LMFDB - Embedded newform 208.3.c.b.207.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [208,3,Mod(207,208)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(208, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("208.207");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 208 = 2^{4} \cdot 13 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	208.c (of order $2$, degree $1$, 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:	$5.66758949869$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{3}, \sqrt{-23})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} + 7x^{2} - 6x + 78 $ x^4 - 2x^3 + 7x^2 - 6*x + 78
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			207.3
Root			$-1.23205 - 2.39792i$ of defining polynomial
Character	$\chi$	$=$	208.207
Dual form			208.3.c.b.207.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.79583i q^{3} -8.30662i q^{5} -8.66025 q^{7} -14.0000 q^{9} -13.8564 q^{11} +(-10.0000 + 8.30662i) q^{13} +39.8372 q^{15} -5.00000 q^{17} -3.46410 q^{19} -41.5331i q^{21} -28.7750i q^{23} -44.0000 q^{25} -23.9792i q^{27} +40.0000 q^{29} +20.7846 q^{31} -66.4530i q^{33} +71.9375i q^{35} +41.5331i q^{37} +(-39.8372 - 47.9583i) q^{39} +43.1625i q^{43} +116.293i q^{45} -77.9423 q^{47} +26.0000 q^{49} -23.9792i q^{51} -10.0000 q^{53} +115.100i q^{55} -16.6132i q^{57} +65.8179 q^{59} -40.0000 q^{61} +121.244 q^{63} +(69.0000 + 83.0662i) q^{65} -103.923 q^{67} +138.000 q^{69} -29.4449 q^{71} -83.0662i q^{73} -211.017i q^{75} +120.000 q^{77} -143.875i q^{79} -11.0000 q^{81} +17.3205 q^{83} +41.5331i q^{85} +191.833i q^{87} +83.0662i q^{89} +(86.6025 - 71.9375i) q^{91} +99.6795i q^{93} +28.7750i q^{95} -99.6795i q^{97} +193.990 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 56 q^{9} - 40 q^{13} - 20 q^{17} - 176 q^{25} + 160 q^{29} + 104 q^{49} - 40 q^{53} - 160 q^{61} + 276 q^{65} + 552 q^{69} + 480 q^{77} - 44 q^{81}+O(q^{100}) $ 4 * q - 56 * q^9 - 40 * q^13 - 20 * q^17 - 176 * q^25 + 160 * q^29 + 104 * q^49 - 40 * q^53 - 160 * q^61 + 276 * q^65 + 552 * q^69 + 480 * q^77 - 44 * q^81

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/208\mathbb{Z}\right)^\times$.

$n$	$53$	$79$	$145$
$\chi(n)$	$1$	$-1$	$-1$

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 $

$2$		0			0
$2$		0			0
$3$			4.79583i			1.59861i	0.600925	+	0.799305i	$0.294799\pi$
$3$			4.79583i			1.59861i	−0.600925	+	0.799305i	$0.705201\pi$
$4$		0			0
$5$		−	8.30662i		−	1.66132i	−0.556776	−	0.830662i	$-0.687962\pi$
$5$		−	8.30662i		−	1.66132i	0.556776	−	0.830662i	$-0.312038\pi$
$6$		0			0
$7$	−8.66025			−1.23718			−0.618590	−	0.785714i	$-0.712296\pi$
$7$	−8.66025			−1.23718			−0.618590	+	0.785714i	$0.712296\pi$
$8$		0			0
$9$	−14.0000			−1.55556
$10$		0			0
$11$	−13.8564			−1.25967			−0.629837	−	0.776728i	$-0.716878\pi$
$11$	−13.8564			−1.25967			−0.629837	+	0.776728i	$0.716878\pi$
$12$		0			0
$13$	−10.0000	+	8.30662i	−0.769231	+	0.638971i
$13$	−10.0000	+	8.30662i	−0.769231	+	0.638971i
$14$		0			0
$15$	39.8372			2.65581
$16$		0			0
$17$	−5.00000			−0.294118			−0.147059	−	0.989128i	$-0.546981\pi$
$17$	−5.00000			−0.294118			−0.147059	+	0.989128i	$0.546981\pi$
$18$		0			0
$19$	−3.46410			−0.182321			−0.0911606	−	0.995836i	$-0.529058\pi$
$19$	−3.46410			−0.182321			−0.0911606	+	0.995836i	$0.529058\pi$
$20$		0			0
$21$		−	41.5331i		−	1.97777i
$22$		0			0
$23$		−	28.7750i		−	1.25109i	−0.780189	−	0.625543i	$-0.784877\pi$
$23$		−	28.7750i		−	1.25109i	0.780189	−	0.625543i	$-0.215123\pi$
$24$		0			0
$25$	−44.0000			−1.76000
$26$		0			0
$27$		−	23.9792i		−	0.888117i
$28$		0			0
$29$	40.0000			1.37931			0.689655	−	0.724138i	$-0.257762\pi$
$29$	40.0000			1.37931			0.689655	+	0.724138i	$0.257762\pi$
$30$		0			0
$31$	20.7846			0.670471			0.335236	−	0.942134i	$-0.391184\pi$
$31$	20.7846			0.670471			0.335236	+	0.942134i	$0.391184\pi$
$32$		0			0
$33$		−	66.4530i		−	2.01373i
$34$		0			0
$35$			71.9375i			2.05536i
$36$		0			0
$37$			41.5331i			1.12252i	0.827641	+	0.561258i	$0.189683\pi$
$37$			41.5331i			1.12252i	−0.827641	+	0.561258i	$0.810317\pi$
$38$		0			0
$39$	−39.8372	−	47.9583i	−1.02147	−	1.22970i
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$			43.1625i			1.00378i	0.864932	+	0.501889i	$0.167362\pi$
$43$			43.1625i			1.00378i	−0.864932	+	0.501889i	$0.832638\pi$
$44$		0			0
$45$			116.293i			2.58428i
$46$		0			0
$47$	−77.9423			−1.65835			−0.829173	−	0.558992i	$-0.811188\pi$
$47$	−77.9423			−1.65835			−0.829173	+	0.558992i	$0.811188\pi$
$48$		0			0
$49$	26.0000			0.530612
$50$		0			0
$51$		−	23.9792i		−	0.470180i
$52$		0			0
$53$	−10.0000			−0.188679			−0.0943396	−	0.995540i	$-0.530074\pi$
$53$	−10.0000			−0.188679			−0.0943396	+	0.995540i	$0.530074\pi$
$54$		0			0
$55$			115.100i			2.09273i
$56$		0			0
$57$		−	16.6132i		−	0.291460i
$58$		0			0
$59$	65.8179			1.11556			0.557779	−	0.829989i	$-0.311654\pi$
$59$	65.8179			1.11556			0.557779	+	0.829989i	$0.311654\pi$
$60$		0			0
$61$	−40.0000			−0.655738			−0.327869	−	0.944723i	$-0.606330\pi$
$61$	−40.0000			−0.655738			−0.327869	+	0.944723i	$0.606330\pi$
$62$		0			0
$63$	121.244			1.92450
$64$		0			0
$65$	69.0000	+	83.0662i	1.06154	+	1.27794i
$66$		0			0
$67$	−103.923			−1.55109			−0.775545	−	0.631292i	$-0.782525\pi$
$67$	−103.923			−1.55109			−0.775545	+	0.631292i	$0.782525\pi$
$68$		0			0
$69$	138.000			2.00000
$70$		0			0
$71$	−29.4449			−0.414716			−0.207358	−	0.978265i	$-0.566487\pi$
$71$	−29.4449			−0.414716			−0.207358	+	0.978265i	$0.566487\pi$
$72$		0			0
$73$		−	83.0662i		−	1.13789i	−0.822374	−	0.568947i	$-0.807351\pi$
$73$		−	83.0662i		−	1.13789i	0.822374	−	0.568947i	$-0.192649\pi$
$74$		0			0
$75$		−	211.017i		−	2.81355i
$76$		0			0
$77$	120.000			1.55844
$78$		0			0
$79$		−	143.875i		−	1.82120i	−0.413287	−	0.910601i	$-0.635619\pi$
$79$		−	143.875i		−	1.82120i	0.413287	−	0.910601i	$-0.364381\pi$
$80$		0			0
$81$	−11.0000			−0.135802
$82$		0			0
$83$	17.3205			0.208681			0.104340	−	0.994542i	$-0.466727\pi$
$83$	17.3205			0.208681			0.104340	+	0.994542i	$0.466727\pi$
$84$		0			0
$85$			41.5331i			0.488625i
$86$		0			0
$87$			191.833i			2.20498i
$88$		0			0
$89$			83.0662i			0.933329i	0.884435	+	0.466664i	$0.154544\pi$
$89$			83.0662i			0.933329i	−0.884435	+	0.466664i	$0.845456\pi$
$90$		0			0
$91$	86.6025	−	71.9375i	0.951676	−	0.790522i
$92$		0			0
$93$			99.6795i			1.07182i
$94$		0			0
$95$			28.7750i			0.302895i
$96$		0			0
$97$		−	99.6795i		−	1.02762i	−0.857903	−	0.513812i	$-0.828233\pi$
$97$		−	99.6795i		−	1.02762i	0.857903	−	0.513812i	$-0.171767\pi$
$98$		0			0
$99$	193.990			1.95949
$100$		0			0
$101$	−52.0000			−0.514851			−0.257426	−	0.966298i	$-0.582874\pi$
$101$	−52.0000			−0.514851			−0.257426	+	0.966298i	$0.582874\pi$
$102$		0			0
$103$			28.7750i			0.279369i	0.990196	+	0.139684i	$0.0446088\pi$
$103$			28.7750i			0.279369i	−0.990196	+	0.139684i	$0.955391\pi$
$104$		0			0
$105$	−345.000			−3.28571
$106$		0			0
$107$			115.100i			1.07570i	0.843040	+	0.537850i	$0.180763\pi$
$107$			115.100i			1.07570i	−0.843040	+	0.537850i	$0.819237\pi$
$108$		0			0
$109$		−	41.5331i		−	0.381038i	−0.981684	−	0.190519i	$-0.938983\pi$
$109$		−	41.5331i		−	0.381038i	0.981684	−	0.190519i	$-0.0610170\pi$
$110$		0			0
$111$	−199.186			−1.79447
$112$		0			0
$113$	10.0000			0.0884956			0.0442478	−	0.999021i	$-0.485911\pi$
$113$	10.0000			0.0884956			0.0442478	+	0.999021i	$0.485911\pi$
$114$		0			0
$115$	−239.023			−2.07846
$116$		0			0
$117$	140.000	−	116.293i	1.19658	−	0.993955i
$118$		0			0
$119$	43.3013			0.363876
$120$		0			0
$121$	71.0000			0.586777
$122$		0			0
$123$		0			0
$124$		0			0
$125$			157.826i			1.26261i
$126$		0			0
$127$			28.7750i			0.226575i	0.993562	+	0.113287i	$0.0361381\pi$
$127$			28.7750i			0.226575i	−0.993562	+	0.113287i	$0.963862\pi$
$128$		0			0
$129$	−207.000			−1.60465
$130$		0			0
$131$			71.9375i			0.549141i	0.961567	+	0.274571i	$0.0885357\pi$
$131$			71.9375i			0.549141i	−0.961567	+	0.274571i	$0.911464\pi$
$132$		0			0
$133$	30.0000			0.225564
$134$		0			0
$135$	−199.186			−1.47545
$136$		0			0
$137$			16.6132i			0.121265i	0.998160	+	0.0606323i	$0.0193117\pi$
$137$			16.6132i			0.121265i	−0.998160	+	0.0606323i	$0.980688\pi$
$138$		0			0
$139$			71.9375i			0.517536i	0.965940	+	0.258768i	$0.0833165\pi$
$139$			71.9375i			0.517536i	−0.965940	+	0.258768i	$0.916684\pi$
$140$		0			0
$141$		−	373.798i		−	2.65105i
$142$		0			0
$143$	138.564	−	115.100i	0.968979	−	0.804895i
$144$		0			0
$145$		−	332.265i		−	2.29148i
$146$		0			0
$147$			124.692i			0.848242i
$148$		0			0
$149$		−	83.0662i		−	0.557492i	−0.960365	−	0.278746i	$-0.910081\pi$
$149$		−	83.0662i		−	0.557492i	0.960365	−	0.278746i	$-0.0899187\pi$
$150$		0			0
$151$	91.7987			0.607938			0.303969	−	0.952682i	$-0.401688\pi$
$151$	91.7987			0.607938			0.303969	+	0.952682i	$0.401688\pi$
$152$		0			0
$153$	70.0000			0.457516
$154$		0			0
$155$		−	172.650i		−	1.11387i
$156$		0			0
$157$	−170.000			−1.08280			−0.541401	−	0.840764i	$-0.682106\pi$
$157$	−170.000			−1.08280			−0.541401	+	0.840764i	$0.682106\pi$
$158$		0			0
$159$		−	47.9583i		−	0.301625i
$160$		0			0
$161$			249.199i			1.54782i
$162$		0			0
$163$	155.885			0.956347			0.478174	−	0.878265i	$-0.341299\pi$
$163$	155.885			0.956347			0.478174	+	0.878265i	$0.341299\pi$
$164$		0			0
$165$	−552.000			−3.34545
$166$		0			0
$167$	−69.2820			−0.414862			−0.207431	−	0.978250i	$-0.566510\pi$
$167$	−69.2820			−0.414862			−0.207431	+	0.978250i	$0.566510\pi$
$168$		0			0
$169$	31.0000	−	166.132i	0.183432	−	0.983032i
$170$		0			0
$171$	48.4974			0.283611
$172$		0			0
$173$	230.000			1.32948			0.664740	−	0.747075i	$-0.268542\pi$
$173$	230.000			1.32948			0.664740	+	0.747075i	$0.268542\pi$
$174$		0			0
$175$	381.051			2.17744
$176$		0			0
$177$			315.652i			1.78334i
$178$		0			0
$179$		−	71.9375i		−	0.401885i	−0.979603	−	0.200943i	$-0.935600\pi$
$179$		−	71.9375i		−	0.401885i	0.979603	−	0.200943i	$-0.0644005\pi$
$180$		0			0
$181$	−170.000			−0.939227			−0.469613	−	0.882872i	$-0.655607\pi$
$181$	−170.000			−0.939227			−0.469613	+	0.882872i	$0.655607\pi$
$182$		0			0
$183$		−	191.833i		−	1.04827i
$184$		0			0
$185$	345.000			1.86486
$186$		0			0
$187$	69.2820			0.370492
$188$		0			0
$189$			207.666i			1.09876i
$190$		0			0
$191$		−	287.750i		−	1.50654i	−0.657709	−	0.753272i	$-0.728474\pi$
$191$		−	287.750i		−	1.50654i	0.657709	−	0.753272i	$-0.271526\pi$
$192$		0			0
$193$			182.746i			0.946869i	0.880829	+	0.473435i	$0.156986\pi$
$193$			182.746i			0.946869i	−0.880829	+	0.473435i	$0.843014\pi$
$194$		0			0
$195$	−398.372	+	330.912i	−2.04293	+	1.69699i
$196$		0			0
$197$			307.345i			1.56013i	0.625700	+	0.780064i	$0.284813\pi$
$197$			307.345i			1.56013i	−0.625700	+	0.780064i	$0.715187\pi$
$198$		0			0
$199$		−	287.750i		−	1.44598i	−0.690859	−	0.722990i	$-0.742767\pi$
$199$		−	287.750i		−	1.44598i	0.690859	−	0.722990i	$-0.257233\pi$
$200$		0			0
$201$		−	498.397i		−	2.47959i
$202$		0			0
$203$	−346.410			−1.70645
$204$		0			0
$205$		0			0
$206$		0			0
$207$			402.850i			1.94613i
$208$		0			0
$209$	48.0000			0.229665
$210$		0			0
$211$		−	215.812i		−	1.02281i	−0.859341	−	0.511404i	$-0.829126\pi$
$211$		−	215.812i		−	1.02281i	0.859341	−	0.511404i	$-0.170874\pi$
$212$		0			0
$213$		−	141.213i		−	0.662970i
$214$		0			0
$215$	358.535			1.66760
$216$		0			0
$217$	−180.000			−0.829493
$218$		0			0
$219$	398.372			1.81905
$220$		0			0
$221$	50.0000	−	41.5331i	0.226244	−	0.187933i
$222$		0			0
$223$	−320.429			−1.43690			−0.718452	−	0.695577i	$-0.755149\pi$
$223$	−320.429			−1.43690			−0.718452	+	0.695577i	$0.755149\pi$
$224$		0			0
$225$	616.000			2.73778
$226$		0			0
$227$	−381.051			−1.67864			−0.839320	−	0.543638i	$-0.817046\pi$
$227$	−381.051			−1.67864			−0.839320	+	0.543638i	$0.817046\pi$
$228$		0			0
$229$		−	41.5331i		−	0.181367i	−0.995880	−	0.0906837i	$-0.971095\pi$
$229$		−	41.5331i		−	0.181367i	0.995880	−	0.0906837i	$-0.0289052\pi$
$230$		0			0
$231$			575.500i			2.49134i
$232$		0			0
$233$	−205.000			−0.879828			−0.439914	−	0.898040i	$-0.644991\pi$
$233$	−205.000			−0.879828			−0.439914	+	0.898040i	$0.644991\pi$
$234$		0			0
$235$			647.437i			2.75505i
$236$		0			0
$237$	690.000			2.91139
$238$		0			0
$239$	64.0859			0.268142			0.134071	−	0.990972i	$-0.457195\pi$
$239$	64.0859			0.268142			0.134071	+	0.990972i	$0.457195\pi$
$240$		0			0
$241$			415.331i			1.72337i	0.507447	+	0.861683i	$0.330589\pi$
$241$			415.331i			1.72337i	−0.507447	+	0.861683i	$0.669411\pi$
$242$		0			0
$243$		−	268.567i		−	1.10521i
$244$		0			0
$245$		−	215.972i		−	0.881519i
$246$		0			0
$247$	34.6410	−	28.7750i	0.140247	−	0.116498i
$248$		0			0
$249$			83.0662i			0.333599i
$250$		0			0
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$		0			0
$253$			398.718i			1.57596i
$254$		0			0
$255$	−199.186			−0.781121
$256$		0			0
$257$	395.000			1.53696			0.768482	−	0.639871i	$-0.221012\pi$
$257$	395.000			1.53696			0.768482	+	0.639871i	$0.221012\pi$
$258$		0			0
$259$		−	359.687i		−	1.38875i
$260$		0			0
$261$	−560.000			−2.14559
$262$		0			0
$263$		−	115.100i		−	0.437642i	−0.975765	−	0.218821i	$-0.929779\pi$
$263$		−	115.100i		−	0.437642i	0.975765	−	0.218821i	$-0.0702211\pi$
$264$		0			0
$265$			83.0662i			0.313458i
$266$		0			0
$267$	−398.372			−1.49203
$268$		0			0
$269$	112.000			0.416357			0.208178	−	0.978091i	$-0.433247\pi$
$269$	112.000			0.416357			0.208178	+	0.978091i	$0.433247\pi$
$270$		0			0
$271$	39.8372			0.147001			0.0735003	−	0.997295i	$-0.476583\pi$
$271$	39.8372			0.147001			0.0735003	+	0.997295i	$0.476583\pi$
$272$		0			0
$273$	345.000	+	415.331i	1.26374	+	1.52136i
$274$		0			0
$275$	609.682			2.21703
$276$		0			0
$277$	−160.000			−0.577617			−0.288809	−	0.957387i	$-0.593259\pi$
$277$	−160.000			−0.577617			−0.288809	+	0.957387i	$0.593259\pi$
$278$		0			0
$279$	−290.985			−1.04296
$280$		0			0
$281$			83.0662i			0.295609i	0.989017	+	0.147805i	$0.0472207\pi$
$281$			83.0662i			0.295609i	−0.989017	+	0.147805i	$0.952779\pi$
$282$		0			0
$283$			115.100i			0.406714i	0.979105	+	0.203357i	$0.0651851\pi$
$283$			115.100i			0.406714i	−0.979105	+	0.203357i	$0.934815\pi$
$284$		0			0
$285$	−138.000			−0.484211
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−264.000			−0.913495
$290$		0			0
$291$	478.046			1.64277
$292$		0			0
$293$		−	456.864i		−	1.55926i	−0.626238	−	0.779632i	$-0.715406\pi$
$293$		−	456.864i		−	1.55926i	0.626238	−	0.779632i	$-0.284594\pi$
$294$		0			0
$295$		−	546.725i		−	1.85330i
$296$		0			0
$297$			332.265i			1.11874i
$298$		0			0
$299$	239.023	+	287.750i	0.799408	+	0.962374i
$300$		0			0
$301$		−	373.798i		−	1.24185i
$302$		0			0
$303$		−	249.383i		−	0.823047i
$304$		0			0
$305$			332.265i			1.08939i
$306$		0			0
$307$	259.808			0.846279			0.423139	−	0.906065i	$-0.360928\pi$
$307$	259.808			0.846279			0.423139	+	0.906065i	$0.360928\pi$
$308$		0			0
$309$	−138.000			−0.446602
$310$		0			0
$311$			143.875i			0.462620i	0.972880	+	0.231310i	$0.0743012\pi$
$311$			143.875i			0.462620i	−0.972880	+	0.231310i	$0.925699\pi$
$312$		0			0
$313$	−365.000			−1.16613			−0.583067	−	0.812424i	$-0.698148\pi$
$313$	−365.000			−1.16613			−0.583067	+	0.812424i	$0.698148\pi$
$314$		0			0
$315$		−	1007.12i		−	3.19722i
$316$		0			0
$317$		−	83.0662i		−	0.262039i	−0.991380	−	0.131019i	$-0.958175\pi$
$317$		−	83.0662i		−	0.262039i	0.991380	−	0.131019i	$-0.0418250\pi$
$318$		0			0
$319$	−554.256			−1.73748
$320$		0			0
$321$	−552.000			−1.71963
$322$		0			0
$323$	17.3205			0.0536239
$324$		0			0
$325$	440.000	−	365.491i	1.35385	−	1.12459i
$326$		0			0
$327$	199.186			0.609131
$328$		0			0
$329$	675.000			2.05167
$330$		0			0
$331$	−135.100			−0.408157			−0.204078	−	0.978955i	$-0.565420\pi$
$331$	−135.100			−0.408157			−0.204078	+	0.978955i	$0.565420\pi$
$332$		0			0
$333$		−	581.464i		−	1.74614i
$334$		0			0
$335$			863.250i			2.57686i
$336$		0			0
$337$	505.000			1.49852			0.749258	−	0.662278i	$-0.230410\pi$
$337$	505.000			1.49852			0.749258	+	0.662278i	$0.230410\pi$
$338$		0			0
$339$			47.9583i			0.141470i
$340$		0			0
$341$	−288.000			−0.844575
$342$		0			0
$343$	199.186			0.580717
$344$		0			0
$345$		−	1146.31i		−	3.32265i
$346$		0			0
$347$		−	330.912i		−	0.953638i	−0.879002	−	0.476819i	$-0.841790\pi$
$347$		−	330.912i		−	0.953638i	0.879002	−	0.476819i	$-0.158210\pi$
$348$		0			0
$349$			373.798i			1.07105i	0.844518	+	0.535527i	$0.179887\pi$
$349$			373.798i			1.07105i	−0.844518	+	0.535527i	$0.820113\pi$
$350$		0			0
$351$	199.186	+	239.792i	0.567481	+	0.683167i
$352$		0			0
$353$			249.199i			0.705945i	0.935634	+	0.352973i	$0.114829\pi$
$353$			249.199i			0.705945i	−0.935634	+	0.352973i	$0.885171\pi$
$354$		0			0
$355$			244.587i			0.688979i
$356$		0			0
$357$			207.666i			0.581696i
$358$		0			0
$359$	187.061			0.521063			0.260531	−	0.965465i	$-0.416102\pi$
$359$	187.061			0.521063			0.260531	+	0.965465i	$0.416102\pi$
$360$		0			0
$361$	−349.000			−0.966759
$362$		0			0
$363$			340.504i			0.938028i
$364$		0			0
$365$	−690.000			−1.89041
$366$		0			0
$367$			460.400i			1.25450i	0.778820	+	0.627248i	$0.215819\pi$
$367$			460.400i			1.25450i	−0.778820	+	0.627248i	$0.784181\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	86.6025			0.233430
$372$		0			0
$373$	−140.000			−0.375335			−0.187668	−	0.982233i	$-0.560093\pi$
$373$	−140.000			−0.375335			−0.187668	+	0.982233i	$0.560093\pi$
$374$		0			0
$375$	−756.906			−2.01842
$376$		0			0
$377$	−400.000	+	332.265i	−1.06101	+	0.881339i
$378$		0			0
$379$	−308.305			−0.813470			−0.406735	−	0.913546i	$-0.633333\pi$
$379$	−308.305			−0.813470			−0.406735	+	0.913546i	$0.633333\pi$
$380$		0			0
$381$	−138.000			−0.362205
$382$		0			0
$383$	−112.583			−0.293951			−0.146976	−	0.989140i	$-0.546954\pi$
$383$	−112.583			−0.293951			−0.146976	+	0.989140i	$0.546954\pi$
$384$		0			0
$385$		−	996.795i		−	2.58908i
$386$		0			0
$387$		−	604.275i		−	1.56143i
$388$		0			0
$389$	−122.000			−0.313625			−0.156812	−	0.987628i	$-0.550122\pi$
$389$	−122.000			−0.313625			−0.156812	+	0.987628i	$0.550122\pi$
$390$		0			0
$391$			143.875i			0.367967i
$392$		0			0
$393$	−345.000			−0.877863
$394$		0			0
$395$	−1195.12			−3.02561
$396$		0			0
$397$			581.464i			1.46464i	0.680959	+	0.732322i	$0.261563\pi$
$397$			581.464i			1.46464i	−0.680959	+	0.732322i	$0.738437\pi$
$398$		0			0
$399$			143.875i			0.360589i
$400$		0			0
$401$		−	415.331i		−	1.03574i	−0.855460	−	0.517869i	$-0.826725\pi$
$401$		−	415.331i		−	1.03574i	0.855460	−	0.517869i	$-0.173275\pi$
$402$		0			0
$403$	−207.846	+	172.650i	−0.515747	+	0.428412i
$404$		0			0
$405$			91.3729i			0.225612i
$406$		0			0
$407$		−	575.500i		−	1.41400i
$408$		0			0
$409$			415.331i			1.01548i	0.861510	+	0.507740i	$0.169519\pi$
$409$			415.331i			1.01548i	−0.861510	+	0.507740i	$0.830481\pi$
$410$		0			0
$411$	−79.6743			−0.193855
$412$		0			0
$413$	−570.000			−1.38015
$414$		0			0
$415$		−	143.875i		−	0.346687i
$416$		0			0
$417$	−345.000			−0.827338
$418$		0			0
$419$			503.562i			1.20182i	0.799317	+	0.600910i	$0.205195\pi$
$419$			503.562i			1.20182i	−0.799317	+	0.600910i	$0.794805\pi$
$420$		0			0
$421$		−	706.063i		−	1.67711i	−0.544817	−	0.838555i	$-0.683401\pi$
$421$		−	706.063i		−	1.67711i	0.544817	−	0.838555i	$-0.316599\pi$
$422$		0			0
$423$	1091.19			2.57965
$424$		0			0
$425$	220.000			0.517647
$426$		0			0
$427$	346.410			0.811265
$428$		0			0
$429$	552.000	+	664.530i	1.28671	+	1.54902i
$430$		0			0
$431$	−351.606			−0.815792			−0.407896	−	0.913028i	$-0.633737\pi$
$431$	−351.606			−0.815792			−0.407896	+	0.913028i	$0.633737\pi$
$432$		0			0
$433$	425.000			0.981524			0.490762	−	0.871294i	$-0.336718\pi$
$433$	425.000			0.981524			0.490762	+	0.871294i	$0.336718\pi$
$434$		0			0
$435$	1593.49			3.66319
$436$		0			0
$437$			99.6795i			0.228100i
$438$		0			0
$439$			143.875i			0.327733i	0.986482	+	0.163867i	$0.0523967\pi$
$439$			143.875i			0.327733i	−0.986482	+	0.163867i	$0.947603\pi$
$440$		0			0
$441$	−364.000			−0.825397
$442$		0			0
$443$			676.212i			1.52644i	0.646140	+	0.763219i	$0.276382\pi$
$443$			676.212i			1.52644i	−0.646140	+	0.763219i	$0.723618\pi$
$444$		0			0
$445$	690.000			1.55056
$446$		0			0
$447$	398.372			0.891212
$448$		0			0
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$			440.251i			0.971857i
$454$		0			0
$455$	−597.558	−	719.375i	−1.31331	−	1.58104i
$456$		0			0
$457$		−	182.746i		−	0.399881i	−0.979808	−	0.199941i	$-0.935925\pi$
$457$		−	182.746i		−	0.399881i	0.979808	−	0.199941i	$-0.0640749\pi$
$458$		0			0
$459$			119.896i			0.261211i
$460$		0			0
$461$		−	373.798i		−	0.810842i	−0.914130	−	0.405421i	$-0.867125\pi$
$461$		−	373.798i		−	0.810842i	0.914130	−	0.405421i	$-0.132875\pi$
$462$		0			0
$463$	34.6410			0.0748186			0.0374093	−	0.999300i	$-0.488089\pi$
$463$	34.6410			0.0748186			0.0374093	+	0.999300i	$0.488089\pi$
$464$		0			0
$465$	828.000			1.78065
$466$		0			0
$467$		−	690.600i		−	1.47880i	−0.673266	−	0.739400i	$-0.735109\pi$
$467$		−	690.600i		−	1.47880i	0.673266	−	0.739400i	$-0.264891\pi$
$468$		0			0
$469$	900.000			1.91898
$470$		0			0
$471$		−	815.291i		−	1.73098i
$472$		0			0
$473$		−	598.077i		−	1.26443i
$474$		0			0
$475$	152.420			0.320885
$476$		0			0
$477$	140.000			0.293501
$478$		0			0
$479$	594.093			1.24028			0.620139	−	0.784492i	$-0.287076\pi$
$479$	594.093			1.24028			0.620139	+	0.784492i	$0.287076\pi$
$480$		0			0
$481$	−345.000	−	415.331i	−0.717256	−	0.863474i
$482$		0			0
$483$	−1195.12			−2.47436
$484$		0			0
$485$	−828.000			−1.70722
$486$		0			0
$487$	69.2820			0.142263			0.0711315	−	0.997467i	$-0.477339\pi$
$487$	69.2820			0.142263			0.0711315	+	0.997467i	$0.477339\pi$
$488$		0			0
$489$			747.596i			1.52883i
$490$		0			0
$491$		−	359.687i		−	0.732561i	−0.930505	−	0.366280i	$-0.880631\pi$
$491$		−	359.687i		−	0.732561i	0.930505	−	0.366280i	$-0.119369\pi$
$492$		0			0
$493$	−200.000			−0.405680
$494$		0			0
$495$		−	1611.40i		−	3.25535i
$496$		0			0
$497$	255.000			0.513078
$498$		0			0
$499$	−713.605			−1.43007			−0.715035	−	0.699089i	$-0.753589\pi$
$499$	−713.605			−1.43007			−0.715035	+	0.699089i	$0.753589\pi$
$500$		0			0
$501$		−	332.265i		−	0.663204i
$502$		0			0
$503$			546.725i			1.08693i	0.839432	+	0.543464i	$0.182888\pi$
$503$			546.725i			1.08693i	−0.839432	+	0.543464i	$0.817112\pi$
$504$		0			0
$505$			431.944i			0.855336i
$506$		0			0
$507$	796.743	+	148.671i	1.57149	+	0.293236i
$508$		0			0
$509$			249.199i			0.489585i	0.969576	+	0.244792i	$0.0787198\pi$
$509$			249.199i			0.489585i	−0.969576	+	0.244792i	$0.921280\pi$
$510$		0			0
$511$			719.375i			1.40778i
$512$		0			0
$513$			83.0662i			0.161922i
$514$		0			0
$515$	239.023			0.464122
$516$		0			0
$517$	1080.00			2.08897
$518$		0			0
$519$			1103.04i			2.12532i
$520$		0			0
$521$	235.000			0.451056			0.225528	−	0.974237i	$-0.427589\pi$
$521$	235.000			0.451056			0.225528	+	0.974237i	$0.427589\pi$
$522$		0			0
$523$			748.150i			1.43050i	0.698870	+	0.715248i	$0.253686\pi$
$523$			748.150i			1.43050i	−0.698870	+	0.715248i	$0.746314\pi$
$524$		0			0
$525$			1827.46i			3.48087i
$526$		0			0
$527$	−103.923			−0.197197
$528$		0			0
$529$	−299.000			−0.565217
$530$		0			0
$531$	−921.451			−1.73531
$532$		0			0
$533$		0			0
$534$		0			0
$535$	956.092			1.78709
$536$		0			0
$537$	345.000			0.642458
$538$		0			0
$539$	−360.267			−0.668398
$540$		0			0
$541$			290.732i			0.537397i	0.963224	+	0.268699i	$0.0865935\pi$
$541$			290.732i			0.537397i	−0.963224	+	0.268699i	$0.913406\pi$
$542$		0			0
$543$		−	815.291i		−	1.50146i
$544$		0			0
$545$	−345.000			−0.633028
$546$		0			0
$547$		−	244.587i		−	0.447143i	−0.974687	−	0.223572i	$-0.928228\pi$
$547$		−	244.587i		−	0.447143i	0.974687	−	0.223572i	$-0.0717717\pi$
$548$		0			0
$549$	560.000			1.02004
$550$		0			0
$551$	−138.564			−0.251477
$552$		0			0
$553$			1245.99i			2.25315i
$554$		0			0
$555$			1654.56i			2.98119i
$556$		0			0
$557$		−	539.931i		−	0.969355i	−0.874693	−	0.484677i	$-0.838937\pi$
$557$		−	539.931i		−	0.969355i	0.874693	−	0.484677i	$-0.161063\pi$
$558$		0			0
$559$	−358.535	−	431.625i	−0.641386	−	0.772137i
$560$		0			0
$561$			332.265i			0.592273i
$562$		0			0
$563$			618.662i			1.09887i	0.835537	+	0.549434i	$0.185156\pi$
$563$			618.662i			1.09887i	−0.835537	+	0.549434i	$0.814844\pi$
$564$		0			0
$565$		−	83.0662i		−	0.147020i
$566$		0			0
$567$	95.2628			0.168012
$568$		0			0
$569$	−775.000			−1.36204			−0.681019	−	0.732265i	$-0.738463\pi$
$569$	−775.000			−1.36204			−0.681019	+	0.732265i	$0.738463\pi$
$570$		0			0
$571$		−	791.312i		−	1.38584i	−0.721017	−	0.692918i	$-0.756325\pi$
$571$		−	791.312i		−	1.38584i	0.721017	−	0.692918i	$-0.243675\pi$
$572$		0			0
$573$	1380.00			2.40838
$574$		0			0
$575$			1266.10i			2.20191i
$576$		0			0
$577$			581.464i			1.00774i	0.863781	+	0.503868i	$0.168090\pi$
$577$			581.464i			1.00774i	−0.863781	+	0.503868i	$0.831910\pi$
$578$		0			0
$579$	−876.418			−1.51367
$580$		0			0
$581$	−150.000			−0.258176
$582$		0			0
$583$	138.564			0.237674
$584$		0			0
$585$	−966.000	−	1162.93i	−1.65128	−	1.98791i
$586$		0			0
$587$	259.808			0.442602			0.221301	−	0.975206i	$-0.428970\pi$
$587$	259.808			0.442602			0.221301	+	0.975206i	$0.428970\pi$
$588$		0			0
$589$	−72.0000			−0.122241
$590$		0			0
$591$	−1473.98			−2.49404
$592$		0			0
$593$			232.585i			0.392218i	0.980582	+	0.196109i	$0.0628307\pi$
$593$			232.585i			0.392218i	−0.980582	+	0.196109i	$0.937169\pi$
$594$		0			0
$595$		−	359.687i		−	0.604517i
$596$		0			0
$597$	1380.00			2.31156
$598$		0			0
$599$			431.625i			0.720576i	0.932841	+	0.360288i	$0.117322\pi$
$599$			431.625i			0.720576i	−0.932841	+	0.360288i	$0.882678\pi$
$600$		0			0
$601$	−527.000			−0.876872			−0.438436	−	0.898762i	$-0.644467\pi$
$601$	−527.000			−0.876872			−0.438436	+	0.898762i	$0.644467\pi$
$602$		0			0
$603$	1454.92			2.41281
$604$		0			0
$605$		−	589.770i		−	0.974827i
$606$		0			0
$607$		−	28.7750i		−	0.0474053i	−0.999719	−	0.0237026i	$-0.992455\pi$
$607$		−	28.7750i		−	0.0474053i	0.999719	−	0.0237026i	$-0.00754549\pi$
$608$		0			0
$609$		−	1661.32i		−	2.72796i
$610$		0			0
$611$	779.423	−	647.437i	1.27565	−	1.05964i
$612$		0			0
$613$		−	847.276i		−	1.38218i	−0.722769	−	0.691089i	$-0.757131\pi$
$613$		−	847.276i		−	1.38218i	0.722769	−	0.691089i	$-0.242869\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	564.850i		−	0.915479i	−0.889086	−	0.457739i	$-0.848659\pi$
$617$		−	564.850i		−	0.915479i	0.889086	−	0.457739i	$-0.151341\pi$
$618$		0			0
$619$	−55.4256			−0.0895406			−0.0447703	−	0.998997i	$-0.514256\pi$
$619$	−55.4256			−0.0895406			−0.0447703	+	0.998997i	$0.514256\pi$
$620$		0			0
$621$	−690.000			−1.11111
$622$		0			0
$623$		−	719.375i		−	1.15469i
$624$		0			0
$625$	211.000			0.337600
$626$		0			0
$627$			230.200i			0.367145i
$628$		0			0
$629$		−	207.666i		−	0.330152i
$630$		0			0
$631$	−1113.71			−1.76499			−0.882495	−	0.470322i	$-0.844138\pi$
$631$	−1113.71			−1.76499			−0.882495	+	0.470322i	$0.844138\pi$
$632$		0			0
$633$	1035.00			1.63507
$634$		0			0
$635$	239.023			0.376414
$636$		0			0
$637$	−260.000	+	215.972i	−0.408163	+	0.339046i
$638$		0			0
$639$	412.228			0.645114
$640$		0			0
$641$	−482.000			−0.751950			−0.375975	−	0.926630i	$-0.622692\pi$
$641$	−482.000			−0.751950			−0.375975	+	0.926630i	$0.622692\pi$
$642$		0			0
$643$	−450.333			−0.700363			−0.350181	−	0.936682i	$-0.613880\pi$
$643$	−450.333			−0.700363			−0.350181	+	0.936682i	$0.613880\pi$
$644$		0			0
$645$			1719.47i			2.66585i
$646$		0			0
$647$		−	1122.22i		−	1.73450i	−0.497869	−	0.867252i	$-0.665884\pi$
$647$		−	1122.22i		−	1.73450i	0.497869	−	0.867252i	$-0.334116\pi$
$648$		0			0
$649$	−912.000			−1.40524
$650$		0			0
$651$		−	863.250i		−	1.32604i
$652$		0			0
$653$	460.000			0.704441			0.352221	−	0.935917i	$-0.385427\pi$
$653$	460.000			0.704441			0.352221	+	0.935917i	$0.385427\pi$
$654$		0			0
$655$	597.558			0.912302
$656$		0			0
$657$			1162.93i			1.77006i
$658$		0			0
$659$		−	287.750i		−	0.436646i	−0.975877	−	0.218323i	$-0.929941\pi$
$659$		−	287.750i		−	0.436646i	0.975877	−	0.218323i	$-0.0700587\pi$
$660$		0			0
$661$			1079.86i			1.63368i	0.576866	+	0.816839i	$0.304276\pi$
$661$			1079.86i			1.63368i	−0.576866	+	0.816839i	$0.695724\pi$
$662$		0			0
$663$	199.186	+	239.792i	0.300431	+	0.361677i
$664$		0			0
$665$		−	249.199i		−	0.374735i
$666$		0			0
$667$		−	1151.00i		−	1.72564i
$668$		0			0
$669$		−	1536.73i		−	2.29705i
$670$		0			0
$671$	554.256			0.826015
$672$		0			0
$673$	535.000			0.794948			0.397474	−	0.917613i	$-0.369887\pi$
$673$	535.000			0.794948			0.397474	+	0.917613i	$0.369887\pi$
$674$		0			0
$675$			1055.08i			1.56309i
$676$		0			0
$677$	−970.000			−1.43279			−0.716396	−	0.697694i	$-0.754210\pi$
$677$	−970.000			−1.43279			−0.716396	+	0.697694i	$0.754210\pi$
$678$		0			0
$679$			863.250i			1.27135i
$680$		0			0
$681$		−	1827.46i		−	2.68349i
$682$		0			0
$683$	155.885			0.228235			0.114118	−	0.993467i	$-0.463596\pi$
$683$	155.885			0.228235			0.114118	+	0.993467i	$0.463596\pi$
$684$		0			0
$685$	138.000			0.201460
$686$		0			0
$687$	199.186			0.289936
$688$		0			0
$689$	100.000	−	83.0662i	0.145138	−	0.120561i
$690$		0			0
$691$	1233.22			1.78469			0.892345	−	0.451355i	$-0.149059\pi$
$691$	1233.22			1.78469			0.892345	+	0.451355i	$0.149059\pi$
$692$		0			0
$693$	−1680.00			−2.42424
$694$		0			0
$695$	597.558			0.859795
$696$		0			0
$697$		0			0
$698$		0			0
$699$		−	983.145i		−	1.40650i
$700$		0			0
$701$	502.000			0.716120			0.358060	−	0.933699i	$-0.383438\pi$
$701$	502.000			0.716120			0.358060	+	0.933699i	$0.383438\pi$
$702$		0			0
$703$		−	143.875i		−	0.204659i
$704$		0			0
$705$	−3105.00			−4.40426
$706$		0			0
$707$	450.333			0.636964
$708$		0			0
$709$			581.464i			0.820118i	0.912059	+	0.410059i	$0.134492\pi$
$709$			581.464i			0.820118i	−0.912059	+	0.410059i	$0.865508\pi$
$710$		0			0
$711$			2014.25i			2.83298i
$712$		0			0
$713$		−	598.077i		−	0.838818i
$714$		0			0
$715$	−956.092	−	1151.00i	−1.33719	−	1.60979i
$716$		0			0
$717$			307.345i			0.428654i
$718$		0			0
$719$			575.500i			0.800417i	0.916424	+	0.400208i	$0.131062\pi$
$719$			575.500i			0.800417i	−0.916424	+	0.400208i	$0.868938\pi$
$720$		0			0
$721$		−	249.199i		−	0.345629i
$722$		0			0
$723$	−1991.86			−2.75499
$724$		0			0
$725$	−1760.00			−2.42759
$726$		0			0
$727$		−	172.650i		−	0.237483i	−0.992925	−	0.118741i	$-0.962114\pi$
$727$		−	172.650i		−	0.237483i	0.992925	−	0.118741i	$-0.0378859\pi$
$728$		0			0
$729$	1189.00			1.63100
$730$		0			0
$731$		−	215.812i		−	0.295229i
$732$		0			0
$733$			473.478i			0.645945i	0.946408	+	0.322972i	$0.104682\pi$
$733$			473.478i			0.645945i	−0.946408	+	0.322972i	$0.895318\pi$
$734$		0			0
$735$	1035.77			1.40921
$736$		0			0
$737$	1440.00			1.95387
$738$		0			0
$739$	921.451			1.24689			0.623445	−	0.781868i	$-0.285733\pi$
$739$	921.451			1.24689			0.623445	+	0.781868i	$0.285733\pi$
$740$		0			0
$741$	138.000	+	166.132i	0.186235	+	0.224200i
$742$		0			0
$743$	−1030.57			−1.38704			−0.693520	−	0.720438i	$-0.743941\pi$
$743$	−1030.57			−1.38704			−0.693520	+	0.720438i	$0.743941\pi$
$744$		0			0
$745$	−690.000			−0.926174
$746$		0			0
$747$	−242.487			−0.324615
$748$		0			0
$749$		−	996.795i		−	1.33083i
$750$		0			0
$751$			575.500i			0.766311i	0.923684	+	0.383156i	$0.125163\pi$
$751$			575.500i			0.766311i	−0.923684	+	0.383156i	$0.874837\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	762.537i		−	1.00998i
$756$		0			0
$757$	−910.000			−1.20211			−0.601057	−	0.799206i	$-0.705253\pi$
$757$	−910.000			−1.20211			−0.601057	+	0.799206i	$0.705253\pi$
$758$		0			0
$759$	−1912.18			−2.51935
$760$		0			0
$761$		−	581.464i		−	0.764078i	−0.924146	−	0.382039i	$-0.875222\pi$
$761$		−	581.464i		−	0.764078i	0.924146	−	0.382039i	$-0.124778\pi$
$762$		0			0
$763$			359.687i			0.471412i
$764$		0			0
$765$		−	581.464i		−	0.760083i
$766$		0			0
$767$	−658.179	+	546.725i	−0.858122	+	0.712809i
$768$		0			0
$769$			332.265i			0.432074i	0.976385	+	0.216037i	$0.0693132\pi$
$769$			332.265i			0.432074i	−0.976385	+	0.216037i	$0.930687\pi$
$770$		0			0
$771$			1894.35i			2.45701i
$772$		0			0
$773$		−	1204.46i		−	1.55816i	−0.626922	−	0.779082i	$-0.715686\pi$
$773$		−	1204.46i		−	1.55816i	0.626922	−	0.779082i	$-0.284314\pi$
$774$		0			0
$775$	−914.523			−1.18003
$776$		0			0
$777$	1725.00			2.22008
$778$		0			0
$779$		0			0
$780$		0			0
$781$	408.000			0.522407
$782$		0			0
$783$		−	959.166i		−	1.22499i
$784$		0			0
$785$			1412.13i			1.79889i
$786$		0			0
$787$	1489.56			1.89271			0.946356	−	0.323127i	$-0.104734\pi$
$787$	1489.56			1.89271			0.946356	+	0.323127i	$0.104734\pi$
$788$		0			0
$789$	552.000			0.699620
$790$		0			0
$791$	−86.6025			−0.109485
$792$		0			0
$793$	400.000	−	332.265i	0.504414	−	0.418997i
$794$		0			0
$795$	−398.372			−0.501096
$796$		0			0
$797$	−530.000			−0.664994			−0.332497	−	0.943104i	$-0.607891\pi$
$797$	−530.000			−0.664994			−0.332497	+	0.943104i	$0.607891\pi$
$798$		0			0
$799$	389.711			0.487749
$800$		0			0
$801$		−	1162.93i		−	1.45184i
$802$		0			0
$803$			1151.00i			1.43337i
$804$		0			0
$805$	2070.00			2.57143
$806$		0			0
$807$			537.133i			0.665592i
$808$		0			0
$809$	−65.0000			−0.0803461			−0.0401731	−	0.999193i	$-0.512791\pi$
$809$	−65.0000			−0.0803461			−0.0401731	+	0.999193i	$0.512791\pi$
$810$		0			0
$811$	886.810			1.09348			0.546739	−	0.837303i	$-0.315869\pi$
$811$	886.810			1.09348			0.546739	+	0.837303i	$0.315869\pi$
$812$		0			0
$813$			191.052i			0.234997i
$814$		0			0
$815$		−	1294.87i		−	1.58880i
$816$		0			0
$817$		−	149.519i		−	0.183010i
$818$		0			0
$819$	−1212.44	+	1007.12i	−1.48039	+	1.22970i
$820$		0			0
$821$		−	539.931i		−	0.657650i	−0.944391	−	0.328825i	$-0.893347\pi$
$821$		−	539.931i		−	0.657650i	0.944391	−	0.328825i	$-0.106653\pi$
$822$		0			0
$823$		−	546.725i		−	0.664307i	−0.943225	−	0.332154i	$-0.892225\pi$
$823$		−	546.725i		−	0.664307i	0.943225	−	0.332154i	$-0.107775\pi$
$824$		0			0
$825$			2923.93i			3.54416i
$826$		0			0
$827$	259.808			0.314157			0.157078	−	0.987586i	$-0.449792\pi$
$827$	259.808			0.314157			0.157078	+	0.987586i	$0.449792\pi$
$828$		0			0
$829$	−1180.00			−1.42340			−0.711701	−	0.702483i	$-0.752075\pi$
$829$	−1180.00			−1.42340			−0.711701	+	0.702483i	$0.752075\pi$
$830$		0			0
$831$		−	767.333i		−	0.923385i
$832$		0			0
$833$	−130.000			−0.156062
$834$		0			0
$835$			575.500i			0.689221i
$836$		0			0
$837$		−	498.397i		−	0.595457i
$838$		0			0
$839$	−394.908			−0.470688			−0.235344	−	0.971912i	$-0.575622\pi$
$839$	−394.908			−0.470688			−0.235344	+	0.971912i	$0.575622\pi$
$840$		0			0
$841$	759.000			0.902497
$842$		0			0
$843$	−398.372			−0.472564
$844$		0			0
$845$	−1380.00	−	257.505i	−1.63314	−	0.304740i
$846$		0			0
$847$	−614.878			−0.725948
$848$		0			0
$849$	−552.000			−0.650177
$850$		0			0
$851$	1195.12			1.40437
$852$		0			0
$853$			622.997i			0.730360i	0.930937	+	0.365180i	$0.118993\pi$
$853$			622.997i			0.730360i	−0.930937	+	0.365180i	$0.881007\pi$
$854$		0			0
$855$		−	402.850i		−	0.471169i
$856$		0			0
$857$	410.000			0.478413			0.239207	−	0.970969i	$-0.423113\pi$
$857$	410.000			0.478413			0.239207	+	0.970969i	$0.423113\pi$
$858$		0			0
$859$			863.250i			1.00495i	0.864593	+	0.502474i	$0.167577\pi$
$859$			863.250i			1.00495i	−0.864593	+	0.502474i	$0.832423\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	8.66025			0.0100351			0.00501753	−	0.999987i	$-0.498403\pi$
$863$	8.66025			0.0100351			0.00501753	+	0.999987i	$0.498403\pi$
$864$		0			0
$865$		−	1910.52i		−	2.20870i
$866$		0			0
$867$		−	1266.10i		−	1.46032i
$868$		0			0
$869$			1993.59i			2.29412i
$870$		0			0
$871$	1039.23	−	863.250i	1.19315	−	0.991102i
$872$		0			0
$873$			1395.51i			1.59853i
$874$		0			0
$875$		−	1366.81i		−	1.56207i
$876$		0			0
$877$			1104.78i			1.25973i	0.776706	+	0.629864i	$0.216889\pi$
$877$			1104.78i			1.25973i	−0.776706	+	0.629864i	$0.783111\pi$
$878$		0			0
$879$	2191.04			2.49266
$880$		0			0
$881$	−25.0000			−0.0283768			−0.0141884	−	0.999899i	$-0.504516\pi$
$881$	−25.0000			−0.0283768			−0.0141884	+	0.999899i	$0.504516\pi$
$882$		0			0
$883$			187.037i			0.211820i	0.994376	+	0.105910i	$0.0337756\pi$
$883$			187.037i			0.211820i	−0.994376	+	0.105910i	$0.966224\pi$
$884$		0			0
$885$	2622.00			2.96271
$886$		0			0
$887$			172.650i			0.194645i	0.995253	+	0.0973224i	$0.0310278\pi$
$887$			172.650i			0.194645i	−0.995253	+	0.0973224i	$0.968972\pi$
$888$		0			0
$889$		−	249.199i		−	0.280314i
$890$		0			0
$891$	152.420			0.171067
$892$		0			0
$893$	270.000			0.302352
$894$		0			0
$895$	−597.558			−0.667662
$896$		0			0
$897$	−1380.00	+	1146.31i	−1.53846	+	1.27794i
$898$		0			0
$899$	831.384			0.924788
$900$		0			0
$901$	50.0000			0.0554939
$902$		0			0
$903$	1792.67			1.98524
$904$		0			0
$905$			1412.13i			1.56036i
$906$		0			0
$907$			963.962i			1.06280i	0.847120	+	0.531401i	$0.178334\pi$
$907$			963.962i			1.06280i	−0.847120	+	0.531401i	$0.821666\pi$
$908$		0			0
$909$	728.000			0.800880
$910$		0			0
$911$			863.250i			0.947585i	0.880637	+	0.473792i	$0.157115\pi$
$911$			863.250i			0.947585i	−0.880637	+	0.473792i	$0.842885\pi$
$912$		0			0
$913$	−240.000			−0.262870
$914$		0			0
$915$	−1593.49			−1.74152
$916$		0			0
$917$		−	622.997i		−	0.679386i
$918$		0			0
$919$		−	863.250i		−	0.939336i	−0.882843	−	0.469668i	$-0.844374\pi$
$919$		−	863.250i		−	0.939336i	0.882843	−	0.469668i	$-0.155626\pi$
$920$		0			0
$921$			1245.99i			1.35287i
$922$		0			0
$923$	294.449	−	244.587i	0.319013	−	0.264992i
$924$		0			0
$925$		−	1827.46i		−	1.97563i
$926$		0			0
$927$		−	402.850i		−	0.434574i
$928$		0			0
$929$			1162.93i			1.25181i	0.779901	+	0.625903i	$0.215269\pi$
$929$			1162.93i			1.25181i	−0.779901	+	0.625903i	$0.784731\pi$
$930$		0			0
$931$	−90.0666			−0.0967418
$932$		0			0
$933$	−690.000			−0.739550
$934$		0			0
$935$		−	575.500i		−	0.615508i
$936$		0			0
$937$	950.000			1.01387			0.506937	−	0.861983i	$-0.330778\pi$
$937$	950.000			1.01387			0.506937	+	0.861983i	$0.330778\pi$
$938$		0			0
$939$		−	1750.48i		−	1.86419i
$940$		0			0
$941$			41.5331i			0.0441372i	0.999756	+	0.0220686i	$0.00702523\pi$
$941$			41.5331i			0.0441372i	−0.999756	+	0.0220686i	$0.992975\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$	1725.00			1.82540
$946$		0			0
$947$	−1662.77			−1.75583			−0.877914	−	0.478819i	$-0.841065\pi$
$947$	−1662.77			−1.75583			−0.877914	+	0.478819i	$0.841065\pi$
$948$		0			0
$949$	690.000	+	830.662i	0.727081	+	0.875303i
$950$		0			0
$951$	398.372			0.418898
$952$		0			0
$953$	1085.00			1.13851			0.569255	−	0.822161i	$-0.307232\pi$
$953$	1085.00			1.13851			0.569255	+	0.822161i	$0.307232\pi$
$954$		0			0
$955$	−2390.23			−2.50286
$956$		0			0
$957$		−	2658.12i		−	2.77755i
$958$		0			0
$959$		−	143.875i		−	0.150026i
$960$		0			0
$961$	−529.000			−0.550468
$962$		0			0
$963$		−	1611.40i		−	1.67331i
$964$		0			0
$965$	1518.00			1.57306
$966$		0			0
$967$	−216.506			−0.223895			−0.111947	−	0.993714i	$-0.535709\pi$
$967$	−216.506			−0.223895			−0.111947	+	0.993714i	$0.535709\pi$
$968$		0			0
$969$			83.0662i			0.0857237i
$970$		0			0
$971$			215.812i			0.222258i	0.993806	+	0.111129i	$0.0354467\pi$
$971$			215.812i			0.222258i	−0.993806	+	0.111129i	$0.964553\pi$
$972$		0			0
$973$		−	622.997i		−	0.640284i
$974$		0			0
$975$	1752.84	+	2110.17i	1.79778	+	2.16427i
$976$		0			0
$977$			498.397i			0.510130i	0.966924	+	0.255065i	$0.0820969\pi$
$977$			498.397i			0.510130i	−0.966924	+	0.255065i	$0.917903\pi$
$978$		0			0
$979$		−	1151.00i		−	1.17569i
$980$		0			0
$981$			581.464i			0.592725i
$982$		0			0
$983$	−320.429			−0.325971			−0.162985	−	0.986628i	$-0.552112\pi$
$983$	−320.429			−0.325971			−0.162985	+	0.986628i	$0.552112\pi$
$984$		0			0
$985$	2553.00			2.59188
$986$		0			0
$987$			3237.19i			3.27982i
$988$		0			0
$989$	1242.00			1.25581
$990$		0			0
$991$		−	1294.87i		−	1.30663i	−0.757084	−	0.653317i	$-0.773377\pi$
$991$		−	1294.87i		−	1.30663i	0.757084	−	0.653317i	$-0.226623\pi$
$992$		0			0
$993$		−	647.917i		−	0.652484i
$994$		0			0
$995$	−2390.23			−2.40224
$996$		0			0
$997$	−1120.00			−1.12337			−0.561685	−	0.827351i	$-0.689847\pi$
$997$	−1120.00			−1.12337			−0.561685	+	0.827351i	$0.689847\pi$
$998$		0			0
$999$	995.929			0.996926

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	208.3.c.b.207.3	yes	4
3.2	odd	2		1872.3.i.k.415.3		4
4.3	odd	2	inner	208.3.c.b.207.1	✓	4
8.3	odd	2		832.3.c.e.831.4		4
8.5	even	2		832.3.c.e.831.2		4
12.11	even	2		1872.3.i.k.415.4		4
13.12	even	2	inner	208.3.c.b.207.4	yes	4
39.38	odd	2		1872.3.i.k.415.2		4
52.51	odd	2	inner	208.3.c.b.207.2	yes	4
104.51	odd	2		832.3.c.e.831.3		4
104.77	even	2		832.3.c.e.831.1		4
156.155	even	2		1872.3.i.k.415.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
208.3.c.b.207.1	✓	4	4.3	odd	2	inner
208.3.c.b.207.2	yes	4	52.51	odd	2	inner
208.3.c.b.207.3	yes	4	1.1	even	1	trivial
208.3.c.b.207.4	yes	4	13.12	even	2	inner
832.3.c.e.831.1		4	104.77	even	2
832.3.c.e.831.2		4	8.5	even	2
832.3.c.e.831.3		4	104.51	odd	2
832.3.c.e.831.4		4	8.3	odd	2
1872.3.i.k.415.1		4	156.155	even	2
1872.3.i.k.415.2		4	39.38	odd	2
1872.3.i.k.415.3		4	3.2	odd	2
1872.3.i.k.415.4		4	12.11	even	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:	\( N \)	\(=\)	\( 208 = 2^{4} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	208.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+4.79583i q^{3} -8.30662i q^{5} -8.66025 q^{7} -14.0000 q^{9} -13.8564 q^{11} +(-10.0000 + 8.30662i) q^{13} +39.8372 q^{15} -5.00000 q^{17} -3.46410 q^{19} -41.5331i q^{21} -28.7750i q^{23} -44.0000 q^{25} -23.9792i q^{27} +40.0000 q^{29} +20.7846 q^{31} -66.4530i q^{33} +71.9375i q^{35} +41.5331i q^{37} +(-39.8372 - 47.9583i) q^{39} +43.1625i q^{43} +116.293i q^{45} -77.9423 q^{47} +26.0000 q^{49} -23.9792i q^{51} -10.0000 q^{53} +115.100i q^{55} -16.6132i q^{57} +65.8179 q^{59} -40.0000 q^{61} +121.244 q^{63} +(69.0000 + 83.0662i) q^{65} -103.923 q^{67} +138.000 q^{69} -29.4449 q^{71} -83.0662i q^{73} -211.017i q^{75} +120.000 q^{77} -143.875i q^{79} -11.0000 q^{81} +17.3205 q^{83} +41.5331i q^{85} +191.833i q^{87} +83.0662i q^{89} +(86.6025 - 71.9375i) q^{91} +99.6795i q^{93} +28.7750i q^{95} -99.6795i q^{97} +193.990 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 56 q^{9} - 40 q^{13} - 20 q^{17} - 176 q^{25} + 160 q^{29} + 104 q^{49} - 40 q^{53} - 160 q^{61} + 276 q^{65} + 552 q^{69} + 480 q^{77} - 44 q^{81}+O(q^{100}) \) 4 * q - 56 * q^9 - 40 * q^13 - 20 * q^17 - 176 * q^25 + 160 * q^29 + 104 * q^49 - 40 * q^53 - 160 * q^61 + 276 * q^65 + 552 * q^69 + 480 * q^77 - 44 * q^81

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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