LMFDB - Embedded newform 3100.3.d.g.1301.14

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:	$22$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1301.14
Character	$\chi$	$=$	3100.1301
Dual form			3100.3.d.g.1301.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.957337i q^{3} +12.5381 q^{7} +8.08351 q^{9} +O(q^{10})$	$q+0.957337i q^{3} +12.5381 q^{7} +8.08351 q^{9} +6.45459i q^{11} +4.62883i q^{13} -27.9754i q^{17} -18.3169 q^{19} +12.0031i q^{21} +17.4106i q^{23} +16.3547i q^{27} +34.0557i q^{29} +(-8.55941 + 29.7949i) q^{31} -6.17922 q^{33} +10.2841i q^{37} -4.43135 q^{39} +34.2069 q^{41} +82.1464i q^{43} +2.35672 q^{47} +108.203 q^{49} +26.7819 q^{51} +63.9636i q^{53} -17.5354i q^{57} -49.0778 q^{59} +44.7523i q^{61} +101.352 q^{63} -54.9369 q^{67} -16.6678 q^{69} +23.8443 q^{71} -123.706i q^{73} +80.9281i q^{77} -94.2940i q^{79} +57.0946 q^{81} -24.1863i q^{83} -32.6028 q^{87} -21.0604i q^{89} +58.0366i q^{91} +(-28.5238 - 8.19424i) q^{93} +115.185 q^{97} +52.1757i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 22 q + 4 q^{7} - 72 q^{9}+O(q^{10}) $ 22 * q + 4 * q^7 - 72 * q^9	$ 22 q + 4 q^{7} - 72 q^{9} - 28 q^{19} - 18 q^{31} - 34 q^{33} - 62 q^{39} + 64 q^{41} - 96 q^{47} + 150 q^{49} - 130 q^{51} + 40 q^{59} - 4 q^{63} + 110 q^{67} + 100 q^{69} + 132 q^{71} + 234 q^{81} - 62 q^{87} + 16 q^{93} + 186 q^{97}+O(q^{100}) $ 22 * q + 4 * q^7 - 72 * q^9 - 28 * q^19 - 18 * q^31 - 34 * q^33 - 62 * q^39 + 64 * q^41 - 96 * q^47 + 150 * q^49 - 130 * q^51 + 40 * q^59 - 4 * q^63 + 110 * q^67 + 100 * q^69 + 132 * q^71 + 234 * q^81 - 62 * q^87 + 16 * q^93 + 186 * q^97

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
$3$			0.957337i			0.319112i	0.987189	+	0.159556i	$0.0510063\pi$
$3$			0.957337i			0.319112i	−0.987189	+	0.159556i	$0.948994\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	12.5381			1.79115			0.895576	−	0.444909i	$-0.146764\pi$
$7$	12.5381			1.79115			0.895576	+	0.444909i	$0.146764\pi$
$8$		0			0
$9$	8.08351			0.898167
$10$		0			0
$11$			6.45459i			0.586781i	0.955993	+	0.293390i	$0.0947836\pi$
$11$			6.45459i			0.586781i	−0.955993	+	0.293390i	$0.905216\pi$
$12$		0			0
$13$			4.62883i			0.356064i	0.984025	+	0.178032i	$0.0569730\pi$
$13$			4.62883i			0.356064i	−0.984025	+	0.178032i	$0.943027\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	27.9754i		−	1.64561i	−0.568323	−	0.822805i	$-0.692408\pi$
$17$		−	27.9754i		−	1.64561i	0.568323	−	0.822805i	$-0.307592\pi$
$18$		0			0
$19$	−18.3169			−0.964047			−0.482023	−	0.876158i	$-0.660098\pi$
$19$	−18.3169			−0.964047			−0.482023	+	0.876158i	$0.660098\pi$
$20$		0			0
$21$			12.0031i			0.571579i
$22$		0			0
$23$			17.4106i			0.756982i	0.925605	+	0.378491i	$0.123557\pi$
$23$			17.4106i			0.756982i	−0.925605	+	0.378491i	$0.876443\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$			16.3547i			0.605729i
$28$		0			0
$29$			34.0557i			1.17433i	0.809466	+	0.587167i	$0.199757\pi$
$29$			34.0557i			1.17433i	−0.809466	+	0.587167i	$0.800243\pi$
$30$		0			0
$31$	−8.55941	+	29.7949i	−0.276110	+	0.961126i
$31$	−8.55941	+	29.7949i	−0.276110	+	0.961126i
$32$		0			0
$33$	−6.17922			−0.187249
$34$		0			0
$35$		0			0
$36$		0			0
$37$			10.2841i			0.277950i	0.990296	+	0.138975i	$0.0443807\pi$
$37$			10.2841i			0.277950i	−0.990296	+	0.138975i	$0.955619\pi$
$38$		0			0
$39$	−4.43135			−0.113624
$40$		0			0
$41$	34.2069			0.834315			0.417157	−	0.908834i	$-0.363026\pi$
$41$	34.2069			0.834315			0.417157	+	0.908834i	$0.363026\pi$
$42$		0			0
$43$			82.1464i			1.91038i	0.295990	+	0.955191i	$0.404350\pi$
$43$			82.1464i			1.91038i	−0.295990	+	0.955191i	$0.595650\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	2.35672			0.0501431			0.0250715	−	0.999686i	$-0.492019\pi$
$47$	2.35672			0.0501431			0.0250715	+	0.999686i	$0.492019\pi$
$48$		0			0
$49$	108.203			2.20822
$50$		0			0
$51$	26.7819			0.525135
$52$		0			0
$53$			63.9636i			1.20686i	0.797416	+	0.603430i	$0.206200\pi$
$53$			63.9636i			1.20686i	−0.797416	+	0.603430i	$0.793800\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		−	17.5354i		−	0.307639i
$58$		0			0
$59$	−49.0778			−0.831827			−0.415914	−	0.909404i	$-0.636538\pi$
$59$	−49.0778			−0.831827			−0.415914	+	0.909404i	$0.636538\pi$
$60$		0			0
$61$			44.7523i			0.733644i	0.930291	+	0.366822i	$0.119554\pi$
$61$			44.7523i			0.733644i	−0.930291	+	0.366822i	$0.880446\pi$
$62$		0			0
$63$	101.352			1.60875
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−54.9369			−0.819954			−0.409977	−	0.912096i	$-0.634463\pi$
$67$	−54.9369			−0.819954			−0.409977	+	0.912096i	$0.634463\pi$
$68$		0			0
$69$	−16.6678			−0.241562
$70$		0			0
$71$	23.8443			0.335836			0.167918	−	0.985801i	$-0.446296\pi$
$71$	23.8443			0.335836			0.167918	+	0.985801i	$0.446296\pi$
$72$		0			0
$73$		−	123.706i		−	1.69460i	−0.531116	−	0.847299i	$-0.678227\pi$
$73$		−	123.706i		−	1.69460i	0.531116	−	0.847299i	$-0.321773\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			80.9281i			1.05101i
$78$		0			0
$79$		−	94.2940i		−	1.19359i	−0.802392	−	0.596797i	$-0.796440\pi$
$79$		−	94.2940i		−	1.19359i	0.802392	−	0.596797i	$-0.203560\pi$
$80$		0			0
$81$	57.0946			0.704872
$82$		0			0
$83$		−	24.1863i		−	0.291401i	−0.989329	−	0.145701i	$-0.953456\pi$
$83$		−	24.1863i		−	0.291401i	0.989329	−	0.145701i	$-0.0465436\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−32.6028			−0.374744
$88$		0			0
$89$		−	21.0604i		−	0.236633i	−0.992976	−	0.118317i	$-0.962250\pi$
$89$		−	21.0604i		−	0.236633i	0.992976	−	0.118317i	$-0.0377498\pi$
$90$		0			0
$91$			58.0366i			0.637765i
$92$		0			0
$93$	−28.5238	−	8.19424i	−0.306707	−	0.0881101i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	115.185			1.18748			0.593739	−	0.804657i	$-0.297651\pi$
$97$	115.185			1.18748			0.593739	+	0.804657i	$0.297651\pi$
$98$		0			0
$99$			52.1757i			0.527027i
$100$		0			0
$101$	−150.722			−1.49230			−0.746149	−	0.665779i	$-0.768099\pi$
$101$	−150.722			−1.49230			−0.746149	+	0.665779i	$0.768099\pi$
$102$		0			0
$103$	−159.678			−1.55027			−0.775136	−	0.631794i	$-0.782319\pi$
$103$	−159.678			−1.55027			−0.775136	+	0.631794i	$0.782319\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	97.5647			0.911820			0.455910	−	0.890026i	$-0.349314\pi$
$107$	97.5647			0.911820			0.455910	+	0.890026i	$0.349314\pi$
$108$		0			0
$109$	78.7375			0.722362			0.361181	−	0.932496i	$-0.382374\pi$
$109$	78.7375			0.722362			0.361181	+	0.932496i	$0.382374\pi$
$110$		0			0
$111$	−9.84539			−0.0886972
$112$		0			0
$113$	10.9312			0.0967367			0.0483683	−	0.998830i	$-0.484598\pi$
$113$	10.9312			0.0967367			0.0483683	+	0.998830i	$0.484598\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$			37.4172i			0.319805i
$118$		0			0
$119$		−	350.757i		−	2.94754i
$120$		0			0
$121$	79.3383			0.655688
$122$		0			0
$123$			32.7475i			0.266240i
$124$		0			0
$125$		0			0
$126$		0			0
$127$			190.929i			1.50338i	0.659517	+	0.751690i	$0.270761\pi$
$127$			190.929i			1.50338i	−0.659517	+	0.751690i	$0.729239\pi$
$128$		0			0
$129$	−78.6418			−0.609626
$130$		0			0
$131$	−72.8934			−0.556438			−0.278219	−	0.960518i	$-0.589744\pi$
$131$	−72.8934			−0.556438			−0.278219	+	0.960518i	$0.589744\pi$
$132$		0			0
$133$	−229.658			−1.72675
$134$		0			0
$135$		0			0
$136$		0			0
$137$			132.645i			0.968215i	0.875008	+	0.484108i	$0.160856\pi$
$137$			132.645i			0.968215i	−0.875008	+	0.484108i	$0.839144\pi$
$138$		0			0
$139$		−	28.6204i		−	0.205902i	−0.994686	−	0.102951i	$-0.967171\pi$
$139$		−	28.6204i		−	0.205902i	0.994686	−	0.102951i	$-0.0328285\pi$
$140$		0			0
$141$			2.25618i			0.0160013i
$142$		0			0
$143$	−29.8772			−0.208932
$144$		0			0
$145$		0			0
$146$		0			0
$147$			103.587i			0.704672i
$148$		0			0
$149$	−200.836			−1.34789			−0.673947	−	0.738779i	$-0.735402\pi$
$149$	−200.836			−1.34789			−0.673947	+	0.738779i	$0.735402\pi$
$150$		0			0
$151$		−	148.996i		−	0.986727i	−0.869823	−	0.493364i	$-0.835767\pi$
$151$		−	148.996i		−	0.986727i	0.869823	−	0.493364i	$-0.164233\pi$
$152$		0			0
$153$		−	226.139i		−	1.47803i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	120.126			0.765133			0.382567	−	0.923928i	$-0.375040\pi$
$157$	120.126			0.765133			0.382567	+	0.923928i	$0.375040\pi$
$158$		0			0
$159$	−61.2347			−0.385124
$160$		0			0
$161$			218.295i			1.35587i
$162$		0			0
$163$	172.918			1.06085			0.530423	−	0.847733i	$-0.322033\pi$
$163$	172.918			1.06085			0.530423	+	0.847733i	$0.322033\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			61.6481i			0.369150i	0.982818	+	0.184575i	$0.0590909\pi$
$167$			61.6481i			0.369150i	−0.982818	+	0.184575i	$0.940909\pi$
$168$		0			0
$169$	147.574			0.873218
$170$		0			0
$171$	−148.065			−0.865875
$172$		0			0
$173$	194.170			1.12237			0.561184	−	0.827691i	$-0.310346\pi$
$173$	194.170			1.12237			0.561184	+	0.827691i	$0.310346\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		−	46.9840i		−	0.265446i
$178$		0			0
$179$		−	326.091i		−	1.82174i	−0.412693	−	0.910870i	$-0.635412\pi$
$179$		−	326.091i		−	1.82174i	0.412693	−	0.910870i	$-0.364588\pi$
$180$		0			0
$181$		−	202.064i		−	1.11637i	−0.829715	−	0.558187i	$-0.811497\pi$
$181$		−	202.064i		−	1.11637i	0.829715	−	0.558187i	$-0.188503\pi$
$182$		0			0
$183$	−42.8430			−0.234115
$184$		0			0
$185$		0			0
$186$		0			0
$187$	180.570			0.965613
$188$		0			0
$189$			205.056i			1.08495i
$190$		0			0
$191$	245.447			1.28506			0.642531	−	0.766260i	$-0.277884\pi$
$191$	245.447			1.28506			0.642531	+	0.766260i	$0.277884\pi$
$192$		0			0
$193$	139.941			0.725085			0.362542	−	0.931967i	$-0.381909\pi$
$193$	139.941			0.725085			0.362542	+	0.931967i	$0.381909\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			264.058i			1.34039i	0.742183	+	0.670197i	$0.233791\pi$
$197$			264.058i			1.34039i	−0.742183	+	0.670197i	$0.766209\pi$
$198$		0			0
$199$			105.641i			0.530859i	0.964130	+	0.265429i	$0.0855137\pi$
$199$			105.641i			0.530859i	−0.964130	+	0.265429i	$0.914486\pi$
$200$		0			0
$201$		−	52.5931i		−	0.261657i
$202$		0			0
$203$			426.992i			2.10341i
$204$		0			0
$205$		0			0
$206$		0			0
$207$			140.739i			0.679897i
$208$		0			0
$209$		−	118.228i		−	0.565684i
$210$		0			0
$211$	232.697			1.10283			0.551414	−	0.834232i	$-0.314088\pi$
$211$	232.697			1.10283			0.551414	+	0.834232i	$0.314088\pi$
$212$		0			0
$213$			22.8271i			0.107169i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−107.318	+	373.570i	−0.494555	+	1.72152i
$218$		0			0
$219$	118.428			0.540767
$220$		0			0
$221$	129.493			0.585943
$222$		0			0
$223$			280.489i			1.25780i	0.777487	+	0.628899i	$0.216494\pi$
$223$			280.489i			1.25780i	−0.777487	+	0.628899i	$0.783506\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−61.4774			−0.270825			−0.135413	−	0.990789i	$-0.543236\pi$
$227$	−61.4774			−0.270825			−0.135413	+	0.990789i	$0.543236\pi$
$228$		0			0
$229$			19.9507i			0.0871211i	0.999051	+	0.0435605i	$0.0138701\pi$
$229$			19.9507i			0.0871211i	−0.999051	+	0.0435605i	$0.986130\pi$
$230$		0			0
$231$	−77.4754			−0.335391
$232$		0			0
$233$	330.871			1.42005			0.710024	−	0.704177i	$-0.248684\pi$
$233$	330.871			1.42005			0.710024	+	0.704177i	$0.248684\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	90.2711			0.380891
$238$		0			0
$239$		−	139.998i		−	0.585766i	−0.956148	−	0.292883i	$-0.905385\pi$
$239$		−	139.998i		−	0.585766i	0.956148	−	0.292883i	$-0.0946147\pi$
$240$		0			0
$241$			246.036i			1.02089i	0.859909	+	0.510447i	$0.170520\pi$
$241$			246.036i			1.02089i	−0.859909	+	0.510447i	$0.829480\pi$
$242$		0			0
$243$			201.851i			0.830662i
$244$		0			0
$245$		0			0
$246$		0			0
$247$		−	84.7858i		−	0.343262i
$248$		0			0
$249$	23.1544			0.0929897
$250$		0			0
$251$			138.762i			0.552835i	0.961038	+	0.276417i	$0.0891472\pi$
$251$			138.762i			0.552835i	−0.961038	+	0.276417i	$0.910853\pi$
$252$		0			0
$253$	−112.378			−0.444183
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−86.1950			−0.335389			−0.167695	−	0.985839i	$-0.553632\pi$
$257$	−86.1950			−0.335389			−0.167695	+	0.985839i	$0.553632\pi$
$258$		0			0
$259$			128.943i			0.497850i
$260$		0			0
$261$			275.289i			1.05475i
$262$		0			0
$263$		−	111.985i		−	0.425799i	−0.977074	−	0.212900i	$-0.931709\pi$
$263$		−	111.985i		−	0.425799i	0.977074	−	0.212900i	$-0.0682907\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	20.1619			0.0755126
$268$		0			0
$269$		−	389.223i		−	1.44693i	−0.690363	−	0.723463i	$-0.742549\pi$
$269$		−	389.223i		−	1.44693i	0.690363	−	0.723463i	$-0.257451\pi$
$270$		0			0
$271$			142.880i			0.527234i	0.964627	+	0.263617i	$0.0849155\pi$
$271$			142.880i			0.527234i	−0.964627	+	0.263617i	$0.915084\pi$
$272$		0			0
$273$	−55.5606			−0.203519
$274$		0			0
$275$		0			0
$276$		0			0
$277$			235.632i			0.850658i	0.905039	+	0.425329i	$0.139842\pi$
$277$			235.632i			0.850658i	−0.905039	+	0.425329i	$0.860158\pi$
$278$		0			0
$279$	−69.1900	+	240.847i	−0.247993	+	0.863252i
$280$		0			0
$281$	316.317			1.12568			0.562842	−	0.826565i	$-0.309708\pi$
$281$	316.317			1.12568			0.562842	+	0.826565i	$0.309708\pi$
$282$		0			0
$283$	−327.084			−1.15577			−0.577887	−	0.816117i	$-0.696123\pi$
$283$	−327.084			−1.15577			−0.577887	+	0.816117i	$0.696123\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	428.888			1.49438
$288$		0			0
$289$	−493.622			−1.70803
$290$		0			0
$291$			110.271i			0.378939i
$292$		0			0
$293$	99.5877			0.339890			0.169945	−	0.985454i	$-0.445641\pi$
$293$	99.5877			0.339890			0.169945	+	0.985454i	$0.445641\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−105.563			−0.355430
$298$		0			0
$299$	−80.5907			−0.269534
$300$		0			0
$301$			1029.96i			3.42178i
$302$		0			0
$303$		−	144.292i		−	0.476211i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−73.6388			−0.239866			−0.119933	−	0.992782i	$-0.538268\pi$
$307$	−73.6388			−0.239866			−0.119933	+	0.992782i	$0.538268\pi$
$308$		0			0
$309$		−	152.866i		−	0.494711i
$310$		0			0
$311$	−554.949			−1.78440			−0.892201	−	0.451640i	$-0.850839\pi$
$311$	−554.949			−1.78440			−0.892201	+	0.451640i	$0.850839\pi$
$312$		0			0
$313$			283.035i			0.904265i	0.891951	+	0.452133i	$0.149337\pi$
$313$			283.035i			0.904265i	−0.891951	+	0.452133i	$0.850663\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−457.252			−1.44243			−0.721217	−	0.692709i	$-0.756417\pi$
$317$	−457.252			−1.44243			−0.721217	+	0.692709i	$0.756417\pi$
$318$		0			0
$319$	−219.816			−0.689077
$320$		0			0
$321$			93.4023i			0.290973i
$322$		0			0
$323$			512.422i			1.58645i
$324$		0			0
$325$		0			0
$326$		0			0
$327$			75.3783i			0.230515i
$328$		0			0
$329$	29.5488			0.0898139
$330$		0			0
$331$		−	203.608i		−	0.615131i	−0.951527	−	0.307565i	$-0.900486\pi$
$331$		−	203.608i		−	0.615131i	0.951527	−	0.307565i	$-0.0995143\pi$
$332$		0			0
$333$			83.1319i			0.249645i
$334$		0			0
$335$		0			0
$336$		0			0
$337$			123.808i			0.367383i	0.982984	+	0.183691i	$0.0588047\pi$
$337$			123.808i			0.367383i	−0.982984	+	0.183691i	$0.941195\pi$
$338$		0			0
$339$			10.4649i			0.0308699i
$340$		0			0
$341$	−192.314	−	55.2475i	−0.563970	−	0.162016i
$342$		0			0
$343$	742.291			2.16411
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	656.462i		−	1.89182i	−0.324428	−	0.945911i	$-0.605172\pi$
$347$		−	656.462i		−	1.89182i	0.324428	−	0.945911i	$-0.394828\pi$
$348$		0			0
$349$	10.8273			0.0310239			0.0155120	−	0.999880i	$-0.495062\pi$
$349$	10.8273			0.0310239			0.0155120	+	0.999880i	$0.495062\pi$
$350$		0			0
$351$	−75.7030			−0.215678
$352$		0			0
$353$			86.1215i			0.243970i	0.992532	+	0.121985i	$0.0389260\pi$
$353$			86.1215i			0.243970i	−0.992532	+	0.121985i	$0.961074\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	335.793			0.940596
$358$		0			0
$359$	−135.320			−0.376937			−0.188468	−	0.982079i	$-0.560352\pi$
$359$	−135.320			−0.376937			−0.188468	+	0.982079i	$0.560352\pi$
$360$		0			0
$361$	−25.4917			−0.0706141
$362$		0			0
$363$			75.9534i			0.209238i
$364$		0			0
$365$		0			0
$366$		0			0
$367$			570.456i			1.55438i	0.629268	+	0.777189i	$0.283355\pi$
$367$			570.456i			1.55438i	−0.629268	+	0.777189i	$0.716645\pi$
$368$		0			0
$369$	276.512			0.749354
$370$		0			0
$371$			801.980i			2.16167i
$372$		0			0
$373$	−151.217			−0.405409			−0.202704	−	0.979240i	$-0.564973\pi$
$373$	−151.217			−0.405409			−0.202704	+	0.979240i	$0.564973\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−157.638			−0.418138
$378$		0			0
$379$	79.6051			0.210040			0.105020	−	0.994470i	$-0.466509\pi$
$379$	79.6051			0.210040			0.105020	+	0.994470i	$0.466509\pi$
$380$		0			0
$381$	−182.784			−0.479747
$382$		0			0
$383$		−	385.132i		−	1.00557i	−0.864413	−	0.502783i	$-0.832309\pi$
$383$		−	385.132i		−	1.00557i	0.864413	−	0.502783i	$-0.167691\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$			664.031i			1.71584i
$388$		0			0
$389$		−	37.5291i		−	0.0964759i	−0.998836	−	0.0482379i	$-0.984639\pi$
$389$		−	37.5291i		−	0.0964759i	0.998836	−	0.0482379i	$-0.0153606\pi$
$390$		0			0
$391$	487.068			1.24570
$392$		0			0
$393$		−	69.7835i		−	0.177566i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	181.996			0.458429			0.229215	−	0.973376i	$-0.426384\pi$
$397$	181.996			0.458429			0.229215	+	0.973376i	$0.426384\pi$
$398$		0			0
$399$		−	219.860i		−	0.551028i
$400$		0			0
$401$		−	199.817i		−	0.498298i	−0.968465	−	0.249149i	$-0.919849\pi$
$401$		−	199.817i		−	0.498298i	0.968465	−	0.249149i	$-0.0801508\pi$
$402$		0			0
$403$	−137.916	−	39.6201i	−0.342222	−	0.0983128i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−66.3799			−0.163096
$408$		0			0
$409$		−	289.651i		−	0.708193i	−0.935209	−	0.354096i	$-0.884788\pi$
$409$		−	289.651i		−	0.708193i	0.935209	−	0.354096i	$-0.115212\pi$
$410$		0			0
$411$	−126.986			−0.308969
$412$		0			0
$413$	−615.341			−1.48993
$414$		0			0
$415$		0			0
$416$		0			0
$417$	27.3994			0.0657060
$418$		0			0
$419$	177.037			0.422523			0.211262	−	0.977430i	$-0.432243\pi$
$419$	177.037			0.422523			0.211262	+	0.977430i	$0.432243\pi$
$420$		0			0
$421$	539.661			1.28186			0.640928	−	0.767601i	$-0.278550\pi$
$421$	539.661			1.28186			0.640928	+	0.767601i	$0.278550\pi$
$422$		0			0
$423$	19.0506			0.0450369
$424$		0			0
$425$		0			0
$426$		0			0
$427$			561.107i			1.31407i
$428$		0			0
$429$		−	28.6026i		−	0.0666726i
$430$		0			0
$431$	−386.168			−0.895981			−0.447991	−	0.894038i	$-0.647860\pi$
$431$	−386.168			−0.895981			−0.447991	+	0.894038i	$0.647860\pi$
$432$		0			0
$433$		−	264.248i		−	0.610273i	−0.952309	−	0.305137i	$-0.901298\pi$
$433$		−	264.248i		−	0.610273i	0.952309	−	0.305137i	$-0.0987021\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	318.908i		−	0.729766i
$438$		0			0
$439$	247.325			0.563382			0.281691	−	0.959505i	$-0.409105\pi$
$439$	247.325			0.563382			0.281691	+	0.959505i	$0.409105\pi$
$440$		0			0
$441$	874.660			1.98336
$442$		0			0
$443$	689.353			1.55610			0.778051	−	0.628202i	$-0.216209\pi$
$443$	689.353			1.55610			0.778051	+	0.628202i	$0.216209\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		−	192.268i		−	0.430130i
$448$		0			0
$449$		−	832.590i		−	1.85432i	−0.374665	−	0.927160i	$-0.622242\pi$
$449$		−	832.590i		−	1.85432i	0.374665	−	0.927160i	$-0.377758\pi$
$450$		0			0
$451$			220.792i			0.489560i
$452$		0			0
$453$	142.639			0.314877
$454$		0			0
$455$		0			0
$456$		0			0
$457$		−	156.740i		−	0.342975i	−0.985186	−	0.171488i	$-0.945143\pi$
$457$		−	156.740i		−	0.342975i	0.985186	−	0.171488i	$-0.0548574\pi$
$458$		0			0
$459$	457.528			0.996793
$460$		0			0
$461$		−	846.609i		−	1.83646i	−0.396046	−	0.918231i	$-0.629618\pi$
$461$		−	846.609i		−	1.83646i	0.396046	−	0.918231i	$-0.370382\pi$
$462$		0			0
$463$		−	104.757i		−	0.226256i	−0.993580	−	0.113128i	$-0.963913\pi$
$463$		−	104.757i		−	0.226256i	0.993580	−	0.113128i	$-0.0360871\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−469.270			−1.00486			−0.502430	−	0.864618i	$-0.667561\pi$
$467$	−469.270			−1.00486			−0.502430	+	0.864618i	$0.667561\pi$
$468$		0			0
$469$	−688.802			−1.46866
$470$		0			0
$471$			115.001i			0.244163i
$472$		0			0
$473$	−530.221			−1.12098
$474$		0			0
$475$		0			0
$476$		0			0
$477$			517.050i			1.08396i
$478$		0			0
$479$	−721.582			−1.50643			−0.753217	−	0.657772i	$-0.771499\pi$
$479$	−721.582			−1.50643			−0.753217	+	0.657772i	$0.771499\pi$
$480$		0			0
$481$	−47.6036			−0.0989679
$482$		0			0
$483$	−208.982			−0.432675
$484$		0			0
$485$		0			0
$486$		0			0
$487$			508.350i			1.04384i	0.852995	+	0.521919i	$0.174784\pi$
$487$			508.350i			1.04384i	−0.852995	+	0.521919i	$0.825216\pi$
$488$		0			0
$489$			165.541i			0.338529i
$490$		0			0
$491$			43.0848i			0.0877492i	0.999037	+	0.0438746i	$0.0139702\pi$
$491$			43.0848i			0.0877492i	−0.999037	+	0.0438746i	$0.986030\pi$
$492$		0			0
$493$	952.721			1.93250
$494$		0			0
$495$		0			0
$496$		0			0
$497$	298.962			0.601533
$498$		0			0
$499$			397.773i			0.797140i	0.917138	+	0.398570i	$0.130493\pi$
$499$			397.773i			0.797140i	−0.917138	+	0.398570i	$0.869507\pi$
$500$		0			0
$501$	−59.0180			−0.117800
$502$		0			0
$503$	−759.610			−1.51016			−0.755080	−	0.655633i	$-0.772402\pi$
$503$	−759.610			−1.51016			−0.755080	+	0.655633i	$0.772402\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$			141.278i			0.278655i
$508$		0			0
$509$		−	705.929i		−	1.38689i	−0.720508	−	0.693447i	$-0.756091\pi$
$509$		−	705.929i		−	1.38689i	0.720508	−	0.693447i	$-0.243909\pi$
$510$		0			0
$511$		−	1551.03i		−	3.03528i
$512$		0			0
$513$		−	299.567i		−	0.583951i
$514$		0			0
$515$		0			0
$516$		0			0
$517$			15.2117i			0.0294230i
$518$		0			0
$519$			185.886i			0.358162i
$520$		0			0
$521$	406.342			0.779927			0.389963	−	0.920830i	$-0.372488\pi$
$521$	406.342			0.779927			0.389963	+	0.920830i	$0.372488\pi$
$522$		0			0
$523$			192.449i			0.367971i	0.982929	+	0.183986i	$0.0589000\pi$
$523$			192.449i			0.367971i	−0.982929	+	0.183986i	$0.941100\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	833.524	+	239.453i	1.58164	+	0.454370i
$528$		0			0
$529$	225.871			0.426978
$530$		0			0
$531$	−396.721			−0.747120
$532$		0			0
$533$			158.338i			0.297069i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	312.179			0.581340
$538$		0			0
$539$			698.406i			1.29574i
$540$		0			0
$541$	−577.722			−1.06788			−0.533939	−	0.845523i	$-0.679289\pi$
$541$	−577.722			−1.06788			−0.533939	+	0.845523i	$0.679289\pi$
$542$		0			0
$543$	193.443			0.356248
$544$		0			0
$545$		0			0
$546$		0			0
$547$	347.000			0.634370			0.317185	−	0.948364i	$-0.397262\pi$
$547$	347.000			0.634370			0.317185	+	0.948364i	$0.397262\pi$
$548$		0			0
$549$			361.755i			0.658935i
$550$		0			0
$551$		−	623.794i		−	1.13211i
$552$		0			0
$553$		−	1182.26i		−	2.13791i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	485.901i		−	0.872354i	−0.899861	−	0.436177i	$-0.856332\pi$
$557$		−	485.901i		−	0.872354i	0.899861	−	0.436177i	$-0.143668\pi$
$558$		0			0
$559$	−380.242			−0.680218
$560$		0			0
$561$			172.866i			0.308139i
$562$		0			0
$563$	943.562			1.67595			0.837977	−	0.545706i	$-0.183738\pi$
$563$	943.562			1.67595			0.837977	+	0.545706i	$0.183738\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	715.856			1.26253
$568$		0			0
$569$		−	1044.94i		−	1.83645i	−0.396062	−	0.918224i	$-0.629623\pi$
$569$		−	1044.94i		−	1.83645i	0.396062	−	0.918224i	$-0.370377\pi$
$570$		0			0
$571$			715.772i			1.25354i	0.779204	+	0.626771i	$0.215624\pi$
$571$			715.772i			1.25354i	−0.779204	+	0.626771i	$0.784376\pi$
$572$		0			0
$573$			234.975i			0.410079i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−995.223			−1.72482			−0.862412	−	0.506207i	$-0.831047\pi$
$577$	−995.223			−1.72482			−0.862412	+	0.506207i	$0.831047\pi$
$578$		0			0
$579$			133.971i			0.231383i
$580$		0			0
$581$		−	303.249i		−	0.521944i
$582$		0			0
$583$	−412.859			−0.708163
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	327.196i		−	0.557403i	−0.960378	−	0.278702i	$-0.910096\pi$
$587$		−	327.196i		−	0.557403i	0.960378	−	0.278702i	$-0.0899041\pi$
$588$		0			0
$589$	156.782	−	545.750i	0.266183	−	0.926570i
$590$		0			0
$591$	−252.792			−0.427736
$592$		0			0
$593$	−845.163			−1.42523			−0.712616	−	0.701554i	$-0.752490\pi$
$593$	−845.163			−1.42523			−0.712616	+	0.701554i	$0.752490\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−101.134			−0.169403
$598$		0			0
$599$	239.373			0.399620			0.199810	−	0.979835i	$-0.435967\pi$
$599$	239.373			0.399620			0.199810	+	0.979835i	$0.435967\pi$
$600$		0			0
$601$			387.358i			0.644523i	0.946651	+	0.322262i	$0.104443\pi$
$601$			387.358i			0.644523i	−0.946651	+	0.322262i	$0.895557\pi$
$602$		0			0
$603$	−444.083			−0.736456
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−728.736			−1.20055			−0.600277	−	0.799792i	$-0.704943\pi$
$607$	−728.736			−1.20055			−0.600277	+	0.799792i	$0.704943\pi$
$608$		0			0
$609$	−408.776			−0.671224
$610$		0			0
$611$			10.9089i			0.0178541i
$612$		0			0
$613$		−	281.663i		−	0.459483i	−0.973252	−	0.229741i	$-0.926212\pi$
$613$		−	281.663i		−	0.459483i	0.973252	−	0.229741i	$-0.0737880\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	743.929			1.20572			0.602860	−	0.797847i	$-0.294028\pi$
$617$	743.929			1.20572			0.602860	+	0.797847i	$0.294028\pi$
$618$		0			0
$619$			1082.36i			1.74856i	0.485418	+	0.874282i	$0.338667\pi$
$619$			1082.36i			1.74856i	−0.485418	+	0.874282i	$0.661333\pi$
$620$		0			0
$621$	−284.745			−0.458526
$622$		0			0
$623$		−	264.056i		−	0.423846i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	113.184			0.180517
$628$		0			0
$629$	287.703			0.457397
$630$		0			0
$631$		−	500.313i		−	0.792889i	−0.918059	−	0.396445i	$-0.870244\pi$
$631$		−	500.313i		−	0.792889i	0.918059	−	0.396445i	$-0.129756\pi$
$632$		0			0
$633$			222.769i			0.351926i
$634$		0			0
$635$		0			0
$636$		0			0
$637$			500.853i			0.786269i
$638$		0			0
$639$	192.746			0.301637
$640$		0			0
$641$			167.638i			0.261525i	0.991414	+	0.130763i	$0.0417426\pi$
$641$			167.638i			0.261525i	−0.991414	+	0.130763i	$0.958257\pi$
$642$		0			0
$643$			602.608i			0.937182i	0.883415	+	0.468591i	$0.155238\pi$
$643$			602.608i			0.937182i	−0.883415	+	0.468591i	$0.844762\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			352.530i			0.544869i	0.962174	+	0.272435i	$0.0878289\pi$
$647$			352.530i			0.544869i	−0.962174	+	0.272435i	$0.912171\pi$
$648$		0			0
$649$		−	316.777i		−	0.488101i
$650$		0			0
$651$	−357.633	−	102.740i	−0.549359	−	0.157819i
$652$		0			0
$653$	1072.69			1.64271			0.821354	−	0.570418i	$-0.193219\pi$
$653$	1072.69			1.64271			0.821354	+	0.570418i	$0.193219\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		−	999.976i		−	1.52203i
$658$		0			0
$659$	−34.8230			−0.0528422			−0.0264211	−	0.999651i	$-0.508411\pi$
$659$	−34.8230			−0.0528422			−0.0264211	+	0.999651i	$0.508411\pi$
$660$		0			0
$661$	−701.878			−1.06184			−0.530922	−	0.847421i	$-0.678154\pi$
$661$	−701.878			−1.06184			−0.530922	+	0.847421i	$0.678154\pi$
$662$		0			0
$663$			123.969i			0.186982i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−592.930			−0.888950
$668$		0			0
$669$	−268.522			−0.401379
$670$		0			0
$671$	−288.858			−0.430488
$672$		0			0
$673$			1080.02i			1.60479i	0.596796	+	0.802393i	$0.296440\pi$
$673$			1080.02i			1.60479i	−0.596796	+	0.802393i	$0.703560\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			651.535i			0.962385i	0.876615	+	0.481193i	$0.159796\pi$
$677$			651.535i			0.962385i	−0.876615	+	0.481193i	$0.840204\pi$
$678$		0			0
$679$	1444.20			2.12695
$680$		0			0
$681$		−	58.8546i		−	0.0864237i
$682$		0			0
$683$	−733.610			−1.07410			−0.537050	−	0.843551i	$-0.680461\pi$
$683$	−733.610			−1.07410			−0.537050	+	0.843551i	$0.680461\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−19.0996			−0.0278014
$688$		0			0
$689$	−296.077			−0.429719
$690$		0			0
$691$	−464.673			−0.672465			−0.336232	−	0.941779i	$-0.609153\pi$
$691$	−464.673			−0.672465			−0.336232	+	0.941779i	$0.609153\pi$
$692$		0			0
$693$			654.182i			0.943986i
$694$		0			0
$695$		0			0
$696$		0			0
$697$		−	956.951i		−	1.37296i
$698$		0			0
$699$			316.755i			0.453155i
$700$		0			0
$701$	1316.24			1.87766			0.938828	−	0.344386i	$-0.111913\pi$
$701$	1316.24			1.87766			0.938828	+	0.344386i	$0.111913\pi$
$702$		0			0
$703$		−	188.373i		−	0.267957i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−1889.76			−2.67293
$708$		0			0
$709$			924.122i			1.30342i	0.758470	+	0.651708i	$0.225947\pi$
$709$			924.122i			1.30342i	−0.758470	+	0.651708i	$0.774053\pi$
$710$		0			0
$711$		−	762.226i		−	1.07205i
$712$		0			0
$713$	−518.747	−	149.024i	−0.727555	−	0.209010i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	134.025			0.186925
$718$		0			0
$719$			988.873i			1.37535i	0.726021	+	0.687673i	$0.241368\pi$
$719$			988.873i			1.37535i	−0.726021	+	0.687673i	$0.758632\pi$
$720$		0			0
$721$	−2002.05			−2.77677
$722$		0			0
$723$	−235.539			−0.325780
$724$		0			0
$725$		0			0
$726$		0			0
$727$	787.202			1.08281			0.541404	−	0.840762i	$-0.317893\pi$
$727$	787.202			1.08281			0.541404	+	0.840762i	$0.317893\pi$
$728$		0			0
$729$	320.612			0.439798
$730$		0			0
$731$	2298.08			3.14375
$732$		0			0
$733$	−153.102			−0.208871			−0.104435	−	0.994532i	$-0.533304\pi$
$733$	−153.102			−0.208871			−0.104435	+	0.994532i	$0.533304\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	354.595i		−	0.481133i
$738$		0			0
$739$			643.904i			0.871318i	0.900112	+	0.435659i	$0.143485\pi$
$739$			643.904i			0.871318i	−0.900112	+	0.435659i	$0.856515\pi$
$740$		0			0
$741$	81.1685			0.109539
$742$		0			0
$743$			831.462i			1.11906i	0.828810	+	0.559530i	$0.189018\pi$
$743$			831.462i			1.11906i	−0.828810	+	0.559530i	$0.810982\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		−	195.510i		−	0.261727i
$748$		0			0
$749$	1223.27			1.63321
$750$		0			0
$751$	1146.18			1.52621			0.763104	−	0.646275i	$-0.223674\pi$
$751$	1146.18			1.52621			0.763104	+	0.646275i	$0.223674\pi$
$752$		0			0
$753$	−132.842			−0.176416
$754$		0			0
$755$		0			0
$756$		0			0
$757$		−	944.831i		−	1.24813i	−0.781374	−	0.624063i	$-0.785481\pi$
$757$		−	944.831i		−	1.24813i	0.781374	−	0.624063i	$-0.214519\pi$
$758$		0			0
$759$		−	107.584i		−	0.141744i
$760$		0			0
$761$			1198.74i			1.57522i	0.616174	+	0.787610i	$0.288682\pi$
$761$			1198.74i			1.57522i	−0.616174	+	0.787610i	$0.711318\pi$
$762$		0			0
$763$	987.216			1.29386
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	227.173i		−	0.296184i
$768$		0			0
$769$	−242.061			−0.314773			−0.157387	−	0.987537i	$-0.550307\pi$
$769$	−242.061			−0.314773			−0.157387	+	0.987537i	$0.550307\pi$
$770$		0			0
$771$		−	82.5176i		−	0.107027i
$772$		0			0
$773$		−	643.478i		−	0.832443i	−0.909263	−	0.416221i	$-0.863354\pi$
$773$		−	643.478i		−	0.832443i	0.909263	−	0.416221i	$-0.136646\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−123.442			−0.158870
$778$		0			0
$779$	−626.564			−0.804318
$780$		0			0
$781$			153.905i			0.197062i
$782$		0			0
$783$	−556.970			−0.711328
$784$		0			0
$785$		0			0
$786$		0			0
$787$		−	788.278i		−	1.00162i	−0.865556	−	0.500812i	$-0.833035\pi$
$787$		−	788.278i		−	1.00162i	0.865556	−	0.500812i	$-0.166965\pi$
$788$		0			0
$789$	107.208			0.135878
$790$		0			0
$791$	137.057			0.173270
$792$		0			0
$793$	−207.151			−0.261224
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	950.849i		−	1.19304i	−0.802600	−	0.596518i	$-0.796551\pi$
$797$		−	950.849i		−	1.19304i	0.802600	−	0.596518i	$-0.203449\pi$
$798$		0			0
$799$		−	65.9303i		−	0.0825160i
$800$		0			0
$801$		−	170.241i		−	0.212536i
$802$		0			0
$803$	798.470			0.994358
$804$		0			0
$805$		0			0
$806$		0			0
$807$	372.618			0.461732
$808$		0			0
$809$			5.08190i			0.00628171i	0.999995	+	0.00314085i	$0.000999767\pi$
$809$			5.08190i			0.00628171i	−0.999995	+	0.00314085i	$0.999000\pi$
$810$		0			0
$811$	918.456			1.13250			0.566249	−	0.824234i	$-0.308394\pi$
$811$	918.456			1.13250			0.566249	+	0.824234i	$0.308394\pi$
$812$		0			0
$813$	−136.785			−0.168247
$814$		0			0
$815$		0			0
$816$		0			0
$817$		−	1504.67i		−	1.84170i
$818$		0			0
$819$			469.139i			0.572819i
$820$		0			0
$821$			979.982i			1.19364i	0.802373	+	0.596822i	$0.203570\pi$
$821$			979.982i			1.19364i	−0.802373	+	0.596822i	$0.796430\pi$
$822$		0			0
$823$			418.294i			0.508255i	0.967171	+	0.254128i	$0.0817883\pi$
$823$			418.294i			0.508255i	−0.967171	+	0.254128i	$0.918212\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	484.546i		−	0.585908i	−0.956127	−	0.292954i	$-0.905362\pi$
$827$		−	484.546i		−	0.585908i	0.956127	−	0.292954i	$-0.0946383\pi$
$828$		0			0
$829$		−	1256.49i		−	1.51567i	−0.652444	−	0.757837i	$-0.726256\pi$
$829$		−	1256.49i		−	1.51567i	0.652444	−	0.757837i	$-0.273744\pi$
$830$		0			0
$831$	−225.579			−0.271455
$832$		0			0
$833$		−	3027.02i		−	3.63388i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−487.286	−	139.986i	−0.582181	−	0.167248i
$838$		0			0
$839$	1599.82			1.90682			0.953408	−	0.301685i	$-0.0975492\pi$
$839$	1599.82			1.90682			0.953408	+	0.301685i	$0.0975492\pi$
$840$		0			0
$841$	−318.790			−0.379061
$842$		0			0
$843$			302.822i			0.359219i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	994.748			1.17444
$848$		0			0
$849$		−	313.130i		−	0.368822i
$850$		0			0
$851$	−179.053			−0.210403
$852$		0			0
$853$	−196.356			−0.230194			−0.115097	−	0.993354i	$-0.536718\pi$
$853$	−196.356			−0.230194			−0.115097	+	0.993354i	$0.536718\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−1226.13			−1.43072			−0.715360	−	0.698756i	$-0.753737\pi$
$857$	−1226.13			−1.43072			−0.715360	+	0.698756i	$0.753737\pi$
$858$		0			0
$859$		−	732.830i		−	0.853120i	−0.904459	−	0.426560i	$-0.859725\pi$
$859$		−	732.830i		−	0.853120i	0.904459	−	0.426560i	$-0.140275\pi$
$860$		0			0
$861$			410.591i			0.476876i
$862$		0			0
$863$			484.751i			0.561704i	0.959751	+	0.280852i	$0.0906170\pi$
$863$			484.751i			0.561704i	−0.959751	+	0.280852i	$0.909383\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		−	472.563i		−	0.545055i
$868$		0			0
$869$	608.629			0.700378
$870$		0			0
$871$		−	254.294i		−	0.291956i
$872$		0			0
$873$	931.102			1.06655
$874$		0			0
$875$		0			0
$876$		0			0
$877$	676.353			0.771212			0.385606	−	0.922664i	$-0.373992\pi$
$877$	676.353			0.771212			0.385606	+	0.922664i	$0.373992\pi$
$878$		0			0
$879$			95.3390i			0.108463i
$880$		0			0
$881$		−	1212.48i		−	1.37626i	−0.725589	−	0.688128i	$-0.758433\pi$
$881$		−	1212.48i		−	1.37626i	0.725589	−	0.688128i	$-0.241567\pi$
$882$		0			0
$883$		−	491.333i		−	0.556437i	−0.960518	−	0.278218i	$-0.910256\pi$
$883$		−	491.333i		−	0.556437i	0.960518	−	0.278218i	$-0.0897439\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	80.9774			0.0912936			0.0456468	−	0.998958i	$-0.485465\pi$
$887$	80.9774			0.0912936			0.0456468	+	0.998958i	$0.485465\pi$
$888$		0			0
$889$			2393.88i			2.69278i
$890$		0			0
$891$			368.522i			0.413605i
$892$		0			0
$893$	−43.1679			−0.0483403
$894$		0			0
$895$		0			0
$896$		0			0
$897$		−	77.1525i		−	0.0860117i
$898$		0			0
$899$	−1014.69	−	291.497i	−1.12868	−	0.324245i
$900$		0			0
$901$	1789.41			1.98602
$902$		0			0
$903$	−986.016			−1.09193
$904$		0			0
$905$		0			0
$906$		0			0
$907$	1053.61			1.16164			0.580819	−	0.814033i	$-0.302732\pi$
$907$	1053.61			1.16164			0.580819	+	0.814033i	$0.302732\pi$
$908$		0			0
$909$	−1218.36			−1.34033
$910$		0			0
$911$			1017.91i			1.11736i	0.829383	+	0.558680i	$0.188692\pi$
$911$			1017.91i			1.11736i	−0.829383	+	0.558680i	$0.811308\pi$
$912$		0			0
$913$	156.113			0.170989
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−913.942			−0.996665
$918$		0			0
$919$	404.861			0.440545			0.220272	−	0.975438i	$-0.429305\pi$
$919$	404.861			0.440545			0.220272	+	0.975438i	$0.429305\pi$
$920$		0			0
$921$		−	70.4971i		−	0.0765441i
$922$		0			0
$923$			110.371i			0.119579i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−1290.76			−1.39240
$928$		0			0
$929$			449.751i			0.484124i	0.970261	+	0.242062i	$0.0778237\pi$
$929$			449.751i			0.484124i	−0.970261	+	0.242062i	$0.922176\pi$
$930$		0			0
$931$	−1981.94			−2.12883
$932$		0			0
$933$		−	531.273i		−	0.569424i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−1095.58			−1.16924			−0.584621	−	0.811306i	$-0.698757\pi$
$937$	−1095.58			−1.16924			−0.584621	+	0.811306i	$0.698757\pi$
$938$		0			0
$939$	−270.960			−0.288562
$940$		0			0
$941$		−	398.859i		−	0.423867i	−0.977284	−	0.211934i	$-0.932024\pi$
$941$		−	398.859i		−	0.423867i	0.977284	−	0.211934i	$-0.0679761\pi$
$942$		0			0
$943$			595.562i			0.631561i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	1697.26i		−	1.79225i	−0.443803	−	0.896125i	$-0.646371\pi$
$947$		−	1697.26i		−	1.79225i	0.443803	−	0.896125i	$-0.353629\pi$
$948$		0			0
$949$	572.613			0.603386
$950$		0			0
$951$		−	437.744i		−	0.460299i
$952$		0			0
$953$		−	1365.25i		−	1.43258i	−0.697801	−	0.716292i	$-0.745838\pi$
$953$		−	1365.25i		−	1.43258i	0.697801	−	0.716292i	$-0.254162\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		−	210.437i		−	0.219893i
$958$		0			0
$959$			1663.12i			1.73422i
$960$		0			0
$961$	−814.473	−	510.054i	−0.847527	−	0.530753i
$962$		0			0
$963$	788.665			0.818967
$964$		0			0
$965$		0			0
$966$		0			0
$967$			809.773i			0.837407i	0.908123	+	0.418704i	$0.137515\pi$
$967$			809.773i			0.837407i	−0.908123	+	0.418704i	$0.862485\pi$
$968$		0			0
$969$	−490.560			−0.506254
$970$		0			0
$971$	−886.258			−0.912727			−0.456363	−	0.889793i	$-0.650848\pi$
$971$	−886.258			−0.912727			−0.456363	+	0.889793i	$0.650848\pi$
$972$		0			0
$973$		−	358.845i		−	0.368802i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	648.945			0.664222			0.332111	−	0.943240i	$-0.392239\pi$
$977$	648.945			0.664222			0.332111	+	0.943240i	$0.392239\pi$
$978$		0			0
$979$	135.936			0.138852
$980$		0			0
$981$	636.475			0.648802
$982$		0			0
$983$		−	652.183i		−	0.663462i	−0.943374	−	0.331731i	$-0.892367\pi$
$983$		−	652.183i		−	0.663462i	0.943374	−	0.331731i	$-0.107633\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$			28.2881i			0.0286607i
$988$		0			0
$989$	−1430.22			−1.44613
$990$		0			0
$991$		−	1282.72i		−	1.29437i	−0.762335	−	0.647183i	$-0.775947\pi$
$991$		−	1282.72i		−	1.29437i	0.762335	−	0.647183i	$-0.224053\pi$
$992$		0			0
$993$	194.922			0.196296
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−1112.81			−1.11616			−0.558080	−	0.829787i	$-0.688462\pi$
$997$	−1112.81			−1.11616			−0.558080	+	0.829787i	$0.688462\pi$
$998$		0			0
$999$	−168.194			−0.168362

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3100.3.d.g.1301.14	yes	22
5.2	odd	4		3100.3.f.d.1549.27		44
5.3	odd	4		3100.3.f.d.1549.18		44
5.4	even	2		3100.3.d.f.1301.9	✓	22
31.30	odd	2	inner	3100.3.d.g.1301.9	yes	22
155.92	even	4		3100.3.f.d.1549.17		44
155.123	even	4		3100.3.f.d.1549.28		44
155.154	odd	2		3100.3.d.f.1301.14	yes	22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3100.3.d.f.1301.9	✓	22	5.4	even	2
3100.3.d.f.1301.14	yes	22	155.154	odd	2
3100.3.d.g.1301.9	yes	22	31.30	odd	2	inner
3100.3.d.g.1301.14	yes	22	1.1	even	1	trivial
3100.3.f.d.1549.17		44	155.92	even	4
3100.3.f.d.1549.18		44	5.3	odd	4
3100.3.f.d.1549.27		44	5.2	odd	4
3100.3.f.d.1549.28		44	155.123	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+0.957337i q^{3} +12.5381 q^{7} +8.08351 q^{9} +O(q^{10})\)	\(q+0.957337i q^{3} +12.5381 q^{7} +8.08351 q^{9} +6.45459i q^{11} +4.62883i q^{13} -27.9754i q^{17} -18.3169 q^{19} +12.0031i q^{21} +17.4106i q^{23} +16.3547i q^{27} +34.0557i q^{29} +(-8.55941 + 29.7949i) q^{31} -6.17922 q^{33} +10.2841i q^{37} -4.43135 q^{39} +34.2069 q^{41} +82.1464i q^{43} +2.35672 q^{47} +108.203 q^{49} +26.7819 q^{51} +63.9636i q^{53} -17.5354i q^{57} -49.0778 q^{59} +44.7523i q^{61} +101.352 q^{63} -54.9369 q^{67} -16.6678 q^{69} +23.8443 q^{71} -123.706i q^{73} +80.9281i q^{77} -94.2940i q^{79} +57.0946 q^{81} -24.1863i q^{83} -32.6028 q^{87} -21.0604i q^{89} +58.0366i q^{91} +(-28.5238 - 8.19424i) q^{93} +115.185 q^{97} +52.1757i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q + 4 q^{7} - 72 q^{9}+O(q^{10}) \) 22 * q + 4 * q^7 - 72 * q^9	\( 22 q + 4 q^{7} - 72 q^{9} - 28 q^{19} - 18 q^{31} - 34 q^{33} - 62 q^{39} + 64 q^{41} - 96 q^{47} + 150 q^{49} - 130 q^{51} + 40 q^{59} - 4 q^{63} + 110 q^{67} + 100 q^{69} + 132 q^{71} + 234 q^{81} - 62 q^{87} + 16 q^{93} + 186 q^{97}+O(q^{100}) \) 22 * q + 4 * q^7 - 72 * q^9 - 28 * q^19 - 18 * q^31 - 34 * q^33 - 62 * q^39 + 64 * q^41 - 96 * q^47 + 150 * q^49 - 130 * q^51 + 40 * q^59 - 4 * q^63 + 110 * q^67 + 100 * q^69 + 132 * q^71 + 234 * q^81 - 62 * q^87 + 16 * q^93 + 186 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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