LMFDB - Embedded newform 4950.2.c.bc.199.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4950,2,Mod(199,4950)]

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

lf = mfeigenbasis(mf)

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")

//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);

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: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$39.5259490005$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{33})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 17x^{2} + 64 $ x^4 + 17*x^2 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 110)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.1
Root			$-3.37228i$ of defining polynomial
Character	$\chi$	$=$	4950.199
Dual form			4950.2.c.bc.199.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{4} -3.37228i q^{7} +1.00000i q^{8} +1.00000 q^{11} +2.00000i q^{13} -3.37228 q^{14} +1.00000 q^{16} +1.37228i q^{17} -0.627719 q^{19} -1.00000i q^{22} -2.74456i q^{23} +2.00000 q^{26} +3.37228i q^{28} +1.37228 q^{29} +3.37228 q^{31} -1.00000i q^{32} +1.37228 q^{34} -9.37228i q^{37} +0.627719i q^{38} +11.4891 q^{41} -4.00000i q^{43} -1.00000 q^{44} -2.74456 q^{46} +2.74456i q^{47} -4.37228 q^{49} -2.00000i q^{52} +4.11684i q^{53} +3.37228 q^{56} -1.37228i q^{58} -2.74456 q^{59} -5.37228 q^{61} -3.37228i q^{62} -1.00000 q^{64} -8.00000i q^{67} -1.37228i q^{68} -10.1168 q^{71} -15.4891i q^{73} -9.37228 q^{74} +0.627719 q^{76} -3.37228i q^{77} +1.25544 q^{79} -11.4891i q^{82} +2.74456i q^{83} -4.00000 q^{86} +1.00000i q^{88} -1.37228 q^{89} +6.74456 q^{91} +2.74456i q^{92} +2.74456 q^{94} +12.7446i q^{97} +4.37228i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4} + 4 q^{11} - 2 q^{14} + 4 q^{16} - 14 q^{19} + 8 q^{26} - 6 q^{29} + 2 q^{31} - 6 q^{34} - 4 q^{44} + 12 q^{46} - 6 q^{49} + 2 q^{56} + 12 q^{59} - 10 q^{61} - 4 q^{64} - 6 q^{71} - 26 q^{74}+ \cdots - 12 q^{94}+O(q^{100}) $ 4 * q - 4 * q^4 + 4 * q^11 - 2 * q^14 + 4 * q^16 - 14 * q^19 + 8 * q^26 - 6 * q^29 + 2 * q^31 - 6 * q^34 - 4 * q^44 + 12 * q^46 - 6 * q^49 + 2 * q^56 + 12 * q^59 - 10 * q^61 - 4 * q^64 - 6 * q^71 - 26 * q^74 + 14 * q^76 + 28 * q^79 - 16 * q^86 + 6 * q^89 + 4 * q^91 - 12 * 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
$7$	− 3.37228i	− 1.27460i	−0.770615	−	0.637301i	$-0.780051\pi$
$7$	− 3.37228i	− 1.27460i	0.770615	−	0.637301i	$-0.219949\pi$
$8$	1.00000i	0.353553i
$9$	0	0
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	2.00000i	0.554700i	0.960769	+	0.277350i	$0.0894562\pi$
$13$	2.00000i	0.554700i	−0.960769	+	0.277350i	$0.910544\pi$
$14$	−3.37228	−0.901280
$15$	0	0
$16$	1.00000	0.250000
$17$	1.37228i	0.332827i	0.986056	+	0.166414i	$0.0532187\pi$
$17$	1.37228i	0.332827i	−0.986056	+	0.166414i	$0.946781\pi$
$18$	0	0
$19$	−0.627719	−0.144009	−0.0720043	−	0.997404i	$-0.522940\pi$
$19$	−0.627719	−0.144009	−0.0720043	+	0.997404i	$0.522940\pi$
$20$	0	0
$21$	0	0
$22$	− 1.00000i	− 0.213201i
$23$	− 2.74456i	− 0.572281i	−0.958188	−	0.286140i	$-0.907628\pi$
$23$	− 2.74456i	− 0.572281i	0.958188	−	0.286140i	$-0.0923724\pi$
$24$	0	0
$25$	0	0
$26$	2.00000	0.392232
$27$	0	0
$28$	3.37228i	0.637301i
$29$	1.37228	0.254826	0.127413	−	0.991850i	$-0.459333\pi$
$29$	1.37228	0.254826	0.127413	+	0.991850i	$0.459333\pi$
$30$	0	0
$31$	3.37228	0.605680	0.302840	−	0.953041i	$-0.402065\pi$
$31$	3.37228	0.605680	0.302840	+	0.953041i	$0.402065\pi$
$32$	− 1.00000i	− 0.176777i
$33$	0	0
$34$	1.37228	0.235344
$35$	0	0
$36$	0	0
$37$	− 9.37228i	− 1.54079i	−0.637565	−	0.770397i	$-0.720058\pi$
$37$	− 9.37228i	− 1.54079i	0.637565	−	0.770397i	$-0.279942\pi$
$38$	0.627719i	0.101829i
$39$	0	0
$40$	0	0
$41$	11.4891	1.79430	0.897150	−	0.441726i	$-0.145634\pi$
$41$	11.4891	1.79430	0.897150	+	0.441726i	$0.145634\pi$
$42$	0	0
$43$	− 4.00000i	− 0.609994i	−0.952353	−	0.304997i	$-0.901344\pi$
$43$	− 4.00000i	− 0.609994i	0.952353	−	0.304997i	$-0.0986555\pi$
$44$	−1.00000	−0.150756
$45$	0	0
$46$	−2.74456	−0.404664
$47$	2.74456i	0.400336i	0.979762	+	0.200168i	$0.0641487\pi$
$47$	2.74456i	0.400336i	−0.979762	+	0.200168i	$0.935851\pi$
$48$	0	0
$49$	−4.37228	−0.624612
$50$	0	0
$51$	0	0
$52$	− 2.00000i	− 0.277350i
$53$	4.11684i	0.565492i	0.959195	+	0.282746i	$0.0912454\pi$
$53$	4.11684i	0.565492i	−0.959195	+	0.282746i	$0.908755\pi$
$54$	0	0
$55$	0	0
$56$	3.37228	0.450640
$57$	0	0
$58$	− 1.37228i	− 0.180189i
$59$	−2.74456	−0.357312	−0.178656	−	0.983912i	$-0.557175\pi$
$59$	−2.74456	−0.357312	−0.178656	+	0.983912i	$0.557175\pi$
$60$	0	0
$61$	−5.37228	−0.687850	−0.343925	−	0.938997i	$-0.611757\pi$
$61$	−5.37228	−0.687850	−0.343925	+	0.938997i	$0.611757\pi$
$62$	− 3.37228i	− 0.428280i
$63$	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	$-0.162521\pi$
$68$	− 1.37228i	− 0.166414i
$69$	0	0
$70$	0	0
$71$	−10.1168	−1.20065	−0.600324	−	0.799757i	$-0.704962\pi$
$71$	−10.1168	−1.20065	−0.600324	+	0.799757i	$0.704962\pi$
$72$	0	0
$73$	− 15.4891i	− 1.81286i	−0.422351	−	0.906432i	$-0.638795\pi$
$73$	− 15.4891i	− 1.81286i	0.422351	−	0.906432i	$-0.361205\pi$
$74$	−9.37228	−1.08951
$75$	0	0
$76$	0.627719	0.0720043
$77$	− 3.37228i	− 0.384307i
$78$	0	0
$79$	1.25544	0.141248	0.0706239	−	0.997503i	$-0.477501\pi$
$79$	1.25544	0.141248	0.0706239	+	0.997503i	$0.477501\pi$
$80$	0	0
$81$	0	0
$82$	− 11.4891i	− 1.26876i
$83$	2.74456i	0.301255i	0.988591	+	0.150627i	$0.0481294\pi$
$83$	2.74456i	0.301255i	−0.988591	+	0.150627i	$0.951871\pi$
$84$	0	0
$85$	0	0
$86$	−4.00000	−0.431331
$87$	0	0
$88$	1.00000i	0.106600i
$89$	−1.37228	−0.145462	−0.0727308	−	0.997352i	$-0.523171\pi$
$89$	−1.37228	−0.145462	−0.0727308	+	0.997352i	$0.523171\pi$
$90$	0	0
$91$	6.74456	0.707022
$92$	2.74456i	0.286140i
$93$	0	0
$94$	2.74456	0.283080
$95$	0	0
$96$	0	0
$97$	12.7446i	1.29401i	0.762484	+	0.647007i	$0.223980\pi$
$97$	12.7446i	1.29401i	−0.762484	+	0.647007i	$0.776020\pi$
$98$	4.37228i	0.441667i
$99$	0	0
$100$	0	0
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	0	0
$103$	− 9.48913i	− 0.934991i	−0.883995	−	0.467496i	$-0.845156\pi$
$103$	− 9.48913i	− 0.934991i	0.883995	−	0.467496i	$-0.154844\pi$
$104$	−2.00000	−0.196116
$105$	0	0
$106$	4.11684	0.399863
$107$	− 12.0000i	− 1.16008i	−0.814587	−	0.580042i	$-0.803036\pi$
$107$	− 12.0000i	− 1.16008i	0.814587	−	0.580042i	$-0.196964\pi$
$108$	0	0
$109$	15.4891	1.48359	0.741795	−	0.670627i	$-0.233975\pi$
$109$	15.4891	1.48359	0.741795	+	0.670627i	$0.233975\pi$
$110$	0	0
$111$	0	0
$112$	− 3.37228i	− 0.318651i
$113$	− 3.25544i	− 0.306246i	−0.988207	−	0.153123i	$-0.951067\pi$
$113$	− 3.25544i	− 0.306246i	0.988207	−	0.153123i	$-0.0489330\pi$
$114$	0	0
$115$	0	0
$116$	−1.37228	−0.127413
$117$	0	0
$118$	2.74456i	0.252657i
$119$	4.62772	0.424222
$120$	0	0
$121$	1.00000	0.0909091
$122$	5.37228i	0.486383i
$123$	0	0
$124$	−3.37228	−0.302840
$125$	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
$131$	−22.1168	−1.93236	−0.966179	−	0.257873i	$-0.916978\pi$
$131$	−22.1168	−1.93236	−0.966179	+	0.257873i	$0.916978\pi$
$132$	0	0
$133$	2.11684i	0.183554i
$134$	−8.00000	−0.691095
$135$	0	0
$136$	−1.37228	−0.117672
$137$	8.74456i	0.747098i	0.927610	+	0.373549i	$0.121859\pi$
$137$	8.74456i	0.747098i	−0.927610	+	0.373549i	$0.878141\pi$
$138$	0	0
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	0	0
$141$	0	0
$142$	10.1168i	0.848987i
$143$	2.00000i	0.167248i
$144$	0	0
$145$	0	0
$146$	−15.4891	−1.28189
$147$	0	0
$148$	9.37228i	0.770397i
$149$	21.6060	1.77003	0.885015	−	0.465563i	$-0.154148\pi$
$149$	21.6060	1.77003	0.885015	+	0.465563i	$0.154148\pi$
$150$	0	0
$151$	−12.2337	−0.995563	−0.497782	−	0.867302i	$-0.665852\pi$
$151$	−12.2337	−0.995563	−0.497782	+	0.867302i	$0.665852\pi$
$152$	− 0.627719i	− 0.0509147i
$153$	0	0
$154$	−3.37228	−0.271746
$155$	0	0
$156$	0	0
$157$	− 9.37228i	− 0.747989i	−0.927431	−	0.373995i	$-0.877988\pi$
$157$	− 9.37228i	− 0.747989i	0.927431	−	0.373995i	$-0.122012\pi$
$158$	− 1.25544i	− 0.0998772i
$159$	0	0
$160$	0	0
$161$	−9.25544	−0.729431
$162$	0	0
$163$	− 5.88316i	− 0.460804i	−0.973095	−	0.230402i	$-0.925996\pi$
$163$	− 5.88316i	− 0.460804i	0.973095	−	0.230402i	$-0.0740042\pi$
$164$	−11.4891	−0.897150
$165$	0	0
$166$	2.74456	0.213019
$167$	− 4.62772i	− 0.358104i	−0.983840	−	0.179052i	$-0.942697\pi$
$167$	− 4.62772i	− 0.358104i	0.983840	−	0.179052i	$-0.0573030\pi$
$168$	0	0
$169$	9.00000	0.692308
$170$	0	0
$171$	0	0
$172$	4.00000i	0.304997i
$173$	6.00000i	0.456172i	0.973641	+	0.228086i	$0.0732467\pi$
$173$	6.00000i	0.456172i	−0.973641	+	0.228086i	$0.926753\pi$
$174$	0	0
$175$	0	0
$176$	1.00000	0.0753778
$177$	0	0
$178$	1.37228i	0.102857i
$179$	−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	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	− 6.74456i	− 0.499940i
$183$	0	0
$184$	2.74456	0.202332
$185$	0	0
$186$	0	0
$187$	1.37228i	0.100351i
$188$	− 2.74456i	− 0.200168i
$189$	0	0
$190$	0	0
$191$	5.48913	0.397179	0.198590	−	0.980083i	$-0.436364\pi$
$191$	5.48913	0.397179	0.198590	+	0.980083i	$0.436364\pi$
$192$	0	0
$193$	14.8614i	1.06975i	0.844932	+	0.534874i	$0.179641\pi$
$193$	14.8614i	1.06975i	−0.844932	+	0.534874i	$0.820359\pi$
$194$	12.7446	0.915006
$195$	0	0
$196$	4.37228	0.312306
$197$	− 20.7446i	− 1.47799i	−0.673712	−	0.738994i	$-0.735301\pi$
$197$	− 20.7446i	− 1.47799i	0.673712	−	0.738994i	$-0.264699\pi$
$198$	0	0
$199$	−18.1168	−1.28427	−0.642135	−	0.766592i	$-0.721951\pi$
$199$	−18.1168	−1.28427	−0.642135	+	0.766592i	$0.721951\pi$
$200$	0	0
$201$	0	0
$202$	6.00000i	0.422159i
$203$	− 4.62772i	− 0.324802i
$204$	0	0
$205$	0	0
$206$	−9.48913	−0.661139
$207$	0	0
$208$	2.00000i	0.138675i
$209$	−0.627719	−0.0434202
$210$	0	0
$211$	6.11684	0.421101	0.210550	−	0.977583i	$-0.432474\pi$
$211$	6.11684	0.421101	0.210550	+	0.977583i	$0.432474\pi$
$212$	− 4.11684i	− 0.282746i
$213$	0	0
$214$	−12.0000	−0.820303
$215$	0	0
$216$	0	0
$217$	− 11.3723i	− 0.772001i
$218$	− 15.4891i	− 1.04906i
$219$	0	0
$220$	0	0
$221$	−2.74456	−0.184619
$222$	0	0
$223$	− 18.7446i	− 1.25523i	−0.778524	−	0.627614i	$-0.784031\pi$
$223$	− 18.7446i	− 1.25523i	0.778524	−	0.627614i	$-0.215969\pi$
$224$	−3.37228	−0.225320
$225$	0	0
$226$	−3.25544	−0.216548
$227$	− 2.74456i	− 0.182163i	−0.995843	−	0.0910815i	$-0.970968\pi$
$227$	− 2.74456i	− 0.182163i	0.995843	−	0.0910815i	$-0.0290324\pi$
$228$	0	0
$229$	10.0000	0.660819	0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000	0.660819	0.330409	+	0.943838i	$0.392813\pi$
$230$	0	0
$231$	0	0
$232$	1.37228i	0.0900947i
$233$	− 1.37228i	− 0.0899011i	−0.998989	−	0.0449506i	$-0.985687\pi$
$233$	− 1.37228i	− 0.0899011i	0.998989	−	0.0449506i	$-0.0143130\pi$
$234$	0	0
$235$	0	0
$236$	2.74456	0.178656
$237$	0	0
$238$	− 4.62772i	− 0.299970i
$239$	−14.7446	−0.953746	−0.476873	−	0.878972i	$-0.658230\pi$
$239$	−14.7446	−0.953746	−0.476873	+	0.878972i	$0.658230\pi$
$240$	0	0
$241$	−22.0000	−1.41714	−0.708572	−	0.705638i	$-0.750660\pi$
$241$	−22.0000	−1.41714	−0.708572	+	0.705638i	$0.750660\pi$
$242$	− 1.00000i	− 0.0642824i
$243$	0	0
$244$	5.37228	0.343925
$245$	0	0
$246$	0	0
$247$	− 1.25544i	− 0.0798816i
$248$	3.37228i	0.214140i
$249$	0	0
$250$	0	0
$251$	−2.74456	−0.173235	−0.0866176	−	0.996242i	$-0.527606\pi$
$251$	−2.74456	−0.173235	−0.0866176	+	0.996242i	$0.527606\pi$
$252$	0	0
$253$	− 2.74456i	− 0.172549i
$254$	−8.00000	−0.501965
$255$	0	0
$256$	1.00000	0.0625000
$257$	18.0000i	1.12281i	0.827541	+	0.561405i	$0.189739\pi$
$257$	18.0000i	1.12281i	−0.827541	+	0.561405i	$0.810261\pi$
$258$	0	0
$259$	−31.6060	−1.96390
$260$	0	0
$261$	0	0
$262$	22.1168i	1.36638i
$263$	24.8614i	1.53302i	0.642232	+	0.766510i	$0.278008\pi$
$263$	24.8614i	1.53302i	−0.642232	+	0.766510i	$0.721992\pi$
$264$	0	0
$265$	0	0
$266$	2.11684	0.129792
$267$	0	0
$268$	8.00000i	0.488678i
$269$	−8.74456	−0.533165	−0.266583	−	0.963812i	$-0.585895\pi$
$269$	−8.74456	−0.533165	−0.266583	+	0.963812i	$0.585895\pi$
$270$	0	0
$271$	−16.0000	−0.971931	−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000	−0.971931	−0.485965	+	0.873978i	$0.661532\pi$
$272$	1.37228i	0.0832068i
$273$	0	0
$274$	8.74456	0.528278
$275$	0	0
$276$	0	0
$277$	12.7446i	0.765747i	0.923801	+	0.382873i	$0.125065\pi$
$277$	12.7446i	0.765747i	−0.923801	+	0.382873i	$0.874935\pi$
$278$	− 4.00000i	− 0.239904i
$279$	0	0
$280$	0	0
$281$	−18.0000	−1.07379	−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000	−1.07379	−0.536895	+	0.843649i	$0.680403\pi$
$282$	0	0
$283$	5.25544i	0.312403i	0.987725	+	0.156202i	$0.0499250\pi$
$283$	5.25544i	0.312403i	−0.987725	+	0.156202i	$0.950075\pi$
$284$	10.1168	0.600324
$285$	0	0
$286$	2.00000	0.118262
$287$	− 38.7446i	− 2.28702i
$288$	0	0
$289$	15.1168	0.889226
$290$	0	0
$291$	0	0
$292$	15.4891i	0.906432i
$293$	− 23.4891i	− 1.37225i	−0.727484	−	0.686125i	$-0.759310\pi$
$293$	− 23.4891i	− 1.37225i	0.727484	−	0.686125i	$-0.240690\pi$
$294$	0	0
$295$	0	0
$296$	9.37228	0.544753
$297$	0	0
$298$	− 21.6060i	− 1.25160i
$299$	5.48913	0.317444
$300$	0	0
$301$	−13.4891	−0.777500
$302$	12.2337i	0.703970i
$303$	0	0
$304$	−0.627719	−0.0360021
$305$	0	0
$306$	0	0
$307$	− 5.25544i	− 0.299944i	−0.988690	−	0.149972i	$-0.952082\pi$
$307$	− 5.25544i	− 0.299944i	0.988690	−	0.149972i	$-0.0479183\pi$
$308$	3.37228i	0.192154i
$309$	0	0
$310$	0	0
$311$	−19.3723	−1.09850	−0.549251	−	0.835658i	$-0.685087\pi$
$311$	−19.3723	−1.09850	−0.549251	+	0.835658i	$0.685087\pi$
$312$	0	0
$313$	− 22.0000i	− 1.24351i	−0.783210	−	0.621757i	$-0.786419\pi$
$313$	− 22.0000i	− 1.24351i	0.783210	−	0.621757i	$-0.213581\pi$
$314$	−9.37228	−0.528908
$315$	0	0
$316$	−1.25544	−0.0706239
$317$	− 24.3505i	− 1.36766i	−0.729640	−	0.683831i	$-0.760313\pi$
$317$	− 24.3505i	− 1.36766i	0.729640	−	0.683831i	$-0.239687\pi$
$318$	0	0
$319$	1.37228	0.0768330
$320$	0	0
$321$	0	0
$322$	9.25544i	0.515785i
$323$	− 0.861407i	− 0.0479299i
$324$	0	0
$325$	0	0
$326$	−5.88316	−0.325838
$327$	0	0
$328$	11.4891i	0.634381i
$329$	9.25544	0.510269
$330$	0	0
$331$	30.9783	1.70272	0.851359	−	0.524583i	$-0.175779\pi$
$331$	30.9783	1.70272	0.851359	+	0.524583i	$0.175779\pi$
$332$	− 2.74456i	− 0.150627i
$333$	0	0
$334$	−4.62772	−0.253217
$335$	0	0
$336$	0	0
$337$	− 24.1168i	− 1.31373i	−0.754009	−	0.656864i	$-0.771883\pi$
$337$	− 24.1168i	− 1.31373i	0.754009	−	0.656864i	$-0.228117\pi$
$338$	− 9.00000i	− 0.489535i
$339$	0	0
$340$	0	0
$341$	3.37228	0.182619
$342$	0	0
$343$	− 8.86141i	− 0.478471i
$344$	4.00000	0.215666
$345$	0	0
$346$	6.00000	0.322562
$347$	− 32.2337i	− 1.73040i	−0.501431	−	0.865198i	$-0.667193\pi$
$347$	− 32.2337i	− 1.73040i	0.501431	−	0.865198i	$-0.332807\pi$
$348$	0	0
$349$	−19.4891	−1.04323	−0.521614	−	0.853181i	$-0.674670\pi$
$349$	−19.4891	−1.04323	−0.521614	+	0.853181i	$0.674670\pi$
$350$	0	0
$351$	0	0
$352$	− 1.00000i	− 0.0533002i
$353$	0.510875i	0.0271911i	0.999908	+	0.0135956i	$0.00432773\pi$
$353$	0.510875i	0.0271911i	−0.999908	+	0.0135956i	$0.995672\pi$
$354$	0	0
$355$	0	0
$356$	1.37228	0.0727308
$357$	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
$361$	−18.6060	−0.979262
$362$	10.0000i	0.525588i
$363$	0	0
$364$	−6.74456	−0.353511
$365$	0	0
$366$	0	0
$367$	4.00000i	0.208798i	0.994535	+	0.104399i	$0.0332919\pi$
$367$	4.00000i	0.208798i	−0.994535	+	0.104399i	$0.966708\pi$
$368$	− 2.74456i	− 0.143070i
$369$	0	0
$370$	0	0
$371$	13.8832	0.720778
$372$	0	0
$373$	31.4891i	1.63045i	0.579148	+	0.815223i	$0.303385\pi$
$373$	31.4891i	1.63045i	−0.579148	+	0.815223i	$0.696615\pi$
$374$	1.37228	0.0709590
$375$	0	0
$376$	−2.74456	−0.141540
$377$	2.74456i	0.141352i
$378$	0	0
$379$	0.233688	0.0120037	0.00600187	−	0.999982i	$-0.498090\pi$
$379$	0.233688	0.0120037	0.00600187	+	0.999982i	$0.498090\pi$
$380$	0	0
$381$	0	0
$382$	− 5.48913i	− 0.280848i
$383$	− 32.2337i	− 1.64706i	−0.567269	−	0.823532i	$-0.692000\pi$
$383$	− 32.2337i	− 1.64706i	0.567269	−	0.823532i	$-0.308000\pi$
$384$	0	0
$385$	0	0
$386$	14.8614	0.756426
$387$	0	0
$388$	− 12.7446i	− 0.647007i
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	0	0
$391$	3.76631	0.190471
$392$	− 4.37228i	− 0.220834i
$393$	0	0
$394$	−20.7446	−1.04510
$395$	0	0
$396$	0	0
$397$	− 24.9783i	− 1.25362i	−0.779171	−	0.626811i	$-0.784360\pi$
$397$	− 24.9783i	− 1.25362i	0.779171	−	0.626811i	$-0.215640\pi$
$398$	18.1168i	0.908115i
$399$	0	0
$400$	0	0
$401$	−13.3723	−0.667780	−0.333890	−	0.942612i	$-0.608361\pi$
$401$	−13.3723	−0.667780	−0.333890	+	0.942612i	$0.608361\pi$
$402$	0	0
$403$	6.74456i	0.335971i
$404$	6.00000	0.298511
$405$	0	0
$406$	−4.62772	−0.229670
$407$	− 9.37228i	− 0.464567i
$408$	0	0
$409$	1.76631	0.0873385	0.0436693	−	0.999046i	$-0.486095\pi$
$409$	1.76631	0.0873385	0.0436693	+	0.999046i	$0.486095\pi$
$410$	0	0
$411$	0	0
$412$	9.48913i	0.467496i
$413$	9.25544i	0.455430i
$414$	0	0
$415$	0	0
$416$	2.00000	0.0980581
$417$	0	0
$418$	0.627719i	0.0307027i
$419$	−12.0000	−0.586238	−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000	−0.586238	−0.293119	+	0.956076i	$0.594693\pi$
$420$	0	0
$421$	10.2337	0.498759	0.249380	−	0.968406i	$-0.419773\pi$
$421$	10.2337	0.498759	0.249380	+	0.968406i	$0.419773\pi$
$422$	− 6.11684i	− 0.297763i
$423$	0	0
$424$	−4.11684	−0.199932
$425$	0	0
$426$	0	0
$427$	18.1168i	0.876736i
$428$	12.0000i	0.580042i
$429$	0	0
$430$	0	0
$431$	34.9783	1.68484	0.842422	−	0.538819i	$-0.181129\pi$
$431$	34.9783	1.68484	0.842422	+	0.538819i	$0.181129\pi$
$432$	0	0
$433$	27.7228i	1.33227i	0.745830	+	0.666137i	$0.232053\pi$
$433$	27.7228i	1.33227i	−0.745830	+	0.666137i	$0.767947\pi$
$434$	−11.3723	−0.545887
$435$	0	0
$436$	−15.4891	−0.741795
$437$	1.72281i	0.0824133i
$438$	0	0
$439$	−18.9783	−0.905782	−0.452891	−	0.891566i	$-0.649607\pi$
$439$	−18.9783	−0.905782	−0.452891	+	0.891566i	$0.649607\pi$
$440$	0	0
$441$	0	0
$442$	2.74456i	0.130546i
$443$	− 29.4891i	− 1.40107i	−0.713618	−	0.700535i	$-0.752945\pi$
$443$	− 29.4891i	− 1.40107i	0.713618	−	0.700535i	$-0.247055\pi$
$444$	0	0
$445$	0	0
$446$	−18.7446	−0.887581
$447$	0	0
$448$	3.37228i	0.159325i
$449$	28.9783	1.36757	0.683784	−	0.729684i	$-0.260333\pi$
$449$	28.9783	1.36757	0.683784	+	0.729684i	$0.260333\pi$
$450$	0	0
$451$	11.4891	0.541002
$452$	3.25544i	0.153123i
$453$	0	0
$454$	−2.74456	−0.128809
$455$	0	0
$456$	0	0
$457$	16.3505i	0.764846i	0.923987	+	0.382423i	$0.124910\pi$
$457$	16.3505i	0.764846i	−0.923987	+	0.382423i	$0.875090\pi$
$458$	− 10.0000i	− 0.467269i
$459$	0	0
$460$	0	0
$461$	−16.1168	−0.750636	−0.375318	−	0.926896i	$-0.622467\pi$
$461$	−16.1168	−0.750636	−0.375318	+	0.926896i	$0.622467\pi$
$462$	0	0
$463$	− 0.233688i	− 0.0108604i	−0.999985	−	0.00543020i	$-0.998272\pi$
$463$	− 0.233688i	− 0.0108604i	0.999985	−	0.00543020i	$-0.00172849\pi$
$464$	1.37228	0.0637066
$465$	0	0
$466$	−1.37228	−0.0635697
$467$	− 19.3723i	− 0.896442i	−0.893923	−	0.448221i	$-0.852058\pi$
$467$	− 19.3723i	− 0.896442i	0.893923	−	0.448221i	$-0.147942\pi$
$468$	0	0
$469$	−26.9783	−1.24574
$470$	0	0
$471$	0	0
$472$	− 2.74456i	− 0.126329i
$473$	− 4.00000i	− 0.183920i
$474$	0	0
$475$	0	0
$476$	−4.62772	−0.212111
$477$	0	0
$478$	14.7446i	0.674401i
$479$	5.48913	0.250805	0.125402	−	0.992106i	$-0.459978\pi$
$479$	5.48913	0.250805	0.125402	+	0.992106i	$0.459978\pi$
$480$	0	0
$481$	18.7446	0.854678
$482$	22.0000i	1.00207i
$483$	0	0
$484$	−1.00000	−0.0454545
$485$	0	0
$486$	0	0
$487$	− 20.0000i	− 0.906287i	−0.891438	−	0.453143i	$-0.850303\pi$
$487$	− 20.0000i	− 0.906287i	0.891438	−	0.453143i	$-0.149697\pi$
$488$	− 5.37228i	− 0.243192i
$489$	0	0
$490$	0	0
$491$	7.37228	0.332706	0.166353	−	0.986066i	$-0.446801\pi$
$491$	7.37228	0.332706	0.166353	+	0.986066i	$0.446801\pi$
$492$	0	0
$493$	1.88316i	0.0848131i
$494$	−1.25544	−0.0564848
$495$	0	0
$496$	3.37228	0.151420
$497$	34.1168i	1.53035i
$498$	0	0
$499$	33.4891	1.49918	0.749590	−	0.661903i	$-0.230251\pi$
$499$	33.4891	1.49918	0.749590	+	0.661903i	$0.230251\pi$
$500$	0	0
$501$	0	0
$502$	2.74456i	0.122496i
$503$	34.9783i	1.55960i	0.626027	+	0.779802i	$0.284680\pi$
$503$	34.9783i	1.55960i	−0.626027	+	0.779802i	$0.715320\pi$
$504$	0	0
$505$	0	0
$506$	−2.74456	−0.122011
$507$	0	0
$508$	8.00000i	0.354943i
$509$	9.76631	0.432884	0.216442	−	0.976295i	$-0.430555\pi$
$509$	9.76631	0.432884	0.216442	+	0.976295i	$0.430555\pi$
$510$	0	0
$511$	−52.2337	−2.31068
$512$	− 1.00000i	− 0.0441942i
$513$	0	0
$514$	18.0000	0.793946
$515$	0	0
$516$	0	0
$517$	2.74456i	0.120706i
$518$	31.6060i	1.38869i
$519$	0	0
$520$	0	0
$521$	−12.5109	−0.548111	−0.274056	−	0.961714i	$-0.588365\pi$
$521$	−12.5109	−0.548111	−0.274056	+	0.961714i	$0.588365\pi$
$522$	0	0
$523$	30.9783i	1.35458i	0.735714	+	0.677292i	$0.236847\pi$
$523$	30.9783i	1.35458i	−0.735714	+	0.677292i	$0.763153\pi$
$524$	22.1168	0.966179
$525$	0	0
$526$	24.8614	1.08401
$527$	4.62772i	0.201587i
$528$	0	0
$529$	15.4674	0.672495
$530$	0	0
$531$	0	0
$532$	− 2.11684i	− 0.0917768i
$533$	22.9783i	0.995299i
$534$	0	0
$535$	0	0
$536$	8.00000	0.345547
$537$	0	0
$538$	8.74456i	0.377005i
$539$	−4.37228	−0.188327
$540$	0	0
$541$	−20.1168	−0.864891	−0.432445	−	0.901660i	$-0.642349\pi$
$541$	−20.1168	−0.864891	−0.432445	+	0.901660i	$0.642349\pi$
$542$	16.0000i	0.687259i
$543$	0	0
$544$	1.37228	0.0588361
$545$	0	0
$546$	0	0
$547$	− 20.0000i	− 0.855138i	−0.903983	−	0.427569i	$-0.859370\pi$
$547$	− 20.0000i	− 0.855138i	0.903983	−	0.427569i	$-0.140630\pi$
$548$	− 8.74456i	− 0.373549i
$549$	0	0
$550$	0	0
$551$	−0.861407	−0.0366972
$552$	0	0
$553$	− 4.23369i	− 0.180035i
$554$	12.7446	0.541465
$555$	0	0
$556$	−4.00000	−0.169638
$557$	4.97825i	0.210935i	0.994423	+	0.105468i	$0.0336339\pi$
$557$	4.97825i	0.210935i	−0.994423	+	0.105468i	$0.966366\pi$
$558$	0	0
$559$	8.00000	0.338364
$560$	0	0
$561$	0	0
$562$	18.0000i	0.759284i
$563$	8.23369i	0.347009i	0.984833	+	0.173504i	$0.0555090\pi$
$563$	8.23369i	0.347009i	−0.984833	+	0.173504i	$0.944491\pi$
$564$	0	0
$565$	0	0
$566$	5.25544	0.220903
$567$	0	0
$568$	− 10.1168i	− 0.424493i
$569$	−15.2554	−0.639541	−0.319771	−	0.947495i	$-0.603606\pi$
$569$	−15.2554	−0.639541	−0.319771	+	0.947495i	$0.603606\pi$
$570$	0	0
$571$	15.3723	0.643310	0.321655	−	0.946857i	$-0.395761\pi$
$571$	15.3723	0.643310	0.321655	+	0.946857i	$0.395761\pi$
$572$	− 2.00000i	− 0.0836242i
$573$	0	0
$574$	−38.7446	−1.61717
$575$	0	0
$576$	0	0
$577$	− 36.9783i	− 1.53942i	−0.638391	−	0.769712i	$-0.720400\pi$
$577$	− 36.9783i	− 1.53942i	0.638391	−	0.769712i	$-0.279600\pi$
$578$	− 15.1168i	− 0.628778i
$579$	0	0
$580$	0	0
$581$	9.25544	0.383980
$582$	0	0
$583$	4.11684i	0.170502i
$584$	15.4891	0.640945
$585$	0	0
$586$	−23.4891	−0.970327
$587$	24.8614i	1.02614i	0.858347	+	0.513070i	$0.171492\pi$
$587$	24.8614i	1.02614i	−0.858347	+	0.513070i	$0.828508\pi$
$588$	0	0
$589$	−2.11684	−0.0872230
$590$	0	0
$591$	0	0
$592$	− 9.37228i	− 0.385198i
$593$	12.5109i	0.513760i	0.966443	+	0.256880i	$0.0826945\pi$
$593$	12.5109i	0.513760i	−0.966443	+	0.256880i	$0.917305\pi$
$594$	0	0
$595$	0	0
$596$	−21.6060	−0.885015
$597$	0	0
$598$	− 5.48913i	− 0.224467i
$599$	−39.6060	−1.61826	−0.809128	−	0.587632i	$-0.800060\pi$
$599$	−39.6060	−1.61826	−0.809128	+	0.587632i	$0.800060\pi$
$600$	0	0
$601$	−16.5109	−0.673493	−0.336746	−	0.941595i	$-0.609326\pi$
$601$	−16.5109	−0.673493	−0.336746	+	0.941595i	$0.609326\pi$
$602$	13.4891i	0.549776i
$603$	0	0
$604$	12.2337	0.497782
$605$	0	0
$606$	0	0
$607$	5.88316i	0.238790i	0.992847	+	0.119395i	$0.0380955\pi$
$607$	5.88316i	0.238790i	−0.992847	+	0.119395i	$0.961905\pi$
$608$	0.627719i	0.0254574i
$609$	0	0
$610$	0	0
$611$	−5.48913	−0.222066
$612$	0	0
$613$	20.5109i	0.828426i	0.910180	+	0.414213i	$0.135943\pi$
$613$	20.5109i	0.828426i	−0.910180	+	0.414213i	$0.864057\pi$
$614$	−5.25544	−0.212092
$615$	0	0
$616$	3.37228	0.135873
$617$	− 2.23369i	− 0.0899249i	−0.998989	−	0.0449624i	$-0.985683\pi$
$617$	− 2.23369i	− 0.0899249i	0.998989	−	0.0449624i	$-0.0143168\pi$
$618$	0	0
$619$	44.4674	1.78729	0.893647	−	0.448770i	$-0.148138\pi$
$619$	44.4674	1.78729	0.893647	+	0.448770i	$0.148138\pi$
$620$	0	0
$621$	0	0
$622$	19.3723i	0.776758i
$623$	4.62772i	0.185406i
$624$	0	0
$625$	0	0
$626$	−22.0000	−0.879297
$627$	0	0
$628$	9.37228i	0.373995i
$629$	12.8614	0.512818
$630$	0	0
$631$	42.1168	1.67665	0.838323	−	0.545175i	$-0.183537\pi$
$631$	42.1168	1.67665	0.838323	+	0.545175i	$0.183537\pi$
$632$	1.25544i	0.0499386i
$633$	0	0
$634$	−24.3505	−0.967083
$635$	0	0
$636$	0	0
$637$	− 8.74456i	− 0.346472i
$638$	− 1.37228i	− 0.0543291i
$639$	0	0
$640$	0	0
$641$	27.0951	1.07019	0.535096	−	0.844791i	$-0.320275\pi$
$641$	27.0951	1.07019	0.535096	+	0.844791i	$0.320275\pi$
$642$	0	0
$643$	− 5.88316i	− 0.232009i	−0.993249	−	0.116005i	$-0.962991\pi$
$643$	− 5.88316i	− 0.232009i	0.993249	−	0.116005i	$-0.0370087\pi$
$644$	9.25544	0.364715
$645$	0	0
$646$	−0.861407	−0.0338916
$647$	− 37.7228i	− 1.48304i	−0.670933	−	0.741518i	$-0.734106\pi$
$647$	− 37.7228i	− 1.48304i	0.670933	−	0.741518i	$-0.265894\pi$
$648$	0	0
$649$	−2.74456	−0.107734
$650$	0	0
$651$	0	0
$652$	5.88316i	0.230402i
$653$	− 10.6277i	− 0.415895i	−0.978140	−	0.207947i	$-0.933322\pi$
$653$	− 10.6277i	− 0.415895i	0.978140	−	0.207947i	$-0.0666783\pi$
$654$	0	0
$655$	0	0
$656$	11.4891	0.448575
$657$	0	0
$658$	− 9.25544i	− 0.360815i
$659$	12.8614	0.501009	0.250505	−	0.968115i	$-0.419403\pi$
$659$	12.8614	0.501009	0.250505	+	0.968115i	$0.419403\pi$
$660$	0	0
$661$	8.51087	0.331035	0.165517	−	0.986207i	$-0.447071\pi$
$661$	8.51087	0.331035	0.165517	+	0.986207i	$0.447071\pi$
$662$	− 30.9783i	− 1.20400i
$663$	0	0
$664$	−2.74456	−0.106510
$665$	0	0
$666$	0	0
$667$	− 3.76631i	− 0.145832i
$668$	4.62772i	0.179052i
$669$	0	0
$670$	0	0
$671$	−5.37228	−0.207395
$672$	0	0
$673$	14.8614i	0.572865i	0.958100	+	0.286433i	$0.0924694\pi$
$673$	14.8614i	0.572865i	−0.958100	+	0.286433i	$0.907531\pi$
$674$	−24.1168	−0.928946
$675$	0	0
$676$	−9.00000	−0.346154
$677$	3.25544i	0.125117i	0.998041	+	0.0625583i	$0.0199259\pi$
$677$	3.25544i	0.125117i	−0.998041	+	0.0625583i	$0.980074\pi$
$678$	0	0
$679$	42.9783	1.64935
$680$	0	0
$681$	0	0
$682$	− 3.37228i	− 0.129131i
$683$	28.6277i	1.09541i	0.836672	+	0.547705i	$0.184498\pi$
$683$	28.6277i	1.09541i	−0.836672	+	0.547705i	$0.815502\pi$
$684$	0	0
$685$	0	0
$686$	−8.86141	−0.338330
$687$	0	0
$688$	− 4.00000i	− 0.152499i
$689$	−8.23369	−0.313679
$690$	0	0
$691$	40.2337	1.53056	0.765281	−	0.643697i	$-0.222600\pi$
$691$	40.2337	1.53056	0.765281	+	0.643697i	$0.222600\pi$
$692$	− 6.00000i	− 0.228086i
$693$	0	0
$694$	−32.2337	−1.22357
$695$	0	0
$696$	0	0
$697$	15.7663i	0.597192i
$698$	19.4891i	0.737674i
$699$	0	0
$700$	0	0
$701$	37.3723	1.41153	0.705766	−	0.708445i	$-0.250603\pi$
$701$	37.3723	1.41153	0.705766	+	0.708445i	$0.250603\pi$
$702$	0	0
$703$	5.88316i	0.221887i
$704$	−1.00000	−0.0376889
$705$	0	0
$706$	0.510875	0.0192270
$707$	20.2337i	0.760966i
$708$	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
$712$	− 1.37228i	− 0.0514284i
$713$	− 9.25544i	− 0.346619i
$714$	0	0
$715$	0	0
$716$	12.0000	0.448461
$717$	0	0
$718$	0	0
$719$	13.8832	0.517754	0.258877	−	0.965910i	$-0.416648\pi$
$719$	13.8832	0.517754	0.258877	+	0.965910i	$0.416648\pi$
$720$	0	0
$721$	−32.0000	−1.19174
$722$	18.6060i	0.692442i
$723$	0	0
$724$	10.0000	0.371647
$725$	0	0
$726$	0	0
$727$	24.2337i	0.898778i	0.893336	+	0.449389i	$0.148358\pi$
$727$	24.2337i	0.898778i	−0.893336	+	0.449389i	$0.851642\pi$
$728$	6.74456i	0.249970i
$729$	0	0
$730$	0	0
$731$	5.48913	0.203023
$732$	0	0
$733$	46.2337i	1.70768i	0.520535	+	0.853840i	$0.325732\pi$
$733$	46.2337i	1.70768i	−0.520535	+	0.853840i	$0.674268\pi$
$734$	4.00000	0.147643
$735$	0	0
$736$	−2.74456	−0.101166
$737$	− 8.00000i	− 0.294684i
$738$	0	0
$739$	20.4674	0.752905	0.376452	−	0.926436i	$-0.377144\pi$
$739$	20.4674	0.752905	0.376452	+	0.926436i	$0.377144\pi$
$740$	0	0
$741$	0	0
$742$	− 13.8832i	− 0.509667i
$743$	4.62772i	0.169775i	0.996391	+	0.0848873i	$0.0270530\pi$
$743$	4.62772i	0.169775i	−0.996391	+	0.0848873i	$0.972947\pi$
$744$	0	0
$745$	0	0
$746$	31.4891	1.15290
$747$	0	0
$748$	− 1.37228i	− 0.0501756i
$749$	−40.4674	−1.47865
$750$	0	0
$751$	8.86141	0.323357	0.161679	−	0.986843i	$-0.448309\pi$
$751$	8.86141	0.323357	0.161679	+	0.986843i	$0.448309\pi$
$752$	2.74456i	0.100084i
$753$	0	0
$754$	2.74456	0.0999511
$755$	0	0
$756$	0	0
$757$	20.9783i	0.762467i	0.924479	+	0.381234i	$0.124501\pi$
$757$	20.9783i	0.762467i	−0.924479	+	0.381234i	$0.875499\pi$
$758$	− 0.233688i	− 0.00848793i
$759$	0	0
$760$	0	0
$761$	−4.97825	−0.180461	−0.0902307	−	0.995921i	$-0.528760\pi$
$761$	−4.97825	−0.180461	−0.0902307	+	0.995921i	$0.528760\pi$
$762$	0	0
$763$	− 52.2337i	− 1.89099i
$764$	−5.48913	−0.198590
$765$	0	0
$766$	−32.2337	−1.16465
$767$	− 5.48913i	− 0.198201i
$768$	0	0
$769$	22.0000	0.793340	0.396670	−	0.917961i	$-0.370166\pi$
$769$	22.0000	0.793340	0.396670	+	0.917961i	$0.370166\pi$
$770$	0	0
$771$	0	0
$772$	− 14.8614i	− 0.534874i
$773$	33.6060i	1.20872i	0.796710	+	0.604361i	$0.206572\pi$
$773$	33.6060i	1.20872i	−0.796710	+	0.604361i	$0.793428\pi$
$774$	0	0
$775$	0	0
$776$	−12.7446	−0.457503
$777$	0	0
$778$	− 6.00000i	− 0.215110i
$779$	−7.21194	−0.258395
$780$	0	0
$781$	−10.1168	−0.362009
$782$	− 3.76631i	− 0.134683i
$783$	0	0
$784$	−4.37228	−0.156153
$785$	0	0
$786$	0	0
$787$	44.4674i	1.58509i	0.609813	+	0.792545i	$0.291245\pi$
$787$	44.4674i	1.58509i	−0.609813	+	0.792545i	$0.708755\pi$
$788$	20.7446i	0.738994i
$789$	0	0
$790$	0	0
$791$	−10.9783	−0.390342
$792$	0	0
$793$	− 10.7446i	− 0.381551i
$794$	−24.9783	−0.886445
$795$	0	0
$796$	18.1168	0.642135
$797$	11.4891i	0.406966i	0.979079	+	0.203483i	$0.0652261\pi$
$797$	11.4891i	0.406966i	−0.979079	+	0.203483i	$0.934774\pi$
$798$	0	0
$799$	−3.76631	−0.133243
$800$	0	0
$801$	0	0
$802$	13.3723i	0.472192i
$803$	− 15.4891i	− 0.546599i
$804$	0	0
$805$	0	0
$806$	6.74456	0.237567
$807$	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
$811$	44.8614	1.57530	0.787649	−	0.616125i	$-0.211298\pi$
$811$	44.8614	1.57530	0.787649	+	0.616125i	$0.211298\pi$
$812$	4.62772i	0.162401i
$813$	0	0
$814$	−9.37228	−0.328498
$815$	0	0
$816$	0	0
$817$	2.51087i	0.0878444i
$818$	− 1.76631i	− 0.0617577i
$819$	0	0
$820$	0	0
$821$	−11.4891	−0.400973	−0.200487	−	0.979696i	$-0.564252\pi$
$821$	−11.4891	−0.400973	−0.200487	+	0.979696i	$0.564252\pi$
$822$	0	0
$823$	− 28.0000i	− 0.976019i	−0.872838	−	0.488009i	$-0.837723\pi$
$823$	− 28.0000i	− 0.976019i	0.872838	−	0.488009i	$-0.162277\pi$
$824$	9.48913	0.330569
$825$	0	0
$826$	9.25544	0.322038
$827$	46.9783i	1.63359i	0.576925	+	0.816797i	$0.304252\pi$
$827$	46.9783i	1.63359i	−0.576925	+	0.816797i	$0.695748\pi$
$828$	0	0
$829$	24.7446	0.859414	0.429707	−	0.902968i	$-0.358617\pi$
$829$	24.7446	0.859414	0.429707	+	0.902968i	$0.358617\pi$
$830$	0	0
$831$	0	0
$832$	− 2.00000i	− 0.0693375i
$833$	− 6.00000i	− 0.207888i
$834$	0	0
$835$	0	0
$836$	0.627719	0.0217101
$837$	0	0
$838$	12.0000i	0.414533i
$839$	−10.9783	−0.379011	−0.189506	−	0.981880i	$-0.560689\pi$
$839$	−10.9783	−0.379011	−0.189506	+	0.981880i	$0.560689\pi$
$840$	0	0
$841$	−27.1168	−0.935064
$842$	− 10.2337i	− 0.352676i
$843$	0	0
$844$	−6.11684	−0.210550
$845$	0	0
$846$	0	0
$847$	− 3.37228i	− 0.115873i
$848$	4.11684i	0.141373i
$849$	0	0
$850$	0	0
$851$	−25.7228	−0.881767
$852$	0	0
$853$	− 38.4674i	− 1.31710i	−0.752538	−	0.658549i	$-0.771171\pi$
$853$	− 38.4674i	− 1.31710i	0.752538	−	0.658549i	$-0.228829\pi$
$854$	18.1168	0.619946
$855$	0	0
$856$	12.0000	0.410152
$857$	36.3505i	1.24171i	0.783925	+	0.620855i	$0.213215\pi$
$857$	36.3505i	1.24171i	−0.783925	+	0.620855i	$0.786785\pi$
$858$	0	0
$859$	42.7446	1.45843	0.729213	−	0.684287i	$-0.239886\pi$
$859$	42.7446	1.45843	0.729213	+	0.684287i	$0.239886\pi$
$860$	0	0
$861$	0	0
$862$	− 34.9783i	− 1.19136i
$863$	− 21.2554i	− 0.723544i	−0.932267	−	0.361772i	$-0.882172\pi$
$863$	− 21.2554i	− 0.723544i	0.932267	−	0.361772i	$-0.117828\pi$
$864$	0	0
$865$	0	0
$866$	27.7228	0.942060
$867$	0	0
$868$	11.3723i	0.386000i
$869$	1.25544	0.0425878
$870$	0	0
$871$	16.0000	0.542139
$872$	15.4891i	0.524528i
$873$	0	0
$874$	1.72281	0.0582750
$875$	0	0
$876$	0	0
$877$	− 36.9783i	− 1.24867i	−0.781158	−	0.624333i	$-0.785371\pi$
$877$	− 36.9783i	− 1.24867i	0.781158	−	0.624333i	$-0.214629\pi$
$878$	18.9783i	0.640485i
$879$	0	0
$880$	0	0
$881$	−18.0000	−0.606435	−0.303218	−	0.952921i	$-0.598061\pi$
$881$	−18.0000	−0.606435	−0.303218	+	0.952921i	$0.598061\pi$
$882$	0	0
$883$	3.37228i	0.113486i	0.998389	+	0.0567432i	$0.0180716\pi$
$883$	3.37228i	0.113486i	−0.998389	+	0.0567432i	$0.981928\pi$
$884$	2.74456	0.0923096
$885$	0	0
$886$	−29.4891	−0.990707
$887$	− 10.9783i	− 0.368614i	−0.982869	−	0.184307i	$-0.940996\pi$
$887$	− 10.9783i	− 0.368614i	0.982869	−	0.184307i	$-0.0590040\pi$
$888$	0	0
$889$	−26.9783	−0.904821
$890$	0	0
$891$	0	0
$892$	18.7446i	0.627614i
$893$	− 1.72281i	− 0.0576517i
$894$	0	0
$895$	0	0
$896$	3.37228	0.112660
$897$	0	0
$898$	− 28.9783i	− 0.967017i
$899$	4.62772	0.154343
$900$	0	0
$901$	−5.64947	−0.188211
$902$	− 11.4891i	− 0.382546i
$903$	0	0
$904$	3.25544	0.108274
$905$	0	0
$906$	0	0
$907$	0.394031i	0.0130836i	0.999979	+	0.00654179i	$0.00208233\pi$
$907$	0.394031i	0.0130836i	−0.999979	+	0.00654179i	$0.997918\pi$
$908$	2.74456i	0.0910815i
$909$	0	0
$910$	0	0
$911$	8.39403	0.278107	0.139053	−	0.990285i	$-0.455594\pi$
$911$	8.39403	0.278107	0.139053	+	0.990285i	$0.455594\pi$
$912$	0	0
$913$	2.74456i	0.0908318i
$914$	16.3505	0.540828
$915$	0	0
$916$	−10.0000	−0.330409
$917$	74.5842i	2.46299i
$918$	0	0
$919$	−18.9783	−0.626035	−0.313017	−	0.949747i	$-0.601340\pi$
$919$	−18.9783	−0.626035	−0.313017	+	0.949747i	$0.601340\pi$
$920$	0	0
$921$	0	0
$922$	16.1168i	0.530780i
$923$	− 20.2337i	− 0.666000i
$924$	0	0
$925$	0	0
$926$	−0.233688	−0.00767946
$927$	0	0
$928$	− 1.37228i	− 0.0450473i
$929$	24.3505	0.798915	0.399458	−	0.916752i	$-0.369199\pi$
$929$	24.3505	0.798915	0.399458	+	0.916752i	$0.369199\pi$
$930$	0	0
$931$	2.74456	0.0899494
$932$	1.37228i	0.0449506i
$933$	0	0
$934$	−19.3723	−0.633880
$935$	0	0
$936$	0	0
$937$	28.5109i	0.931410i	0.884940	+	0.465705i	$0.154199\pi$
$937$	28.5109i	0.931410i	−0.884940	+	0.465705i	$0.845801\pi$
$938$	26.9783i	0.880871i
$939$	0	0
$940$	0	0
$941$	15.0951	0.492086	0.246043	−	0.969259i	$-0.420870\pi$
$941$	15.0951	0.492086	0.246043	+	0.969259i	$0.420870\pi$
$942$	0	0
$943$	− 31.5326i	− 1.02684i
$944$	−2.74456	−0.0893279
$945$	0	0
$946$	−4.00000	−0.130051
$947$	48.8614i	1.58778i	0.608060	+	0.793891i	$0.291948\pi$
$947$	48.8614i	1.58778i	−0.608060	+	0.793891i	$0.708052\pi$
$948$	0	0
$949$	30.9783	1.00560
$950$	0	0
$951$	0	0
$952$	4.62772i	0.149985i
$953$	− 40.1168i	− 1.29951i	−0.760143	−	0.649756i	$-0.774871\pi$
$953$	− 40.1168i	− 1.29951i	0.760143	−	0.649756i	$-0.225129\pi$
$954$	0	0
$955$	0	0
$956$	14.7446	0.476873
$957$	0	0
$958$	− 5.48913i	− 0.177346i
$959$	29.4891	0.952254
$960$	0	0
$961$	−19.6277	−0.633152
$962$	− 18.7446i	− 0.604349i
$963$	0	0
$964$	22.0000	0.708572
$965$	0	0
$966$	0	0
$967$	− 47.6060i	− 1.53090i	−0.643493	−	0.765452i	$-0.722515\pi$
$967$	− 47.6060i	− 1.53090i	0.643493	−	0.765452i	$-0.277485\pi$
$968$	1.00000i	0.0321412i
$969$	0	0
$970$	0	0
$971$	1.02175	0.0327895	0.0163947	−	0.999866i	$-0.494781\pi$
$971$	1.02175	0.0327895	0.0163947	+	0.999866i	$0.494781\pi$
$972$	0	0
$973$	− 13.4891i	− 0.432442i
$974$	−20.0000	−0.640841
$975$	0	0
$976$	−5.37228	−0.171963
$977$	14.2337i	0.455376i	0.973734	+	0.227688i	$0.0731166\pi$
$977$	14.2337i	0.455376i	−0.973734	+	0.227688i	$0.926883\pi$
$978$	0	0
$979$	−1.37228	−0.0438583
$980$	0	0
$981$	0	0
$982$	− 7.37228i	− 0.235259i
$983$	13.7228i	0.437690i	0.975760	+	0.218845i	$0.0702289\pi$
$983$	13.7228i	0.437690i	−0.975760	+	0.218845i	$0.929771\pi$
$984$	0	0
$985$	0	0
$986$	1.88316	0.0599719
$987$	0	0
$988$	1.25544i	0.0399408i
$989$	−10.9783	−0.349088
$990$	0	0
$991$	8.00000	0.254128	0.127064	−	0.991894i	$-0.459445\pi$
$991$	8.00000	0.254128	0.127064	+	0.991894i	$0.459445\pi$
$992$	− 3.37228i	− 0.107070i
$993$	0	0
$994$	34.1168	1.08212
$995$	0	0
$996$	0	0
$997$	− 22.2337i	− 0.704148i	−0.935972	−	0.352074i	$-0.885477\pi$
$997$	− 22.2337i	− 0.704148i	0.935972	−	0.352074i	$-0.114523\pi$
$998$	− 33.4891i	− 1.06008i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4950.2.c.bc.199.1		4
3.2	odd	2		550.2.b.f.199.3		4
5.2	odd	4		990.2.a.m.1.2		2
5.3	odd	4		4950.2.a.bw.1.1		2
5.4	even	2	inner	4950.2.c.bc.199.4		4
12.11	even	2		4400.2.b.p.4049.4		4
15.2	even	4		110.2.a.d.1.1	✓	2
15.8	even	4		550.2.a.n.1.2		2
15.14	odd	2		550.2.b.f.199.2		4
20.7	even	4		7920.2.a.bq.1.1		2
60.23	odd	4		4400.2.a.bl.1.1		2
60.47	odd	4		880.2.a.n.1.2		2
60.59	even	2		4400.2.b.p.4049.1		4
105.62	odd	4		5390.2.a.bp.1.2		2
120.77	even	4		3520.2.a.bq.1.2		2
120.107	odd	4		3520.2.a.bj.1.1		2
165.32	odd	4		1210.2.a.r.1.1		2
165.98	odd	4		6050.2.a.cb.1.2		2
660.527	even	4		9680.2.a.bt.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
110.2.a.d.1.1	✓	2	15.2	even	4
550.2.a.n.1.2		2	15.8	even	4
550.2.b.f.199.2		4	15.14	odd	2
550.2.b.f.199.3		4	3.2	odd	2
880.2.a.n.1.2		2	60.47	odd	4
990.2.a.m.1.2		2	5.2	odd	4
1210.2.a.r.1.1		2	165.32	odd	4
3520.2.a.bj.1.1		2	120.107	odd	4
3520.2.a.bq.1.2		2	120.77	even	4
4400.2.a.bl.1.1		2	60.23	odd	4
4400.2.b.p.4049.1		4	60.59	even	2
4400.2.b.p.4049.4		4	12.11	even	2
4950.2.a.bw.1.1		2	5.3	odd	4
4950.2.c.bc.199.1		4	1.1	even	1	trivial
4950.2.c.bc.199.4		4	5.4	even	2	inner
5390.2.a.bp.1.2		2	105.62	odd	4
6050.2.a.cb.1.2		2	165.98	odd	4
7920.2.a.bq.1.1		2	20.7	even	4
9680.2.a.bt.1.2		2	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 $\F_{q}$

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)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} -3.37228i q^{7} +1.00000i q^{8} +1.00000 q^{11} +2.00000i q^{13} -3.37228 q^{14} +1.00000 q^{16} +1.37228i q^{17} -0.627719 q^{19} -1.00000i q^{22} -2.74456i q^{23} +2.00000 q^{26} +3.37228i q^{28} +1.37228 q^{29} +3.37228 q^{31} -1.00000i q^{32} +1.37228 q^{34} -9.37228i q^{37} +0.627719i q^{38} +11.4891 q^{41} -4.00000i q^{43} -1.00000 q^{44} -2.74456 q^{46} +2.74456i q^{47} -4.37228 q^{49} -2.00000i q^{52} +4.11684i q^{53} +3.37228 q^{56} -1.37228i q^{58} -2.74456 q^{59} -5.37228 q^{61} -3.37228i q^{62} -1.00000 q^{64} -8.00000i q^{67} -1.37228i q^{68} -10.1168 q^{71} -15.4891i q^{73} -9.37228 q^{74} +0.627719 q^{76} -3.37228i q^{77} +1.25544 q^{79} -11.4891i q^{82} +2.74456i q^{83} -4.00000 q^{86} +1.00000i q^{88} -1.37228 q^{89} +6.74456 q^{91} +2.74456i q^{92} +2.74456 q^{94} +12.7446i q^{97} +4.37228i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4} + 4 q^{11} - 2 q^{14} + 4 q^{16} - 14 q^{19} + 8 q^{26} - 6 q^{29} + 2 q^{31} - 6 q^{34} - 4 q^{44} + 12 q^{46} - 6 q^{49} + 2 q^{56} + 12 q^{59} - 10 q^{61} - 4 q^{64} - 6 q^{71} - 26 q^{74}+ \cdots - 12 q^{94}+O(q^{100}) \) 4 * q - 4 * q^4 + 4 * q^11 - 2 * q^14 + 4 * q^16 - 14 * q^19 + 8 * q^26 - 6 * q^29 + 2 * q^31 - 6 * q^34 - 4 * q^44 + 12 * q^46 - 6 * q^49 + 2 * q^56 + 12 * q^59 - 10 * q^61 - 4 * q^64 - 6 * q^71 - 26 * q^74 + 14 * q^76 + 28 * q^79 - 16 * q^86 + 6 * q^89 + 4 * q^91 - 12 * q^94

Introduction

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