LMFDB - Embedded newform 425.2.b.a.324.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [425,2,Mod(324,425)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(425, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("425.324");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 425 = 5^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	425.b (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:	$3.39364208590$
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			324.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	425.324
Dual form			425.2.b.a.324.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000i q^{2} -1.00000i q^{3} +1.00000 q^{4} -1.00000 q^{6} -1.00000i q^{7} -3.00000i q^{8} +2.00000 q^{9} -4.00000 q^{11} -1.00000i q^{12} -1.00000i q^{13} -1.00000 q^{14} -1.00000 q^{16} -1.00000i q^{17} -2.00000i q^{18} +6.00000 q^{19} -1.00000 q^{21} +4.00000i q^{22} -3.00000 q^{24} -1.00000 q^{26} -5.00000i q^{27} -1.00000i q^{28} -7.00000 q^{31} -5.00000i q^{32} +4.00000i q^{33} -1.00000 q^{34} +2.00000 q^{36} +4.00000i q^{37} -6.00000i q^{38} -1.00000 q^{39} -2.00000 q^{41} +1.00000i q^{42} +4.00000i q^{43} -4.00000 q^{44} +6.00000i q^{47} +1.00000i q^{48} +6.00000 q^{49} -1.00000 q^{51} -1.00000i q^{52} +11.0000i q^{53} -5.00000 q^{54} -3.00000 q^{56} -6.00000i q^{57} -8.00000 q^{59} +10.0000 q^{61} +7.00000i q^{62} -2.00000i q^{63} -7.00000 q^{64} +4.00000 q^{66} -8.00000i q^{67} -1.00000i q^{68} +7.00000 q^{71} -6.00000i q^{72} +4.00000i q^{73} +4.00000 q^{74} +6.00000 q^{76} +4.00000i q^{77} +1.00000i q^{78} +11.0000 q^{79} +1.00000 q^{81} +2.00000i q^{82} -8.00000i q^{83} -1.00000 q^{84} +4.00000 q^{86} +12.0000i q^{88} +6.00000 q^{89} -1.00000 q^{91} +7.00000i q^{93} +6.00000 q^{94} -5.00000 q^{96} +16.0000i q^{97} -6.00000i q^{98} -8.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{4} - 2 q^{6} + 4 q^{9} - 8 q^{11} - 2 q^{14} - 2 q^{16} + 12 q^{19} - 2 q^{21} - 6 q^{24} - 2 q^{26} - 14 q^{31} - 2 q^{34} + 4 q^{36} - 2 q^{39} - 4 q^{41} - 8 q^{44} + 12 q^{49} - 2 q^{51}+ \cdots - 16 q^{99}+O(q^{100}) $ 2 * q + 2 * q^4 - 2 * q^6 + 4 * q^9 - 8 * q^11 - 2 * q^14 - 2 * q^16 + 12 * q^19 - 2 * q^21 - 6 * q^24 - 2 * q^26 - 14 * q^31 - 2 * q^34 + 4 * q^36 - 2 * q^39 - 4 * q^41 - 8 * q^44 + 12 * q^49 - 2 * q^51 - 10 * q^54 - 6 * q^56 - 16 * q^59 + 20 * q^61 - 14 * q^64 + 8 * q^66 + 14 * q^71 + 8 * q^74 + 12 * q^76 + 22 * q^79 + 2 * q^81 - 2 * q^84 + 8 * q^86 + 12 * q^89 - 2 * q^91 + 12 * q^94 - 10 * q^96 - 16 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/425\mathbb{Z}\right)^\times$.

$n$	$52$	$326$
$\chi(n)$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	− 1.00000i	− 0.707107i	−0.935414	−	0.353553i	$-0.884973\pi$
$2$	− 1.00000i	− 0.707107i	0.935414	−	0.353553i	$-0.115027\pi$
$3$	− 1.00000i	− 0.577350i	−0.957427	−	0.288675i	$-0.906785\pi$
$3$	− 1.00000i	− 0.577350i	0.957427	−	0.288675i	$-0.0932147\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−1.00000	−0.408248
$7$	− 1.00000i	− 0.377964i	−0.981981	−	0.188982i	$-0.939481\pi$
$7$	− 1.00000i	− 0.377964i	0.981981	−	0.188982i	$-0.0605189\pi$
$8$	− 3.00000i	− 1.06066i
$9$	2.00000	0.666667
$10$	0	0
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	− 1.00000i	− 0.288675i
$13$	− 1.00000i	− 0.277350i	−0.990338	−	0.138675i	$-0.955716\pi$
$13$	− 1.00000i	− 0.277350i	0.990338	−	0.138675i	$-0.0442844\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	−1.00000	−0.250000
$17$	− 1.00000i	− 0.242536i
$17$	− 1.00000i	− 0.242536i
$18$	− 2.00000i	− 0.471405i
$19$	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
$21$	−1.00000	−0.218218
$22$	4.00000i	0.852803i
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	−3.00000	−0.612372
$25$	0	0
$26$	−1.00000	−0.196116
$27$	− 5.00000i	− 0.962250i
$28$	− 1.00000i	− 0.188982i
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	−7.00000	−1.25724	−0.628619	−	0.777714i	$-0.716379\pi$
$31$	−7.00000	−1.25724	−0.628619	+	0.777714i	$0.716379\pi$
$32$	− 5.00000i	− 0.883883i
$33$	4.00000i	0.696311i
$34$	−1.00000	−0.171499
$35$	0	0
$36$	2.00000	0.333333
$37$	4.00000i	0.657596i	0.944400	+	0.328798i	$0.106644\pi$
$37$	4.00000i	0.657596i	−0.944400	+	0.328798i	$0.893356\pi$
$38$	− 6.00000i	− 0.973329i
$39$	−1.00000	−0.160128
$40$	0	0
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	1.00000i	0.154303i
$43$	4.00000i	0.609994i	0.952353	+	0.304997i	$0.0986555\pi$
$43$	4.00000i	0.609994i	−0.952353	+	0.304997i	$0.901344\pi$
$44$	−4.00000	−0.603023
$45$	0	0
$46$	0	0
$47$	6.00000i	0.875190i	0.899172	+	0.437595i	$0.144170\pi$
$47$	6.00000i	0.875190i	−0.899172	+	0.437595i	$0.855830\pi$
$48$	1.00000i	0.144338i
$49$	6.00000	0.857143
$50$	0	0
$51$	−1.00000	−0.140028
$52$	− 1.00000i	− 0.138675i
$53$	11.0000i	1.51097i	0.655168	+	0.755483i	$0.272598\pi$
$53$	11.0000i	1.51097i	−0.655168	+	0.755483i	$0.727402\pi$
$54$	−5.00000	−0.680414
$55$	0	0
$56$	−3.00000	−0.400892
$57$	− 6.00000i	− 0.794719i
$58$	0	0
$59$	−8.00000	−1.04151	−0.520756	−	0.853706i	$-0.674350\pi$
$59$	−8.00000	−1.04151	−0.520756	+	0.853706i	$0.674350\pi$
$60$	0	0
$61$	10.0000	1.28037	0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000	1.28037	0.640184	+	0.768221i	$0.278858\pi$
$62$	7.00000i	0.889001i
$63$	− 2.00000i	− 0.251976i
$64$	−7.00000	−0.875000
$65$	0	0
$66$	4.00000	0.492366
$67$	− 8.00000i	− 0.977356i	−0.872464	−	0.488678i	$-0.837479\pi$
$67$	− 8.00000i	− 0.977356i	0.872464	−	0.488678i	$-0.162521\pi$
$68$	− 1.00000i	− 0.121268i
$69$	0	0
$70$	0	0
$71$	7.00000	0.830747	0.415374	−	0.909651i	$-0.363651\pi$
$71$	7.00000	0.830747	0.415374	+	0.909651i	$0.363651\pi$
$72$	− 6.00000i	− 0.707107i
$73$	4.00000i	0.468165i	0.972217	+	0.234082i	$0.0752085\pi$
$73$	4.00000i	0.468165i	−0.972217	+	0.234082i	$0.924791\pi$
$74$	4.00000	0.464991
$75$	0	0
$76$	6.00000	0.688247
$77$	4.00000i	0.455842i
$78$	1.00000i	0.113228i
$79$	11.0000	1.23760	0.618798	−	0.785550i	$-0.287620\pi$
$79$	11.0000	1.23760	0.618798	+	0.785550i	$0.287620\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	2.00000i	0.220863i
$83$	− 8.00000i	− 0.878114i	−0.898459	−	0.439057i	$-0.855313\pi$
$83$	− 8.00000i	− 0.878114i	0.898459	−	0.439057i	$-0.144687\pi$
$84$	−1.00000	−0.109109
$85$	0	0
$86$	4.00000	0.431331
$87$	0	0
$88$	12.0000i	1.27920i
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	7.00000i	0.725866i
$94$	6.00000	0.618853
$95$	0	0
$96$	−5.00000	−0.510310
$97$	16.0000i	1.62455i	0.583272	+	0.812277i	$0.301772\pi$
$97$	16.0000i	1.62455i	−0.583272	+	0.812277i	$0.698228\pi$
$98$	− 6.00000i	− 0.606092i
$99$	−8.00000	−0.804030
$100$	0	0
$101$	15.0000	1.49256	0.746278	−	0.665635i	$-0.231839\pi$
$101$	15.0000	1.49256	0.746278	+	0.665635i	$0.231839\pi$
$102$	1.00000i	0.0990148i
$103$	6.00000i	0.591198i	0.955312	+	0.295599i	$0.0955191\pi$
$103$	6.00000i	0.591198i	−0.955312	+	0.295599i	$0.904481\pi$
$104$	−3.00000	−0.294174
$105$	0	0
$106$	11.0000	1.06841
$107$	19.0000i	1.83680i	0.395654	+	0.918400i	$0.370518\pi$
$107$	19.0000i	1.83680i	−0.395654	+	0.918400i	$0.629482\pi$
$108$	− 5.00000i	− 0.481125i
$109$	4.00000	0.383131	0.191565	−	0.981480i	$-0.438644\pi$
$109$	4.00000	0.383131	0.191565	+	0.981480i	$0.438644\pi$
$110$	0	0
$111$	4.00000	0.379663
$112$	1.00000i	0.0944911i
$113$	− 12.0000i	− 1.12887i	−0.825479	−	0.564433i	$-0.809095\pi$
$113$	− 12.0000i	− 1.12887i	0.825479	−	0.564433i	$-0.190905\pi$
$114$	−6.00000	−0.561951
$115$	0	0
$116$	0	0
$117$	− 2.00000i	− 0.184900i
$118$	8.00000i	0.736460i
$119$	−1.00000	−0.0916698
$120$	0	0
$121$	5.00000	0.454545
$122$	− 10.0000i	− 0.905357i
$123$	2.00000i	0.180334i
$124$	−7.00000	−0.628619
$125$	0	0
$126$	−2.00000	−0.178174
$127$	16.0000i	1.41977i	0.704317	+	0.709885i	$0.251253\pi$
$127$	16.0000i	1.41977i	−0.704317	+	0.709885i	$0.748747\pi$
$128$	− 3.00000i	− 0.265165i
$129$	4.00000	0.352180
$130$	0	0
$131$	−15.0000	−1.31056	−0.655278	−	0.755388i	$-0.727449\pi$
$131$	−15.0000	−1.31056	−0.655278	+	0.755388i	$0.727449\pi$
$132$	4.00000i	0.348155i
$133$	− 6.00000i	− 0.520266i
$134$	−8.00000	−0.691095
$135$	0	0
$136$	−3.00000	−0.257248
$137$	11.0000i	0.939793i	0.882721	+	0.469897i	$0.155709\pi$
$137$	11.0000i	0.939793i	−0.882721	+	0.469897i	$0.844291\pi$
$138$	0	0
$139$	13.0000	1.10265	0.551323	−	0.834292i	$-0.314123\pi$
$139$	13.0000	1.10265	0.551323	+	0.834292i	$0.314123\pi$
$140$	0	0
$141$	6.00000	0.505291
$142$	− 7.00000i	− 0.587427i
$143$	4.00000i	0.334497i
$144$	−2.00000	−0.166667
$145$	0	0
$146$	4.00000	0.331042
$147$	− 6.00000i	− 0.494872i
$148$	4.00000i	0.328798i
$149$	−21.0000	−1.72039	−0.860194	−	0.509968i	$-0.829657\pi$
$149$	−21.0000	−1.72039	−0.860194	+	0.509968i	$0.829657\pi$
$150$	0	0
$151$	−18.0000	−1.46482	−0.732410	−	0.680864i	$-0.761604\pi$
$151$	−18.0000	−1.46482	−0.732410	+	0.680864i	$0.761604\pi$
$152$	− 18.0000i	− 1.45999i
$153$	− 2.00000i	− 0.161690i
$154$	4.00000	0.322329
$155$	0	0
$156$	−1.00000	−0.0800641
$157$	− 21.0000i	− 1.67598i	−0.545684	−	0.837991i	$-0.683730\pi$
$157$	− 21.0000i	− 1.67598i	0.545684	−	0.837991i	$-0.316270\pi$
$158$	− 11.0000i	− 0.875113i
$159$	11.0000	0.872357
$160$	0	0
$161$	0	0
$162$	− 1.00000i	− 0.0785674i
$163$	− 11.0000i	− 0.861586i	−0.902451	−	0.430793i	$-0.858234\pi$
$163$	− 11.0000i	− 0.861586i	0.902451	−	0.430793i	$-0.141766\pi$
$164$	−2.00000	−0.156174
$165$	0	0
$166$	−8.00000	−0.620920
$167$	− 8.00000i	− 0.619059i	−0.950890	−	0.309529i	$-0.899829\pi$
$167$	− 8.00000i	− 0.619059i	0.950890	−	0.309529i	$-0.100171\pi$
$168$	3.00000i	0.231455i
$169$	12.0000	0.923077
$170$	0	0
$171$	12.0000	0.917663
$172$	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
$176$	4.00000	0.301511
$177$	8.00000i	0.601317i
$178$	− 6.00000i	− 0.449719i
$179$	−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
$181$	−22.0000	−1.63525	−0.817624	−	0.575753i	$-0.804709\pi$
$181$	−22.0000	−1.63525	−0.817624	+	0.575753i	$0.804709\pi$
$182$	1.00000i	0.0741249i
$183$	− 10.0000i	− 0.739221i
$184$	0	0
$185$	0	0
$186$	7.00000	0.513265
$187$	4.00000i	0.292509i
$188$	6.00000i	0.437595i
$189$	−5.00000	−0.363696
$190$	0	0
$191$	−10.0000	−0.723575	−0.361787	−	0.932261i	$-0.617833\pi$
$191$	−10.0000	−0.723575	−0.361787	+	0.932261i	$0.617833\pi$
$192$	7.00000i	0.505181i
$193$	24.0000i	1.72756i	0.503871	+	0.863779i	$0.331909\pi$
$193$	24.0000i	1.72756i	−0.503871	+	0.863779i	$0.668091\pi$
$194$	16.0000	1.14873
$195$	0	0
$196$	6.00000	0.428571
$197$	− 14.0000i	− 0.997459i	−0.866758	−	0.498729i	$-0.833800\pi$
$197$	− 14.0000i	− 0.997459i	0.866758	−	0.498729i	$-0.166200\pi$
$198$	8.00000i	0.568535i
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	0	0
$201$	−8.00000	−0.564276
$202$	− 15.0000i	− 1.05540i
$203$	0	0
$204$	−1.00000	−0.0700140
$205$	0	0
$206$	6.00000	0.418040
$207$	0	0
$208$	1.00000i	0.0693375i
$209$	−24.0000	−1.66011
$210$	0	0
$211$	−25.0000	−1.72107	−0.860535	−	0.509390i	$-0.829871\pi$
$211$	−25.0000	−1.72107	−0.860535	+	0.509390i	$0.829871\pi$
$212$	11.0000i	0.755483i
$213$	− 7.00000i	− 0.479632i
$214$	19.0000	1.29881
$215$	0	0
$216$	−15.0000	−1.02062
$217$	7.00000i	0.475191i
$218$	− 4.00000i	− 0.270914i
$219$	4.00000	0.270295
$220$	0	0
$221$	−1.00000	−0.0672673
$222$	− 4.00000i	− 0.268462i
$223$	− 2.00000i	− 0.133930i	−0.997755	−	0.0669650i	$-0.978668\pi$
$223$	− 2.00000i	− 0.133930i	0.997755	−	0.0669650i	$-0.0213316\pi$
$224$	−5.00000	−0.334077
$225$	0	0
$226$	−12.0000	−0.798228
$227$	− 25.0000i	− 1.65931i	−0.558278	−	0.829654i	$-0.688538\pi$
$227$	− 25.0000i	− 1.65931i	0.558278	−	0.829654i	$-0.311462\pi$
$228$	− 6.00000i	− 0.397360i
$229$	−1.00000	−0.0660819	−0.0330409	−	0.999454i	$-0.510519\pi$
$229$	−1.00000	−0.0660819	−0.0330409	+	0.999454i	$0.510519\pi$
$230$	0	0
$231$	4.00000	0.263181
$232$	0	0
$233$	− 8.00000i	− 0.524097i	−0.965055	−	0.262049i	$-0.915602\pi$
$233$	− 8.00000i	− 0.524097i	0.965055	−	0.262049i	$-0.0843981\pi$
$234$	−2.00000	−0.130744
$235$	0	0
$236$	−8.00000	−0.520756
$237$	− 11.0000i	− 0.714527i
$238$	1.00000i	0.0648204i
$239$	8.00000	0.517477	0.258738	−	0.965947i	$-0.416693\pi$
$239$	8.00000	0.517477	0.258738	+	0.965947i	$0.416693\pi$
$240$	0	0
$241$	18.0000	1.15948	0.579741	−	0.814801i	$-0.303154\pi$
$241$	18.0000	1.15948	0.579741	+	0.814801i	$0.303154\pi$
$242$	− 5.00000i	− 0.321412i
$243$	− 16.0000i	− 1.02640i
$244$	10.0000	0.640184
$245$	0	0
$246$	2.00000	0.127515
$247$	− 6.00000i	− 0.381771i
$248$	21.0000i	1.33350i
$249$	−8.00000	−0.506979
$250$	0	0
$251$	−8.00000	−0.504956	−0.252478	−	0.967603i	$-0.581245\pi$
$251$	−8.00000	−0.504956	−0.252478	+	0.967603i	$0.581245\pi$
$252$	− 2.00000i	− 0.125988i
$253$	0	0
$254$	16.0000	1.00393
$255$	0	0
$256$	−17.0000	−1.06250
$257$	− 19.0000i	− 1.18519i	−0.805502	−	0.592594i	$-0.798104\pi$
$257$	− 19.0000i	− 1.18519i	0.805502	−	0.592594i	$-0.201896\pi$
$258$	− 4.00000i	− 0.249029i
$259$	4.00000	0.248548
$260$	0	0
$261$	0	0
$262$	15.0000i	0.926703i
$263$	16.0000i	0.986602i	0.869859	+	0.493301i	$0.164210\pi$
$263$	16.0000i	0.986602i	−0.869859	+	0.493301i	$0.835790\pi$
$264$	12.0000	0.738549
$265$	0	0
$266$	−6.00000	−0.367884
$267$	− 6.00000i	− 0.367194i
$268$	− 8.00000i	− 0.488678i
$269$	−10.0000	−0.609711	−0.304855	−	0.952399i	$-0.598608\pi$
$269$	−10.0000	−0.609711	−0.304855	+	0.952399i	$0.598608\pi$
$270$	0	0
$271$	2.00000	0.121491	0.0607457	−	0.998153i	$-0.480652\pi$
$271$	2.00000	0.121491	0.0607457	+	0.998153i	$0.480652\pi$
$272$	1.00000i	0.0606339i
$273$	1.00000i	0.0605228i
$274$	11.0000	0.664534
$275$	0	0
$276$	0	0
$277$	30.0000i	1.80253i	0.433273	+	0.901263i	$0.357359\pi$
$277$	30.0000i	1.80253i	−0.433273	+	0.901263i	$0.642641\pi$
$278$	− 13.0000i	− 0.779688i
$279$	−14.0000	−0.838158
$280$	0	0
$281$	9.00000	0.536895	0.268447	−	0.963294i	$-0.413489\pi$
$281$	9.00000	0.536895	0.268447	+	0.963294i	$0.413489\pi$
$282$	− 6.00000i	− 0.357295i
$283$	− 24.0000i	− 1.42665i	−0.700832	−	0.713326i	$-0.747188\pi$
$283$	− 24.0000i	− 1.42665i	0.700832	−	0.713326i	$-0.252812\pi$
$284$	7.00000	0.415374
$285$	0	0
$286$	4.00000	0.236525
$287$	2.00000i	0.118056i
$288$	− 10.0000i	− 0.589256i
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	16.0000	0.937937
$292$	4.00000i	0.234082i
$293$	6.00000i	0.350524i	0.984522	+	0.175262i	$0.0560772\pi$
$293$	6.00000i	0.350524i	−0.984522	+	0.175262i	$0.943923\pi$
$294$	−6.00000	−0.349927
$295$	0	0
$296$	12.0000	0.697486
$297$	20.0000i	1.16052i
$298$	21.0000i	1.21650i
$299$	0	0
$300$	0	0
$301$	4.00000	0.230556
$302$	18.0000i	1.03578i
$303$	− 15.0000i	− 0.861727i
$304$	−6.00000	−0.344124
$305$	0	0
$306$	−2.00000	−0.114332
$307$	− 20.0000i	− 1.14146i	−0.821138	−	0.570730i	$-0.806660\pi$
$307$	− 20.0000i	− 1.14146i	0.821138	−	0.570730i	$-0.193340\pi$
$308$	4.00000i	0.227921i
$309$	6.00000	0.341328
$310$	0	0
$311$	−27.0000	−1.53103	−0.765515	−	0.643418i	$-0.777516\pi$
$311$	−27.0000	−1.53103	−0.765515	+	0.643418i	$0.777516\pi$
$312$	3.00000i	0.169842i
$313$	− 10.0000i	− 0.565233i	−0.959233	−	0.282617i	$-0.908798\pi$
$313$	− 10.0000i	− 0.565233i	0.959233	−	0.282617i	$-0.0912024\pi$
$314$	−21.0000	−1.18510
$315$	0	0
$316$	11.0000	0.618798
$317$	14.0000i	0.786318i	0.919470	+	0.393159i	$0.128618\pi$
$317$	14.0000i	0.786318i	−0.919470	+	0.393159i	$0.871382\pi$
$318$	− 11.0000i	− 0.616849i
$319$	0	0
$320$	0	0
$321$	19.0000	1.06048
$322$	0	0
$323$	− 6.00000i	− 0.333849i
$324$	1.00000	0.0555556
$325$	0	0
$326$	−11.0000	−0.609234
$327$	− 4.00000i	− 0.221201i
$328$	6.00000i	0.331295i
$329$	6.00000	0.330791
$330$	0	0
$331$	−14.0000	−0.769510	−0.384755	−	0.923019i	$-0.625714\pi$
$331$	−14.0000	−0.769510	−0.384755	+	0.923019i	$0.625714\pi$
$332$	− 8.00000i	− 0.439057i
$333$	8.00000i	0.438397i
$334$	−8.00000	−0.437741
$335$	0	0
$336$	1.00000	0.0545545
$337$	24.0000i	1.30736i	0.756770	+	0.653682i	$0.226776\pi$
$337$	24.0000i	1.30736i	−0.756770	+	0.653682i	$0.773224\pi$
$338$	− 12.0000i	− 0.652714i
$339$	−12.0000	−0.651751
$340$	0	0
$341$	28.0000	1.51629
$342$	− 12.0000i	− 0.648886i
$343$	− 13.0000i	− 0.701934i
$344$	12.0000	0.646997
$345$	0	0
$346$	6.00000	0.322562
$347$	− 7.00000i	− 0.375780i	−0.982190	−	0.187890i	$-0.939835\pi$
$347$	− 7.00000i	− 0.375780i	0.982190	−	0.187890i	$-0.0601648\pi$
$348$	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
$351$	−5.00000	−0.266880
$352$	20.0000i	1.06600i
$353$	19.0000i	1.01127i	0.862748	+	0.505634i	$0.168741\pi$
$353$	19.0000i	1.01127i	−0.862748	+	0.505634i	$0.831259\pi$
$354$	8.00000	0.425195
$355$	0	0
$356$	6.00000	0.317999
$357$	1.00000i	0.0529256i
$358$	6.00000i	0.317110i
$359$	10.0000	0.527780	0.263890	−	0.964553i	$-0.414994\pi$
$359$	10.0000	0.527780	0.263890	+	0.964553i	$0.414994\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	22.0000i	1.15629i
$363$	− 5.00000i	− 0.262432i
$364$	−1.00000	−0.0524142
$365$	0	0
$366$	−10.0000	−0.522708
$367$	− 23.0000i	− 1.20059i	−0.799779	−	0.600295i	$-0.795050\pi$
$367$	− 23.0000i	− 1.20059i	0.799779	−	0.600295i	$-0.204950\pi$
$368$	0	0
$369$	−4.00000	−0.208232
$370$	0	0
$371$	11.0000	0.571092
$372$	7.00000i	0.362933i
$373$	19.0000i	0.983783i	0.870657	+	0.491891i	$0.163694\pi$
$373$	19.0000i	0.983783i	−0.870657	+	0.491891i	$0.836306\pi$
$374$	4.00000	0.206835
$375$	0	0
$376$	18.0000	0.928279
$377$	0	0
$378$	5.00000i	0.257172i
$379$	−25.0000	−1.28416	−0.642082	−	0.766636i	$-0.721929\pi$
$379$	−25.0000	−1.28416	−0.642082	+	0.766636i	$0.721929\pi$
$380$	0	0
$381$	16.0000	0.819705
$382$	10.0000i	0.511645i
$383$	6.00000i	0.306586i	0.988181	+	0.153293i	$0.0489878\pi$
$383$	6.00000i	0.306586i	−0.988181	+	0.153293i	$0.951012\pi$
$384$	−3.00000	−0.153093
$385$	0	0
$386$	24.0000	1.22157
$387$	8.00000i	0.406663i
$388$	16.0000i	0.812277i
$389$	34.0000	1.72387	0.861934	−	0.507020i	$-0.169253\pi$
$389$	34.0000	1.72387	0.861934	+	0.507020i	$0.169253\pi$
$390$	0	0
$391$	0	0
$392$	− 18.0000i	− 0.909137i
$393$	15.0000i	0.756650i
$394$	−14.0000	−0.705310
$395$	0	0
$396$	−8.00000	−0.402015
$397$	− 4.00000i	− 0.200754i	−0.994949	−	0.100377i	$-0.967995\pi$
$397$	− 4.00000i	− 0.200754i	0.994949	−	0.100377i	$-0.0320049\pi$
$398$	0	0
$399$	−6.00000	−0.300376
$400$	0	0
$401$	6.00000	0.299626	0.149813	−	0.988714i	$-0.452133\pi$
$401$	6.00000	0.299626	0.149813	+	0.988714i	$0.452133\pi$
$402$	8.00000i	0.399004i
$403$	7.00000i	0.348695i
$404$	15.0000	0.746278
$405$	0	0
$406$	0	0
$407$	− 16.0000i	− 0.793091i
$408$	3.00000i	0.148522i
$409$	−19.0000	−0.939490	−0.469745	−	0.882802i	$-0.655654\pi$
$409$	−19.0000	−0.939490	−0.469745	+	0.882802i	$0.655654\pi$
$410$	0	0
$411$	11.0000	0.542590
$412$	6.00000i	0.295599i
$413$	8.00000i	0.393654i
$414$	0	0
$415$	0	0
$416$	−5.00000	−0.245145
$417$	− 13.0000i	− 0.636613i
$418$	24.0000i	1.17388i
$419$	−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
$421$	−25.0000	−1.21843	−0.609213	−	0.793007i	$-0.708514\pi$
$421$	−25.0000	−1.21843	−0.609213	+	0.793007i	$0.708514\pi$
$422$	25.0000i	1.21698i
$423$	12.0000i	0.583460i
$424$	33.0000	1.60262
$425$	0	0
$426$	−7.00000	−0.339151
$427$	− 10.0000i	− 0.483934i
$428$	19.0000i	0.918400i
$429$	4.00000	0.193122
$430$	0	0
$431$	31.0000	1.49322	0.746609	−	0.665263i	$-0.231681\pi$
$431$	31.0000	1.49322	0.746609	+	0.665263i	$0.231681\pi$
$432$	5.00000i	0.240563i
$433$	− 34.0000i	− 1.63394i	−0.576683	−	0.816968i	$-0.695653\pi$
$433$	− 34.0000i	− 1.63394i	0.576683	−	0.816968i	$-0.304347\pi$
$434$	7.00000	0.336011
$435$	0	0
$436$	4.00000	0.191565
$437$	0	0
$438$	− 4.00000i	− 0.191127i
$439$	33.0000	1.57500	0.787502	−	0.616312i	$-0.211374\pi$
$439$	33.0000	1.57500	0.787502	+	0.616312i	$0.211374\pi$
$440$	0	0
$441$	12.0000	0.571429
$442$	1.00000i	0.0475651i
$443$	6.00000i	0.285069i	0.989790	+	0.142534i	$0.0455251\pi$
$443$	6.00000i	0.285069i	−0.989790	+	0.142534i	$0.954475\pi$
$444$	4.00000	0.189832
$445$	0	0
$446$	−2.00000	−0.0947027
$447$	21.0000i	0.993266i
$448$	7.00000i	0.330719i
$449$	−36.0000	−1.69895	−0.849473	−	0.527633i	$-0.823080\pi$
$449$	−36.0000	−1.69895	−0.849473	+	0.527633i	$0.823080\pi$
$450$	0	0
$451$	8.00000	0.376705
$452$	− 12.0000i	− 0.564433i
$453$	18.0000i	0.845714i
$454$	−25.0000	−1.17331
$455$	0	0
$456$	−18.0000	−0.842927
$457$	− 3.00000i	− 0.140334i	−0.997535	−	0.0701670i	$-0.977647\pi$
$457$	− 3.00000i	− 0.140334i	0.997535	−	0.0701670i	$-0.0223532\pi$
$458$	1.00000i	0.0467269i
$459$	−5.00000	−0.233380
$460$	0	0
$461$	−14.0000	−0.652045	−0.326023	−	0.945362i	$-0.605709\pi$
$461$	−14.0000	−0.652045	−0.326023	+	0.945362i	$0.605709\pi$
$462$	− 4.00000i	− 0.186097i
$463$	12.0000i	0.557687i	0.960337	+	0.278844i	$0.0899511\pi$
$463$	12.0000i	0.557687i	−0.960337	+	0.278844i	$0.910049\pi$
$464$	0	0
$465$	0	0
$466$	−8.00000	−0.370593
$467$	− 22.0000i	− 1.01804i	−0.860755	−	0.509019i	$-0.830008\pi$
$467$	− 22.0000i	− 1.01804i	0.860755	−	0.509019i	$-0.169992\pi$
$468$	− 2.00000i	− 0.0924500i
$469$	−8.00000	−0.369406
$470$	0	0
$471$	−21.0000	−0.967629
$472$	24.0000i	1.10469i
$473$	− 16.0000i	− 0.735681i
$474$	−11.0000	−0.505247
$475$	0	0
$476$	−1.00000	−0.0458349
$477$	22.0000i	1.00731i
$478$	− 8.00000i	− 0.365911i
$479$	12.0000	0.548294	0.274147	−	0.961688i	$-0.411605\pi$
$479$	12.0000	0.548294	0.274147	+	0.961688i	$0.411605\pi$
$480$	0	0
$481$	4.00000	0.182384
$482$	− 18.0000i	− 0.819878i
$483$	0	0
$484$	5.00000	0.227273
$485$	0	0
$486$	−16.0000	−0.725775
$487$	− 4.00000i	− 0.181257i	−0.995885	−	0.0906287i	$-0.971112\pi$
$487$	− 4.00000i	− 0.181257i	0.995885	−	0.0906287i	$-0.0288876\pi$
$488$	− 30.0000i	− 1.35804i
$489$	−11.0000	−0.497437
$490$	0	0
$491$	20.0000	0.902587	0.451294	−	0.892375i	$-0.350963\pi$
$491$	20.0000	0.902587	0.451294	+	0.892375i	$0.350963\pi$
$492$	2.00000i	0.0901670i
$493$	0	0
$494$	−6.00000	−0.269953
$495$	0	0
$496$	7.00000	0.314309
$497$	− 7.00000i	− 0.313993i
$498$	8.00000i	0.358489i
$499$	−13.0000	−0.581960	−0.290980	−	0.956729i	$-0.593981\pi$
$499$	−13.0000	−0.581960	−0.290980	+	0.956729i	$0.593981\pi$
$500$	0	0
$501$	−8.00000	−0.357414
$502$	8.00000i	0.357057i
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	−6.00000	−0.267261
$505$	0	0
$506$	0	0
$507$	− 12.0000i	− 0.532939i
$508$	16.0000i	0.709885i
$509$	−5.00000	−0.221621	−0.110811	−	0.993842i	$-0.535345\pi$
$509$	−5.00000	−0.221621	−0.110811	+	0.993842i	$0.535345\pi$
$510$	0	0
$511$	4.00000	0.176950
$512$	11.0000i	0.486136i
$513$	− 30.0000i	− 1.32453i
$514$	−19.0000	−0.838054
$515$	0	0
$516$	4.00000	0.176090
$517$	− 24.0000i	− 1.05552i
$518$	− 4.00000i	− 0.175750i
$519$	6.00000	0.263371
$520$	0	0
$521$	18.0000	0.788594	0.394297	−	0.918983i	$-0.370988\pi$
$521$	18.0000	0.788594	0.394297	+	0.918983i	$0.370988\pi$
$522$	0	0
$523$	20.0000i	0.874539i	0.899331	+	0.437269i	$0.144054\pi$
$523$	20.0000i	0.874539i	−0.899331	+	0.437269i	$0.855946\pi$
$524$	−15.0000	−0.655278
$525$	0	0
$526$	16.0000	0.697633
$527$	7.00000i	0.304925i
$528$	− 4.00000i	− 0.174078i
$529$	23.0000	1.00000
$530$	0	0
$531$	−16.0000	−0.694341
$532$	− 6.00000i	− 0.260133i
$533$	2.00000i	0.0866296i
$534$	−6.00000	−0.259645
$535$	0	0
$536$	−24.0000	−1.03664
$537$	6.00000i	0.258919i
$538$	10.0000i	0.431131i
$539$	−24.0000	−1.03375
$540$	0	0
$541$	−8.00000	−0.343947	−0.171973	−	0.985102i	$-0.555014\pi$
$541$	−8.00000	−0.343947	−0.171973	+	0.985102i	$0.555014\pi$
$542$	− 2.00000i	− 0.0859074i
$543$	22.0000i	0.944110i
$544$	−5.00000	−0.214373
$545$	0	0
$546$	1.00000	0.0427960
$547$	− 5.00000i	− 0.213785i	−0.994271	−	0.106892i	$-0.965910\pi$
$547$	− 5.00000i	− 0.213785i	0.994271	−	0.106892i	$-0.0340900\pi$
$548$	11.0000i	0.469897i
$549$	20.0000	0.853579
$550$	0	0
$551$	0	0
$552$	0	0
$553$	− 11.0000i	− 0.467768i
$554$	30.0000	1.27458
$555$	0	0
$556$	13.0000	0.551323
$557$	37.0000i	1.56774i	0.620925	+	0.783870i	$0.286757\pi$
$557$	37.0000i	1.56774i	−0.620925	+	0.783870i	$0.713243\pi$
$558$	14.0000i	0.592667i
$559$	4.00000	0.169182
$560$	0	0
$561$	4.00000	0.168880
$562$	− 9.00000i	− 0.379642i
$563$	− 26.0000i	− 1.09577i	−0.836554	−	0.547885i	$-0.815433\pi$
$563$	− 26.0000i	− 1.09577i	0.836554	−	0.547885i	$-0.184567\pi$
$564$	6.00000	0.252646
$565$	0	0
$566$	−24.0000	−1.00880
$567$	− 1.00000i	− 0.0419961i
$568$	− 21.0000i	− 0.881140i
$569$	14.0000	0.586911	0.293455	−	0.955973i	$-0.405195\pi$
$569$	14.0000	0.586911	0.293455	+	0.955973i	$0.405195\pi$
$570$	0	0
$571$	−3.00000	−0.125546	−0.0627730	−	0.998028i	$-0.519994\pi$
$571$	−3.00000	−0.125546	−0.0627730	+	0.998028i	$0.519994\pi$
$572$	4.00000i	0.167248i
$573$	10.0000i	0.417756i
$574$	2.00000	0.0834784
$575$	0	0
$576$	−14.0000	−0.583333
$577$	− 30.0000i	− 1.24892i	−0.781058	−	0.624458i	$-0.785320\pi$
$577$	− 30.0000i	− 1.24892i	0.781058	−	0.624458i	$-0.214680\pi$
$578$	1.00000i	0.0415945i
$579$	24.0000	0.997406
$580$	0	0
$581$	−8.00000	−0.331896
$582$	− 16.0000i	− 0.663221i
$583$	− 44.0000i	− 1.82229i
$584$	12.0000	0.496564
$585$	0	0
$586$	6.00000	0.247858
$587$	0	0	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	− 6.00000i	− 0.247436i
$589$	−42.0000	−1.73058
$590$	0	0
$591$	−14.0000	−0.575883
$592$	− 4.00000i	− 0.164399i
$593$	9.00000i	0.369586i	0.982777	+	0.184793i	$0.0591614\pi$
$593$	9.00000i	0.369586i	−0.982777	+	0.184793i	$0.940839\pi$
$594$	20.0000	0.820610
$595$	0	0
$596$	−21.0000	−0.860194
$597$	0	0
$598$	0	0
$599$	44.0000	1.79779	0.898896	−	0.438163i	$-0.144371\pi$
$599$	44.0000	1.79779	0.898896	+	0.438163i	$0.144371\pi$
$600$	0	0
$601$	−8.00000	−0.326327	−0.163163	−	0.986599i	$-0.552170\pi$
$601$	−8.00000	−0.326327	−0.163163	+	0.986599i	$0.552170\pi$
$602$	− 4.00000i	− 0.163028i
$603$	− 16.0000i	− 0.651570i
$604$	−18.0000	−0.732410
$605$	0	0
$606$	−15.0000	−0.609333
$607$	27.0000i	1.09590i	0.836512	+	0.547948i	$0.184591\pi$
$607$	27.0000i	1.09590i	−0.836512	+	0.547948i	$0.815409\pi$
$608$	− 30.0000i	− 1.21666i
$609$	0	0
$610$	0	0
$611$	6.00000	0.242734
$612$	− 2.00000i	− 0.0808452i
$613$	− 47.0000i	− 1.89831i	−0.314806	−	0.949156i	$-0.601939\pi$
$613$	− 47.0000i	− 1.89831i	0.314806	−	0.949156i	$-0.398061\pi$
$614$	−20.0000	−0.807134
$615$	0	0
$616$	12.0000	0.483494
$617$	− 36.0000i	− 1.44931i	−0.689114	−	0.724653i	$-0.742000\pi$
$617$	− 36.0000i	− 1.44931i	0.689114	−	0.724653i	$-0.258000\pi$
$618$	− 6.00000i	− 0.241355i
$619$	−4.00000	−0.160774	−0.0803868	−	0.996764i	$-0.525616\pi$
$619$	−4.00000	−0.160774	−0.0803868	+	0.996764i	$0.525616\pi$
$620$	0	0
$621$	0	0
$622$	27.0000i	1.08260i
$623$	− 6.00000i	− 0.240385i
$624$	1.00000	0.0400320
$625$	0	0
$626$	−10.0000	−0.399680
$627$	24.0000i	0.958468i
$628$	− 21.0000i	− 0.837991i
$629$	4.00000	0.159490
$630$	0	0
$631$	−28.0000	−1.11466	−0.557331	−	0.830290i	$-0.688175\pi$
$631$	−28.0000	−1.11466	−0.557331	+	0.830290i	$0.688175\pi$
$632$	− 33.0000i	− 1.31267i
$633$	25.0000i	0.993661i
$634$	14.0000	0.556011
$635$	0	0
$636$	11.0000	0.436178
$637$	− 6.00000i	− 0.237729i
$638$	0	0
$639$	14.0000	0.553831
$640$	0	0
$641$	−12.0000	−0.473972	−0.236986	−	0.971513i	$-0.576159\pi$
$641$	−12.0000	−0.473972	−0.236986	+	0.971513i	$0.576159\pi$
$642$	− 19.0000i	− 0.749870i
$643$	1.00000i	0.0394362i	0.999806	+	0.0197181i	$0.00627687\pi$
$643$	1.00000i	0.0394362i	−0.999806	+	0.0197181i	$0.993723\pi$
$644$	0	0
$645$	0	0
$646$	−6.00000	−0.236067
$647$	− 4.00000i	− 0.157256i	−0.996904	−	0.0786281i	$-0.974946\pi$
$647$	− 4.00000i	− 0.157256i	0.996904	−	0.0786281i	$-0.0250540\pi$
$648$	− 3.00000i	− 0.117851i
$649$	32.0000	1.25611
$650$	0	0
$651$	7.00000	0.274352
$652$	− 11.0000i	− 0.430793i
$653$	4.00000i	0.156532i	0.996933	+	0.0782660i	$0.0249384\pi$
$653$	4.00000i	0.156532i	−0.996933	+	0.0782660i	$0.975062\pi$
$654$	−4.00000	−0.156412
$655$	0	0
$656$	2.00000	0.0780869
$657$	8.00000i	0.312110i
$658$	− 6.00000i	− 0.233904i
$659$	2.00000	0.0779089	0.0389545	−	0.999241i	$-0.487597\pi$
$659$	2.00000	0.0779089	0.0389545	+	0.999241i	$0.487597\pi$
$660$	0	0
$661$	−5.00000	−0.194477	−0.0972387	−	0.995261i	$-0.531001\pi$
$661$	−5.00000	−0.194477	−0.0972387	+	0.995261i	$0.531001\pi$
$662$	14.0000i	0.544125i
$663$	1.00000i	0.0388368i
$664$	−24.0000	−0.931381
$665$	0	0
$666$	8.00000	0.309994
$667$	0	0
$668$	− 8.00000i	− 0.309529i
$669$	−2.00000	−0.0773245
$670$	0	0
$671$	−40.0000	−1.54418
$672$	5.00000i	0.192879i
$673$	− 38.0000i	− 1.46479i	−0.680879	−	0.732396i	$-0.738402\pi$
$673$	− 38.0000i	− 1.46479i	0.680879	−	0.732396i	$-0.261598\pi$
$674$	24.0000	0.924445
$675$	0	0
$676$	12.0000	0.461538
$677$	36.0000i	1.38359i	0.722093	+	0.691796i	$0.243180\pi$
$677$	36.0000i	1.38359i	−0.722093	+	0.691796i	$0.756820\pi$
$678$	12.0000i	0.460857i
$679$	16.0000	0.614024
$680$	0	0
$681$	−25.0000	−0.958002
$682$	− 28.0000i	− 1.07218i
$683$	9.00000i	0.344375i	0.985064	+	0.172188i	$0.0550836\pi$
$683$	9.00000i	0.344375i	−0.985064	+	0.172188i	$0.944916\pi$
$684$	12.0000	0.458831
$685$	0	0
$686$	−13.0000	−0.496342
$687$	1.00000i	0.0381524i
$688$	− 4.00000i	− 0.152499i
$689$	11.0000	0.419067
$690$	0	0
$691$	35.0000	1.33146	0.665731	−	0.746191i	$-0.268120\pi$
$691$	35.0000	1.33146	0.665731	+	0.746191i	$0.268120\pi$
$692$	6.00000i	0.228086i
$693$	8.00000i	0.303895i
$694$	−7.00000	−0.265716
$695$	0	0
$696$	0	0
$697$	2.00000i	0.0757554i
$698$	19.0000i	0.719161i
$699$	−8.00000	−0.302588
$700$	0	0
$701$	30.0000	1.13308	0.566542	−	0.824033i	$-0.308281\pi$
$701$	30.0000	1.13308	0.566542	+	0.824033i	$0.308281\pi$
$702$	5.00000i	0.188713i
$703$	24.0000i	0.905177i
$704$	28.0000	1.05529
$705$	0	0
$706$	19.0000	0.715074
$707$	− 15.0000i	− 0.564133i
$708$	8.00000i	0.300658i
$709$	6.00000	0.225335	0.112667	−	0.993633i	$-0.464061\pi$
$709$	6.00000	0.225335	0.112667	+	0.993633i	$0.464061\pi$
$710$	0	0
$711$	22.0000	0.825064
$712$	− 18.0000i	− 0.674579i
$713$	0	0
$714$	1.00000	0.0374241
$715$	0	0
$716$	−6.00000	−0.224231
$717$	− 8.00000i	− 0.298765i
$718$	− 10.0000i	− 0.373197i
$719$	−17.0000	−0.633993	−0.316997	−	0.948427i	$-0.602674\pi$
$719$	−17.0000	−0.633993	−0.316997	+	0.948427i	$0.602674\pi$
$720$	0	0
$721$	6.00000	0.223452
$722$	− 17.0000i	− 0.632674i
$723$	− 18.0000i	− 0.669427i
$724$	−22.0000	−0.817624
$725$	0	0
$726$	−5.00000	−0.185567
$727$	− 18.0000i	− 0.667583i	−0.942647	−	0.333792i	$-0.891672\pi$
$727$	− 18.0000i	− 0.667583i	0.942647	−	0.333792i	$-0.108328\pi$
$728$	3.00000i	0.111187i
$729$	−13.0000	−0.481481
$730$	0	0
$731$	4.00000	0.147945
$732$	− 10.0000i	− 0.369611i
$733$	− 5.00000i	− 0.184679i	−0.995728	−	0.0923396i	$-0.970565\pi$
$733$	− 5.00000i	− 0.184679i	0.995728	−	0.0923396i	$-0.0294345\pi$
$734$	−23.0000	−0.848945
$735$	0	0
$736$	0	0
$737$	32.0000i	1.17874i
$738$	4.00000i	0.147242i
$739$	2.00000	0.0735712	0.0367856	−	0.999323i	$-0.488288\pi$
$739$	2.00000	0.0735712	0.0367856	+	0.999323i	$0.488288\pi$
$740$	0	0
$741$	−6.00000	−0.220416
$742$	− 11.0000i	− 0.403823i
$743$	21.0000i	0.770415i	0.922830	+	0.385208i	$0.125870\pi$
$743$	21.0000i	0.770415i	−0.922830	+	0.385208i	$0.874130\pi$
$744$	21.0000	0.769897
$745$	0	0
$746$	19.0000	0.695639
$747$	− 16.0000i	− 0.585409i
$748$	4.00000i	0.146254i
$749$	19.0000	0.694245
$750$	0	0
$751$	−32.0000	−1.16770	−0.583848	−	0.811863i	$-0.698454\pi$
$751$	−32.0000	−1.16770	−0.583848	+	0.811863i	$0.698454\pi$
$752$	− 6.00000i	− 0.218797i
$753$	8.00000i	0.291536i
$754$	0	0
$755$	0	0
$756$	−5.00000	−0.181848
$757$	− 38.0000i	− 1.38113i	−0.723269	−	0.690567i	$-0.757361\pi$
$757$	− 38.0000i	− 1.38113i	0.723269	−	0.690567i	$-0.242639\pi$
$758$	25.0000i	0.908041i
$759$	0	0
$760$	0	0
$761$	−39.0000	−1.41375	−0.706874	−	0.707339i	$-0.749895\pi$
$761$	−39.0000	−1.41375	−0.706874	+	0.707339i	$0.749895\pi$
$762$	− 16.0000i	− 0.579619i
$763$	− 4.00000i	− 0.144810i
$764$	−10.0000	−0.361787
$765$	0	0
$766$	6.00000	0.216789
$767$	8.00000i	0.288863i
$768$	17.0000i	0.613435i
$769$	−33.0000	−1.19001	−0.595005	−	0.803722i	$-0.702850\pi$
$769$	−33.0000	−1.19001	−0.595005	+	0.803722i	$0.702850\pi$
$770$	0	0
$771$	−19.0000	−0.684268
$772$	24.0000i	0.863779i
$773$	45.0000i	1.61854i	0.587439	+	0.809269i	$0.300136\pi$
$773$	45.0000i	1.61854i	−0.587439	+	0.809269i	$0.699864\pi$
$774$	8.00000	0.287554
$775$	0	0
$776$	48.0000	1.72310
$777$	− 4.00000i	− 0.143499i
$778$	− 34.0000i	− 1.21896i
$779$	−12.0000	−0.429945
$780$	0	0
$781$	−28.0000	−1.00192
$782$	0	0
$783$	0	0
$784$	−6.00000	−0.214286
$785$	0	0
$786$	15.0000	0.535032
$787$	− 15.0000i	− 0.534692i	−0.963601	−	0.267346i	$-0.913853\pi$
$787$	− 15.0000i	− 0.534692i	0.963601	−	0.267346i	$-0.0861467\pi$
$788$	− 14.0000i	− 0.498729i
$789$	16.0000	0.569615
$790$	0	0
$791$	−12.0000	−0.426671
$792$	24.0000i	0.852803i
$793$	− 10.0000i	− 0.355110i
$794$	−4.00000	−0.141955
$795$	0	0
$796$	0	0
$797$	17.0000i	0.602171i	0.953597	+	0.301085i	$0.0973489\pi$
$797$	17.0000i	0.602171i	−0.953597	+	0.301085i	$0.902651\pi$
$798$	6.00000i	0.212398i
$799$	6.00000	0.212265
$800$	0	0
$801$	12.0000	0.423999
$802$	− 6.00000i	− 0.211867i
$803$	− 16.0000i	− 0.564628i
$804$	−8.00000	−0.282138
$805$	0	0
$806$	7.00000	0.246564
$807$	10.0000i	0.352017i
$808$	− 45.0000i	− 1.58309i
$809$	22.0000	0.773479	0.386739	−	0.922189i	$-0.373601\pi$
$809$	22.0000	0.773479	0.386739	+	0.922189i	$0.373601\pi$
$810$	0	0
$811$	7.00000	0.245803	0.122902	−	0.992419i	$-0.460780\pi$
$811$	7.00000	0.245803	0.122902	+	0.992419i	$0.460780\pi$
$812$	0	0
$813$	− 2.00000i	− 0.0701431i
$814$	−16.0000	−0.560800
$815$	0	0
$816$	1.00000	0.0350070
$817$	24.0000i	0.839654i
$818$	19.0000i	0.664319i
$819$	−2.00000	−0.0698857
$820$	0	0
$821$	−10.0000	−0.349002	−0.174501	−	0.984657i	$-0.555831\pi$
$821$	−10.0000	−0.349002	−0.174501	+	0.984657i	$0.555831\pi$
$822$	− 11.0000i	− 0.383669i
$823$	53.0000i	1.84746i	0.383040	+	0.923732i	$0.374877\pi$
$823$	53.0000i	1.84746i	−0.383040	+	0.923732i	$0.625123\pi$
$824$	18.0000	0.627060
$825$	0	0
$826$	8.00000	0.278356
$827$	36.0000i	1.25184i	0.779886	+	0.625921i	$0.215277\pi$
$827$	36.0000i	1.25184i	−0.779886	+	0.625921i	$0.784723\pi$
$828$	0	0
$829$	−25.0000	−0.868286	−0.434143	−	0.900844i	$-0.642949\pi$
$829$	−25.0000	−0.868286	−0.434143	+	0.900844i	$0.642949\pi$
$830$	0	0
$831$	30.0000	1.04069
$832$	7.00000i	0.242681i
$833$	− 6.00000i	− 0.207888i
$834$	−13.0000	−0.450153
$835$	0	0
$836$	−24.0000	−0.830057
$837$	35.0000i	1.20978i
$838$	12.0000i	0.414533i
$839$	−23.0000	−0.794048	−0.397024	−	0.917808i	$-0.629957\pi$
$839$	−23.0000	−0.794048	−0.397024	+	0.917808i	$0.629957\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	25.0000i	0.861557i
$843$	− 9.00000i	− 0.309976i
$844$	−25.0000	−0.860535
$845$	0	0
$846$	12.0000	0.412568
$847$	− 5.00000i	− 0.171802i
$848$	− 11.0000i	− 0.377742i
$849$	−24.0000	−0.823678
$850$	0	0
$851$	0	0
$852$	− 7.00000i	− 0.239816i
$853$	− 14.0000i	− 0.479351i	−0.970853	−	0.239675i	$-0.922959\pi$
$853$	− 14.0000i	− 0.479351i	0.970853	−	0.239675i	$-0.0770410\pi$
$854$	−10.0000	−0.342193
$855$	0	0
$856$	57.0000	1.94822
$857$	20.0000i	0.683187i	0.939848	+	0.341593i	$0.110967\pi$
$857$	20.0000i	0.683187i	−0.939848	+	0.341593i	$0.889033\pi$
$858$	− 4.00000i	− 0.136558i
$859$	−18.0000	−0.614152	−0.307076	−	0.951685i	$-0.599351\pi$
$859$	−18.0000	−0.614152	−0.307076	+	0.951685i	$0.599351\pi$
$860$	0	0
$861$	2.00000	0.0681598
$862$	− 31.0000i	− 1.05586i
$863$	56.0000i	1.90626i	0.302558	+	0.953131i	$0.402160\pi$
$863$	56.0000i	1.90626i	−0.302558	+	0.953131i	$0.597840\pi$
$864$	−25.0000	−0.850517
$865$	0	0
$866$	−34.0000	−1.15537
$867$	1.00000i	0.0339618i
$868$	7.00000i	0.237595i
$869$	−44.0000	−1.49260
$870$	0	0
$871$	−8.00000	−0.271070
$872$	− 12.0000i	− 0.406371i
$873$	32.0000i	1.08304i
$874$	0	0
$875$	0	0
$876$	4.00000	0.135147
$877$	− 16.0000i	− 0.540282i	−0.962821	−	0.270141i	$-0.912930\pi$
$877$	− 16.0000i	− 0.540282i	0.962821	−	0.270141i	$-0.0870703\pi$
$878$	− 33.0000i	− 1.11370i
$879$	6.00000	0.202375
$880$	0	0
$881$	30.0000	1.01073	0.505363	−	0.862907i	$-0.331359\pi$
$881$	30.0000	1.01073	0.505363	+	0.862907i	$0.331359\pi$
$882$	− 12.0000i	− 0.404061i
$883$	− 2.00000i	− 0.0673054i	−0.999434	−	0.0336527i	$-0.989286\pi$
$883$	− 2.00000i	− 0.0673054i	0.999434	−	0.0336527i	$-0.0107140\pi$
$884$	−1.00000	−0.0336336
$885$	0	0
$886$	6.00000	0.201574
$887$	43.0000i	1.44380i	0.691998	+	0.721899i	$0.256731\pi$
$887$	43.0000i	1.44380i	−0.691998	+	0.721899i	$0.743269\pi$
$888$	− 12.0000i	− 0.402694i
$889$	16.0000	0.536623
$890$	0	0
$891$	−4.00000	−0.134005
$892$	− 2.00000i	− 0.0669650i
$893$	36.0000i	1.20469i
$894$	21.0000	0.702345
$895$	0	0
$896$	−3.00000	−0.100223
$897$	0	0
$898$	36.0000i	1.20134i
$899$	0	0
$900$	0	0
$901$	11.0000	0.366463
$902$	− 8.00000i	− 0.266371i
$903$	− 4.00000i	− 0.133112i
$904$	−36.0000	−1.19734
$905$	0	0
$906$	18.0000	0.598010
$907$	36.0000i	1.19536i	0.801735	+	0.597680i	$0.203911\pi$
$907$	36.0000i	1.19536i	−0.801735	+	0.597680i	$0.796089\pi$
$908$	− 25.0000i	− 0.829654i
$909$	30.0000	0.995037
$910$	0	0
$911$	1.00000	0.0331315	0.0165657	−	0.999863i	$-0.494727\pi$
$911$	1.00000	0.0331315	0.0165657	+	0.999863i	$0.494727\pi$
$912$	6.00000i	0.198680i
$913$	32.0000i	1.05905i
$914$	−3.00000	−0.0992312
$915$	0	0
$916$	−1.00000	−0.0330409
$917$	15.0000i	0.495344i
$918$	5.00000i	0.165025i
$919$	44.0000	1.45143	0.725713	−	0.687998i	$-0.241510\pi$
$919$	44.0000	1.45143	0.725713	+	0.687998i	$0.241510\pi$
$920$	0	0
$921$	−20.0000	−0.659022
$922$	14.0000i	0.461065i
$923$	− 7.00000i	− 0.230408i
$924$	4.00000	0.131590
$925$	0	0
$926$	12.0000	0.394344
$927$	12.0000i	0.394132i
$928$	0	0
$929$	56.0000	1.83730	0.918650	−	0.395072i	$-0.129280\pi$
$929$	56.0000	1.83730	0.918650	+	0.395072i	$0.129280\pi$
$930$	0	0
$931$	36.0000	1.17985
$932$	− 8.00000i	− 0.262049i
$933$	27.0000i	0.883940i
$934$	−22.0000	−0.719862
$935$	0	0
$936$	−6.00000	−0.196116
$937$	− 25.0000i	− 0.816714i	−0.912822	−	0.408357i	$-0.866102\pi$
$937$	− 25.0000i	− 0.816714i	0.912822	−	0.408357i	$-0.133898\pi$
$938$	8.00000i	0.261209i
$939$	−10.0000	−0.326338
$940$	0	0
$941$	34.0000	1.10837	0.554184	−	0.832394i	$-0.313030\pi$
$941$	34.0000	1.10837	0.554184	+	0.832394i	$0.313030\pi$
$942$	21.0000i	0.684217i
$943$	0	0
$944$	8.00000	0.260378
$945$	0	0
$946$	−16.0000	−0.520205
$947$	4.00000i	0.129983i	0.997886	+	0.0649913i	$0.0207020\pi$
$947$	4.00000i	0.129983i	−0.997886	+	0.0649913i	$0.979298\pi$
$948$	− 11.0000i	− 0.357263i
$949$	4.00000	0.129845
$950$	0	0
$951$	14.0000	0.453981
$952$	3.00000i	0.0972306i
$953$	9.00000i	0.291539i	0.989319	+	0.145769i	$0.0465657\pi$
$953$	9.00000i	0.291539i	−0.989319	+	0.145769i	$0.953434\pi$
$954$	22.0000	0.712276
$955$	0	0
$956$	8.00000	0.258738
$957$	0	0
$958$	− 12.0000i	− 0.387702i
$959$	11.0000	0.355209
$960$	0	0
$961$	18.0000	0.580645
$962$	− 4.00000i	− 0.128965i
$963$	38.0000i	1.22453i
$964$	18.0000	0.579741
$965$	0	0
$966$	0	0
$967$	4.00000i	0.128631i	0.997930	+	0.0643157i	$0.0204865\pi$
$967$	4.00000i	0.128631i	−0.997930	+	0.0643157i	$0.979514\pi$
$968$	− 15.0000i	− 0.482118i
$969$	−6.00000	−0.192748
$970$	0	0
$971$	−32.0000	−1.02693	−0.513464	−	0.858111i	$-0.671638\pi$
$971$	−32.0000	−1.02693	−0.513464	+	0.858111i	$0.671638\pi$
$972$	− 16.0000i	− 0.513200i
$973$	− 13.0000i	− 0.416761i
$974$	−4.00000	−0.128168
$975$	0	0
$976$	−10.0000	−0.320092
$977$	− 13.0000i	− 0.415907i	−0.978139	−	0.207953i	$-0.933320\pi$
$977$	− 13.0000i	− 0.415907i	0.978139	−	0.207953i	$-0.0666802\pi$
$978$	11.0000i	0.351741i
$979$	−24.0000	−0.767043
$980$	0	0
$981$	8.00000	0.255420
$982$	− 20.0000i	− 0.638226i
$983$	− 21.0000i	− 0.669796i	−0.942254	−	0.334898i	$-0.891298\pi$
$983$	− 21.0000i	− 0.669796i	0.942254	−	0.334898i	$-0.108702\pi$
$984$	6.00000	0.191273
$985$	0	0
$986$	0	0
$987$	− 6.00000i	− 0.190982i
$988$	− 6.00000i	− 0.190885i
$989$	0	0
$990$	0	0
$991$	5.00000	0.158830	0.0794151	−	0.996842i	$-0.474695\pi$
$991$	5.00000	0.158830	0.0794151	+	0.996842i	$0.474695\pi$
$992$	35.0000i	1.11125i
$993$	14.0000i	0.444277i
$994$	−7.00000	−0.222027
$995$	0	0
$996$	−8.00000	−0.253490
$997$	− 4.00000i	− 0.126681i	−0.997992	−	0.0633406i	$-0.979825\pi$
$997$	− 4.00000i	− 0.126681i	0.997992	−	0.0633406i	$-0.0201755\pi$
$998$	13.0000i	0.411508i
$999$	20.0000	0.632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	425.2.b.a.324.1		2
5.2	odd	4		425.2.a.c.1.1	yes	1
5.3	odd	4		425.2.a.b.1.1	✓	1
5.4	even	2	inner	425.2.b.a.324.2		2
15.2	even	4		3825.2.a.f.1.1		1
15.8	even	4		3825.2.a.k.1.1		1
20.3	even	4		6800.2.a.i.1.1		1
20.7	even	4		6800.2.a.q.1.1		1
85.33	odd	4		7225.2.a.b.1.1		1
85.67	odd	4		7225.2.a.h.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
425.2.a.b.1.1	✓	1	5.3	odd	4
425.2.a.c.1.1	yes	1	5.2	odd	4
425.2.b.a.324.1		2	1.1	even	1	trivial
425.2.b.a.324.2		2	5.4	even	2	inner
3825.2.a.f.1.1		1	15.2	even	4
3825.2.a.k.1.1		1	15.8	even	4
6800.2.a.i.1.1		1	20.3	even	4
6800.2.a.q.1.1		1	20.7	even	4
7225.2.a.b.1.1		1	85.33	odd	4
7225.2.a.h.1.1		1	85.67	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 425 = 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	425.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000i q^{3} +1.00000 q^{4} -1.00000 q^{6} -1.00000i q^{7} -3.00000i q^{8} +2.00000 q^{9} -4.00000 q^{11} -1.00000i q^{12} -1.00000i q^{13} -1.00000 q^{14} -1.00000 q^{16} -1.00000i q^{17} -2.00000i q^{18} +6.00000 q^{19} -1.00000 q^{21} +4.00000i q^{22} -3.00000 q^{24} -1.00000 q^{26} -5.00000i q^{27} -1.00000i q^{28} -7.00000 q^{31} -5.00000i q^{32} +4.00000i q^{33} -1.00000 q^{34} +2.00000 q^{36} +4.00000i q^{37} -6.00000i q^{38} -1.00000 q^{39} -2.00000 q^{41} +1.00000i q^{42} +4.00000i q^{43} -4.00000 q^{44} +6.00000i q^{47} +1.00000i q^{48} +6.00000 q^{49} -1.00000 q^{51} -1.00000i q^{52} +11.0000i q^{53} -5.00000 q^{54} -3.00000 q^{56} -6.00000i q^{57} -8.00000 q^{59} +10.0000 q^{61} +7.00000i q^{62} -2.00000i q^{63} -7.00000 q^{64} +4.00000 q^{66} -8.00000i q^{67} -1.00000i q^{68} +7.00000 q^{71} -6.00000i q^{72} +4.00000i q^{73} +4.00000 q^{74} +6.00000 q^{76} +4.00000i q^{77} +1.00000i q^{78} +11.0000 q^{79} +1.00000 q^{81} +2.00000i q^{82} -8.00000i q^{83} -1.00000 q^{84} +4.00000 q^{86} +12.0000i q^{88} +6.00000 q^{89} -1.00000 q^{91} +7.00000i q^{93} +6.00000 q^{94} -5.00000 q^{96} +16.0000i q^{97} -6.00000i q^{98} -8.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{4} - 2 q^{6} + 4 q^{9} - 8 q^{11} - 2 q^{14} - 2 q^{16} + 12 q^{19} - 2 q^{21} - 6 q^{24} - 2 q^{26} - 14 q^{31} - 2 q^{34} + 4 q^{36} - 2 q^{39} - 4 q^{41} - 8 q^{44} + 12 q^{49} - 2 q^{51}+ \cdots - 16 q^{99}+O(q^{100}) \) 2 * q + 2 * q^4 - 2 * q^6 + 4 * q^9 - 8 * q^11 - 2 * q^14 - 2 * q^16 + 12 * q^19 - 2 * q^21 - 6 * q^24 - 2 * q^26 - 14 * q^31 - 2 * q^34 + 4 * q^36 - 2 * q^39 - 4 * q^41 - 8 * q^44 + 12 * q^49 - 2 * q^51 - 10 * q^54 - 6 * q^56 - 16 * q^59 + 20 * q^61 - 14 * q^64 + 8 * q^66 + 14 * q^71 + 8 * q^74 + 12 * q^76 + 22 * q^79 + 2 * q^81 - 2 * q^84 + 8 * q^86 + 12 * q^89 - 2 * q^91 + 12 * q^94 - 10 * q^96 - 16 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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