LMFDB - Embedded newform 3100.3.d.g.1301.3

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			1301.3
Character	$\chi$	$=$	3100.1301
Dual form			3100.3.d.g.1301.20

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.50171i q^{3} +3.34932 q^{7} -11.2654 q^{9} +O(q^{10})$	$q-4.50171i q^{3} +3.34932 q^{7} -11.2654 q^{9} +19.8180i q^{11} -24.1835i q^{13} -10.8394i q^{17} -20.7925 q^{19} -15.0777i q^{21} +36.3471i q^{23} +10.1982i q^{27} -16.7084i q^{29} +(-5.58212 + 30.4933i) q^{31} +89.2148 q^{33} -34.2374i q^{37} -108.867 q^{39} +20.2476 q^{41} -31.2128i q^{43} -66.9619 q^{47} -37.7821 q^{49} -48.7957 q^{51} -70.0365i q^{53} +93.6018i q^{57} -32.5693 q^{59} +84.0097i q^{61} -37.7315 q^{63} +52.1513 q^{67} +163.624 q^{69} +53.3886 q^{71} +114.268i q^{73} +66.3767i q^{77} +148.249i q^{79} -55.4792 q^{81} +113.192i q^{83} -75.2165 q^{87} -34.7475i q^{89} -80.9983i q^{91} +(137.272 + 25.1291i) q^{93} -120.153 q^{97} -223.258i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$		−	4.50171i		−	1.50057i	−0.661114	−	0.750285i	$-0.729916\pi$
$3$		−	4.50171i		−	1.50057i	0.661114	−	0.750285i	$-0.270084\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	3.34932			0.478474			0.239237	−	0.970961i	$-0.423103\pi$
$7$	3.34932			0.478474			0.239237	+	0.970961i	$0.423103\pi$
$8$		0			0
$9$	−11.2654			−1.25171
$10$		0			0
$11$			19.8180i			1.80163i	0.434200	+	0.900817i	$0.357031\pi$
$11$			19.8180i			1.80163i	−0.434200	+	0.900817i	$0.642969\pi$
$12$		0			0
$13$		−	24.1835i		−	1.86027i	−0.367220	−	0.930134i	$-0.619690\pi$
$13$		−	24.1835i		−	1.86027i	0.367220	−	0.930134i	$-0.380310\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	10.8394i		−	0.637610i	−0.947820	−	0.318805i	$-0.896718\pi$
$17$		−	10.8394i		−	0.637610i	0.947820	−	0.318805i	$-0.103282\pi$
$18$		0			0
$19$	−20.7925			−1.09434			−0.547171	−	0.837021i	$-0.684295\pi$
$19$	−20.7925			−1.09434			−0.547171	+	0.837021i	$0.684295\pi$
$20$		0			0
$21$		−	15.0777i		−	0.717985i
$22$		0			0
$23$			36.3471i			1.58031i	0.612908	+	0.790154i	$0.289999\pi$
$23$			36.3471i			1.58031i	−0.612908	+	0.790154i	$0.710001\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$			10.1982i			0.377713i
$28$		0			0
$29$		−	16.7084i		−	0.576152i	−0.957607	−	0.288076i	$-0.906984\pi$
$29$		−	16.7084i		−	0.576152i	0.957607	−	0.288076i	$-0.0930156\pi$
$30$		0			0
$31$	−5.58212	+	30.4933i	−0.180068	+	0.983654i
$31$	−5.58212	+	30.4933i	−0.180068	+	0.983654i
$32$		0			0
$33$	89.2148			2.70348
$34$		0			0
$35$		0			0
$36$		0			0
$37$		−	34.2374i		−	0.925334i	−0.886532	−	0.462667i	$-0.846893\pi$
$37$		−	34.2374i		−	0.925334i	0.886532	−	0.462667i	$-0.153107\pi$
$38$		0			0
$39$	−108.867			−2.79146
$40$		0			0
$41$	20.2476			0.493844			0.246922	−	0.969035i	$-0.420581\pi$
$41$	20.2476			0.493844			0.246922	+	0.969035i	$0.420581\pi$
$42$		0			0
$43$		−	31.2128i		−	0.725880i	−0.931813	−	0.362940i	$-0.881773\pi$
$43$		−	31.2128i		−	0.725880i	0.931813	−	0.362940i	$-0.118227\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−66.9619			−1.42472			−0.712361	−	0.701813i	$-0.752374\pi$
$47$	−66.9619			−1.42472			−0.712361	+	0.701813i	$0.752374\pi$
$48$		0			0
$49$	−37.7821			−0.771062
$50$		0			0
$51$	−48.7957			−0.956779
$52$		0			0
$53$		−	70.0365i		−	1.32144i	−0.750631	−	0.660722i	$-0.770250\pi$
$53$		−	70.0365i		−	1.32144i	0.750631	−	0.660722i	$-0.229750\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$			93.6018i			1.64214i
$58$		0			0
$59$	−32.5693			−0.552023			−0.276011	−	0.961154i	$-0.589013\pi$
$59$	−32.5693			−0.552023			−0.276011	+	0.961154i	$0.589013\pi$
$60$		0			0
$61$			84.0097i			1.37721i	0.725137	+	0.688604i	$0.241776\pi$
$61$			84.0097i			1.37721i	−0.725137	+	0.688604i	$0.758224\pi$
$62$		0			0
$63$	−37.7315			−0.598912
$64$		0			0
$65$		0			0
$66$		0			0
$67$	52.1513			0.778377			0.389189	−	0.921158i	$-0.372755\pi$
$67$	52.1513			0.778377			0.389189	+	0.921158i	$0.372755\pi$
$68$		0			0
$69$	163.624			2.37136
$70$		0			0
$71$	53.3886			0.751952			0.375976	−	0.926629i	$-0.377308\pi$
$71$	53.3886			0.751952			0.375976	+	0.926629i	$0.377308\pi$
$72$		0			0
$73$			114.268i			1.56531i	0.622453	+	0.782657i	$0.286136\pi$
$73$			114.268i			1.56531i	−0.622453	+	0.782657i	$0.713864\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			66.3767i			0.862035i
$78$		0			0
$79$			148.249i			1.87657i	0.345863	+	0.938285i	$0.387586\pi$
$79$			148.249i			1.87657i	−0.345863	+	0.938285i	$0.612414\pi$
$80$		0			0
$81$	−55.4792			−0.684928
$82$		0			0
$83$			113.192i			1.36376i	0.731466	+	0.681878i	$0.238836\pi$
$83$			113.192i			1.36376i	−0.731466	+	0.681878i	$0.761164\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−75.2165			−0.864558
$88$		0			0
$89$		−	34.7475i		−	0.390421i	−0.980761	−	0.195210i	$-0.937461\pi$
$89$		−	34.7475i		−	0.390421i	0.980761	−	0.195210i	$-0.0625390\pi$
$90$		0			0
$91$		−	80.9983i		−	0.890091i
$92$		0			0
$93$	137.272	+	25.1291i	1.47604	+	0.270205i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−120.153			−1.23869			−0.619345	−	0.785119i	$-0.712602\pi$
$97$	−120.153			−1.23869			−0.619345	+	0.785119i	$0.712602\pi$
$98$		0			0
$99$		−	223.258i		−	2.25513i
$100$		0			0
$101$	91.5966			0.906897			0.453448	−	0.891283i	$-0.350194\pi$
$101$	91.5966			0.906897			0.453448	+	0.891283i	$0.350194\pi$
$102$		0			0
$103$	−157.177			−1.52599			−0.762997	−	0.646402i	$-0.776273\pi$
$103$	−157.177			−1.52599			−0.762997	+	0.646402i	$0.776273\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−126.709			−1.18420			−0.592098	−	0.805866i	$-0.701700\pi$
$107$	−126.709			−1.18420			−0.592098	+	0.805866i	$0.701700\pi$
$108$		0			0
$109$	−102.729			−0.942472			−0.471236	−	0.882007i	$-0.656192\pi$
$109$	−102.729			−0.942472			−0.471236	+	0.882007i	$0.656192\pi$
$110$		0			0
$111$	−154.127			−1.38853
$112$		0			0
$113$	45.7264			0.404658			0.202329	−	0.979318i	$-0.435149\pi$
$113$	45.7264			0.404658			0.202329	+	0.979318i	$0.435149\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$			272.437i			2.32852i
$118$		0			0
$119$		−	36.3045i		−	0.305080i
$120$		0			0
$121$	−271.752			−2.24588
$122$		0			0
$123$		−	91.1489i		−	0.741048i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		−	7.26979i		−	0.0572425i	−0.999590	−	0.0286212i	$-0.990888\pi$
$127$		−	7.26979i		−	0.0572425i	0.999590	−	0.0286212i	$-0.00911167\pi$
$128$		0			0
$129$	−140.511			−1.08923
$130$		0			0
$131$	−15.2024			−0.116049			−0.0580244	−	0.998315i	$-0.518480\pi$
$131$	−15.2024			−0.116049			−0.0580244	+	0.998315i	$0.518480\pi$
$132$		0			0
$133$	−69.6407			−0.523614
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	90.3799i		−	0.659707i	−0.944032	−	0.329854i	$-0.893001\pi$
$137$		−	90.3799i		−	0.659707i	0.944032	−	0.329854i	$-0.106999\pi$
$138$		0			0
$139$		−	32.0861i		−	0.230835i	−0.993317	−	0.115418i	$-0.963179\pi$
$139$		−	32.0861i		−	0.230835i	0.993317	−	0.115418i	$-0.0368206\pi$
$140$		0			0
$141$			301.443i			2.13790i
$142$		0			0
$143$	479.268			3.35152
$144$		0			0
$145$		0			0
$146$		0			0
$147$			170.084i			1.15703i
$148$		0			0
$149$	−43.6641			−0.293048			−0.146524	−	0.989207i	$-0.546808\pi$
$149$	−43.6641			−0.293048			−0.146524	+	0.989207i	$0.546808\pi$
$150$		0			0
$151$		−	161.751i		−	1.07120i	−0.844472	−	0.535600i	$-0.820086\pi$
$151$		−	161.751i		−	1.07120i	0.844472	−	0.535600i	$-0.179914\pi$
$152$		0			0
$153$			122.110i			0.798105i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	139.170			0.886432			0.443216	−	0.896415i	$-0.353838\pi$
$157$	139.170			0.886432			0.443216	+	0.896415i	$0.353838\pi$
$158$		0			0
$159$	−315.284			−1.98292
$160$		0			0
$161$			121.738i			0.756137i
$162$		0			0
$163$	−284.803			−1.74726			−0.873629	−	0.486592i	$-0.838240\pi$
$163$	−284.803			−1.74726			−0.873629	+	0.486592i	$0.838240\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			269.071i			1.61120i	0.592458	+	0.805601i	$0.298158\pi$
$167$			269.071i			1.61120i	−0.592458	+	0.805601i	$0.701842\pi$
$168$		0			0
$169$	−415.841			−2.46060
$170$		0			0
$171$	234.236			1.36980
$172$		0			0
$173$	73.5210			0.424977			0.212488	−	0.977164i	$-0.431843\pi$
$173$	73.5210			0.424977			0.212488	+	0.977164i	$0.431843\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$			146.618i			0.828349i
$178$		0			0
$179$			320.566i			1.79087i	0.445192	+	0.895435i	$0.353135\pi$
$179$			320.566i			1.79087i	−0.445192	+	0.895435i	$0.646865\pi$
$180$		0			0
$181$			128.981i			0.712604i	0.934371	+	0.356302i	$0.115963\pi$
$181$			128.981i			0.712604i	−0.934371	+	0.356302i	$0.884037\pi$
$182$		0			0
$183$	378.188			2.06660
$184$		0			0
$185$		0			0
$186$		0			0
$187$	214.814			1.14874
$188$		0			0
$189$			34.1572i			0.180726i
$190$		0			0
$191$	196.427			1.02841			0.514206	−	0.857667i	$-0.328087\pi$
$191$	196.427			1.02841			0.514206	+	0.857667i	$0.328087\pi$
$192$		0			0
$193$	36.3915			0.188557			0.0942786	−	0.995546i	$-0.469946\pi$
$193$	36.3915			0.188557			0.0942786	+	0.995546i	$0.469946\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			230.269i			1.16888i	0.811438	+	0.584438i	$0.198685\pi$
$197$			230.269i			1.16888i	−0.811438	+	0.584438i	$0.801315\pi$
$198$		0			0
$199$		−	45.5525i		−	0.228907i	−0.993429	−	0.114454i	$-0.963488\pi$
$199$		−	45.5525i		−	0.228907i	0.993429	−	0.114454i	$-0.0365117\pi$
$200$		0			0
$201$		−	234.770i		−	1.16801i
$202$		0			0
$203$		−	55.9619i		−	0.275674i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		−	409.465i		−	1.97809i
$208$		0			0
$209$		−	412.065i		−	1.97160i
$210$		0			0
$211$	290.748			1.37795			0.688977	−	0.724783i	$-0.258060\pi$
$211$	290.748			1.37795			0.688977	+	0.724783i	$0.258060\pi$
$212$		0			0
$213$		−	240.340i		−	1.12836i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−18.6963	+	102.132i	−0.0861581	+	0.470653i
$218$		0			0
$219$	514.401			2.34886
$220$		0			0
$221$	−262.134			−1.18613
$222$		0			0
$223$			376.047i			1.68631i	0.537670	+	0.843156i	$0.319305\pi$
$223$			376.047i			1.68631i	−0.537670	+	0.843156i	$0.680695\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−236.786			−1.04311			−0.521556	−	0.853217i	$-0.674648\pi$
$227$	−236.786			−1.04311			−0.521556	+	0.853217i	$0.674648\pi$
$228$		0			0
$229$		−	237.809i		−	1.03847i	−0.854632	−	0.519234i	$-0.826217\pi$
$229$		−	237.809i		−	1.03847i	0.854632	−	0.519234i	$-0.173783\pi$
$230$		0			0
$231$	298.809			1.29355
$232$		0			0
$233$	−119.851			−0.514381			−0.257190	−	0.966361i	$-0.582797\pi$
$233$	−119.851			−0.514381			−0.257190	+	0.966361i	$0.582797\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	667.374			2.81593
$238$		0			0
$239$		−	392.130i		−	1.64071i	−0.571855	−	0.820355i	$-0.693776\pi$
$239$		−	392.130i		−	1.64071i	0.571855	−	0.820355i	$-0.306224\pi$
$240$		0			0
$241$			239.660i			0.994442i	0.867624	+	0.497221i	$0.165646\pi$
$241$			239.660i			0.994442i	−0.867624	+	0.497221i	$0.834354\pi$
$242$		0			0
$243$			341.535i			1.40550i
$244$		0			0
$245$		0			0
$246$		0			0
$247$			502.835i			2.03577i
$248$		0			0
$249$	509.557			2.04641
$250$		0			0
$251$			252.413i			1.00563i	0.864394	+	0.502815i	$0.167702\pi$
$251$			252.413i			1.00563i	−0.864394	+	0.502815i	$0.832298\pi$
$252$		0			0
$253$	−720.325			−2.84714
$254$		0			0
$255$		0			0
$256$		0			0
$257$	298.418			1.16116			0.580581	−	0.814203i	$-0.302826\pi$
$257$	298.418			1.16116			0.580581	+	0.814203i	$0.302826\pi$
$258$		0			0
$259$		−	114.672i		−	0.442749i
$260$		0			0
$261$			188.227i			0.721177i
$262$		0			0
$263$		−	260.325i		−	0.989831i	−0.868941	−	0.494915i	$-0.835199\pi$
$263$		−	260.325i		−	0.989831i	0.868941	−	0.494915i	$-0.164801\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−156.423			−0.585854
$268$		0			0
$269$		−	428.653i		−	1.59351i	−0.604305	−	0.796753i	$-0.706549\pi$
$269$		−	428.653i		−	1.59351i	0.604305	−	0.796753i	$-0.293451\pi$
$270$		0			0
$271$			297.053i			1.09614i	0.836433	+	0.548069i	$0.184637\pi$
$271$			297.053i			1.09614i	−0.836433	+	0.548069i	$0.815363\pi$
$272$		0			0
$273$	−364.631			−1.33564
$274$		0			0
$275$		0			0
$276$		0			0
$277$		−	274.081i		−	0.989461i	−0.869046	−	0.494730i	$-0.835267\pi$
$277$		−	274.081i		−	0.989461i	0.869046	−	0.494730i	$-0.164733\pi$
$278$		0			0
$279$	62.8849	−	343.519i	0.225394	−	1.23125i
$280$		0			0
$281$	27.8906			0.0992549			0.0496274	−	0.998768i	$-0.484197\pi$
$281$	27.8906			0.0992549			0.0496274	+	0.998768i	$0.484197\pi$
$282$		0			0
$283$	139.234			0.491993			0.245997	−	0.969271i	$-0.420885\pi$
$283$	139.234			0.491993			0.245997	+	0.969271i	$0.420885\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	67.8157			0.236292
$288$		0			0
$289$	171.508			0.593453
$290$		0			0
$291$			540.894i			1.85874i
$292$		0			0
$293$	−193.747			−0.661252			−0.330626	−	0.943762i	$-0.607260\pi$
$293$	−193.747			−0.661252			−0.330626	+	0.943762i	$0.607260\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−202.108			−0.680500
$298$		0			0
$299$	878.999			2.93980
$300$		0			0
$301$		−	104.542i		−	0.347315i
$302$		0			0
$303$		−	412.341i		−	1.36086i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−89.8510			−0.292674			−0.146337	−	0.989235i	$-0.546748\pi$
$307$	−89.8510			−0.292674			−0.146337	+	0.989235i	$0.546748\pi$
$308$		0			0
$309$			707.567i			2.28986i
$310$		0			0
$311$	189.714			0.610013			0.305006	−	0.952350i	$-0.401341\pi$
$311$	189.714			0.610013			0.305006	+	0.952350i	$0.401341\pi$
$312$		0			0
$313$		−	192.385i		−	0.614650i	−0.951605	−	0.307325i	$-0.900566\pi$
$313$		−	192.385i		−	0.614650i	0.951605	−	0.307325i	$-0.0994339\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	598.717			1.88870			0.944349	−	0.328944i	$-0.106693\pi$
$317$	598.717			1.88870			0.944349	+	0.328944i	$0.106693\pi$
$318$		0			0
$319$	331.127			1.03802
$320$		0			0
$321$			570.407i			1.77697i
$322$		0			0
$323$			225.377i			0.697763i
$324$		0			0
$325$		0			0
$326$		0			0
$327$			462.459i			1.41425i
$328$		0			0
$329$	−224.277			−0.681693
$330$		0			0
$331$		−	579.499i		−	1.75075i	−0.483443	−	0.875376i	$-0.660614\pi$
$331$		−	579.499i		−	1.75075i	0.483443	−	0.875376i	$-0.339386\pi$
$332$		0			0
$333$			385.698i			1.15825i
$334$		0			0
$335$		0			0
$336$		0			0
$337$			630.359i			1.87050i	0.353987	+	0.935250i	$0.384826\pi$
$337$			630.359i			1.87050i	−0.353987	+	0.935250i	$0.615174\pi$
$338$		0			0
$339$		−	205.847i		−	0.607218i
$340$		0			0
$341$	−604.315	−	110.626i	−1.77218	−	0.324417i
$342$		0			0
$343$	−290.661			−0.847408
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	118.326i		−	0.340997i	−0.985358	−	0.170498i	$-0.945462\pi$
$347$		−	118.326i		−	0.340997i	0.985358	−	0.170498i	$-0.0545377\pi$
$348$		0			0
$349$	−13.8119			−0.0395756			−0.0197878	−	0.999804i	$-0.506299\pi$
$349$	−13.8119			−0.0395756			−0.0197878	+	0.999804i	$0.506299\pi$
$350$		0			0
$351$	246.629			0.702647
$352$		0			0
$353$			202.652i			0.574084i	0.957918	+	0.287042i	$0.0926719\pi$
$353$			202.652i			0.574084i	−0.957918	+	0.287042i	$0.907328\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−163.433			−0.457794
$358$		0			0
$359$	−256.533			−0.714577			−0.357288	−	0.933994i	$-0.616299\pi$
$359$	−256.533			−0.714577			−0.357288	+	0.933994i	$0.616299\pi$
$360$		0			0
$361$	71.3272			0.197582
$362$		0			0
$363$			1223.35i			3.37011i
$364$		0			0
$365$		0			0
$366$		0			0
$367$			231.586i			0.631025i	0.948921	+	0.315513i	$0.102176\pi$
$367$			231.586i			0.631025i	−0.948921	+	0.315513i	$0.897824\pi$
$368$		0			0
$369$	−228.098			−0.618151
$370$		0			0
$371$		−	234.575i		−	0.632277i
$372$		0			0
$373$	−318.142			−0.852928			−0.426464	−	0.904505i	$-0.640241\pi$
$373$	−318.142			−0.852928			−0.426464	+	0.904505i	$0.640241\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−404.068			−1.07180
$378$		0			0
$379$	528.420			1.39425			0.697125	−	0.716950i	$-0.254462\pi$
$379$	528.420			1.39425			0.697125	+	0.716950i	$0.254462\pi$
$380$		0			0
$381$	−32.7265			−0.0858964
$382$		0			0
$383$		−	80.3520i		−	0.209796i	−0.994483	−	0.104898i	$-0.966548\pi$
$383$		−	80.3520i		−	0.209796i	0.994483	−	0.104898i	$-0.0334517\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$			351.626i			0.908593i
$388$		0			0
$389$			1.15036i			0.00295723i	0.999999	+	0.00147861i	$0.000470658\pi$
$389$			1.15036i			0.00295723i	−0.999999	+	0.00147861i	$0.999529\pi$
$390$		0			0
$391$	393.980			1.00762
$392$		0			0
$393$			68.4368i			0.174139i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	107.852			0.271668			0.135834	−	0.990732i	$-0.456629\pi$
$397$	107.852			0.271668			0.135834	+	0.990732i	$0.456629\pi$
$398$		0			0
$399$			313.502i			0.785720i
$400$		0			0
$401$		−	147.442i		−	0.367687i	−0.982956	−	0.183843i	$-0.941146\pi$
$401$		−	147.442i		−	0.367687i	0.982956	−	0.183843i	$-0.0588540\pi$
$402$		0			0
$403$	737.434	+	134.995i	1.82986	+	0.334976i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	678.515			1.66711
$408$		0			0
$409$			350.458i			0.856864i	0.903574	+	0.428432i	$0.140934\pi$
$409$			350.458i			0.856864i	−0.903574	+	0.428432i	$0.859066\pi$
$410$		0			0
$411$	−406.864			−0.989937
$412$		0			0
$413$	−109.085			−0.264129
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−144.442			−0.346385
$418$		0			0
$419$	−97.3529			−0.232346			−0.116173	−	0.993229i	$-0.537063\pi$
$419$	−97.3529			−0.232346			−0.116173	+	0.993229i	$0.537063\pi$
$420$		0			0
$421$	305.264			0.725093			0.362546	−	0.931966i	$-0.381907\pi$
$421$	305.264			0.725093			0.362546	+	0.931966i	$0.381907\pi$
$422$		0			0
$423$	754.354			1.78334
$424$		0			0
$425$		0			0
$426$		0			0
$427$			281.375i			0.658959i
$428$		0			0
$429$		−	2157.53i		−	5.02920i
$430$		0			0
$431$	−0.151901			−0.000352438			−0.000176219	−	1.00000i	$-0.500056\pi$
$431$	−0.151901			−0.000352438			−0.000176219		1.00000i	$0.500056\pi$
$432$		0			0
$433$			68.0420i			0.157141i	0.996909	+	0.0785705i	$0.0250356\pi$
$433$			68.0420i			0.157141i	−0.996909	+	0.0785705i	$0.974964\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	755.746i		−	1.72940i
$438$		0			0
$439$	−295.971			−0.674195			−0.337097	−	0.941470i	$-0.609445\pi$
$439$	−295.971			−0.674195			−0.337097	+	0.941470i	$0.609445\pi$
$440$		0			0
$441$	425.630			0.965148
$442$		0			0
$443$	−319.367			−0.720918			−0.360459	−	0.932775i	$-0.617380\pi$
$443$	−319.367			−0.720918			−0.360459	+	0.932775i	$0.617380\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$			196.563i			0.439739i
$448$		0			0
$449$			124.643i			0.277601i	0.990320	+	0.138801i	$0.0443247\pi$
$449$			124.643i			0.277601i	−0.990320	+	0.138801i	$0.955675\pi$
$450$		0			0
$451$			401.266i			0.889726i
$452$		0			0
$453$	−728.157			−1.60741
$454$		0			0
$455$		0			0
$456$		0			0
$457$		−	198.423i		−	0.434187i	−0.976151	−	0.217093i	$-0.930342\pi$
$457$		−	198.423i		−	0.434187i	0.976151	−	0.217093i	$-0.0696576\pi$
$458$		0			0
$459$	110.543			0.240833
$460$		0			0
$461$			692.581i			1.50235i	0.660105	+	0.751173i	$0.270512\pi$
$461$			692.581i			1.50235i	−0.660105	+	0.751173i	$0.729488\pi$
$462$		0			0
$463$		−	34.3206i		−	0.0741265i	−0.999313	−	0.0370633i	$-0.988200\pi$
$463$		−	34.3206i		−	0.0741265i	0.999313	−	0.0370633i	$-0.0118003\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	72.8349			0.155963			0.0779817	−	0.996955i	$-0.475152\pi$
$467$	72.8349			0.155963			0.0779817	+	0.996955i	$0.475152\pi$
$468$		0			0
$469$	174.671			0.372434
$470$		0			0
$471$		−	626.502i		−	1.33015i
$472$		0			0
$473$	618.575			1.30777
$474$		0			0
$475$		0			0
$476$		0			0
$477$			788.991i			1.65407i
$478$		0			0
$479$	135.709			0.283317			0.141659	−	0.989916i	$-0.454756\pi$
$479$	135.709			0.283317			0.141659	+	0.989916i	$0.454756\pi$
$480$		0			0
$481$	−827.979			−1.72137
$482$		0			0
$483$	548.030			1.13464
$484$		0			0
$485$		0			0
$486$		0			0
$487$		−	167.937i		−	0.344841i	−0.985023	−	0.172420i	$-0.944841\pi$
$487$		−	167.937i		−	0.344841i	0.985023	−	0.172420i	$-0.0551588\pi$
$488$		0			0
$489$			1282.10i			2.62188i
$490$		0			0
$491$		−	774.563i		−	1.57752i	−0.614701	−	0.788760i	$-0.710723\pi$
$491$		−	774.563i		−	1.57752i	0.614701	−	0.788760i	$-0.289277\pi$
$492$		0			0
$493$	−181.109			−0.367361
$494$		0			0
$495$		0			0
$496$		0			0
$497$	178.815			0.359790
$498$		0			0
$499$			833.266i			1.66987i	0.550348	+	0.834935i	$0.314495\pi$
$499$			833.266i			1.66987i	−0.550348	+	0.834935i	$0.685505\pi$
$500$		0			0
$501$	1211.28			2.41772
$502$		0			0
$503$	−755.835			−1.50265			−0.751327	−	0.659930i	$-0.770586\pi$
$503$	−755.835			−1.50265			−0.751327	+	0.659930i	$0.770586\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$			1872.00i			3.69230i
$508$		0			0
$509$			680.815i			1.33755i	0.743463	+	0.668777i	$0.233182\pi$
$509$			680.815i			1.33755i	−0.743463	+	0.668777i	$0.766818\pi$
$510$		0			0
$511$			382.720i			0.748963i
$512$		0			0
$513$		−	212.047i		−	0.413347i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		−	1327.05i		−	2.56683i
$518$		0			0
$519$		−	330.970i		−	0.637708i
$520$		0			0
$521$	−578.082			−1.10956			−0.554781	−	0.831996i	$-0.687198\pi$
$521$	−578.082			−1.10956			−0.554781	+	0.831996i	$0.687198\pi$
$522$		0			0
$523$		−	552.158i		−	1.05575i	−0.849321	−	0.527876i	$-0.822989\pi$
$523$		−	552.158i		−	1.05575i	0.849321	−	0.527876i	$-0.177011\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	330.528	+	60.5067i	0.627188	+	0.114813i
$528$		0			0
$529$	−792.111			−1.49737
$530$		0			0
$531$	366.907			0.690974
$532$		0			0
$533$		−	489.658i		−	0.918683i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	1443.10			2.68733
$538$		0			0
$539$		−	748.763i		−	1.38917i
$540$		0			0
$541$	−418.616			−0.773782			−0.386891	−	0.922126i	$-0.626451\pi$
$541$	−418.616			−0.773782			−0.386891	+	0.922126i	$0.626451\pi$
$542$		0			0
$543$	580.637			1.06931
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−590.701			−1.07989			−0.539946	−	0.841699i	$-0.681555\pi$
$547$	−590.701			−1.07989			−0.539946	+	0.841699i	$0.681555\pi$
$548$		0			0
$549$		−	946.404i		−	1.72387i
$550$		0			0
$551$			347.409i			0.630507i
$552$		0			0
$553$			496.534i			0.897891i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	791.506i		−	1.42102i	−0.703689	−	0.710508i	$-0.748465\pi$
$557$		−	791.506i		−	1.42102i	0.703689	−	0.710508i	$-0.251535\pi$
$558$		0			0
$559$	−754.835			−1.35033
$560$		0			0
$561$		−	967.032i		−	1.72376i
$562$		0			0
$563$	−535.930			−0.951919			−0.475960	−	0.879467i	$-0.657899\pi$
$563$	−535.930			−0.951919			−0.475960	+	0.879467i	$0.657899\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−185.818			−0.327720
$568$		0			0
$569$			285.025i			0.500923i	0.968127	+	0.250461i	$0.0805823\pi$
$569$			285.025i			0.500923i	−0.968127	+	0.250461i	$0.919418\pi$
$570$		0			0
$571$		−	177.095i		−	0.310149i	−0.987903	−	0.155075i	$-0.950438\pi$
$571$		−	177.095i		−	0.310149i	0.987903	−	0.155075i	$-0.0495618\pi$
$572$		0			0
$573$		−	884.257i		−	1.54321i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−124.186			−0.215226			−0.107613	−	0.994193i	$-0.534321\pi$
$577$	−124.186			−0.215226			−0.107613	+	0.994193i	$0.534321\pi$
$578$		0			0
$579$		−	163.824i		−	0.282943i
$580$		0			0
$581$			379.115i			0.652522i
$582$		0			0
$583$	1387.98			2.38076
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	276.763i		−	0.471487i	−0.971815	−	0.235744i	$-0.924247\pi$
$587$		−	276.763i		−	0.471487i	0.971815	−	0.235744i	$-0.0757526\pi$
$588$		0			0
$589$	116.066	−	634.031i	0.197056	−	1.07645i
$590$		0			0
$591$	1036.60			1.75398
$592$		0			0
$593$	−40.6659			−0.0685765			−0.0342883	−	0.999412i	$-0.510916\pi$
$593$	−40.6659			−0.0685765			−0.0342883	+	0.999412i	$0.510916\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−205.064			−0.343491
$598$		0			0
$599$	−199.924			−0.333762			−0.166881	−	0.985977i	$-0.553370\pi$
$599$	−199.924			−0.333762			−0.166881	+	0.985977i	$0.553370\pi$
$600$		0			0
$601$			80.0981i			0.133275i	0.997777	+	0.0666373i	$0.0212270\pi$
$601$			80.0981i			0.133275i	−0.997777	+	0.0666373i	$0.978773\pi$
$602$		0			0
$603$	−587.506			−0.974305
$604$		0			0
$605$		0			0
$606$		0			0
$607$	771.702			1.27134			0.635669	−	0.771962i	$-0.280724\pi$
$607$	771.702			1.27134			0.635669	+	0.771962i	$0.280724\pi$
$608$		0			0
$609$	−251.924			−0.413669
$610$		0			0
$611$			1619.37i			2.65037i
$612$		0			0
$613$		−	479.875i		−	0.782831i	−0.920214	−	0.391415i	$-0.871986\pi$
$613$		−	479.875i		−	0.782831i	0.920214	−	0.391415i	$-0.128014\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−920.118			−1.49128			−0.745639	−	0.666350i	$-0.767856\pi$
$617$	−920.118			−1.49128			−0.745639	+	0.666350i	$0.767856\pi$
$618$		0			0
$619$			274.488i			0.443439i	0.975111	+	0.221719i	$0.0711669\pi$
$619$			274.488i			0.443439i	−0.975111	+	0.221719i	$0.928833\pi$
$620$		0			0
$621$	−370.676			−0.596903
$622$		0			0
$623$		−	116.380i		−	0.186806i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−1855.00			−2.95853
$628$		0			0
$629$	−371.111			−0.590002
$630$		0			0
$631$		−	171.922i		−	0.272459i	−0.990677	−	0.136230i	$-0.956502\pi$
$631$		−	171.922i		−	0.272459i	0.990677	−	0.136230i	$-0.0434985\pi$
$632$		0			0
$633$		−	1308.87i		−	2.06772i
$634$		0			0
$635$		0			0
$636$		0			0
$637$			913.702i			1.43438i
$638$		0			0
$639$	−601.444			−0.941227
$640$		0			0
$641$		−	857.296i		−	1.33744i	−0.743516	−	0.668718i	$-0.766843\pi$
$641$		−	857.296i		−	1.33744i	0.743516	−	0.668718i	$-0.233157\pi$
$642$		0			0
$643$			720.650i			1.12076i	0.828235	+	0.560381i	$0.189345\pi$
$643$			720.650i			1.12076i	−0.828235	+	0.560381i	$0.810655\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	961.945i		−	1.48678i	−0.668859	−	0.743389i	$-0.733217\pi$
$647$		−	961.945i		−	1.48678i	0.668859	−	0.743389i	$-0.266783\pi$
$648$		0			0
$649$		−	645.458i		−	0.994542i
$650$		0			0
$651$	459.768	+	84.1654i	0.706249	+	0.129286i
$652$		0			0
$653$	26.6424			0.0408000			0.0204000	−	0.999792i	$-0.493506\pi$
$653$	26.6424			0.0408000			0.0204000	+	0.999792i	$0.493506\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		−	1287.28i		−	1.95932i
$658$		0			0
$659$	375.223			0.569382			0.284691	−	0.958619i	$-0.408109\pi$
$659$	375.223			0.569382			0.284691	+	0.958619i	$0.408109\pi$
$660$		0			0
$661$	−897.110			−1.35720			−0.678601	−	0.734507i	$-0.737413\pi$
$661$	−897.110			−1.35720			−0.678601	+	0.734507i	$0.737413\pi$
$662$		0			0
$663$			1180.05i			1.77987i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	607.302			0.910498
$668$		0			0
$669$	1692.86			2.53043
$670$		0			0
$671$	−1664.90			−2.48122
$672$		0			0
$673$			4.19846i			0.00623842i	0.999995	+	0.00311921i	$0.000992877\pi$
$673$			4.19846i			0.00623842i	−0.999995	+	0.00311921i	$0.999007\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	134.034i		−	0.197983i	−0.995088	−	0.0989914i	$-0.968438\pi$
$677$		−	134.034i		−	0.197983i	0.995088	−	0.0989914i	$-0.0315616\pi$
$678$		0			0
$679$	−402.431			−0.592681
$680$		0			0
$681$			1065.94i			1.56526i
$682$		0			0
$683$	−609.762			−0.892770			−0.446385	−	0.894841i	$-0.647289\pi$
$683$	−609.762			−0.892770			−0.446385	+	0.894841i	$0.647289\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−1070.55			−1.55830
$688$		0			0
$689$	−1693.73			−2.45824
$690$		0			0
$691$	−410.497			−0.594063			−0.297031	−	0.954868i	$-0.595997\pi$
$691$	−410.497			−0.594063			−0.297031	+	0.954868i	$0.595997\pi$
$692$		0			0
$693$		−	747.761i		−	1.07902i
$694$		0			0
$695$		0			0
$696$		0			0
$697$		−	219.471i		−	0.314880i
$698$		0			0
$699$			539.534i			0.771865i
$700$		0			0
$701$	−1057.13			−1.50803			−0.754017	−	0.656855i	$-0.771886\pi$
$701$	−1057.13			−1.50803			−0.754017	+	0.656855i	$0.771886\pi$
$702$		0			0
$703$			711.880i			1.01263i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	306.786			0.433927
$708$		0			0
$709$			559.057i			0.788515i	0.919000	+	0.394257i	$0.128998\pi$
$709$			559.057i			0.788515i	−0.919000	+	0.394257i	$0.871002\pi$
$710$		0			0
$711$		−	1670.09i		−	2.34893i
$712$		0			0
$713$	−1108.34	−	202.894i	−1.55448	−	0.284564i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−1765.26			−2.46200
$718$		0			0
$719$			290.646i			0.404236i	0.979361	+	0.202118i	$0.0647824\pi$
$719$			290.646i			0.404236i	−0.979361	+	0.202118i	$0.935218\pi$
$720$		0			0
$721$	−526.437			−0.730149
$722$		0			0
$723$	1078.88			1.49223
$724$		0			0
$725$		0			0
$726$		0			0
$727$	284.203			0.390925			0.195463	−	0.980711i	$-0.437379\pi$
$727$	284.203			0.390925			0.195463	+	0.980711i	$0.437379\pi$
$728$		0			0
$729$	1038.18			1.42412
$730$		0			0
$731$	−338.328			−0.462828
$732$		0			0
$733$	742.774			1.01333			0.506667	−	0.862142i	$-0.330877\pi$
$733$	742.774			1.01333			0.506667	+	0.862142i	$0.330877\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			1033.53i			1.40235i
$738$		0			0
$739$		−	235.851i		−	0.319149i	−0.987186	−	0.159574i	$-0.948988\pi$
$739$		−	235.851i		−	0.319149i	0.987186	−	0.159574i	$-0.0510122\pi$
$740$		0			0
$741$	2263.62			3.05481
$742$		0			0
$743$		−	978.456i		−	1.31690i	−0.752625	−	0.658450i	$-0.771213\pi$
$743$		−	978.456i		−	1.31690i	0.752625	−	0.658450i	$-0.228787\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		−	1275.15i		−	1.70703i
$748$		0			0
$749$	−424.389			−0.566607
$750$		0			0
$751$	−385.028			−0.512687			−0.256344	−	0.966586i	$-0.582518\pi$
$751$	−385.028			−0.512687			−0.256344	+	0.966586i	$0.582518\pi$
$752$		0			0
$753$	1136.29			1.50902
$754$		0			0
$755$		0			0
$756$		0			0
$757$			468.448i			0.618822i	0.950928	+	0.309411i	$0.100132\pi$
$757$			468.448i			0.618822i	−0.950928	+	0.309411i	$0.899868\pi$
$758$		0			0
$759$			3242.70i			4.27233i
$760$		0			0
$761$		−	332.033i		−	0.436311i	−0.975914	−	0.218156i	$-0.929996\pi$
$761$		−	332.033i		−	0.436311i	0.975914	−	0.218156i	$-0.0700041\pi$
$762$		0			0
$763$	−344.074			−0.450949
$764$		0			0
$765$		0			0
$766$		0			0
$767$			787.640i			1.02691i
$768$		0			0
$769$	598.984			0.778913			0.389456	−	0.921045i	$-0.372663\pi$
$769$	598.984			0.778913			0.389456	+	0.921045i	$0.372663\pi$
$770$		0			0
$771$		−	1343.39i		−	1.74240i
$772$		0			0
$773$			680.178i			0.879919i	0.898018	+	0.439960i	$0.145007\pi$
$773$			680.178i			0.879919i	−0.898018	+	0.439960i	$0.854993\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−516.220			−0.664376
$778$		0			0
$779$	−420.998			−0.540434
$780$		0			0
$781$			1058.05i			1.35474i
$782$		0			0
$783$	170.397			0.217620
$784$		0			0
$785$		0			0
$786$		0			0
$787$		−	982.381i		−	1.24826i	−0.781320	−	0.624130i	$-0.785453\pi$
$787$		−	982.381i		−	1.24826i	0.781320	−	0.624130i	$-0.214547\pi$
$788$		0			0
$789$	−1171.91			−1.48531
$790$		0			0
$791$	153.152			0.193619
$792$		0			0
$793$	2031.65			2.56198
$794$		0			0
$795$		0			0
$796$		0			0
$797$			977.489i			1.22646i	0.789904	+	0.613230i	$0.210130\pi$
$797$			977.489i			1.22646i	−0.789904	+	0.613230i	$0.789870\pi$
$798$		0			0
$799$			725.825i			0.908417i
$800$		0			0
$801$			391.445i			0.488695i
$802$		0			0
$803$	−2264.56			−2.82012
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−1929.67			−2.39117
$808$		0			0
$809$		−	351.893i		−	0.434973i	−0.976063	−	0.217487i	$-0.930214\pi$
$809$		−	351.893i		−	0.434973i	0.976063	−	0.217487i	$-0.0697859\pi$
$810$		0			0
$811$	−1125.82			−1.38819			−0.694095	−	0.719883i	$-0.744195\pi$
$811$	−1125.82			−1.38819			−0.694095	+	0.719883i	$0.744195\pi$
$812$		0			0
$813$	1337.25			1.64483
$814$		0			0
$815$		0			0
$816$		0			0
$817$			648.992i			0.794360i
$818$		0			0
$819$			912.479i			1.11414i
$820$		0			0
$821$		−	277.087i		−	0.337500i	−0.985659	−	0.168750i	$-0.946027\pi$
$821$		−	277.087i		−	0.337500i	0.985659	−	0.168750i	$-0.0539730\pi$
$822$		0			0
$823$		−	52.9450i		−	0.0643317i	−0.999483	−	0.0321659i	$-0.989760\pi$
$823$		−	52.9450i		−	0.0643317i	0.999483	−	0.0321659i	$-0.0102405\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	1507.53i		−	1.82289i	−0.411424	−	0.911444i	$-0.634969\pi$
$827$		−	1507.53i		−	1.82289i	0.411424	−	0.911444i	$-0.365031\pi$
$828$		0			0
$829$		−	184.129i		−	0.222110i	−0.993814	−	0.111055i	$-0.964577\pi$
$829$		−	184.129i		−	0.222110i	0.993814	−	0.111055i	$-0.0354230\pi$
$830$		0			0
$831$	−1233.83			−1.48476
$832$		0			0
$833$			409.534i			0.491637i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−310.978	−	56.9278i	−0.371539	−	0.0680141i
$838$		0			0
$839$	−647.481			−0.771729			−0.385865	−	0.922555i	$-0.626097\pi$
$839$	−647.481			−0.771729			−0.385865	+	0.922555i	$0.626097\pi$
$840$		0			0
$841$	561.829			0.668048
$842$		0			0
$843$		−	125.556i		−	0.148939i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−910.184			−1.07460
$848$		0			0
$849$		−	626.792i		−	0.738271i
$850$		0			0
$851$	1244.43			1.46231
$852$		0			0
$853$	154.031			0.180576			0.0902879	−	0.995916i	$-0.471221\pi$
$853$	154.031			0.180576			0.0902879	+	0.995916i	$0.471221\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	87.5404			0.102147			0.0510737	−	0.998695i	$-0.483736\pi$
$857$	87.5404			0.102147			0.0510737	+	0.998695i	$0.483736\pi$
$858$		0			0
$859$		−	58.8859i		−	0.0685517i	−0.999412	−	0.0342759i	$-0.989088\pi$
$859$		−	58.8859i		−	0.0685517i	0.999412	−	0.0342759i	$-0.0109125\pi$
$860$		0			0
$861$		−	305.287i		−	0.354573i
$862$		0			0
$863$			1419.16i			1.64445i	0.569166	+	0.822223i	$0.307266\pi$
$863$			1419.16i			1.64445i	−0.569166	+	0.822223i	$0.692734\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		−	772.080i		−	0.890519i
$868$		0			0
$869$	−2937.99			−3.38089
$870$		0			0
$871$		−	1261.20i		−	1.44799i
$872$		0			0
$873$	1353.57			1.55048
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−193.862			−0.221051			−0.110525	−	0.993873i	$-0.535253\pi$
$877$	−193.862			−0.221051			−0.110525	+	0.993873i	$0.535253\pi$
$878$		0			0
$879$			872.192i			0.992255i
$880$		0			0
$881$			721.196i			0.818610i	0.912398	+	0.409305i	$0.134229\pi$
$881$			721.196i			0.818610i	−0.912398	+	0.409305i	$0.865771\pi$
$882$		0			0
$883$		−	714.021i		−	0.808630i	−0.914620	−	0.404315i	$-0.867510\pi$
$883$		−	714.021i		−	0.808630i	0.914620	−	0.404315i	$-0.132490\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−160.966			−0.181473			−0.0907364	−	0.995875i	$-0.528922\pi$
$887$	−160.966			−0.181473			−0.0907364	+	0.995875i	$0.528922\pi$
$888$		0			0
$889$		−	24.3489i		−	0.0273891i
$890$		0			0
$891$		−	1099.48i		−	1.23399i
$892$		0			0
$893$	1392.30			1.55913
$894$		0			0
$895$		0			0
$896$		0			0
$897$		−	3957.00i		−	4.41137i
$898$		0			0
$899$	509.494	+	93.2684i	0.566735	+	0.103747i
$900$		0			0
$901$	−759.152			−0.842566
$902$		0			0
$903$	−470.617			−0.521171
$904$		0			0
$905$		0			0
$906$		0			0
$907$	732.962			0.808117			0.404058	−	0.914733i	$-0.367599\pi$
$907$	732.962			0.808117			0.404058	+	0.914733i	$0.367599\pi$
$908$		0			0
$909$	−1031.87			−1.13517
$910$		0			0
$911$			975.273i			1.07055i	0.844677	+	0.535276i	$0.179792\pi$
$911$			975.273i			1.07055i	−0.844677	+	0.535276i	$0.820208\pi$
$912$		0			0
$913$	−2243.23			−2.45699
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−50.9177			−0.0555264
$918$		0			0
$919$	−1000.07			−1.08822			−0.544109	−	0.839014i	$-0.683132\pi$
$919$	−1000.07			−1.08822			−0.544109	+	0.839014i	$0.683132\pi$
$920$		0			0
$921$			404.484i			0.439179i
$922$		0			0
$923$		−	1291.12i		−	1.39883i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	1770.67			1.91011
$928$		0			0
$929$		−	690.475i		−	0.743246i	−0.928384	−	0.371623i	$-0.878801\pi$
$929$		−	690.475i		−	0.743246i	0.928384	−	0.371623i	$-0.121199\pi$
$930$		0			0
$931$	785.583			0.843805
$932$		0			0
$933$		−	854.037i		−	0.915367i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	312.165			0.333153			0.166577	−	0.986028i	$-0.446729\pi$
$937$	312.165			0.333153			0.166577	+	0.986028i	$0.446729\pi$
$938$		0			0
$939$	−866.064			−0.922326
$940$		0			0
$941$			991.756i			1.05394i	0.849884	+	0.526969i	$0.176672\pi$
$941$			991.756i			1.05394i	−0.849884	+	0.526969i	$0.823328\pi$
$942$		0			0
$943$			735.942i			0.780426i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			57.3610i			0.0605713i	0.999541	+	0.0302857i	$0.00964170\pi$
$947$			57.3610i			0.0605713i	−0.999541	+	0.0302857i	$0.990358\pi$
$948$		0			0
$949$	2763.40			2.91190
$950$		0			0
$951$		−	2695.25i		−	2.83413i
$952$		0			0
$953$		−	1254.16i		−	1.31601i	−0.753013	−	0.658006i	$-0.771400\pi$
$953$		−	1254.16i		−	1.31601i	0.753013	−	0.658006i	$-0.228600\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		−	1490.64i		−	1.55762i
$958$		0			0
$959$		−	302.711i		−	0.315653i
$960$		0			0
$961$	−898.680	−	340.434i	−0.935151	−	0.354250i
$962$		0			0
$963$	1427.43			1.48227
$964$		0			0
$965$		0			0
$966$		0			0
$967$		−	570.613i		−	0.590086i	−0.955484	−	0.295043i	$-0.904666\pi$
$967$		−	570.613i		−	0.590086i	0.955484	−	0.295043i	$-0.0953340\pi$
$968$		0			0
$969$	1014.58			1.04704
$970$		0			0
$971$	−1755.24			−1.80767			−0.903833	−	0.427884i	$-0.859259\pi$
$971$	−1755.24			−1.80767			−0.903833	+	0.427884i	$0.859259\pi$
$972$		0			0
$973$		−	107.467i		−	0.110449i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1663.66			1.70283			0.851413	−	0.524496i	$-0.175746\pi$
$977$	1663.66			1.70283			0.851413	+	0.524496i	$0.175746\pi$
$978$		0			0
$979$	688.624			0.703395
$980$		0			0
$981$	1157.29			1.17970
$982$		0			0
$983$		−	907.363i		−	0.923055i	−0.887126	−	0.461527i	$-0.847302\pi$
$983$		−	907.363i		−	0.923055i	0.887126	−	0.461527i	$-0.152698\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$			1009.63i			1.02293i
$988$		0			0
$989$	1134.50			1.14711
$990$		0			0
$991$		−	1648.07i		−	1.66304i	−0.555496	−	0.831519i	$-0.687472\pi$
$991$		−	1648.07i		−	1.66304i	0.555496	−	0.831519i	$-0.312528\pi$
$992$		0			0
$993$	−2608.74			−2.62713
$994$		0			0
$995$		0			0
$996$		0			0
$997$	196.896			0.197488			0.0987441	−	0.995113i	$-0.468517\pi$
$997$	196.896			0.197488			0.0987441	+	0.995113i	$0.468517\pi$
$998$		0			0
$999$	349.161			0.349511

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3100.3.d.g.1301.3	yes	22
5.2	odd	4		3100.3.f.d.1549.6		44
5.3	odd	4		3100.3.f.d.1549.39		44
5.4	even	2		3100.3.d.f.1301.20	yes	22
31.30	odd	2	inner	3100.3.d.g.1301.20	yes	22
155.92	even	4		3100.3.f.d.1549.40		44
155.123	even	4		3100.3.f.d.1549.5		44
155.154	odd	2		3100.3.d.f.1301.3	✓	22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3100.3.d.f.1301.3	✓	22	155.154	odd	2
3100.3.d.f.1301.20	yes	22	5.4	even	2
3100.3.d.g.1301.3	yes	22	1.1	even	1	trivial
3100.3.d.g.1301.20	yes	22	31.30	odd	2	inner
3100.3.f.d.1549.5		44	155.123	even	4
3100.3.f.d.1549.6		44	5.2	odd	4
3100.3.f.d.1549.39		44	5.3	odd	4
3100.3.f.d.1549.40		44	155.92	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)

\(f(q)\)	\(=\)	\(q-4.50171i q^{3} +3.34932 q^{7} -11.2654 q^{9} +O(q^{10})\)	\(q-4.50171i q^{3} +3.34932 q^{7} -11.2654 q^{9} +19.8180i q^{11} -24.1835i q^{13} -10.8394i q^{17} -20.7925 q^{19} -15.0777i q^{21} +36.3471i q^{23} +10.1982i q^{27} -16.7084i q^{29} +(-5.58212 + 30.4933i) q^{31} +89.2148 q^{33} -34.2374i q^{37} -108.867 q^{39} +20.2476 q^{41} -31.2128i q^{43} -66.9619 q^{47} -37.7821 q^{49} -48.7957 q^{51} -70.0365i q^{53} +93.6018i q^{57} -32.5693 q^{59} +84.0097i q^{61} -37.7315 q^{63} +52.1513 q^{67} +163.624 q^{69} +53.3886 q^{71} +114.268i q^{73} +66.3767i q^{77} +148.249i q^{79} -55.4792 q^{81} +113.192i q^{83} -75.2165 q^{87} -34.7475i q^{89} -80.9983i q^{91} +(137.272 + 25.1291i) q^{93} -120.153 q^{97} -223.258i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q + 4 q^{7} - 72 q^{9}+O(q^{10}) \) 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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