^{41} + 8 q^{44} - 18 q^{46} - 2 q^{49} + 2 q^{56} - 22 q^{59} + 26 q^{61} - 2 q^{64} - 16 q^{71} - 20 q^{74} - 12 q^{76} - 8 q^{79} - 26 q^{86} - 28 q^{89} + 6 q^{91} - 4 q^{94}+O(q^{100}) $ 2 * q - 2 * q^4 - 8 * q^11 - 2 * q^14 + 2 * q^16 + 12 * q^19 + 6 * q^26 + 6 * q^29 - 14 * q^31 + 14 * q^34 + 2 * q^41 + 8 * q^44 - 18 * q^46 - 2 * q^49 + 2 * q^56 - 22 * q^59 + 26 * q^61 - 2 * q^64 - 16 * q^71 - 20 * q^74 - 12 * q^76 - 8 * q^79 - 26 * q^86 - 28 * q^89 + 6 * q^91 - 4 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3150\mathbb{Z}\right)^\times$.

$n$	$127$	$451$	$2801$
$\chi(n)$	$-1$	$1$	$1$

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
$7$	− 1.00000i	− 0.377964i
$7$	− 1.00000i	− 0.377964i
$8$	1.00000i	0.353553i
$9$	0	0
$10$	0	0
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	0	0
$13$	3.00000i	0.832050i	0.909353	+	0.416025i	$0.136577\pi$
$13$	3.00000i	0.832050i	−0.909353	+	0.416025i	$0.863423\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	7.00000i	1.69775i	0.528594	+	0.848875i	$0.322719\pi$
$17$	7.00000i	1.69775i	−0.528594	+	0.848875i	$0.677281\pi$
$18$	0	0
$19$	6.00000	1.37649	0.688247	−	0.725476i	$-0.258380\pi$
$19$	6.00000	1.37649	0.688247	+	0.725476i	$0.258380\pi$
$20$	0	0
$21$	0	0
$22$	4.00000i	0.852803i
$23$	− 9.00000i	− 1.87663i	−0.345782	−	0.938315i	$-0.612386\pi$
$23$	− 9.00000i	− 1.87663i	0.345782	−	0.938315i	$-0.387614\pi$
$24$	0	0
$25$	0	0
$26$	3.00000	0.588348
$27$	0	0
$28$	1.00000i	0.188982i
$29$	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
$31$	−7.00000	−1.25724	−0.628619	−	0.777714i	$-0.716379\pi$
$31$	−7.00000	−1.25724	−0.628619	+	0.777714i	$0.716379\pi$
$32$	− 1.00000i	− 0.176777i
$33$	0	0
$34$	7.00000	1.20049
$35$	0	0
$36$	0	0
$37$	− 10.0000i	− 1.64399i	−0.569495	−	0.821995i	$-0.692861\pi$
$37$	− 10.0000i	− 1.64399i	0.569495	−	0.821995i	$-0.307139\pi$
$38$	− 6.00000i	− 0.973329i
$39$	0	0
$40$	0	0
$41$	1.00000	0.156174	0.0780869	−	0.996947i	$-0.475119\pi$
$41$	1.00000	0.156174	0.0780869	+	0.996947i	$0.475119\pi$
$42$	0	0
$43$	− 13.0000i	− 1.98248i	−0.132068	−	0.991241i	$-0.542162\pi$
$43$	− 13.0000i	− 1.98248i	0.132068	−	0.991241i	$-0.457838\pi$
$44$	4.00000	0.603023
$45$	0	0
$46$	−9.00000	−1.32698
$47$	− 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
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	− 3.00000i	− 0.416025i
$53$	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
$56$	1.00000	0.133631
$57$	0	0
$58$	− 3.00000i	− 0.393919i
$59$	−11.0000	−1.43208	−0.716039	−	0.698060i	$-0.754047\pi$
$59$	−11.0000	−1.43208	−0.716039	+	0.698060i	$0.754047\pi$
$60$	0	0
$61$	13.0000	1.66448	0.832240	−	0.554416i	$-0.187058\pi$
$61$	13.0000	1.66448	0.832240	+	0.554416i	$0.187058\pi$
$62$	7.00000i	0.889001i
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	− 7.00000i	− 0.848875i
$69$	0	0
$70$	0	0
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	0	0
$73$	− 8.00000i	− 0.936329i	−0.883641	−	0.468165i	$-0.844915\pi$
$73$	− 8.00000i	− 0.936329i	0.883641	−	0.468165i	$-0.155085\pi$
$74$	−10.0000	−1.16248
$75$	0	0
$76$	−6.00000	−0.688247
$77$	4.00000i	0.455842i
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	0	0
$81$	0	0
$82$	− 1.00000i	− 0.110432i
$83$	− 7.00000i	− 0.768350i	−0.923260	−	0.384175i	$-0.874486\pi$
$83$	− 7.00000i	− 0.768350i	0.923260	−	0.384175i	$-0.125514\pi$
$84$	0	0
$85$	0	0
$86$	−13.0000	−1.40183
$87$	0	0
$88$	− 4.00000i	− 0.426401i
$89$	−14.0000	−1.48400	−0.741999	−	0.670402i	$-0.766122\pi$
$89$	−14.0000	−1.48400	−0.741999	+	0.670402i	$0.766122\pi$
$90$	0	0
$91$	3.00000	0.314485
$92$	9.00000i	0.938315i
$93$	0	0
$94$	−2.00000	−0.206284
$95$	0	0
$96$	0	0
$97$	8.00000i	0.812277i	0.913812	+	0.406138i	$0.133125\pi$
$97$	8.00000i	0.812277i	−0.913812	+	0.406138i	$0.866875\pi$
$98$	1.00000i	0.101015i
$99$	0	0
$100$	0	0
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	0	0
$103$	11.0000i	1.08386i	0.840423	+	0.541931i	$0.182307\pi$
$103$	11.0000i	1.08386i	−0.840423	+	0.541931i	$0.817693\pi$
$104$	−3.00000	−0.294174
$105$	0	0
$106$	1.00000	0.0971286
$107$	− 6.00000i	− 0.580042i	−0.957020	−	0.290021i	$-0.906338\pi$
$107$	− 6.00000i	− 0.580042i	0.957020	−	0.290021i	$-0.0936623\pi$
$108$	0	0
$109$	6.00000	0.574696	0.287348	−	0.957826i	$-0.407226\pi$
$109$	6.00000	0.574696	0.287348	+	0.957826i	$0.407226\pi$
$110$	0	0
$111$	0	0
$112$	− 1.00000i	− 0.0944911i
$113$	− 16.0000i	− 1.50515i	−0.658505	−	0.752577i	$-0.728811\pi$
$113$	− 16.0000i	− 1.50515i	0.658505	−	0.752577i	$-0.271189\pi$
$114$	0	0
$115$	0	0
$116$	−3.00000	−0.278543
$117$	0	0
$118$	11.0000i	1.01263i
$119$	7.00000	0.641689
$120$	0	0
$121$	5.00000	0.454545
$122$	− 13.0000i	− 1.17696i
$123$	0	0
$124$	7.00000	0.628619
$125$	0	0
$126$	0	0
$127$	− 14.0000i	− 1.24230i	−0.783692	−	0.621150i	$-0.786666\pi$
$127$	− 14.0000i	− 1.24230i	0.783692	−	0.621150i	$-0.213334\pi$
$128$	1.00000i	0.0883883i
$129$	0	0
$130$	0	0
$131$	−20.0000	−1.74741	−0.873704	−	0.486458i	$-0.838289\pi$
$131$	−20.0000	−1.74741	−0.873704	+	0.486458i	$0.838289\pi$
$132$	0	0
$133$	− 6.00000i	− 0.520266i
$134$	0	0
$135$	0	0
$136$	−7.00000	−0.600245
$137$	12.0000i	1.02523i	0.858619	+	0.512615i	$0.171323\pi$
$137$	12.0000i	1.02523i	−0.858619	+	0.512615i	$0.828677\pi$
$138$	0	0
$139$	8.00000	0.678551	0.339276	−	0.940687i	$-0.389818\pi$
$139$	8.00000	0.678551	0.339276	+	0.940687i	$0.389818\pi$
$140$	0	0
$141$	0	0
$142$	8.00000i	0.671345i
$143$	− 12.0000i	− 1.00349i
$144$	0	0
$145$	0	0
$146$	−8.00000	−0.662085
$147$	0	0
$148$	10.0000i	0.821995i
$149$	−9.00000	−0.737309	−0.368654	−	0.929567i	$-0.620181\pi$
$149$	−9.00000	−0.737309	−0.368654	+	0.929567i	$0.620181\pi$
$150$	0	0
$151$	2.00000	0.162758	0.0813788	−	0.996683i	$-0.474068\pi$
$151$	2.00000	0.162758	0.0813788	+	0.996683i	$0.474068\pi$
$152$	6.00000i	0.486664i
$153$	0	0
$154$	4.00000	0.322329
$155$	0	0
$156$	0	0
$157$	− 2.00000i	− 0.159617i	−0.996810	−	0.0798087i	$-0.974569\pi$
$157$	− 2.00000i	− 0.159617i	0.996810	−	0.0798087i	$-0.0254309\pi$
$158$	4.00000i	0.318223i
$159$	0	0
$160$	0	0
$161$	−9.00000	−0.709299
$162$	0	0
$163$	− 11.0000i	− 0.861586i	−0.902451	−	0.430793i	$-0.858234\pi$
$163$	− 11.0000i	− 0.861586i	0.902451	−	0.430793i	$-0.141766\pi$
$164$	−1.00000	−0.0780869
$165$	0	0
$166$	−7.00000	−0.543305
$167$	12.0000i	0.928588i	0.885681	+	0.464294i	$0.153692\pi$
$167$	12.0000i	0.928588i	−0.885681	+	0.464294i	$0.846308\pi$
$168$	0	0
$169$	4.00000	0.307692
$170$	0	0
$171$	0	0
$172$	13.0000i	0.991241i
$173$	− 10.0000i	− 0.760286i	−0.924928	−	0.380143i	$-0.875875\pi$
$173$	− 10.0000i	− 0.760286i	0.924928	−	0.380143i	$-0.124125\pi$
$174$	0	0
$175$	0	0
$176$	−4.00000	−0.301511
$177$	0	0
$178$	14.0000i	1.04934i
$179$	−6.00000	−0.448461	−0.224231	−	0.974536i	$-0.571987\pi$
$179$	−6.00000	−0.448461	−0.224231	+	0.974536i	$0.571987\pi$
$180$	0	0
$181$	−2.00000	−0.148659	−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000	−0.148659	−0.0743294	+	0.997234i	$0.523682\pi$
$182$	− 3.00000i	− 0.222375i
$183$	0	0
$184$	9.00000	0.663489
$185$	0	0
$186$	0	0
$187$	− 28.0000i	− 2.04756i
$188$	2.00000i	0.145865i
$189$	0	0
$190$	0	0
$191$	21.0000	1.51951	0.759753	−	0.650211i	$-0.225320\pi$
$191$	21.0000	1.51951	0.759753	+	0.650211i	$0.225320\pi$
$192$	0	0
$193$	− 14.0000i	− 1.00774i	−0.863779	−	0.503871i	$-0.831909\pi$
$193$	− 14.0000i	− 1.00774i	0.863779	−	0.503871i	$-0.168091\pi$
$194$	8.00000	0.574367
$195$	0	0
$196$	1.00000	0.0714286
$197$	− 5.00000i	− 0.356235i	−0.984009	−	0.178118i	$-0.942999\pi$
$197$	− 5.00000i	− 0.356235i	0.984009	−	0.178118i	$-0.0570008\pi$
$198$	0	0
$199$	−8.00000	−0.567105	−0.283552	−	0.958957i	$-0.591513\pi$
$199$	−8.00000	−0.567105	−0.283552	+	0.958957i	$0.591513\pi$
$200$	0	0
$201$	0	0
$202$	− 6.00000i	− 0.422159i
$203$	− 3.00000i	− 0.210559i
$204$	0	0
$205$	0	0
$206$	11.0000	0.766406
$207$	0	0
$208$	3.00000i	0.208013i
$209$	−24.0000	−1.66011
$210$	0	0
$211$	13.0000	0.894957	0.447478	−	0.894295i	$-0.352322\pi$
$211$	13.0000	0.894957	0.447478	+	0.894295i	$0.352322\pi$
$212$	− 1.00000i	− 0.0686803i
$213$	0	0
$214$	−6.00000	−0.410152
$215$	0	0
$216$	0	0
$217$	7.00000i	0.475191i
$218$	− 6.00000i	− 0.406371i
$219$	0	0
$220$	0	0
$221$	−21.0000	−1.41261
$222$	0	0
$223$	− 13.0000i	− 0.870544i	−0.900299	−	0.435272i	$-0.856652\pi$
$223$	− 13.0000i	− 0.870544i	0.900299	−	0.435272i	$-0.143348\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	−16.0000	−1.06430
$227$	− 11.0000i	− 0.730096i	−0.930989	−	0.365048i	$-0.881053\pi$
$227$	− 11.0000i	− 0.730096i	0.930989	−	0.365048i	$-0.118947\pi$
$228$	0	0
$229$	−22.0000	−1.45380	−0.726900	−	0.686743i	$-0.759040\pi$
$229$	−22.0000	−1.45380	−0.726900	+	0.686743i	$0.759040\pi$
$230$	0	0
$231$	0	0
$232$	3.00000i	0.196960i
$233$	− 18.0000i	− 1.17922i	−0.807688	−	0.589610i	$-0.799282\pi$
$233$	− 18.0000i	− 1.17922i	0.807688	−	0.589610i	$-0.200718\pi$
$234$	0	0
$235$	0	0
$236$	11.0000	0.716039
$237$	0	0
$238$	− 7.00000i	− 0.453743i
$239$	−8.00000	−0.517477	−0.258738	−	0.965947i	$-0.583307\pi$
$239$	−8.00000	−0.517477	−0.258738	+	0.965947i	$0.583307\pi$
$240$	0	0
$241$	22.0000	1.41714	0.708572	−	0.705638i	$-0.249340\pi$
$241$	22.0000	1.41714	0.708572	+	0.705638i	$0.249340\pi$
$242$	− 5.00000i	− 0.321412i
$243$	0	0
$244$	−13.0000	−0.832240
$245$	0	0
$246$	0	0
$247$	18.0000i	1.14531i
$248$	− 7.00000i	− 0.444500i
$249$	0	0
$250$	0	0
$251$	−21.0000	−1.32551	−0.662754	−	0.748837i	$-0.730613\pi$
$251$	−21.0000	−1.32551	−0.662754	+	0.748837i	$0.730613\pi$
$252$	0	0
$253$	36.0000i	2.26330i
$254$	−14.0000	−0.878438
$255$	0	0
$256$	1.00000	0.0625000
$257$	− 17.0000i	− 1.06043i	−0.847863	−	0.530215i	$-0.822111\pi$
$257$	− 17.0000i	− 1.06043i	0.847863	−	0.530215i	$-0.177889\pi$
$258$	0	0
$259$	−10.0000	−0.621370
$260$	0	0
$261$	0	0
$262$	20.0000i	1.23560i
$263$	− 13.0000i	− 0.801614i	−0.916162	−	0.400807i	$-0.868730\pi$
$263$	− 13.0000i	− 0.801614i	0.916162	−	0.400807i	$-0.131270\pi$
$264$	0	0
$265$	0	0
$266$	−6.00000	−0.367884
$267$	0	0
$268$	0	0
$269$	6.00000	0.365826	0.182913	−	0.983129i	$-0.441447\pi$
$269$	6.00000	0.365826	0.182913	+	0.983129i	$0.441447\pi$
$270$	0	0
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	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	$0.882604\pi$
$278$	− 8.00000i	− 0.479808i
$279$	0	0
$280$	0	0
$281$	−32.0000	−1.90896	−0.954480	−	0.298275i	$-0.903589\pi$
$281$	−32.0000	−1.90896	−0.954480	+	0.298275i	$0.903589\pi$
$282$	0	0
$283$	− 14.0000i	− 0.832214i	−0.909316	−	0.416107i	$-0.863394\pi$
$283$	− 14.0000i	− 0.832214i	0.909316	−	0.416107i	$-0.136606\pi$
$284$	8.00000	0.474713
$285$	0	0
$286$	−12.0000	−0.709575
$287$	− 1.00000i	− 0.0590281i
$288$	0	0
$289$	−32.0000	−1.88235
$290$	0	0
$291$	0	0
$292$	8.00000i	0.468165i
$293$	28.0000i	1.63578i	0.575376	+	0.817889i	$0.304856\pi$
$293$	28.0000i	1.63578i	−0.575376	+	0.817889i	$0.695144\pi$
$294$	0	0
$295$	0	0
$296$	10.0000	0.581238
$297$	0	0
$298$	9.00000i	0.521356i
$299$	27.0000	1.56145
$300$	0	0
$301$	−13.0000	−0.749308
$302$	− 2.00000i	− 0.115087i
$303$	0	0
$304$	6.00000	0.344124
$305$	0	0
$306$	0	0
$307$	− 2.00000i	− 0.114146i	−0.998370	−	0.0570730i	$-0.981823\pi$
$307$	− 2.00000i	− 0.114146i	0.998370	−	0.0570730i	$-0.0181768\pi$
$308$	− 4.00000i	− 0.227921i
$309$	0	0
$310$	0	0
$311$	30.0000	1.70114	0.850572	−	0.525859i	$-0.176256\pi$
$311$	30.0000	1.70114	0.850572	+	0.525859i	$0.176256\pi$
$312$	0	0
$313$	16.0000i	0.904373i	0.891923	+	0.452187i	$0.149356\pi$
$313$	16.0000i	0.904373i	−0.891923	+	0.452187i	$0.850644\pi$
$314$	−2.00000	−0.112867
$315$	0	0
$316$	4.00000	0.225018
$317$	11.0000i	0.617822i	0.951091	+	0.308911i	$0.0999645\pi$
$317$	11.0000i	0.617822i	−0.951091	+	0.308911i	$0.900036\pi$
$318$	0	0
$319$	−12.0000	−0.671871
$320$	0	0
$321$	0	0
$322$	9.00000i	0.501550i
$323$	42.0000i	2.33694i
$324$	0	0
$325$	0	0
$326$	−11.0000	−0.609234
$327$	0	0
$328$	1.00000i	0.0552158i
$329$	−2.00000	−0.110264
$330$	0	0
$331$	−5.00000	−0.274825	−0.137412	−	0.990514i	$-0.543879\pi$
$331$	−5.00000	−0.274825	−0.137412	+	0.990514i	$0.543879\pi$
$332$	7.00000i	0.384175i
$333$	0	0
$334$	12.0000	0.656611
$335$	0	0
$336$	0	0
$337$	− 5.00000i	− 0.272367i	−0.990684	−	0.136184i	$-0.956516\pi$
$337$	− 5.00000i	− 0.272367i	0.990684	−	0.136184i	$-0.0434837\pi$
$338$	− 4.00000i	− 0.217571i
$339$	0	0
$340$	0	0
$341$	28.0000	1.51629
$342$	0	0
$343$	1.00000i	0.0539949i
$344$	13.0000	0.700913
$345$	0	0
$346$	−10.0000	−0.537603
$347$	32.0000i	1.71785i	0.512101	+	0.858925i	$0.328867\pi$
$347$	32.0000i	1.71785i	−0.512101	+	0.858925i	$0.671133\pi$
$348$	0	0
$349$	−19.0000	−1.01705	−0.508523	−	0.861048i	$-0.669808\pi$
$349$	−19.0000	−1.01705	−0.508523	+	0.861048i	$0.669808\pi$
$350$	0	0
$351$	0	0
$352$	4.00000i	0.213201i
$353$	− 30.0000i	− 1.59674i	−0.602168	−	0.798369i	$-0.705696\pi$
$353$	− 30.0000i	− 1.59674i	0.602168	−	0.798369i	$-0.294304\pi$
$354$	0	0
$355$	0	0
$356$	14.0000	0.741999
$357$	0	0
$358$	6.00000i	0.317110i
$359$	11.0000	0.580558	0.290279	−	0.956942i	$-0.406252\pi$
$359$	11.0000	0.580558	0.290279	+	0.956942i	$0.406252\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	2.00000i	0.105118i
$363$	0	0
$364$	−3.00000	−0.157243
$365$	0	0
$366$	0	0
$367$	− 17.0000i	− 0.887393i	−0.896177	−	0.443696i	$-0.853667\pi$
$367$	− 17.0000i	− 0.887393i	0.896177	−	0.443696i	$-0.146333\pi$
$368$	− 9.00000i	− 0.469157i
$369$	0	0
$370$	0	0
$371$	1.00000	0.0519174
$372$	0	0
$373$	− 14.0000i	− 0.724893i	−0.932005	−	0.362446i	$-0.881942\pi$
$373$	− 14.0000i	− 0.724893i	0.932005	−	0.362446i	$-0.118058\pi$
$374$	−28.0000	−1.44785
$375$	0	0
$376$	2.00000	0.103142
$377$	9.00000i	0.463524i
$378$	0	0
$379$	15.0000	0.770498	0.385249	−	0.922813i	$-0.374116\pi$
$379$	15.0000	0.770498	0.385249	+	0.922813i	$0.374116\pi$
$380$	0	0
$381$	0	0
$382$	− 21.0000i	− 1.07445i
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	0	0
$385$	0	0
$386$	−14.0000	−0.712581
$387$	0	0
$388$	− 8.00000i	− 0.406138i
$389$	26.0000	1.31825	0.659126	−	0.752032i	$-0.270926\pi$
$389$	26.0000	1.31825	0.659126	+	0.752032i	$0.270926\pi$
$390$	0	0
$391$	63.0000	3.18605
$392$	− 1.00000i	− 0.0505076i
$393$	0	0
$394$	−5.00000	−0.251896
$395$	0	0
$396$	0	0
$397$	19.0000i	0.953583i	0.879017	+	0.476791i	$0.158200\pi$
$397$	19.0000i	0.953583i	−0.879017	+	0.476791i	$0.841800\pi$
$398$	8.00000i	0.401004i
$399$	0	0
$400$	0	0
$401$	−4.00000	−0.199750	−0.0998752	−	0.995000i	$-0.531844\pi$
$401$	−4.00000	−0.199750	−0.0998752	+	0.995000i	$0.531844\pi$
$402$	0	0
$403$	− 21.0000i	− 1.04608i
$404$	−6.00000	−0.298511
$405$	0	0
$406$	−3.00000	−0.148888
$407$	40.0000i	1.98273i
$408$	0	0
$409$	0	0		−	1.00000i	$-0.5\pi$
$409$	0	0			1.00000i	$0.5\pi$
$410$	0	0
$411$	0	0
$412$	− 11.0000i	− 0.541931i
$413$	11.0000i	0.541275i
$414$	0	0
$415$	0	0
$416$	3.00000	0.147087
$417$	0	0
$418$	24.0000i	1.17388i
$419$	−9.00000	−0.439679	−0.219839	−	0.975536i	$-0.570553\pi$
$419$	−9.00000	−0.439679	−0.219839	+	0.975536i	$0.570553\pi$
$420$	0	0
$421$	8.00000	0.389896	0.194948	−	0.980814i	$-0.437546\pi$
$421$	8.00000	0.389896	0.194948	+	0.980814i	$0.437546\pi$
$422$	− 13.0000i	− 0.632830i
$423$	0	0
$424$	−1.00000	−0.0485643
$425$	0	0
$426$	0	0
$427$	− 13.0000i	− 0.629114i
$428$	6.00000i	0.290021i
$429$	0	0
$430$	0	0
$431$	21.0000	1.01153	0.505767	−	0.862670i	$-0.331209\pi$
$431$	21.0000	1.01153	0.505767	+	0.862670i	$0.331209\pi$
$432$	0	0
$433$	− 2.00000i	− 0.0961139i	−0.998845	−	0.0480569i	$-0.984697\pi$
$433$	− 2.00000i	− 0.0961139i	0.998845	−	0.0480569i	$-0.0153029\pi$
$434$	7.00000	0.336011
$435$	0	0
$436$	−6.00000	−0.287348
$437$	− 54.0000i	− 2.58317i
$438$	0	0
$439$	35.0000	1.67046	0.835229	−	0.549902i	$-0.185335\pi$
$439$	35.0000	1.67046	0.835229	+	0.549902i	$0.185335\pi$
$440$	0	0
$441$	0	0
$442$	21.0000i	0.998868i
$443$	− 2.00000i	− 0.0950229i	−0.998871	−	0.0475114i	$-0.984871\pi$
$443$	− 2.00000i	− 0.0950229i	0.998871	−	0.0475114i	$-0.0151291\pi$
$444$	0	0
$445$	0	0
$446$	−13.0000	−0.615568
$447$	0	0
$448$	1.00000i	0.0472456i
$449$	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
$451$	−4.00000	−0.188353
$452$	16.0000i	0.752577i
$453$	0	0
$454$	−11.0000	−0.516256
$455$	0	0
$456$	0	0
$457$	− 17.0000i	− 0.795226i	−0.917553	−	0.397613i	$-0.869839\pi$
$457$	− 17.0000i	− 0.795226i	0.917553	−	0.397613i	$-0.130161\pi$
$458$	22.0000i	1.02799i
$459$	0	0
$460$	0	0
$461$	14.0000	0.652045	0.326023	−	0.945362i	$-0.394291\pi$
$461$	14.0000	0.652045	0.326023	+	0.945362i	$0.394291\pi$
$462$	0	0
$463$	16.0000i	0.743583i	0.928316	+	0.371792i	$0.121256\pi$
$463$	16.0000i	0.743583i	−0.928316	+	0.371792i	$0.878744\pi$
$464$	3.00000	0.139272
$465$	0	0
$466$	−18.0000	−0.833834
$467$	− 5.00000i	− 0.231372i	−0.993286	−	0.115686i	$-0.963093\pi$
$467$	− 5.00000i	− 0.231372i	0.993286	−	0.115686i	$-0.0369067\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	− 11.0000i	− 0.506316i
$473$	52.0000i	2.39096i
$474$	0	0
$475$	0	0
$476$	−7.00000	−0.320844
$477$	0	0
$478$	8.00000i	0.365911i
$479$	−30.0000	−1.37073	−0.685367	−	0.728197i	$-0.740358\pi$
$479$	−30.0000	−1.37073	−0.685367	+	0.728197i	$0.740358\pi$
$480$	0	0
$481$	30.0000	1.36788
$482$	− 22.0000i	− 1.00207i
$483$	0	0
$484$	−5.00000	−0.227273
$485$	0	0
$486$	0	0
$487$	− 4.00000i	− 0.181257i	−0.995885	−	0.0906287i	$-0.971112\pi$
$487$	− 4.00000i	− 0.181257i	0.995885	−	0.0906287i	$-0.0288876\pi$
$488$	13.0000i	0.588482i
$489$	0	0
$490$	0	0
$491$	−6.00000	−0.270776	−0.135388	−	0.990793i	$-0.543228\pi$
$491$	−6.00000	−0.270776	−0.135388	+	0.990793i	$0.543228\pi$
$492$	0	0
$493$	21.0000i	0.945792i
$494$	18.0000	0.809858
$495$	0	0
$496$	−7.00000	−0.314309
$497$	8.00000i	0.358849i
$498$	0	0
$499$	21.0000	0.940089	0.470045	−	0.882643i	$-0.344238\pi$
$499$	21.0000	0.940089	0.470045	+	0.882643i	$0.344238\pi$
$500$	0	0
$501$	0	0
$502$	21.0000i	0.937276i
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0	0
$505$	0	0
$506$	36.0000	1.60040
$507$	0	0
$508$	14.0000i	0.621150i
$509$	−26.0000	−1.15243	−0.576215	−	0.817298i	$-0.695471\pi$
$509$	−26.0000	−1.15243	−0.576215	+	0.817298i	$0.695471\pi$
$510$	0	0
$511$	−8.00000	−0.353899
$512$	− 1.00000i	− 0.0441942i
$513$	0	0
$514$	−17.0000	−0.749838
$515$	0	0
$516$	0	0
$517$	8.00000i	0.351840i
$518$	10.0000i	0.439375i
$519$	0	0
$520$	0	0
$521$	43.0000	1.88386	0.941932	−	0.335803i	$-0.109008\pi$
$521$	43.0000	1.88386	0.941932	+	0.335803i	$0.109008\pi$
$522$	0	0
$523$	14.0000i	0.612177i	0.952003	+	0.306089i	$0.0990204\pi$
$523$	14.0000i	0.612177i	−0.952003	+	0.306089i	$0.900980\pi$
$524$	20.0000	0.873704
$525$	0	0
$526$	−13.0000	−0.566827
$527$	− 49.0000i	− 2.13447i
$528$	0	0
$529$	−58.0000	−2.52174
$530$	0	0
$531$	0	0
$532$	6.00000i	0.260133i
$533$	3.00000i	0.129944i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	− 6.00000i	− 0.258678i
$539$	4.00000	0.172292
$540$	0	0
$541$	−42.0000	−1.80572	−0.902861	−	0.429934i	$-0.858537\pi$
$541$	−42.0000	−1.80572	−0.902861	+	0.429934i	$0.858537\pi$
$542$	0	0
$543$	0	0
$544$	7.00000	0.300123
$545$	0	0
$546$	0	0
$547$	− 9.00000i	− 0.384812i	−0.981315	−	0.192406i	$-0.938371\pi$
$547$	− 9.00000i	− 0.384812i	0.981315	−	0.192406i	$-0.0616291\pi$
$548$	− 12.0000i	− 0.512615i
$549$	0	0
$550$	0	0
$551$	18.0000	0.766826
$552$	0	0
$553$	4.00000i	0.170097i
$554$	12.0000	0.509831
$555$	0	0
$556$	−8.00000	−0.339276
$557$	2.00000i	0.0847427i	0.999102	+	0.0423714i	$0.0134913\pi$
$557$	2.00000i	0.0847427i	−0.999102	+	0.0423714i	$0.986509\pi$
$558$	0	0
$559$	39.0000	1.64952
$560$	0	0
$561$	0	0
$562$	32.0000i	1.34984i
$563$	− 19.0000i	− 0.800755i	−0.916350	−	0.400377i	$-0.868879\pi$
$563$	− 19.0000i	− 0.800755i	0.916350	−	0.400377i	$-0.131121\pi$
$564$	0	0
$565$	0	0
$566$	−14.0000	−0.588464
$567$	0	0
$568$	− 8.00000i	− 0.335673i
$569$	24.0000	1.00613	0.503066	−	0.864248i	$-0.332205\pi$
$569$	24.0000	1.00613	0.503066	+	0.864248i	$0.332205\pi$
$570$	0	0
$571$	−39.0000	−1.63210	−0.816050	−	0.577982i	$-0.803840\pi$
$571$	−39.0000	−1.63210	−0.816050	+	0.577982i	$0.803840\pi$
$572$	12.0000i	0.501745i
$573$	0	0
$574$	−1.00000	−0.0417392
$575$	0	0
$576$	0	0
$577$	46.0000i	1.91501i	0.288425	+	0.957503i	$0.406868\pi$
$577$	46.0000i	1.91501i	−0.288425	+	0.957503i	$0.593132\pi$
$578$	32.0000i	1.33102i
$579$	0	0
$580$	0	0
$581$	−7.00000	−0.290409
$582$	0	0
$583$	− 4.00000i	− 0.165663i
$584$	8.00000	0.331042
$585$	0	0
$586$	28.0000	1.15667
$587$	7.00000i	0.288921i	0.989511	+	0.144460i	$0.0461446\pi$
$587$	7.00000i	0.288921i	−0.989511	+	0.144460i	$0.953855\pi$
$588$	0	0
$589$	−42.0000	−1.73058
$590$	0	0
$591$	0	0
$592$	− 10.0000i	− 0.410997i
$593$	− 18.0000i	− 0.739171i	−0.929197	−	0.369586i	$-0.879500\pi$
$593$	− 18.0000i	− 0.739171i	0.929197	−	0.369586i	$-0.120500\pi$
$594$	0	0
$595$	0	0
$596$	9.00000	0.368654
$597$	0	0
$598$	− 27.0000i	− 1.10411i
$599$	−21.0000	−0.858037	−0.429018	−	0.903296i	$-0.641140\pi$
$599$	−21.0000	−0.858037	−0.429018	+	0.903296i	$0.641140\pi$
$600$	0	0
$601$	30.0000	1.22373	0.611863	−	0.790964i	$-0.290420\pi$
$601$	30.0000	1.22373	0.611863	+	0.790964i	$0.290420\pi$
$602$	13.0000i	0.529840i
$603$	0	0
$604$	−2.00000	−0.0813788
$605$	0	0
$606$	0	0
$607$	− 8.00000i	− 0.324710i	−0.986732	−	0.162355i	$-0.948091\pi$
$607$	− 8.00000i	− 0.324710i	0.986732	−	0.162355i	$-0.0519090\pi$
$608$	− 6.00000i	− 0.243332i
$609$	0	0
$610$	0	0
$611$	6.00000	0.242734
$612$	0	0
$613$	− 42.0000i	− 1.69636i	−0.529705	−	0.848182i	$-0.677697\pi$
$613$	− 42.0000i	− 1.69636i	0.529705	−	0.848182i	$-0.322303\pi$
$614$	−2.00000	−0.0807134
$615$	0	0
$616$	−4.00000	−0.161165
$617$	− 26.0000i	− 1.04672i	−0.852111	−	0.523360i	$-0.824678\pi$
$617$	− 26.0000i	− 1.04672i	0.852111	−	0.523360i	$-0.175322\pi$
$618$	0	0
$619$	−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
$621$	0	0
$622$	− 30.0000i	− 1.20289i
$623$	14.0000i	0.560898i
$624$	0	0
$625$	0	0
$626$	16.0000	0.639489
$627$	0	0
$628$	2.00000i	0.0798087i
$629$	70.0000	2.79108
$630$	0	0
$631$	−22.0000	−0.875806	−0.437903	−	0.899022i	$-0.644279\pi$
$631$	−22.0000	−0.875806	−0.437903	+	0.899022i	$0.644279\pi$
$632$	− 4.00000i	− 0.159111i
$633$	0	0
$634$	11.0000	0.436866
$635$	0	0
$636$	0	0
$637$	− 3.00000i	− 0.118864i
$638$	12.0000i	0.475085i
$639$	0	0
$640$	0	0
$641$	−14.0000	−0.552967	−0.276483	−	0.961019i	$-0.589169\pi$
$641$	−14.0000	−0.552967	−0.276483	+	0.961019i	$0.589169\pi$
$642$	0	0
$643$	8.00000i	0.315489i	0.987480	+	0.157745i	$0.0504223\pi$
$643$	8.00000i	0.315489i	−0.987480	+	0.157745i	$0.949578\pi$
$644$	9.00000	0.354650
$645$	0	0
$646$	42.0000	1.65247
$647$	− 28.0000i	− 1.10079i	−0.834903	−	0.550397i	$-0.814476\pi$
$647$	− 28.0000i	− 1.10079i	0.834903	−	0.550397i	$-0.185524\pi$
$648$	0	0
$649$	44.0000	1.72715
$650$	0	0
$651$	0	0
$652$	11.0000i	0.430793i
$653$	14.0000i	0.547862i	0.961749	+	0.273931i	$0.0883240\pi$
$653$	14.0000i	0.547862i	−0.961749	+	0.273931i	$0.911676\pi$
$654$	0	0
$655$	0	0
$656$	1.00000	0.0390434
$657$	0	0
$658$	2.00000i	0.0779681i
$659$	16.0000	0.623272	0.311636	−	0.950202i	$-0.399123\pi$
$659$	16.0000	0.623272	0.311636	+	0.950202i	$0.399123\pi$
$660$	0	0
$661$	−2.00000	−0.0777910	−0.0388955	−	0.999243i	$-0.512384\pi$
$661$	−2.00000	−0.0777910	−0.0388955	+	0.999243i	$0.512384\pi$
$662$	5.00000i	0.194331i
$663$	0	0
$664$	7.00000	0.271653
$665$	0	0
$666$	0	0
$667$	− 27.0000i	− 1.04544i
$668$	− 12.0000i	− 0.464294i
$669$	0	0
$670$	0	0
$671$	−52.0000	−2.00744
$672$	0	0
$673$	− 31.0000i	− 1.19496i	−0.801883	−	0.597481i	$-0.796168\pi$
$673$	− 31.0000i	− 1.19496i	0.801883	−	0.597481i	$-0.203832\pi$
$674$	−5.00000	−0.192593
$675$	0	0
$676$	−4.00000	−0.153846
$677$	18.0000i	0.691796i	0.938272	+	0.345898i	$0.112426\pi$
$677$	18.0000i	0.691796i	−0.938272	+	0.345898i	$0.887574\pi$
$678$	0	0
$679$	8.00000	0.307012
$680$	0	0
$681$	0	0
$682$	− 28.0000i	− 1.07218i
$683$	38.0000i	1.45403i	0.686622	+	0.727015i	$0.259093\pi$
$683$	38.0000i	1.45403i	−0.686622	+	0.727015i	$0.740907\pi$
$684$	0	0
$685$	0	0
$686$	1.00000	0.0381802
$687$	0	0
$688$	− 13.0000i	− 0.495620i
$689$	−3.00000	−0.114291
$690$	0	0
$691$	−8.00000	−0.304334	−0.152167	−	0.988355i	$-0.548625\pi$
$691$	−8.00000	−0.304334	−0.152167	+	0.988355i	$0.548625\pi$
$692$	10.0000i	0.380143i
$693$	0	0
$694$	32.0000	1.21470
$695$	0	0
$696$	0	0
$697$	7.00000i	0.265144i
$698$	19.0000i	0.719161i
$699$	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
$703$	− 60.0000i	− 2.26294i
$704$	4.00000	0.150756
$705$	0	0
$706$	−30.0000	−1.12906
$707$	− 6.00000i	− 0.225653i
$708$	0	0
$709$	−28.0000	−1.05156	−0.525781	−	0.850620i	$-0.676227\pi$
$709$	−28.0000	−1.05156	−0.525781	+	0.850620i	$0.676227\pi$
$710$	0	0
$711$	0	0
$712$	− 14.0000i	− 0.524672i
$713$	63.0000i	2.35937i
$714$	0	0
$715$	0	0
$716$	6.00000	0.224231
$717$	0	0
$718$	− 11.0000i	− 0.410516i
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	11.0000	0.409661
$722$	− 17.0000i	− 0.632674i
$723$	0	0
$724$	2.00000	0.0743294
$725$	0	0
$726$	0	0
$727$	− 3.00000i	− 0.111264i	−0.998451	−	0.0556319i	$-0.982283\pi$
$727$	− 3.00000i	− 0.111264i	0.998451	−	0.0556319i	$-0.0177173\pi$
$728$	3.00000i	0.111187i
$729$	0	0
$730$	0	0
$731$	91.0000	3.36576
$732$	0	0
$733$	41.0000i	1.51437i	0.653201	+	0.757185i	$0.273426\pi$
$733$	41.0000i	1.51437i	−0.653201	+	0.757185i	$0.726574\pi$
$734$	−17.0000	−0.627481
$735$	0	0
$736$	−9.00000	−0.331744
$737$	0	0
$738$	0	0
$739$	47.0000	1.72892	0.864461	−	0.502699i	$-0.167660\pi$
$739$	47.0000	1.72892	0.864461	+	0.502699i	$0.167660\pi$
$740$	0	0
$741$	0	0
$742$	− 1.00000i	− 0.0367112i
$743$	45.0000i	1.65089i	0.564483	+	0.825445i	$0.309076\pi$
$743$	45.0000i	1.65089i	−0.564483	+	0.825445i	$0.690924\pi$
$744$	0	0
$745$	0	0
$746$	−14.0000	−0.512576
$747$	0	0
$748$	28.0000i	1.02378i
$749$	−6.00000	−0.219235
$750$	0	0
$751$	−10.0000	−0.364905	−0.182453	−	0.983215i	$-0.558404\pi$
$751$	−10.0000	−0.364905	−0.182453	+	0.983215i	$0.558404\pi$
$752$	− 2.00000i	− 0.0729325i
$753$	0	0
$754$	9.00000	0.327761
$755$	0	0
$756$	0	0
$757$	− 20.0000i	− 0.726912i	−0.931611	−	0.363456i	$-0.881597\pi$
$757$	− 20.0000i	− 0.726912i	0.931611	−	0.363456i	$-0.118403\pi$
$758$	− 15.0000i	− 0.544825i
$759$	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	$0.224586\pi$
$762$	0	0
$763$	− 6.00000i	− 0.217215i
$764$	−21.0000	−0.759753
$765$	0	0
$766$	0	0
$767$	− 33.0000i	− 1.19156i
$768$	0	0
$769$	26.0000	0.937584	0.468792	−	0.883309i	$-0.344689\pi$
$769$	26.0000	0.937584	0.468792	+	0.883309i	$0.344689\pi$
$770$	0	0
$771$	0	0
$772$	14.0000i	0.503871i
$773$	− 6.00000i	− 0.215805i	−0.994161	−	0.107903i	$-0.965587\pi$
$773$	− 6.00000i	− 0.215805i	0.994161	−	0.107903i	$-0.0344134\pi$
$774$	0	0
$775$	0	0
$776$	−8.00000	−0.287183
$777$	0	0
$778$	− 26.0000i	− 0.932145i
$779$	6.00000	0.214972
$780$	0	0
$781$	32.0000	1.14505
$782$	− 63.0000i	− 2.25288i
$783$	0	0
$784$	−1.00000	−0.0357143
$785$	0	0
$786$	0	0
$787$	22.0000i	0.784215i	0.919919	+	0.392108i	$0.128254\pi$
$787$	22.0000i	0.784215i	−0.919919	+	0.392108i	$0.871746\pi$
$788$	5.00000i	0.178118i
$789$	0	0
$790$	0	0
$791$	−16.0000	−0.568895
$792$	0	0
$793$	39.0000i	1.38493i
$794$	19.0000	0.674285
$795$	0	0
$796$	8.00000	0.283552
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	0	0
$799$	14.0000	0.495284
$800$	0	0
$801$	0	0
$802$	4.00000i	0.141245i
$803$	32.0000i	1.12926i
$804$	0	0
$805$	0	0
$806$	−21.0000	−0.739693
$807$	0	0
$808$	6.00000i	0.211079i
$809$	18.0000	0.632846	0.316423	−	0.948618i	$-0.397518\pi$
$809$	18.0000	0.632846	0.316423	+	0.948618i	$0.397518\pi$
$810$	0	0
$811$	−40.0000	−1.40459	−0.702295	−	0.711886i	$-0.747841\pi$
$811$	−40.0000	−1.40459	−0.702295	+	0.711886i	$0.747841\pi$
$812$	3.00000i	0.105279i
$813$	0	0
$814$	40.0000	1.40200
$815$	0	0
$816$	0	0
$817$	− 78.0000i	− 2.72887i
$818$	0	0
$819$	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
$823$	− 4.00000i	− 0.139431i	−0.997567	−	0.0697156i	$-0.977791\pi$
$823$	− 4.00000i	− 0.139431i	0.997567	−	0.0697156i	$-0.0222092\pi$
$824$	−11.0000	−0.383203
$825$	0	0
$826$	11.0000	0.382739
$827$	34.0000i	1.18230i	0.806563	+	0.591148i	$0.201325\pi$
$827$	34.0000i	1.18230i	−0.806563	+	0.591148i	$0.798675\pi$
$828$	0	0
$829$	25.0000	0.868286	0.434143	−	0.900844i	$-0.357051\pi$
$829$	25.0000	0.868286	0.434143	+	0.900844i	$0.357051\pi$
$830$	0	0
$831$	0	0
$832$	− 3.00000i	− 0.104006i
$833$	− 7.00000i	− 0.242536i
$834$	0	0
$835$	0	0
$836$	24.0000	0.830057
$837$	0	0
$838$	9.00000i	0.310900i
$839$	32.0000	1.10476	0.552381	−	0.833592i	$-0.313719\pi$
$839$	32.0000	1.10476	0.552381	+	0.833592i	$0.313719\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	− 8.00000i	− 0.275698i
$843$	0	0
$844$	−13.0000	−0.447478
$845$	0	0
$846$	0	0
$847$	− 5.00000i	− 0.171802i
$848$	1.00000i	0.0343401i
$849$	0	0
$850$	0	0
$851$	−90.0000	−3.08516
$852$	0	0
$853$	13.0000i	0.445112i	0.974920	+	0.222556i	$0.0714399\pi$
$853$	13.0000i	0.445112i	−0.974920	+	0.222556i	$0.928560\pi$
$854$	−13.0000	−0.444851
$855$	0	0
$856$	6.00000	0.205076
$857$	− 6.00000i	− 0.204956i	−0.994735	−	0.102478i	$-0.967323\pi$
$857$	− 6.00000i	− 0.204956i	0.994735	−	0.102478i	$-0.0326771\pi$
$858$	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
$862$	− 21.0000i	− 0.715263i
$863$	28.0000i	0.953131i	0.879139	+	0.476566i	$0.158119\pi$
$863$	28.0000i	0.953131i	−0.879139	+	0.476566i	$0.841881\pi$
$864$	0	0
$865$	0	0
$866$	−2.00000	−0.0679628
$867$	0	0
$868$	− 7.00000i	− 0.237595i
$869$	16.0000	0.542763
$870$	0	0
$871$	0	0
$872$	6.00000i	0.203186i
$873$	0	0
$874$	−54.0000	−1.82658
$875$	0	0
$876$	0	0
$877$	8.00000i	0.270141i	0.990836	+	0.135070i	$0.0431261\pi$
$877$	8.00000i	0.270141i	−0.990836	+	0.135070i	$0.956874\pi$
$878$	− 35.0000i	− 1.18119i
$879$	0	0
$880$	0	0
$881$	13.0000	0.437981	0.218991	−	0.975727i	$-0.429724\pi$
$881$	13.0000	0.437981	0.218991	+	0.975727i	$0.429724\pi$
$882$	0	0
$883$	21.0000i	0.706706i	0.935490	+	0.353353i	$0.114959\pi$
$883$	21.0000i	0.706706i	−0.935490	+	0.353353i	$0.885041\pi$
$884$	21.0000	0.706306
$885$	0	0
$886$	−2.00000	−0.0671913
$887$	− 38.0000i	− 1.27592i	−0.770072	−	0.637958i	$-0.779780\pi$
$887$	− 38.0000i	− 1.27592i	0.770072	−	0.637958i	$-0.220220\pi$
$888$	0	0
$889$	−14.0000	−0.469545
$890$	0	0
$891$	0	0
$892$	13.0000i	0.435272i
$893$	− 12.0000i	− 0.401565i
$894$	0	0
$895$	0	0
$896$	1.00000	0.0334077
$897$	0	0
$898$	− 30.0000i	− 1.00111i
$899$	−21.0000	−0.700389
$900$	0	0
$901$	−7.00000	−0.233204
$902$	4.00000i	0.133185i
$903$	0	0
$904$	16.0000	0.532152
$905$	0	0
$906$	0	0
$907$	47.0000i	1.56061i	0.625400	+	0.780305i	$0.284936\pi$
$907$	47.0000i	1.56061i	−0.625400	+	0.780305i	$0.715064\pi$
$908$	11.0000i	0.365048i
$909$	0	0
$910$	0	0
$911$	−21.0000	−0.695761	−0.347881	−	0.937539i	$-0.613099\pi$
$911$	−21.0000	−0.695761	−0.347881	+	0.937539i	$0.613099\pi$
$912$	0	0
$913$	28.0000i	0.926665i
$914$	−17.0000	−0.562310
$915$	0	0
$916$	22.0000	0.726900
$917$	20.0000i	0.660458i
$918$	0	0
$919$	10.0000	0.329870	0.164935	−	0.986304i	$-0.447259\pi$
$919$	10.0000	0.329870	0.164935	+	0.986304i	$0.447259\pi$
$920$	0	0
$921$	0	0
$922$	− 14.0000i	− 0.461065i
$923$	− 24.0000i	− 0.789970i
$924$	0	0
$925$	0	0
$926$	16.0000	0.525793
$927$	0	0
$928$	− 3.00000i	− 0.0984798i
$929$	31.0000	1.01708	0.508539	−	0.861039i	$-0.330186\pi$
$929$	31.0000	1.01708	0.508539	+	0.861039i	$0.330186\pi$
$930$	0	0
$931$	−6.00000	−0.196642
$932$	18.0000i	0.589610i
$933$	0	0
$934$	−5.00000	−0.163605
$935$	0	0
$936$	0	0
$937$	38.0000i	1.24141i	0.784046	+	0.620703i	$0.213153\pi$
$937$	38.0000i	1.24141i	−0.784046	+	0.620703i	$0.786847\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	40.0000	1.30396	0.651981	−	0.758235i	$-0.273938\pi$
$941$	40.0000	1.30396	0.651981	+	0.758235i	$0.273938\pi$
$942$	0	0
$943$	− 9.00000i	− 0.293080i
$944$	−11.0000	−0.358020
$945$	0	0
$946$	52.0000	1.69067
$947$	− 2.00000i	− 0.0649913i	−0.999472	−	0.0324956i	$-0.989654\pi$
$947$	− 2.00000i	− 0.0649913i	0.999472	−	0.0324956i	$-0.0103455\pi$
$948$	0	0
$949$	24.0000	0.779073
$950$	0	0
$951$	0	0
$952$	7.00000i	0.226871i
$953$	12.0000i	0.388718i	0.980930	+	0.194359i	$0.0622627\pi$
$953$	12.0000i	0.388718i	−0.980930	+	0.194359i	$0.937737\pi$
$954$	0	0
$955$	0	0
$956$	8.00000	0.258738
$957$	0	0
$958$	30.0000i	0.969256i
$959$	12.0000	0.387500
$960$	0	0
$961$	18.0000	0.580645
$962$	− 30.0000i	− 0.967239i
$963$	0	0
$964$	−22.0000	−0.708572
$965$	0	0
$966$	0	0
$967$	− 26.0000i	− 0.836104i	−0.908423	−	0.418052i	$-0.862713\pi$
$967$	− 26.0000i	− 0.836104i	0.908423	−	0.418052i	$-0.137287\pi$
$968$	5.00000i	0.160706i
$969$	0	0
$970$	0	0
$971$	36.0000	1.15529	0.577647	−	0.816286i	$-0.303971\pi$
$971$	36.0000	1.15529	0.577647	+	0.816286i	$0.303971\pi$
$972$	0	0
$973$	− 8.00000i	− 0.256468i
$974$	−4.00000	−0.128168
$975$	0	0
$976$	13.0000	0.416120
$977$	− 30.0000i	− 0.959785i	−0.877327	−	0.479893i	$-0.840676\pi$
$977$	− 30.0000i	− 0.959785i	0.877327	−	0.479893i	$-0.159324\pi$
$978$	0	0
$979$	56.0000	1.78977
$980$	0	0
$981$	0	0
$982$	6.00000i	0.191468i
$983$	− 4.00000i	− 0.127580i	−0.997963	−	0.0637901i	$-0.979681\pi$
$983$	− 4.00000i	− 0.127580i	0.997963	−	0.0637901i	$-0.0203188\pi$
$984$	0	0
$985$	0	0
$986$	21.0000	0.668776
$987$	0	0
$988$	− 18.0000i	− 0.572656i
$989$	−117.000	−3.72038
$990$	0	0
$991$	−26.0000	−0.825917	−0.412959	−	0.910750i	$-0.635505\pi$
$991$	−26.0000	−0.825917	−0.412959	+	0.910750i	$0.635505\pi$
$992$	7.00000i	0.222250i
$993$	0	0
$994$	8.00000	0.253745
$995$	0	0
$996$	0	0
$997$	14.0000i	0.443384i	0.975117	+	0.221692i	$0.0711580\pi$
$997$	14.0000i	0.443384i	−0.975117	+	0.221692i	$0.928842\pi$
$998$	− 21.0000i	− 0.664743i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3150.2.g.d.2899.1		2
3.2	odd	2		3150.2.g.u.2899.2		2
5.2	odd	4		3150.2.a.bi.1.1	yes	1
5.3	odd	4		3150.2.a.b.1.1	✓	1
5.4	even	2	inner	3150.2.g.d.2899.2		2
15.2	even	4		3150.2.a.u.1.1	yes	1
15.8	even	4		3150.2.a.bf.1.1	yes	1
15.14	odd	2		3150.2.g.u.2899.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3150.2.a.b.1.1	✓	1	5.3	odd	4
3150.2.a.u.1.1	yes	1	15.2	even	4
3150.2.a.bf.1.1	yes	1	15.8	even	4
3150.2.a.bi.1.1	yes	1	5.2	odd	4
3150.2.g.d.2899.1		2	1.1	even	1	trivial
3150.2.g.d.2899.2		2	5.4	even	2	inner
3150.2.g.u.2899.1		2	15.14	odd	2
3150.2.g.u.2899.2		2	3.2	odd	2

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:	\( N \)	\(=\)	\( 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3150.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} -1.00000i q^{7} +1.00000i q^{8} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000 q^{4} -1.00000i q^{7} +1.00000i q^{8} -4.00000 q^{11} +3.00000i q^{13} -1.00000 q^{14} +1.00000 q^{16} +7.00000i q^{17} +6.00000 q^{19} +4.00000i q^{22} -9.00000i q^{23} +3.00000 q^{26} +1.00000i q^{28} +3.00000 q^{29} -7.00000 q^{31} -1.00000i q^{32} +7.00000 q^{34} -10.0000i q^{37} -6.00000i q^{38} +1.00000 q^{41} -13.0000i q^{43} +4.00000 q^{44} -9.00000 q^{46} -2.00000i q^{47} -1.00000 q^{49} -3.00000i q^{52} +1.00000i q^{53} +1.00000 q^{56} -3.00000i q^{58} -11.0000 q^{59} +13.0000 q^{61} +7.00000i q^{62} -1.00000 q^{64} -7.00000i q^{68} -8.00000 q^{71} -8.00000i q^{73} -10.0000 q^{74} -6.00000 q^{76} +4.00000i q^{77} -4.00000 q^{79} -1.00000i q^{82} -7.00000i q^{83} -13.0000 q^{86} -4.00000i q^{88} -14.0000 q^{89} +3.00000 q^{91} +9.00000i q^{92} -2.00000 q^{94} +8.00000i q^{97} +1.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4}+O(q^{10}) \) 2 * q - 2 * q^4	\( 2 q - 2 q^{4} - 8 q^{11} - 2 q^{14} + 2 q^{16} + 12 q^{19} + 6 q^{26} + 6 q^{29} - 14 q^{31} + 14 q^{34} + 2 q^{41} + 8 q^{44} - 18 q^{46} - 2 q^{49} + 2 q^{56} - 22 q^{59} + 26 q^{61} - 2 q^{64} - 16 q^{71} - 20 q^{74} - 12 q^{76} - 8 q^{79} - 26 q^{86} - 28 q^{89} + 6 q^{91} - 4 q^{94}+O(q^{100}) \) 2 * q - 2 * q^4 - 8 * q^11 - 2 * q^14 + 2 * q^16 + 12 * q^19 + 6 * q^26 + 6 * q^29 - 14 * q^31 + 14 * q^34 + 2 * q^41 + 8 * q^44 - 18 * q^46 - 2 * q^49 + 2 * q^56 - 22 * q^59 + 26 * q^61 - 2 * q^64 - 16 * q^71 - 20 * q^74 - 12 * q^76 - 8 * q^79 - 26 * q^86 - 28 * q^89 + 6 * q^91 - 4 * q^94

Introduction

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