LMFDB - Embedded newform 3100.3.d.h.1301.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3100,3,Mod(1301,3100)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3100, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("3100.1301");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 3100 = 2^{2} \cdot 5^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	3100.d (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:	$84.4688819517$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 620)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1301.4
Character	$\chi$	$=$	3100.1301
Dual form			3100.3.d.h.1301.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.03111i q^{3} +12.6370 q^{7} -0.187623 q^{9} +O(q^{10})$	$q+3.03111i q^{3} +12.6370 q^{7} -0.187623 q^{9} -17.2333i q^{11} -14.9646i q^{13} +2.62351i q^{17} +16.9683 q^{19} +38.3042i q^{21} -0.547465i q^{23} +26.7113i q^{27} +33.2290i q^{29} +(-30.1720 - 7.11683i) q^{31} +52.2360 q^{33} -62.3249i q^{37} +45.3594 q^{39} -10.8590 q^{41} -13.8424i q^{43} +38.8232 q^{47} +110.694 q^{49} -7.95215 q^{51} -53.5576i q^{53} +51.4327i q^{57} -44.2110 q^{59} +71.3500i q^{61} -2.37099 q^{63} +55.3337 q^{67} +1.65943 q^{69} -61.5833 q^{71} +24.5806i q^{73} -217.778i q^{77} -130.554i q^{79} -82.6534 q^{81} -122.621i q^{83} -100.721 q^{87} -149.213i q^{89} -189.108i q^{91} +(21.5719 - 91.4547i) q^{93} -25.3091 q^{97} +3.23336i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 44 q^{9}+O(q^{10}) $ 24 * q - 44 * q^9	$ 24 q - 44 q^{9} - 8 q^{19} - 24 q^{31} + 280 q^{39} - 248 q^{41} + 644 q^{49} - 100 q^{51} - 152 q^{59} + 288 q^{69} - 352 q^{71} - 368 q^{81}+O(q^{100}) $ 24 * q - 44 * q^9 - 8 * q^19 - 24 * q^31 + 280 * q^39 - 248 * q^41 + 644 * q^49 - 100 * q^51 - 152 * q^59 + 288 * q^69 - 352 * q^71 - 368 * q^81

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3100\mathbb{Z}\right)^\times$.

$n$	$1551$	$1801$	$2977$
$\chi(n)$	$1$	$-1$	$1$

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
$3$			3.03111i			1.01037i	0.863011	+	0.505185i	$0.168576\pi$
$3$			3.03111i			1.01037i	−0.863011	+	0.505185i	$0.831424\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	12.6370			1.80529			0.902645	−	0.430387i	$-0.141623\pi$
$7$	12.6370			1.80529			0.902645	+	0.430387i	$0.141623\pi$
$8$		0			0
$9$	−0.187623			−0.0208470
$10$		0			0
$11$		−	17.2333i		−	1.56666i	−0.621604	−	0.783331i	$-0.713519\pi$
$11$		−	17.2333i		−	1.56666i	0.621604	−	0.783331i	$-0.286481\pi$
$12$		0			0
$13$		−	14.9646i		−	1.15112i	−0.817758	−	0.575562i	$-0.804783\pi$
$13$		−	14.9646i		−	1.15112i	0.817758	−	0.575562i	$-0.195217\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			2.62351i			0.154324i	0.997019	+	0.0771621i	$0.0245859\pi$
$17$			2.62351i			0.154324i	−0.997019	+	0.0771621i	$0.975414\pi$
$18$		0			0
$19$	16.9683			0.893067			0.446533	−	0.894767i	$-0.352658\pi$
$19$	16.9683			0.893067			0.446533	+	0.894767i	$0.352658\pi$
$20$		0			0
$21$			38.3042i			1.82401i
$22$		0			0
$23$		−	0.547465i		−	0.0238028i	−0.999929	−	0.0119014i	$-0.996212\pi$
$23$		−	0.547465i		−	0.0238028i	0.999929	−	0.0119014i	$-0.00378843\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$			26.7113i			0.989307i
$28$		0			0
$29$			33.2290i			1.14583i	0.819615	+	0.572914i	$0.194187\pi$
$29$			33.2290i			1.14583i	−0.819615	+	0.572914i	$0.805813\pi$
$30$		0			0
$31$	−30.1720	−	7.11683i	−0.973291	−	0.229575i
$31$	−30.1720	−	7.11683i	−0.973291	−	0.229575i
$32$		0			0
$33$	52.2360			1.58291
$34$		0			0
$35$		0			0
$36$		0			0
$37$		−	62.3249i		−	1.68446i	−0.539121	−	0.842228i	$-0.681244\pi$
$37$		−	62.3249i		−	1.68446i	0.539121	−	0.842228i	$-0.318756\pi$
$38$		0			0
$39$	45.3594			1.16306
$40$		0			0
$41$	−10.8590			−0.264854			−0.132427	−	0.991193i	$-0.542277\pi$
$41$	−10.8590			−0.264854			−0.132427	+	0.991193i	$0.542277\pi$
$42$		0			0
$43$		−	13.8424i		−	0.321915i	−0.986961	−	0.160958i	$-0.948542\pi$
$43$		−	13.8424i		−	0.321915i	0.986961	−	0.160958i	$-0.0514583\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	38.8232			0.826025			0.413012	−	0.910725i	$-0.364477\pi$
$47$	38.8232			0.826025			0.413012	+	0.910725i	$0.364477\pi$
$48$		0			0
$49$	110.694			2.25907
$50$		0			0
$51$	−7.95215			−0.155924
$52$		0			0
$53$		−	53.5576i		−	1.01052i	−0.862967	−	0.505261i	$-0.831396\pi$
$53$		−	53.5576i		−	1.01052i	0.862967	−	0.505261i	$-0.168604\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$			51.4327i			0.902328i
$58$		0			0
$59$	−44.2110			−0.749339			−0.374669	−	0.927158i	$-0.622244\pi$
$59$	−44.2110			−0.749339			−0.374669	+	0.927158i	$0.622244\pi$
$60$		0			0
$61$			71.3500i			1.16967i	0.811151	+	0.584836i	$0.198841\pi$
$61$			71.3500i			1.16967i	−0.811151	+	0.584836i	$0.801159\pi$
$62$		0			0
$63$	−2.37099			−0.0376348
$64$		0			0
$65$		0			0
$66$		0			0
$67$	55.3337			0.825877			0.412938	−	0.910759i	$-0.364502\pi$
$67$	55.3337			0.825877			0.412938	+	0.910759i	$0.364502\pi$
$68$		0			0
$69$	1.65943			0.0240497
$70$		0			0
$71$	−61.5833			−0.867371			−0.433685	−	0.901064i	$-0.642787\pi$
$71$	−61.5833			−0.867371			−0.433685	+	0.901064i	$0.642787\pi$
$72$		0			0
$73$			24.5806i			0.336720i	0.985726	+	0.168360i	$0.0538471\pi$
$73$			24.5806i			0.336720i	−0.985726	+	0.168360i	$0.946153\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	217.778i		−	2.82828i
$78$		0			0
$79$		−	130.554i		−	1.65258i	−0.563245	−	0.826290i	$-0.690447\pi$
$79$		−	130.554i		−	1.65258i	0.563245	−	0.826290i	$-0.309553\pi$
$80$		0			0
$81$	−82.6534			−1.02041
$82$		0			0
$83$		−	122.621i		−	1.47736i	−0.674058	−	0.738678i	$-0.735450\pi$
$83$		−	122.621i		−	1.47736i	0.674058	−	0.738678i	$-0.264550\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−100.721			−1.15771
$88$		0			0
$89$		−	149.213i		−	1.67655i	−0.545244	−	0.838277i	$-0.683563\pi$
$89$		−	149.213i		−	1.67655i	0.545244	−	0.838277i	$-0.316437\pi$
$90$		0			0
$91$		−	189.108i		−	2.07811i
$92$		0			0
$93$	21.5719	−	91.4547i	0.231956	−	0.983384i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−25.3091			−0.260919			−0.130460	−	0.991454i	$-0.541645\pi$
$97$	−25.3091			−0.260919			−0.130460	+	0.991454i	$0.541645\pi$
$98$		0			0
$99$			3.23336i			0.0326602i
$100$		0			0
$101$	165.922			1.64279			0.821396	−	0.570358i	$-0.193196\pi$
$101$	165.922			1.64279			0.821396	+	0.570358i	$0.193196\pi$
$102$		0			0
$103$	−112.617			−1.09337			−0.546685	−	0.837338i	$-0.684110\pi$
$103$	−112.617			−1.09337			−0.546685	+	0.837338i	$0.684110\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−179.840			−1.68075			−0.840374	−	0.542008i	$-0.817664\pi$
$107$	−179.840			−1.68075			−0.840374	+	0.542008i	$0.817664\pi$
$108$		0			0
$109$	186.122			1.70754			0.853772	−	0.520647i	$-0.174309\pi$
$109$	186.122			1.70754			0.853772	+	0.520647i	$0.174309\pi$
$110$		0			0
$111$	188.914			1.70192
$112$		0			0
$113$	−68.1282			−0.602905			−0.301452	−	0.953481i	$-0.597471\pi$
$113$	−68.1282			−0.602905			−0.301452	+	0.953481i	$0.597471\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$			2.80770i			0.0239975i
$118$		0			0
$119$			33.1534i			0.278600i
$120$		0			0
$121$	−175.986			−1.45443
$122$		0			0
$123$		−	32.9148i		−	0.267600i
$124$		0			0
$125$		0			0
$126$		0			0
$127$			137.718i			1.08439i	0.840252	+	0.542196i	$0.182407\pi$
$127$			137.718i			1.08439i	−0.840252	+	0.542196i	$0.817593\pi$
$128$		0			0
$129$	41.9577			0.325254
$130$		0			0
$131$	169.467			1.29365			0.646823	−	0.762641i	$-0.276097\pi$
$131$	169.467			1.29365			0.646823	+	0.762641i	$0.276097\pi$
$132$		0			0
$133$	214.428			1.61224
$134$		0			0
$135$		0			0
$136$		0			0
$137$			82.4262i			0.601651i	0.953679	+	0.300825i	$0.0972621\pi$
$137$			82.4262i			0.601651i	−0.953679	+	0.300825i	$0.902738\pi$
$138$		0			0
$139$			14.1552i			0.101836i	0.998703	+	0.0509178i	$0.0162147\pi$
$139$			14.1552i			0.101836i	−0.998703	+	0.0509178i	$0.983785\pi$
$140$		0			0
$141$			117.677i			0.834590i
$142$		0			0
$143$	−257.890			−1.80342
$144$		0			0
$145$		0			0
$146$		0			0
$147$			335.527i			2.28250i
$148$		0			0
$149$	62.8370			0.421725			0.210863	−	0.977516i	$-0.432373\pi$
$149$	62.8370			0.421725			0.210863	+	0.977516i	$0.432373\pi$
$150$		0			0
$151$			29.1590i			0.193106i	0.995328	+	0.0965529i	$0.0307817\pi$
$151$			29.1590i			0.193106i	−0.995328	+	0.0965529i	$0.969218\pi$
$152$		0			0
$153$		−	0.492230i		−	0.00321719i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−30.4164			−0.193735			−0.0968675	−	0.995297i	$-0.530882\pi$
$157$	−30.4164			−0.193735			−0.0968675	+	0.995297i	$0.530882\pi$
$158$		0			0
$159$	162.339			1.02100
$160$		0			0
$161$		−	6.91833i		−	0.0429710i
$162$		0			0
$163$	−40.0086			−0.245452			−0.122726	−	0.992441i	$-0.539164\pi$
$163$	−40.0086			−0.245452			−0.122726	+	0.992441i	$0.539164\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	297.432i		−	1.78103i	−0.454953	−	0.890516i	$-0.650344\pi$
$167$		−	297.432i		−	1.78103i	0.454953	−	0.890516i	$-0.349656\pi$
$168$		0			0
$169$	−54.9400			−0.325089
$170$		0			0
$171$	−3.18363			−0.0186177
$172$		0			0
$173$	−79.8258			−0.461421			−0.230710	−	0.973022i	$-0.574105\pi$
$173$	−79.8258			−0.461421			−0.230710	+	0.973022i	$0.574105\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		−	134.008i		−	0.757109i
$178$		0			0
$179$			183.941i			1.02761i	0.857908	+	0.513803i	$0.171764\pi$
$179$			183.941i			1.02761i	−0.857908	+	0.513803i	$0.828236\pi$
$180$		0			0
$181$		−	237.728i		−	1.31341i	−0.754147	−	0.656706i	$-0.771949\pi$
$181$		−	237.728i		−	1.31341i	0.754147	−	0.656706i	$-0.228051\pi$
$182$		0			0
$183$	−216.270			−1.18180
$184$		0			0
$185$		0			0
$186$		0			0
$187$	45.2117			0.241774
$188$		0			0
$189$			337.551i			1.78598i
$190$		0			0
$191$	70.2816			0.367967			0.183983	−	0.982929i	$-0.441101\pi$
$191$	70.2816			0.367967			0.183983	+	0.982929i	$0.441101\pi$
$192$		0			0
$193$	374.846			1.94221			0.971104	−	0.238655i	$-0.0767066\pi$
$193$	374.846			1.94221			0.971104	+	0.238655i	$0.0767066\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			96.1079i			0.487858i	0.969793	+	0.243929i	$0.0784363\pi$
$197$			96.1079i			0.487858i	−0.969793	+	0.243929i	$0.921564\pi$
$198$		0			0
$199$			369.081i			1.85468i	0.374225	+	0.927338i	$0.377909\pi$
$199$			369.081i			1.85468i	−0.374225	+	0.927338i	$0.622091\pi$
$200$		0			0
$201$			167.723i			0.834441i
$202$		0			0
$203$			419.916i			2.06855i
$204$		0			0
$205$		0			0
$206$		0			0
$207$			0.102717i			0.000496217i
$208$		0			0
$209$		−	292.419i		−	1.39913i
$210$		0			0
$211$	221.582			1.05015			0.525077	−	0.851055i	$-0.324037\pi$
$211$	221.582			1.05015			0.525077	+	0.851055i	$0.324037\pi$
$212$		0			0
$213$		−	186.666i		−	0.876365i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−381.285	−	89.9356i	−1.75707	−	0.414450i
$218$		0			0
$219$	−74.5064			−0.340212
$220$		0			0
$221$	39.2599			0.177646
$222$		0			0
$223$			64.3966i			0.288774i	0.989521	+	0.144387i	$0.0461210\pi$
$223$			64.3966i			0.288774i	−0.989521	+	0.144387i	$0.953879\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	262.898			1.15814			0.579070	−	0.815278i	$-0.303416\pi$
$227$	262.898			1.15814			0.579070	+	0.815278i	$0.303416\pi$
$228$		0			0
$229$		−	230.765i		−	1.00771i	−0.863789	−	0.503854i	$-0.831915\pi$
$229$		−	230.765i		−	1.00771i	0.863789	−	0.503854i	$-0.168085\pi$
$230$		0			0
$231$	660.107			2.85761
$232$		0			0
$233$	−312.427			−1.34089			−0.670443	−	0.741961i	$-0.733896\pi$
$233$	−312.427			−1.34089			−0.670443	+	0.741961i	$0.733896\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	395.723			1.66972
$238$		0			0
$239$		−	143.331i		−	0.599710i	−0.953985	−	0.299855i	$-0.903062\pi$
$239$		−	143.331i		−	0.599710i	0.953985	−	0.299855i	$-0.0969383\pi$
$240$		0			0
$241$		−	225.766i		−	0.936788i	−0.883520	−	0.468394i	$-0.844833\pi$
$241$		−	225.766i		−	0.936788i	0.883520	−	0.468394i	$-0.155167\pi$
$242$		0			0
$243$		−	10.1300i		−	0.0416872i
$244$		0			0
$245$		0			0
$246$		0			0
$247$		−	253.924i		−	1.02803i
$248$		0			0
$249$	371.676			1.49268
$250$		0			0
$251$			243.419i			0.969796i	0.874571	+	0.484898i	$0.161143\pi$
$251$			243.419i			0.969796i	−0.874571	+	0.484898i	$0.838857\pi$
$252$		0			0
$253$	−9.43463			−0.0372910
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−258.288			−1.00501			−0.502505	−	0.864574i	$-0.667588\pi$
$257$	−258.288			−1.00501			−0.502505	+	0.864574i	$0.667588\pi$
$258$		0			0
$259$		−	787.601i		−	3.04093i
$260$		0			0
$261$		−	6.23452i		−	0.0238871i
$262$		0			0
$263$			228.581i			0.869131i	0.900640	+	0.434565i	$0.143098\pi$
$263$			228.581i			0.869131i	−0.900640	+	0.434565i	$0.856902\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	452.282			1.69394
$268$		0			0
$269$			392.997i			1.46095i	0.682937	+	0.730477i	$0.260702\pi$
$269$			392.997i			1.46095i	−0.682937	+	0.730477i	$0.739298\pi$
$270$		0			0
$271$			41.7275i			0.153976i	0.997032	+	0.0769881i	$0.0245303\pi$
$271$			41.7275i			0.153976i	−0.997032	+	0.0769881i	$0.975470\pi$
$272$		0			0
$273$	573.208			2.09966
$274$		0			0
$275$		0			0
$276$		0			0
$277$			110.025i			0.397203i	0.980080	+	0.198602i	$0.0636400\pi$
$277$			110.025i			0.397203i	−0.980080	+	0.198602i	$0.936360\pi$
$278$		0			0
$279$	5.66096	+	1.33528i	0.0202902	+	0.00478595i
$280$		0			0
$281$	−116.027			−0.412906			−0.206453	−	0.978457i	$-0.566192\pi$
$281$	−116.027			−0.412906			−0.206453	+	0.978457i	$0.566192\pi$
$282$		0			0
$283$	−0.224672			−0.000793896			−0.000396948	−	1.00000i	$-0.500126\pi$
$283$	−0.224672			−0.000793896			−0.000396948		1.00000i	$0.500126\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−137.225			−0.478137
$288$		0			0
$289$	282.117			0.976184
$290$		0			0
$291$		−	76.7148i		−	0.263625i
$292$		0			0
$293$	−218.773			−0.746665			−0.373333	−	0.927698i	$-0.621785\pi$
$293$	−218.773			−0.746665			−0.373333	+	0.927698i	$0.621785\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	460.323			1.54991
$298$		0			0
$299$	−8.19261			−0.0274000
$300$		0			0
$301$		−	174.926i		−	0.581150i
$302$		0			0
$303$			502.928i			1.65983i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	262.356			0.854578			0.427289	−	0.904115i	$-0.359469\pi$
$307$	262.356			0.854578			0.427289	+	0.904115i	$0.359469\pi$
$308$		0			0
$309$		−	341.355i		−	1.10471i
$310$		0			0
$311$	−203.126			−0.653137			−0.326569	−	0.945173i	$-0.605892\pi$
$311$	−203.126			−0.653137			−0.326569	+	0.945173i	$0.605892\pi$
$312$		0			0
$313$		−	435.858i		−	1.39252i	−0.717791	−	0.696259i	$-0.754847\pi$
$313$		−	435.858i		−	1.39252i	0.717791	−	0.696259i	$-0.245153\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	0.559661			0.00176549			0.000882745	−	1.00000i	$-0.499719\pi$
$317$	0.559661			0.00176549			0.000882745		1.00000i	$0.499719\pi$
$318$		0			0
$319$	572.645			1.79513
$320$		0			0
$321$		−	545.115i		−	1.69818i
$322$		0			0
$323$			44.5164i			0.137822i
$324$		0			0
$325$		0			0
$326$		0			0
$327$			564.157i			1.72525i
$328$		0			0
$329$	490.609			1.49121
$330$		0			0
$331$			150.254i			0.453940i	0.973902	+	0.226970i	$0.0728819\pi$
$331$			150.254i			0.453940i	−0.973902	+	0.226970i	$0.927118\pi$
$332$		0			0
$333$			11.6936i			0.0351158i
$334$		0			0
$335$		0			0
$336$		0			0
$337$		−	30.0663i		−	0.0892174i	−0.999005	−	0.0446087i	$-0.985796\pi$
$337$		−	30.0663i		−	0.0892174i	0.999005	−	0.0446087i	$-0.0142041\pi$
$338$		0			0
$339$		−	206.504i		−	0.609157i
$340$		0			0
$341$	−122.646	+	519.963i	−0.359667	+	1.52482i
$342$		0			0
$343$	779.634			2.27299
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	76.6888i		−	0.221005i	−0.993876	−	0.110503i	$-0.964754\pi$
$347$		−	76.6888i		−	0.221005i	0.993876	−	0.110503i	$-0.0352460\pi$
$348$		0			0
$349$	−185.444			−0.531358			−0.265679	−	0.964062i	$-0.585596\pi$
$349$	−185.444			−0.531358			−0.265679	+	0.964062i	$0.585596\pi$
$350$		0			0
$351$	399.724			1.13882
$352$		0			0
$353$			356.386i			1.00959i	0.863239	+	0.504796i	$0.168432\pi$
$353$			356.386i			1.00959i	−0.863239	+	0.504796i	$0.831568\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−100.491			−0.281489
$358$		0			0
$359$	−115.055			−0.320486			−0.160243	−	0.987078i	$-0.551228\pi$
$359$	−115.055			−0.320486			−0.160243	+	0.987078i	$0.551228\pi$
$360$		0			0
$361$	−73.0778			−0.202431
$362$		0			0
$363$		−	533.434i		−	1.46951i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		−	284.273i		−	0.774587i	−0.921956	−	0.387293i	$-0.873410\pi$
$367$		−	284.273i		−	0.774587i	0.921956	−	0.387293i	$-0.126590\pi$
$368$		0			0
$369$	2.03739			0.00552140
$370$		0			0
$371$		−	676.809i		−	1.82428i
$372$		0			0
$373$	200.493			0.537514			0.268757	−	0.963208i	$-0.413387\pi$
$373$	200.493			0.537514			0.268757	+	0.963208i	$0.413387\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	497.260			1.31899
$378$		0			0
$379$	475.405			1.25437			0.627183	−	0.778872i	$-0.284208\pi$
$379$	475.405			1.25437			0.627183	+	0.778872i	$0.284208\pi$
$380$		0			0
$381$	−417.438			−1.09564
$382$		0			0
$383$		−	490.353i		−	1.28029i	−0.768252	−	0.640147i	$-0.778873\pi$
$383$		−	490.353i		−	1.28029i	0.768252	−	0.640147i	$-0.221127\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$			2.59714i			0.00671096i
$388$		0			0
$389$			313.964i			0.807106i	0.914956	+	0.403553i	$0.132225\pi$
$389$			313.964i			0.807106i	−0.914956	+	0.403553i	$0.867775\pi$
$390$		0			0
$391$	1.43628			0.00367335
$392$		0			0
$393$			513.674i			1.30706i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−256.843			−0.646960			−0.323480	−	0.946235i	$-0.604853\pi$
$397$	−256.843			−0.646960			−0.323480	+	0.946235i	$0.604853\pi$
$398$		0			0
$399$			649.956i			1.62896i
$400$		0			0
$401$			567.989i			1.41643i	0.705996	+	0.708216i	$0.250499\pi$
$401$			567.989i			1.41643i	−0.705996	+	0.708216i	$0.749501\pi$
$402$		0			0
$403$	−106.501	+	451.513i	−0.264270	+	1.12038i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−1074.06			−2.63898
$408$		0			0
$409$		−	511.380i		−	1.25032i	−0.780497	−	0.625159i	$-0.785034\pi$
$409$		−	511.380i		−	1.25032i	0.780497	−	0.625159i	$-0.214966\pi$
$410$		0			0
$411$	−249.843			−0.607890
$412$		0			0
$413$	−558.696			−1.35277
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−42.9058			−0.102892
$418$		0			0
$419$	393.577			0.939325			0.469663	−	0.882846i	$-0.344376\pi$
$419$	393.577			0.939325			0.469663	+	0.882846i	$0.344376\pi$
$420$		0			0
$421$	−244.177			−0.579994			−0.289997	−	0.957028i	$-0.593654\pi$
$421$	−244.177			−0.579994			−0.289997	+	0.957028i	$0.593654\pi$
$422$		0			0
$423$	−7.28411			−0.0172201
$424$		0			0
$425$		0			0
$426$		0			0
$427$			901.652i			2.11160i
$428$		0			0
$429$		−	781.692i		−	1.82213i
$430$		0			0
$431$	318.209			0.738304			0.369152	−	0.929369i	$-0.379648\pi$
$431$	318.209			0.738304			0.369152	+	0.929369i	$0.379648\pi$
$432$		0			0
$433$			494.720i			1.14254i	0.820762	+	0.571271i	$0.193549\pi$
$433$			494.720i			1.14254i	−0.820762	+	0.571271i	$0.806451\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	9.28954i		−	0.0212575i
$438$		0			0
$439$	525.555			1.19716			0.598582	−	0.801062i	$-0.295731\pi$
$439$	525.555			1.19716			0.598582	+	0.801062i	$0.295731\pi$
$440$		0			0
$441$	−20.7688			−0.0470948
$442$		0			0
$443$	381.481			0.861130			0.430565	−	0.902560i	$-0.358314\pi$
$443$	381.481			0.861130			0.430565	+	0.902560i	$0.358314\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$			190.466i			0.426098i
$448$		0			0
$449$			522.611i			1.16394i	0.813209	+	0.581972i	$0.197719\pi$
$449$			522.611i			1.16394i	−0.813209	+	0.581972i	$0.802281\pi$
$450$		0			0
$451$			187.136i			0.414936i
$452$		0			0
$453$	−88.3841			−0.195108
$454$		0			0
$455$		0			0
$456$		0			0
$457$			7.68966i			0.0168264i	0.999965	+	0.00841320i	$0.00267804\pi$
$457$			7.68966i			0.0168264i	−0.999965	+	0.00841320i	$0.997322\pi$
$458$		0			0
$459$	−70.0773			−0.152674
$460$		0			0
$461$			17.9518i			0.0389410i	0.999810	+	0.0194705i	$0.00619805\pi$
$461$			17.9518i			0.0389410i	−0.999810	+	0.0194705i	$0.993802\pi$
$462$		0			0
$463$			485.364i			1.04830i	0.851625	+	0.524152i	$0.175617\pi$
$463$			485.364i			1.04830i	−0.851625	+	0.524152i	$0.824383\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	386.478			0.827576			0.413788	−	0.910373i	$-0.364205\pi$
$467$	386.478			0.827576			0.413788	+	0.910373i	$0.364205\pi$
$468$		0			0
$469$	699.254			1.49095
$470$		0			0
$471$		−	92.1954i		−	0.195744i
$472$		0			0
$473$	−238.549			−0.504333
$474$		0			0
$475$		0			0
$476$		0			0
$477$			10.0486i			0.0210663i
$478$		0			0
$479$	−87.7891			−0.183276			−0.0916379	−	0.995792i	$-0.529210\pi$
$479$	−87.7891			−0.183276			−0.0916379	+	0.995792i	$0.529210\pi$
$480$		0			0
$481$	−932.669			−1.93902
$482$		0			0
$483$	20.9702			0.0434166
$484$		0			0
$485$		0			0
$486$		0			0
$487$			182.759i			0.375276i	0.982238	+	0.187638i	$0.0600832\pi$
$487$			182.759i			0.375276i	−0.982238	+	0.187638i	$0.939917\pi$
$488$		0			0
$489$		−	121.270i		−	0.247997i
$490$		0			0
$491$			45.7608i			0.0931992i	0.998914	+	0.0465996i	$0.0148385\pi$
$491$			45.7608i			0.0931992i	−0.998914	+	0.0465996i	$0.985162\pi$
$492$		0			0
$493$	−87.1767			−0.176829
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−778.230			−1.56586
$498$		0			0
$499$			775.048i			1.55320i	0.629993	+	0.776601i	$0.283058\pi$
$499$			775.048i			1.55320i	−0.629993	+	0.776601i	$0.716942\pi$
$500$		0			0
$501$	901.549			1.79950
$502$		0			0
$503$	−35.4011			−0.0703800			−0.0351900	−	0.999381i	$-0.511204\pi$
$503$	−35.4011			−0.0703800			−0.0351900	+	0.999381i	$0.511204\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		−	166.529i		−	0.328460i
$508$		0			0
$509$			441.931i			0.868234i	0.900857	+	0.434117i	$0.142940\pi$
$509$			441.931i			0.868234i	−0.900857	+	0.434117i	$0.857060\pi$
$510$		0			0
$511$			310.625i			0.607877i
$512$		0			0
$513$			453.244i			0.883517i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		−	669.051i		−	1.29410i
$518$		0			0
$519$		−	241.961i		−	0.466205i
$520$		0			0
$521$	−3.82481			−0.00734129			−0.00367064	−	0.999993i	$-0.501168\pi$
$521$	−3.82481			−0.00734129			−0.00367064	+	0.999993i	$0.501168\pi$
$522$		0			0
$523$		−	222.757i		−	0.425921i	−0.977061	−	0.212961i	$-0.931689\pi$
$523$		−	222.757i		−	0.425921i	0.977061	−	0.212961i	$-0.0683106\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	18.6711	−	79.1566i	0.0354290	−	0.150202i
$528$		0			0
$529$	528.700			0.999433
$530$		0			0
$531$	8.29499			0.0156215
$532$		0			0
$533$			162.501i			0.304880i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−557.546			−1.03826
$538$		0			0
$539$		−	1907.63i		−	3.53920i
$540$		0			0
$541$	−72.2331			−0.133518			−0.0667589	−	0.997769i	$-0.521266\pi$
$541$	−72.2331			−0.133518			−0.0667589	+	0.997769i	$0.521266\pi$
$542$		0			0
$543$	720.578			1.32703
$544$		0			0
$545$		0			0
$546$		0			0
$547$	37.7149			0.0689487			0.0344744	−	0.999406i	$-0.489024\pi$
$547$	37.7149			0.0689487			0.0344744	+	0.999406i	$0.489024\pi$
$548$		0			0
$549$		−	13.3869i		−	0.0243841i
$550$		0			0
$551$			563.839i			1.02330i
$552$		0			0
$553$		−	1649.81i		−	2.98339i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	136.072i		−	0.244295i	−0.992512	−	0.122147i	$-0.961022\pi$
$557$		−	136.072i		−	0.244295i	0.992512	−	0.122147i	$-0.0389781\pi$
$558$		0			0
$559$	−207.146			−0.370565
$560$		0			0
$561$			137.042i			0.244281i
$562$		0			0
$563$	264.828			0.470388			0.235194	−	0.971948i	$-0.424428\pi$
$563$	264.828			0.470388			0.235194	+	0.971948i	$0.424428\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−1044.49			−1.84214
$568$		0			0
$569$		−	53.5741i		−	0.0941548i	−0.998891	−	0.0470774i	$-0.985009\pi$
$569$		−	53.5741i		−	0.0941548i	0.998891	−	0.0470774i	$-0.0149907\pi$
$570$		0			0
$571$			621.926i			1.08919i	0.838700	+	0.544594i	$0.183316\pi$
$571$			621.926i			1.08919i	−0.838700	+	0.544594i	$0.816684\pi$
$572$		0			0
$573$			213.031i			0.371782i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−16.3136			−0.0282731			−0.0141366	−	0.999900i	$-0.504500\pi$
$577$	−16.3136			−0.0282731			−0.0141366	+	0.999900i	$0.504500\pi$
$578$		0			0
$579$			1136.20i			1.96235i
$580$		0			0
$581$		−	1549.56i		−	2.66706i
$582$		0			0
$583$	−922.974			−1.58315
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	239.436i		−	0.407898i	−0.978981	−	0.203949i	$-0.934622\pi$
$587$		−	239.436i		−	0.407898i	0.978981	−	0.203949i	$-0.0653777\pi$
$588$		0			0
$589$	−511.967	−	120.760i	−0.869214	−	0.205026i
$590$		0			0
$591$	−291.314			−0.492916
$592$		0			0
$593$	−219.702			−0.370493			−0.185246	−	0.982692i	$-0.559308\pi$
$593$	−219.702			−0.370493			−0.185246	+	0.982692i	$0.559308\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−1118.72			−1.87391
$598$		0			0
$599$	−403.870			−0.674240			−0.337120	−	0.941462i	$-0.609453\pi$
$599$	−403.870			−0.674240			−0.337120	+	0.941462i	$0.609453\pi$
$600$		0			0
$601$			406.825i			0.676914i	0.940982	+	0.338457i	$0.109905\pi$
$601$			406.825i			0.676914i	−0.940982	+	0.338457i	$0.890095\pi$
$602$		0			0
$603$	−10.3819			−0.0172170
$604$		0			0
$605$		0			0
$606$		0			0
$607$	271.236			0.446847			0.223423	−	0.974721i	$-0.428277\pi$
$607$	271.236			0.446847			0.223423	+	0.974721i	$0.428277\pi$
$608$		0			0
$609$	−1272.81			−2.09000
$610$		0			0
$611$		−	580.974i		−	0.950858i
$612$		0			0
$613$		−	431.338i		−	0.703651i	−0.936066	−	0.351825i	$-0.885561\pi$
$613$		−	431.338i		−	0.703651i	0.936066	−	0.351825i	$-0.114439\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−44.3973			−0.0719568			−0.0359784	−	0.999353i	$-0.511455\pi$
$617$	−44.3973			−0.0719568			−0.0359784	+	0.999353i	$0.511455\pi$
$618$		0			0
$619$			1027.51i			1.65995i	0.557799	+	0.829976i	$0.311646\pi$
$619$			1027.51i			1.65995i	−0.557799	+	0.829976i	$0.688354\pi$
$620$		0			0
$621$	14.6235			0.0235483
$622$		0			0
$623$		−	1885.61i		−	3.02667i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	886.354			1.41364
$628$		0			0
$629$	163.510			0.259952
$630$		0			0
$631$		−	381.333i		−	0.604331i	−0.953255	−	0.302165i	$-0.902290\pi$
$631$		−	381.333i		−	0.604331i	0.953255	−	0.302165i	$-0.0977095\pi$
$632$		0			0
$633$			671.640i			1.06104i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		−	1656.50i		−	2.60047i
$638$		0			0
$639$	11.5544			0.0180821
$640$		0			0
$641$			712.516i			1.11157i	0.831326	+	0.555785i	$0.187582\pi$
$641$			712.516i			1.11157i	−0.831326	+	0.555785i	$0.812418\pi$
$642$		0			0
$643$			486.737i			0.756978i	0.925606	+	0.378489i	$0.123556\pi$
$643$			486.737i			0.756978i	−0.925606	+	0.378489i	$0.876444\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			1113.31i			1.72073i	0.509676	+	0.860366i	$0.329765\pi$
$647$			1113.31i			1.72073i	−0.509676	+	0.860366i	$0.670235\pi$
$648$		0			0
$649$			761.901i			1.17396i
$650$		0			0
$651$	272.604	−	1155.72i	0.418747	−	1.77529i
$652$		0			0
$653$	−917.109			−1.40445			−0.702227	−	0.711953i	$-0.747811\pi$
$653$	−917.109			−1.40445			−0.702227	+	0.711953i	$0.747811\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		−	4.61187i		−	0.00701960i
$658$		0			0
$659$	602.233			0.913859			0.456929	−	0.889503i	$-0.348949\pi$
$659$	602.233			0.913859			0.456929	+	0.889503i	$0.348949\pi$
$660$		0			0
$661$	500.175			0.756694			0.378347	−	0.925664i	$-0.376493\pi$
$661$	500.175			0.756694			0.378347	+	0.925664i	$0.376493\pi$
$662$		0			0
$663$			119.001i			0.179489i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	18.1917			0.0272740
$668$		0			0
$669$	−195.193			−0.291768
$670$		0			0
$671$	1229.60			1.83248
$672$		0			0
$673$		−	830.738i		−	1.23438i	−0.786814	−	0.617190i	$-0.788271\pi$
$673$		−	830.738i		−	1.23438i	0.786814	−	0.617190i	$-0.211729\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			498.118i			0.735772i	0.929871	+	0.367886i	$0.119918\pi$
$677$			498.118i			0.735772i	−0.929871	+	0.367886i	$0.880082\pi$
$678$		0			0
$679$	−319.832			−0.471034
$680$		0			0
$681$			796.871i			1.17015i
$682$		0			0
$683$	−928.620			−1.35962			−0.679810	−	0.733388i	$-0.737938\pi$
$683$	−928.620			−1.35962			−0.679810	+	0.733388i	$0.737938\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	699.475			1.01816
$688$		0			0
$689$	−801.470			−1.16324
$690$		0			0
$691$	772.022			1.11725			0.558627	−	0.829419i	$-0.311329\pi$
$691$	772.022			1.11725			0.558627	+	0.829419i	$0.311329\pi$
$692$		0			0
$693$			40.8600i			0.0589611i
$694$		0			0
$695$		0			0
$696$		0			0
$697$		−	28.4887i		−	0.0408733i
$698$		0			0
$699$		−	946.999i		−	1.35479i
$700$		0			0
$701$	−373.767			−0.533191			−0.266595	−	0.963809i	$-0.585899\pi$
$701$	−373.767			−0.533191			−0.266595	+	0.963809i	$0.585899\pi$
$702$		0			0
$703$		−	1057.55i		−	1.50433i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	2096.76			2.96572
$708$		0			0
$709$		−	506.220i		−	0.713991i	−0.934106	−	0.356996i	$-0.883801\pi$
$709$		−	506.220i		−	0.713991i	0.934106	−	0.356996i	$-0.116199\pi$
$710$		0			0
$711$			24.4949i			0.0344513i
$712$		0			0
$713$	−3.89622	+	16.5181i	−0.00546454	+	0.0231671i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	434.451			0.605929
$718$		0			0
$719$			244.535i			0.340105i	0.985435	+	0.170052i	$0.0543937\pi$
$719$			244.535i			0.340105i	−0.985435	+	0.170052i	$0.945606\pi$
$720$		0			0
$721$	−1423.15			−1.97385
$722$		0			0
$723$	684.321			0.946502
$724$		0			0
$725$		0			0
$726$		0			0
$727$	449.202			0.617884			0.308942	−	0.951081i	$-0.400025\pi$
$727$	449.202			0.617884			0.308942	+	0.951081i	$0.400025\pi$
$728$		0			0
$729$	−713.176			−0.978293
$730$		0			0
$731$	36.3156			0.0496793
$732$		0			0
$733$	−979.538			−1.33634			−0.668171	−	0.744008i	$-0.732922\pi$
$733$	−979.538			−1.33634			−0.668171	+	0.744008i	$0.732922\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	953.582i		−	1.29387i
$738$		0			0
$739$		−	858.011i		−	1.16104i	−0.814245	−	0.580522i	$-0.802849\pi$
$739$		−	858.011i		−	1.16104i	0.814245	−	0.580522i	$-0.197151\pi$
$740$		0			0
$741$	769.671			1.03869
$742$		0			0
$743$			445.813i			0.600017i	0.953936	+	0.300009i	$0.0969896\pi$
$743$			445.813i			0.600017i	−0.953936	+	0.300009i	$0.903010\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$			23.0064i			0.0307984i
$748$		0			0
$749$	−2272.64			−3.03423
$750$		0			0
$751$	−742.500			−0.988682			−0.494341	−	0.869268i	$-0.664591\pi$
$751$	−742.500			−0.988682			−0.494341	+	0.869268i	$0.664591\pi$
$752$		0			0
$753$	−737.829			−0.979852
$754$		0			0
$755$		0			0
$756$		0			0
$757$			1414.38i			1.86840i	0.356747	+	0.934201i	$0.383886\pi$
$757$			1414.38i			1.86840i	−0.356747	+	0.934201i	$0.616114\pi$
$758$		0			0
$759$		−	28.5974i		−	0.0376777i
$760$		0			0
$761$		−	528.815i		−	0.694894i	−0.937700	−	0.347447i	$-0.887049\pi$
$761$		−	528.815i		−	0.694894i	0.937700	−	0.347447i	$-0.112951\pi$
$762$		0			0
$763$	2352.03			3.08261
$764$		0			0
$765$		0			0
$766$		0			0
$767$			661.601i			0.862583i
$768$		0			0
$769$	−78.3658			−0.101906			−0.0509530	−	0.998701i	$-0.516226\pi$
$769$	−78.3658			−0.101906			−0.0509530	+	0.998701i	$0.516226\pi$
$770$		0			0
$771$		−	782.899i		−	1.01543i
$772$		0			0
$773$		−	455.033i		−	0.588659i	−0.955704	−	0.294330i	$-0.904904\pi$
$773$		−	455.033i		−	0.588659i	0.955704	−	0.294330i	$-0.0950963\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	2387.31			3.07246
$778$		0			0
$779$	−184.258			−0.236532
$780$		0			0
$781$			1061.28i			1.35888i
$782$		0			0
$783$	−887.589			−1.13358
$784$		0			0
$785$		0			0
$786$		0			0
$787$		−	1364.15i		−	1.73336i	−0.498868	−	0.866678i	$-0.666251\pi$
$787$		−	1364.15i		−	1.73336i	0.498868	−	0.866678i	$-0.333749\pi$
$788$		0			0
$789$	−692.855			−0.878143
$790$		0			0
$791$	−860.938			−1.08842
$792$		0			0
$793$	1067.73			1.34644
$794$		0			0
$795$		0			0
$796$		0			0
$797$			276.950i			0.347490i	0.984791	+	0.173745i	$0.0555869\pi$
$797$			276.950i			0.347490i	−0.984791	+	0.173745i	$0.944413\pi$
$798$		0			0
$799$			101.853i			0.127476i
$800$		0			0
$801$			27.9958i			0.0349511i
$802$		0			0
$803$	423.604			0.527527
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−1191.22			−1.47610
$808$		0			0
$809$			874.241i			1.08064i	0.841458	+	0.540322i	$0.181698\pi$
$809$			874.241i			1.08064i	−0.841458	+	0.540322i	$0.818302\pi$
$810$		0			0
$811$	555.744			0.685257			0.342629	−	0.939471i	$-0.388683\pi$
$811$	555.744			0.685257			0.342629	+	0.939471i	$0.388683\pi$
$812$		0			0
$813$	−126.481			−0.155573
$814$		0			0
$815$		0			0
$816$		0			0
$817$		−	234.881i		−	0.287492i
$818$		0			0
$819$			35.4810i			0.0433224i
$820$		0			0
$821$		−	366.443i		−	0.446337i	−0.974780	−	0.223168i	$-0.928360\pi$
$821$		−	366.443i		−	0.446337i	0.974780	−	0.223168i	$-0.0716400\pi$
$822$		0			0
$823$		−	663.118i		−	0.805732i	−0.915259	−	0.402866i	$-0.868014\pi$
$823$		−	663.118i		−	0.805732i	0.915259	−	0.402866i	$-0.131986\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	1113.53i		−	1.34646i	−0.739431	−	0.673232i	$-0.764905\pi$
$827$		−	1113.53i		−	1.34646i	0.739431	−	0.673232i	$-0.235095\pi$
$828$		0			0
$829$		−	292.757i		−	0.353144i	−0.984288	−	0.176572i	$-0.943499\pi$
$829$		−	292.757i		−	0.353144i	0.984288	−	0.176572i	$-0.0565009\pi$
$830$		0			0
$831$	−333.499			−0.401322
$832$		0			0
$833$			290.408i			0.348629i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	190.100	−	805.933i	0.227120	−	0.962883i
$838$		0			0
$839$	−424.914			−0.506453			−0.253227	−	0.967407i	$-0.581492\pi$
$839$	−424.914			−0.506453			−0.253227	+	0.967407i	$0.581492\pi$
$840$		0			0
$841$	−263.168			−0.312922
$842$		0			0
$843$		−	351.689i		−	0.417187i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−2223.94			−2.62567
$848$		0			0
$849$		−	0.681007i		−	0.000802128i
$850$		0			0
$851$	−34.1207			−0.0400949
$852$		0			0
$853$	813.025			0.953136			0.476568	−	0.879138i	$-0.341881\pi$
$853$	813.025			0.953136			0.476568	+	0.879138i	$0.341881\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−1205.01			−1.40608			−0.703040	−	0.711151i	$-0.748174\pi$
$857$	−1205.01			−1.40608			−0.703040	+	0.711151i	$0.748174\pi$
$858$		0			0
$859$			31.0682i			0.0361678i	0.999836	+	0.0180839i	$0.00575660\pi$
$859$			31.0682i			0.0361678i	−0.999836	+	0.0180839i	$0.994243\pi$
$860$		0			0
$861$		−	415.945i		−	0.483095i
$862$		0			0
$863$			1003.58i			1.16290i	0.813583	+	0.581449i	$0.197514\pi$
$863$			1003.58i			1.16290i	−0.813583	+	0.581449i	$0.802486\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$			855.128i			0.986307i
$868$		0			0
$869$	−2249.87			−2.58904
$870$		0			0
$871$		−	828.049i		−	0.950687i
$872$		0			0
$873$	4.74857			0.00543937
$874$		0			0
$875$		0			0
$876$		0			0
$877$	1215.09			1.38551			0.692755	−	0.721173i	$-0.256397\pi$
$877$	1215.09			1.38551			0.692755	+	0.721173i	$0.256397\pi$
$878$		0			0
$879$		−	663.124i		−	0.754408i
$880$		0			0
$881$		−	770.265i		−	0.874308i	−0.899387	−	0.437154i	$-0.855986\pi$
$881$		−	770.265i		−	0.874308i	0.899387	−	0.437154i	$-0.144014\pi$
$882$		0			0
$883$			1336.86i			1.51400i	0.653413	+	0.757002i	$0.273336\pi$
$883$			1336.86i			1.51400i	−0.653413	+	0.757002i	$0.726664\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	1032.96			1.16455			0.582275	−	0.812992i	$-0.302163\pi$
$887$	1032.96			1.16455			0.582275	+	0.812992i	$0.302163\pi$
$888$		0			0
$889$			1740.34i			1.95764i
$890$		0			0
$891$			1424.39i			1.59864i
$892$		0			0
$893$	658.762			0.737695
$894$		0			0
$895$		0			0
$896$		0			0
$897$		−	24.8327i		−	0.0276842i
$898$		0			0
$899$	236.485	−	1002.59i	0.263054	−	1.11522i
$900$		0			0
$901$	140.509			0.155948
$902$		0			0
$903$	530.221			0.587177
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−1575.76			−1.73733			−0.868665	−	0.495399i	$-0.835022\pi$
$907$	−1575.76			−1.73733			−0.868665	+	0.495399i	$0.835022\pi$
$908$		0			0
$909$	−31.1308			−0.0342473
$910$		0			0
$911$		−	269.186i		−	0.295484i	−0.989026	−	0.147742i	$-0.952799\pi$
$911$		−	269.186i		−	0.295484i	0.989026	−	0.147742i	$-0.0472006\pi$
$912$		0			0
$913$	−2113.16			−2.31452
$914$		0			0
$915$		0			0
$916$		0			0
$917$	2141.57			2.33540
$918$		0			0
$919$	802.258			0.872968			0.436484	−	0.899712i	$-0.356224\pi$
$919$	802.258			0.872968			0.436484	+	0.899712i	$0.356224\pi$
$920$		0			0
$921$			795.228i			0.863440i
$922$		0			0
$923$			921.572i			0.998452i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	21.1295			0.0227935
$928$		0			0
$929$			843.856i			0.908349i	0.890913	+	0.454174i	$0.150066\pi$
$929$			843.856i			0.908349i	−0.890913	+	0.454174i	$0.849934\pi$
$930$		0			0
$931$	1878.29			2.01750
$932$		0			0
$933$		−	615.696i		−	0.659910i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	297.497			0.317500			0.158750	−	0.987319i	$-0.449254\pi$
$937$	297.497			0.317500			0.158750	+	0.987319i	$0.449254\pi$
$938$		0			0
$939$	1321.13			1.40696
$940$		0			0
$941$			1112.43i			1.18218i	0.806605	+	0.591091i	$0.201303\pi$
$941$			1112.43i			1.18218i	−0.806605	+	0.591091i	$0.798697\pi$
$942$		0			0
$943$			5.94492i			0.00630427i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	1271.66i		−	1.34283i	−0.741082	−	0.671414i	$-0.765687\pi$
$947$		−	1271.66i		−	1.34283i	0.741082	−	0.671414i	$-0.234313\pi$
$948$		0			0
$949$	367.839			0.387607
$950$		0			0
$951$			1.69639i			0.00178380i
$952$		0			0
$953$			321.088i			0.336924i	0.985708	+	0.168462i	$0.0538800\pi$
$953$			321.088i			0.336924i	−0.985708	+	0.168462i	$0.946120\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$			1735.75i			1.81374i
$958$		0			0
$959$			1041.62i			1.08615i
$960$		0			0
$961$	859.701	+	429.458i	0.894591	+	0.446887i
$962$		0			0
$963$	33.7421			0.0350385
$964$		0			0
$965$		0			0
$966$		0			0
$967$		−	1510.14i		−	1.56167i	−0.624735	−	0.780836i	$-0.714793\pi$
$967$		−	1510.14i		−	1.56167i	0.624735	−	0.780836i	$-0.285207\pi$
$968$		0			0
$969$	−134.934			−0.139251
$970$		0			0
$971$	−295.871			−0.304708			−0.152354	−	0.988326i	$-0.548685\pi$
$971$	−295.871			−0.304708			−0.152354	+	0.988326i	$0.548685\pi$
$972$		0			0
$973$			178.879i			0.183843i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−1086.22			−1.11179			−0.555895	−	0.831253i	$-0.687624\pi$
$977$	−1086.22			−1.11179			−0.555895	+	0.831253i	$0.687624\pi$
$978$		0			0
$979$	−2571.44			−2.62660
$980$		0			0
$981$	−34.9208			−0.0355971
$982$		0			0
$983$		−	1136.59i		−	1.15625i	−0.815949	−	0.578124i	$-0.803785\pi$
$983$		−	1136.59i		−	1.15625i	0.815949	−	0.578124i	$-0.196215\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$			1487.09i			1.50668i
$988$		0			0
$989$	−7.57821			−0.00766250
$990$		0			0
$991$			1661.00i			1.67608i	0.545609	+	0.838040i	$0.316299\pi$
$991$			1661.00i			1.67608i	−0.545609	+	0.838040i	$0.683701\pi$
$992$		0			0
$993$	−455.436			−0.458647
$994$		0			0
$995$		0			0
$996$		0			0
$997$	660.502			0.662489			0.331245	−	0.943545i	$-0.392531\pi$
$997$	660.502			0.662489			0.331245	+	0.943545i	$0.392531\pi$
$998$		0			0
$999$	1664.78			1.66644

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3100.3.d.h.1301.4		24
5.2	odd	4		620.3.f.c.309.20	yes	24
5.3	odd	4		620.3.f.c.309.5	✓	24
5.4	even	2	inner	3100.3.d.h.1301.21		24
31.30	odd	2	inner	3100.3.d.h.1301.3		24
155.92	even	4		620.3.f.c.309.6	yes	24
155.123	even	4		620.3.f.c.309.19	yes	24
155.154	odd	2	inner	3100.3.d.h.1301.22		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
620.3.f.c.309.5	✓	24	5.3	odd	4
620.3.f.c.309.6	yes	24	155.92	even	4
620.3.f.c.309.19	yes	24	155.123	even	4
620.3.f.c.309.20	yes	24	5.2	odd	4
3100.3.d.h.1301.3		24	31.30	odd	2	inner
3100.3.d.h.1301.4		24	1.1	even	1	trivial
3100.3.d.h.1301.21		24	5.4	even	2	inner
3100.3.d.h.1301.22		24	155.154	odd	2	inner

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:	\( N \)	\(=\)	\( 3100 = 2^{2} \cdot 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	3100.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+3.03111i q^{3} +12.6370 q^{7} -0.187623 q^{9} +O(q^{10})\)	\(q+3.03111i q^{3} +12.6370 q^{7} -0.187623 q^{9} -17.2333i q^{11} -14.9646i q^{13} +2.62351i q^{17} +16.9683 q^{19} +38.3042i q^{21} -0.547465i q^{23} +26.7113i q^{27} +33.2290i q^{29} +(-30.1720 - 7.11683i) q^{31} +52.2360 q^{33} -62.3249i q^{37} +45.3594 q^{39} -10.8590 q^{41} -13.8424i q^{43} +38.8232 q^{47} +110.694 q^{49} -7.95215 q^{51} -53.5576i q^{53} +51.4327i q^{57} -44.2110 q^{59} +71.3500i q^{61} -2.37099 q^{63} +55.3337 q^{67} +1.65943 q^{69} -61.5833 q^{71} +24.5806i q^{73} -217.778i q^{77} -130.554i q^{79} -82.6534 q^{81} -122.621i q^{83} -100.721 q^{87} -149.213i q^{89} -189.108i q^{91} +(21.5719 - 91.4547i) q^{93} -25.3091 q^{97} +3.23336i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 44 q^{9}+O(q^{10}) \) 24 * q - 44 * q^9	\( 24 q - 44 q^{9} - 8 q^{19} - 24 q^{31} + 280 q^{39} - 248 q^{41} + 644 q^{49} - 100 q^{51} - 152 q^{59} + 288 q^{69} - 352 q^{71} - 368 q^{81}+O(q^{100}) \) 24 * q - 44 * q^9 - 8 * q^19 - 24 * q^31 + 280 * q^39 - 248 * q^41 + 644 * q^49 - 100 * q^51 - 152 * q^59 + 288 * q^69 - 352 * q^71 - 368 * q^81

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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