LMFDB - Embedded newform 3100.3.d.g.1301.1

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			1301.1
Character	$\chi$	$=$	3100.1301
Dual form			3100.3.d.g.1301.22

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.52468i q^{3} -8.23275 q^{7} -21.5221 q^{9} +O(q^{10})$	$q-5.52468i q^{3} -8.23275 q^{7} -21.5221 q^{9} -13.8742i q^{11} -4.20384i q^{13} -14.5610i q^{17} -9.90677 q^{19} +45.4833i q^{21} -42.3608i q^{23} +69.1808i q^{27} -28.8373i q^{29} +(-18.7205 + 24.7092i) q^{31} -76.6504 q^{33} -7.59231i q^{37} -23.2249 q^{39} -21.0098 q^{41} -78.7761i q^{43} -76.1891 q^{47} +18.7781 q^{49} -80.4449 q^{51} +62.4653i q^{53} +54.7318i q^{57} -32.7981 q^{59} -60.6004i q^{61} +177.186 q^{63} +43.3426 q^{67} -234.030 q^{69} +67.0600 q^{71} -29.7143i q^{73} +114.223i q^{77} +15.9669i q^{79} +188.503 q^{81} -81.7651i q^{83} -159.317 q^{87} +79.5780i q^{89} +34.6092i q^{91} +(136.510 + 103.425i) q^{93} +151.011 q^{97} +298.602i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$		−	5.52468i		−	1.84156i	−0.390081	−	0.920781i	$-0.627553\pi$
$3$		−	5.52468i		−	1.84156i	0.390081	−	0.920781i	$-0.372447\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−8.23275			−1.17611			−0.588053	−	0.808822i	$-0.700106\pi$
$7$	−8.23275			−1.17611			−0.588053	+	0.808822i	$0.700106\pi$
$8$		0			0
$9$	−21.5221			−2.39135
$10$		0			0
$11$		−	13.8742i		−	1.26129i	−0.776072	−	0.630644i	$-0.782791\pi$
$11$		−	13.8742i		−	1.26129i	0.776072	−	0.630644i	$-0.217209\pi$
$12$		0			0
$13$		−	4.20384i		−	0.323373i	−0.986842	−	0.161686i	$-0.948307\pi$
$13$		−	4.20384i		−	0.323373i	0.986842	−	0.161686i	$-0.0516933\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	14.5610i		−	0.856529i	−0.903653	−	0.428264i	$-0.859125\pi$
$17$		−	14.5610i		−	0.856529i	0.903653	−	0.428264i	$-0.140875\pi$
$18$		0			0
$19$	−9.90677			−0.521409			−0.260705	−	0.965419i	$-0.583955\pi$
$19$	−9.90677			−0.521409			−0.260705	+	0.965419i	$0.583955\pi$
$20$		0			0
$21$			45.4833i			2.16587i
$22$		0			0
$23$		−	42.3608i		−	1.84177i	−0.389832	−	0.920886i	$-0.627467\pi$
$23$		−	42.3608i		−	1.84177i	0.389832	−	0.920886i	$-0.372533\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$			69.1808i			2.56225i
$28$		0			0
$29$		−	28.8373i		−	0.994389i	−0.867639	−	0.497195i	$-0.834363\pi$
$29$		−	28.8373i		−	0.994389i	0.867639	−	0.497195i	$-0.165637\pi$
$30$		0			0
$31$	−18.7205	+	24.7092i	−0.603886	+	0.797071i
$31$	−18.7205	+	24.7092i	−0.603886	+	0.797071i
$32$		0			0
$33$	−76.6504			−2.32274
$34$		0			0
$35$		0			0
$36$		0			0
$37$		−	7.59231i		−	0.205198i	−0.994723	−	0.102599i	$-0.967284\pi$
$37$		−	7.59231i		−	0.205198i	0.994723	−	0.102599i	$-0.0327158\pi$
$38$		0			0
$39$	−23.2249			−0.595510
$40$		0			0
$41$	−21.0098			−0.512435			−0.256217	−	0.966619i	$-0.582476\pi$
$41$	−21.0098			−0.512435			−0.256217	+	0.966619i	$0.582476\pi$
$42$		0			0
$43$		−	78.7761i		−	1.83200i	−0.401176	−	0.916001i	$-0.631398\pi$
$43$		−	78.7761i		−	1.83200i	0.401176	−	0.916001i	$-0.368602\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−76.1891			−1.62105			−0.810523	−	0.585707i	$-0.800817\pi$
$47$	−76.1891			−1.62105			−0.810523	+	0.585707i	$0.800817\pi$
$48$		0			0
$49$	18.7781			0.383227
$50$		0			0
$51$	−80.4449			−1.57735
$52$		0			0
$53$			62.4653i			1.17859i	0.807918	+	0.589295i	$0.200595\pi$
$53$			62.4653i			1.17859i	−0.807918	+	0.589295i	$0.799405\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$			54.7318i			0.960207i
$58$		0			0
$59$	−32.7981			−0.555900			−0.277950	−	0.960596i	$-0.589655\pi$
$59$	−32.7981			−0.555900			−0.277950	+	0.960596i	$0.589655\pi$
$60$		0			0
$61$		−	60.6004i		−	0.993449i	−0.867908	−	0.496725i	$-0.834536\pi$
$61$		−	60.6004i		−	0.993449i	0.867908	−	0.496725i	$-0.165464\pi$
$62$		0			0
$63$	177.186			2.81248
$64$		0			0
$65$		0			0
$66$		0			0
$67$	43.3426			0.646905			0.323452	−	0.946244i	$-0.395156\pi$
$67$	43.3426			0.646905			0.323452	+	0.946244i	$0.395156\pi$
$68$		0			0
$69$	−234.030			−3.39174
$70$		0			0
$71$	67.0600			0.944507			0.472253	−	0.881463i	$-0.343441\pi$
$71$	67.0600			0.944507			0.472253	+	0.881463i	$0.343441\pi$
$72$		0			0
$73$		−	29.7143i		−	0.407045i	−0.979070	−	0.203522i	$-0.934761\pi$
$73$		−	29.7143i		−	0.407045i	0.979070	−	0.203522i	$-0.0652390\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			114.223i			1.48341i
$78$		0			0
$79$			15.9669i			0.202112i	0.994881	+	0.101056i	$0.0322221\pi$
$79$			15.9669i			0.202112i	−0.994881	+	0.101056i	$0.967778\pi$
$80$		0			0
$81$	188.503			2.32720
$82$		0			0
$83$		−	81.7651i		−	0.985122i	−0.870278	−	0.492561i	$-0.836061\pi$
$83$		−	81.7651i		−	0.985122i	0.870278	−	0.492561i	$-0.163939\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−159.317			−1.83123
$88$		0			0
$89$			79.5780i			0.894135i	0.894500	+	0.447068i	$0.147532\pi$
$89$			79.5780i			0.894135i	−0.894500	+	0.447068i	$0.852468\pi$
$90$		0			0
$91$			34.6092i			0.380321i
$92$		0			0
$93$	136.510	+	103.425i	1.46785	+	1.11209i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	151.011			1.55681			0.778405	−	0.627762i	$-0.216029\pi$
$97$	151.011			1.55681			0.778405	+	0.627762i	$0.216029\pi$
$98$		0			0
$99$			298.602i			3.01618i
$100$		0			0
$101$	−143.679			−1.42257			−0.711283	−	0.702906i	$-0.751886\pi$
$101$	−143.679			−1.42257			−0.711283	+	0.702906i	$0.751886\pi$
$102$		0			0
$103$	137.445			1.33442			0.667210	−	0.744869i	$-0.267488\pi$
$103$	137.445			1.33442			0.667210	+	0.744869i	$0.267488\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	201.292			1.88123			0.940616	−	0.339471i	$-0.110248\pi$
$107$	201.292			1.88123			0.940616	+	0.339471i	$0.110248\pi$
$108$		0			0
$109$	27.3289			0.250724			0.125362	−	0.992111i	$-0.459991\pi$
$109$	27.3289			0.250724			0.125362	+	0.992111i	$0.459991\pi$
$110$		0			0
$111$	−41.9451			−0.377884
$112$		0			0
$113$	102.441			0.906559			0.453279	−	0.891368i	$-0.350254\pi$
$113$	102.441			0.906559			0.453279	+	0.891368i	$0.350254\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$			90.4757i			0.773296i
$118$		0			0
$119$			119.877i			1.00737i
$120$		0			0
$121$	−71.4925			−0.590847
$122$		0			0
$123$			116.073i			0.943680i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		−	11.9510i		−	0.0941021i	−0.998892	−	0.0470510i	$-0.985018\pi$
$127$		−	11.9510i		−	0.0941021i	0.998892	−	0.0470510i	$-0.0149823\pi$
$128$		0			0
$129$	−435.213			−3.37374
$130$		0			0
$131$	−64.3061			−0.490887			−0.245443	−	0.969411i	$-0.578934\pi$
$131$	−64.3061			−0.490887			−0.245443	+	0.969411i	$0.578934\pi$
$132$		0			0
$133$	81.5600			0.613233
$134$		0			0
$135$		0			0
$136$		0			0
$137$			256.241i			1.87037i	0.354157	+	0.935186i	$0.384768\pi$
$137$			256.241i			1.87037i	−0.354157	+	0.935186i	$0.615232\pi$
$138$		0			0
$139$		−	176.303i		−	1.26837i	−0.773182	−	0.634184i	$-0.781336\pi$
$139$		−	176.303i		−	1.26837i	0.773182	−	0.634184i	$-0.218664\pi$
$140$		0			0
$141$			420.921i			2.98525i
$142$		0			0
$143$	−58.3248			−0.407866
$144$		0			0
$145$		0			0
$146$		0			0
$147$		−	103.743i		−	0.705737i
$148$		0			0
$149$	−106.493			−0.714716			−0.357358	−	0.933967i	$-0.616322\pi$
$149$	−106.493			−0.714716			−0.357358	+	0.933967i	$0.616322\pi$
$150$		0			0
$151$		−	28.5375i		−	0.188990i	−0.995525	−	0.0944949i	$-0.969876\pi$
$151$		−	28.5375i		−	0.188990i	0.995525	−	0.0944949i	$-0.0301236\pi$
$152$		0			0
$153$			313.384i			2.04826i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−153.037			−0.974756			−0.487378	−	0.873191i	$-0.662047\pi$
$157$	−153.037			−0.974756			−0.487378	+	0.873191i	$0.662047\pi$
$158$		0			0
$159$	345.101			2.17045
$160$		0			0
$161$			348.745i			2.16612i
$162$		0			0
$163$	198.126			1.21550			0.607748	−	0.794130i	$-0.292073\pi$
$163$	198.126			1.21550			0.607748	+	0.794130i	$0.292073\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	60.6314i		−	0.363063i	−0.983385	−	0.181531i	$-0.941895\pi$
$167$		−	60.6314i		−	0.363063i	0.983385	−	0.181531i	$-0.0581054\pi$
$168$		0			0
$169$	151.328			0.895430
$170$		0			0
$171$	213.215			1.24687
$172$		0			0
$173$	−200.091			−1.15660			−0.578298	−	0.815826i	$-0.696283\pi$
$173$	−200.091			−1.15660			−0.578298	+	0.815826i	$0.696283\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$			181.199i			1.02372i
$178$		0			0
$179$			32.6086i			0.182171i	0.995843	+	0.0910855i	$0.0290337\pi$
$179$			32.6086i			0.182171i	−0.995843	+	0.0910855i	$0.970966\pi$
$180$		0			0
$181$		−	137.796i		−	0.761306i	−0.924718	−	0.380653i	$-0.875699\pi$
$181$		−	137.796i		−	0.761306i	0.924718	−	0.380653i	$-0.124301\pi$
$182$		0			0
$183$	−334.798			−1.82950
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−202.022			−1.08033
$188$		0			0
$189$		−	569.548i		−	3.01348i
$190$		0			0
$191$	221.655			1.16050			0.580249	−	0.814439i	$-0.302955\pi$
$191$	221.655			1.16050			0.580249	+	0.814439i	$0.302955\pi$
$192$		0			0
$193$	333.323			1.72706			0.863531	−	0.504296i	$-0.168248\pi$
$193$	333.323			1.72706			0.863531	+	0.504296i	$0.168248\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	134.559i		−	0.683038i	−0.939875	−	0.341519i	$-0.889059\pi$
$197$		−	134.559i		−	0.683038i	0.939875	−	0.341519i	$-0.110941\pi$
$198$		0			0
$199$		−	209.351i		−	1.05201i	−0.850480	−	0.526007i	$-0.823689\pi$
$199$		−	209.351i		−	1.05201i	0.850480	−	0.526007i	$-0.176311\pi$
$200$		0			0
$201$		−	239.454i		−	1.19131i
$202$		0			0
$203$			237.410i			1.16951i
$204$		0			0
$205$		0			0
$206$		0			0
$207$			911.694i			4.40432i
$208$		0			0
$209$			137.448i			0.657647i
$210$		0			0
$211$	−61.0693			−0.289428			−0.144714	−	0.989474i	$-0.546226\pi$
$211$	−61.0693			−0.289428			−0.144714	+	0.989474i	$0.546226\pi$
$212$		0			0
$213$		−	370.485i		−	1.73937i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	154.121	−	203.425i	0.710235	−	0.937440i
$218$		0			0
$219$	−164.162			−0.749598
$220$		0			0
$221$	−61.2121			−0.276978
$222$		0			0
$223$		−	33.1834i		−	0.148804i	−0.997228	−	0.0744022i	$-0.976295\pi$
$223$		−	33.1834i		−	0.148804i	0.997228	−	0.0744022i	$-0.0237049\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−108.467			−0.477829			−0.238915	−	0.971041i	$-0.576792\pi$
$227$	−108.467			−0.477829			−0.238915	+	0.971041i	$0.576792\pi$
$228$		0			0
$229$		−	214.012i		−	0.934550i	−0.884112	−	0.467275i	$-0.845236\pi$
$229$		−	214.012i		−	0.934550i	0.884112	−	0.467275i	$-0.154764\pi$
$230$		0			0
$231$	631.043			2.73179
$232$		0			0
$233$	2.62939			0.0112849			0.00564247	−	0.999984i	$-0.498204\pi$
$233$	2.62939			0.0112849			0.00564247	+	0.999984i	$0.498204\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	88.2118			0.372202
$238$		0			0
$239$		−	3.84285i		−	0.0160789i	−0.999968	−	0.00803943i	$-0.997441\pi$
$239$		−	3.84285i		−	0.0160789i	0.999968	−	0.00803943i	$-0.00255906\pi$
$240$		0			0
$241$			304.247i			1.26244i	0.775606	+	0.631218i	$0.217445\pi$
$241$			304.247i			1.26244i	−0.775606	+	0.631218i	$0.782555\pi$
$242$		0			0
$243$		−	418.792i		−	1.72342i
$244$		0			0
$245$		0			0
$246$		0			0
$247$			41.6465i			0.168609i
$248$		0			0
$249$	−451.726			−1.81416
$250$		0			0
$251$			205.103i			0.817142i	0.912726	+	0.408571i	$0.133973\pi$
$251$			205.103i			0.817142i	−0.912726	+	0.408571i	$0.866027\pi$
$252$		0			0
$253$	−587.720			−2.32300
$254$		0			0
$255$		0			0
$256$		0			0
$257$	351.710			1.36852			0.684261	−	0.729237i	$-0.260125\pi$
$257$	351.710			1.36852			0.684261	+	0.729237i	$0.260125\pi$
$258$		0			0
$259$			62.5056i			0.241334i
$260$		0			0
$261$			620.640i			2.37793i
$262$		0			0
$263$		−	400.984i		−	1.52466i	−0.647191	−	0.762328i	$-0.724056\pi$
$263$		−	400.984i		−	1.52466i	0.647191	−	0.762328i	$-0.275944\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	439.644			1.64660
$268$		0			0
$269$		−	136.703i		−	0.508190i	−0.967179	−	0.254095i	$-0.918222\pi$
$269$		−	136.703i		−	0.508190i	0.967179	−	0.254095i	$-0.0817777\pi$
$270$		0			0
$271$		−	50.6028i		−	0.186726i	−0.995632	−	0.0933631i	$-0.970238\pi$
$271$		−	50.6028i		−	0.186726i	0.995632	−	0.0933631i	$-0.0297617\pi$
$272$		0			0
$273$	191.205			0.700384
$274$		0			0
$275$		0			0
$276$		0			0
$277$		−	278.380i		−	1.00498i	−0.864582	−	0.502492i	$-0.832417\pi$
$277$		−	278.380i		−	1.00498i	0.864582	−	0.502492i	$-0.167583\pi$
$278$		0			0
$279$	402.904	−	531.794i	1.44410	−	1.90607i
$280$		0			0
$281$	−4.55633			−0.0162147			−0.00810734	−	0.999967i	$-0.502581\pi$
$281$	−4.55633			−0.0162147			−0.00810734	+	0.999967i	$0.502581\pi$
$282$		0			0
$283$	−145.425			−0.513869			−0.256934	−	0.966429i	$-0.582712\pi$
$283$	−145.425			−0.513869			−0.256934	+	0.966429i	$0.582712\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	172.969			0.602678
$288$		0			0
$289$	76.9776			0.266358
$290$		0			0
$291$		−	834.286i		−	2.86696i
$292$		0			0
$293$	−486.377			−1.65999			−0.829995	−	0.557771i	$-0.811657\pi$
$293$	−486.377			−1.65999			−0.829995	+	0.557771i	$0.811657\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	959.826			3.23174
$298$		0			0
$299$	−178.078			−0.595579
$300$		0			0
$301$			648.544i			2.15463i
$302$		0			0
$303$			793.782i			2.61974i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	323.953			1.05522			0.527611	−	0.849486i	$-0.323088\pi$
$307$	323.953			1.05522			0.527611	+	0.849486i	$0.323088\pi$
$308$		0			0
$309$		−	759.342i		−	2.45742i
$310$		0			0
$311$	−16.4250			−0.0528136			−0.0264068	−	0.999651i	$-0.508407\pi$
$311$	−16.4250			−0.0528136			−0.0264068	+	0.999651i	$0.508407\pi$
$312$		0			0
$313$			590.168i			1.88552i	0.333471	+	0.942760i	$0.391780\pi$
$313$			590.168i			1.88552i	−0.333471	+	0.942760i	$0.608220\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−73.1487			−0.230753			−0.115377	−	0.993322i	$-0.536807\pi$
$317$	−73.1487			−0.230753			−0.115377	+	0.993322i	$0.536807\pi$
$318$		0			0
$319$	−400.093			−1.25421
$320$		0			0
$321$		−	1112.07i		−	3.46441i
$322$		0			0
$323$			144.252i			0.446602i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		−	150.983i		−	0.461723i
$328$		0			0
$329$	627.246			1.90652
$330$		0			0
$331$		−	83.4178i		−	0.252017i	−0.992029	−	0.126009i	$-0.959783\pi$
$331$		−	83.4178i		−	0.252017i	0.992029	−	0.126009i	$-0.0402167\pi$
$332$		0			0
$333$			163.403i			0.490699i
$334$		0			0
$335$		0			0
$336$		0			0
$337$		−	283.390i		−	0.840920i	−0.907311	−	0.420460i	$-0.861869\pi$
$337$		−	283.390i		−	0.840920i	0.907311	−	0.420460i	$-0.138131\pi$
$338$		0			0
$339$		−	565.955i		−	1.66948i
$340$		0			0
$341$	342.819	+	259.731i	1.00534	+	0.761674i
$342$		0			0
$343$	248.809			0.725390
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	345.858i		−	0.996708i	−0.866974	−	0.498354i	$-0.833938\pi$
$347$		−	345.858i		−	0.996708i	0.866974	−	0.498354i	$-0.166062\pi$
$348$		0			0
$349$	6.16152			0.0176548			0.00882739	−	0.999961i	$-0.497190\pi$
$349$	6.16152			0.0176548			0.00882739	+	0.999961i	$0.497190\pi$
$350$		0			0
$351$	290.825			0.828562
$352$		0			0
$353$		−	340.064i		−	0.963355i	−0.876349	−	0.481678i	$-0.840028\pi$
$353$		−	340.064i		−	0.963355i	0.876349	−	0.481678i	$-0.159972\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	662.282			1.85513
$358$		0			0
$359$	464.348			1.29345			0.646724	−	0.762724i	$-0.276138\pi$
$359$	464.348			1.29345			0.646724	+	0.762724i	$0.276138\pi$
$360$		0			0
$361$	−262.856			−0.728133
$362$		0			0
$363$			394.973i			1.08808i
$364$		0			0
$365$		0			0
$366$		0			0
$367$			399.829i			1.08945i	0.838614	+	0.544726i	$0.183367\pi$
$367$			399.829i			1.08945i	−0.838614	+	0.544726i	$0.816633\pi$
$368$		0			0
$369$	452.176			1.22541
$370$		0			0
$371$		−	514.261i		−	1.38615i
$372$		0			0
$373$	−690.566			−1.85138			−0.925692	−	0.378278i	$-0.876516\pi$
$373$	−690.566			−1.85138			−0.925692	+	0.378278i	$0.876516\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−121.227			−0.321558
$378$		0			0
$379$	239.799			0.632715			0.316358	−	0.948640i	$-0.397540\pi$
$379$	239.799			0.632715			0.316358	+	0.948640i	$0.397540\pi$
$380$		0			0
$381$	−66.0253			−0.173295
$382$		0			0
$383$			102.360i			0.267259i	0.991031	+	0.133629i	$0.0426631\pi$
$383$			102.360i			0.267259i	−0.991031	+	0.133629i	$0.957337\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$			1695.43i			4.38095i
$388$		0			0
$389$			503.773i			1.29505i	0.762046	+	0.647523i	$0.224195\pi$
$389$			503.773i			1.29505i	−0.762046	+	0.647523i	$0.775805\pi$
$390$		0			0
$391$	−616.815			−1.57753
$392$		0			0
$393$			355.271i			0.903998i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	720.730			1.81544			0.907720	−	0.419576i	$-0.137821\pi$
$397$	720.730			1.81544			0.907720	+	0.419576i	$0.137821\pi$
$398$		0			0
$399$		−	450.593i		−	1.12931i
$400$		0			0
$401$			129.598i			0.323186i	0.986857	+	0.161593i	$0.0516632\pi$
$401$			129.598i			0.323186i	−0.986857	+	0.161593i	$0.948337\pi$
$402$		0			0
$403$	103.874	+	78.6979i	0.257751	+	0.195280i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−105.337			−0.258813
$408$		0			0
$409$			607.500i			1.48533i	0.669663	+	0.742666i	$0.266439\pi$
$409$			607.500i			1.48533i	−0.669663	+	0.742666i	$0.733561\pi$
$410$		0			0
$411$	1415.65			3.44440
$412$		0			0
$413$	270.019			0.653798
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−974.019			−2.33578
$418$		0			0
$419$	−82.8624			−0.197762			−0.0988811	−	0.995099i	$-0.531526\pi$
$419$	−82.8624			−0.197762			−0.0988811	+	0.995099i	$0.531526\pi$
$420$		0			0
$421$	−736.256			−1.74883			−0.874413	−	0.485183i	$-0.838753\pi$
$421$	−736.256			−1.74883			−0.874413	+	0.485183i	$0.838753\pi$
$422$		0			0
$423$	1639.75			3.87648
$424$		0			0
$425$		0			0
$426$		0			0
$427$			498.908i			1.16840i
$428$		0			0
$429$			322.226i			0.751110i
$430$		0			0
$431$	736.844			1.70961			0.854807	−	0.518946i	$-0.173675\pi$
$431$	736.844			1.70961			0.854807	+	0.518946i	$0.173675\pi$
$432$		0			0
$433$			376.292i			0.869034i	0.900664	+	0.434517i	$0.143081\pi$
$433$			376.292i			0.869034i	−0.900664	+	0.434517i	$0.856919\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			419.658i			0.960317i
$438$		0			0
$439$	345.018			0.785919			0.392959	−	0.919556i	$-0.371451\pi$
$439$	345.018			0.785919			0.392959	+	0.919556i	$0.371451\pi$
$440$		0			0
$441$	−404.146			−0.916430
$442$		0			0
$443$	−58.8902			−0.132935			−0.0664675	−	0.997789i	$-0.521173\pi$
$443$	−58.8902			−0.132935			−0.0664675	+	0.997789i	$0.521173\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$			588.339i			1.31619i
$448$		0			0
$449$		−	97.0967i		−	0.216251i	−0.994137	−	0.108125i	$-0.965515\pi$
$449$		−	97.0967i		−	0.216251i	0.994137	−	0.108125i	$-0.0344848\pi$
$450$		0			0
$451$			291.494i			0.646328i
$452$		0			0
$453$	−157.660			−0.348036
$454$		0			0
$455$		0			0
$456$		0			0
$457$		−	688.267i		−	1.50605i	−0.657989	−	0.753027i	$-0.728593\pi$
$457$		−	688.267i		−	1.50605i	0.657989	−	0.753027i	$-0.271407\pi$
$458$		0			0
$459$	1007.34			2.19464
$460$		0			0
$461$			14.5385i			0.0315369i	0.999876	+	0.0157685i	$0.00501946\pi$
$461$			14.5385i			0.0315369i	−0.999876	+	0.0157685i	$0.994981\pi$
$462$		0			0
$463$			598.558i			1.29278i	0.763006	+	0.646391i	$0.223722\pi$
$463$			598.558i			1.29278i	−0.763006	+	0.646391i	$0.776278\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	567.702			1.21564			0.607818	−	0.794076i	$-0.292045\pi$
$467$	567.702			1.21564			0.607818	+	0.794076i	$0.292045\pi$
$468$		0			0
$469$	−356.829			−0.760829
$470$		0			0
$471$			845.480i			1.79507i
$472$		0			0
$473$	−1092.95			−2.31068
$474$		0			0
$475$		0			0
$476$		0			0
$477$		−	1344.39i		−	2.81842i
$478$		0			0
$479$	−813.288			−1.69789			−0.848944	−	0.528483i	$-0.822761\pi$
$479$	−813.288			−1.69789			−0.848944	+	0.528483i	$0.822761\pi$
$480$		0			0
$481$	−31.9169			−0.0663553
$482$		0			0
$483$	1926.71			3.98904
$484$		0			0
$485$		0			0
$486$		0			0
$487$		−	732.058i		−	1.50320i	−0.659619	−	0.751600i	$-0.729282\pi$
$487$		−	732.058i		−	1.50320i	0.659619	−	0.751600i	$-0.270718\pi$
$488$		0			0
$489$		−	1094.58i		−	2.23841i
$490$		0			0
$491$		−	446.146i		−	0.908648i	−0.890837	−	0.454324i	$-0.849881\pi$
$491$		−	446.146i		−	0.908648i	0.890837	−	0.454324i	$-0.150119\pi$
$492$		0			0
$493$	−419.899			−0.851723
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−552.088			−1.11084
$498$		0			0
$499$		−	180.742i		−	0.362209i	−0.983464	−	0.181104i	$-0.942033\pi$
$499$		−	180.742i		−	0.362209i	0.983464	−	0.181104i	$-0.0579672\pi$
$500$		0			0
$501$	−334.970			−0.668602
$502$		0			0
$503$	−420.174			−0.835336			−0.417668	−	0.908600i	$-0.637152\pi$
$503$	−420.174			−0.835336			−0.417668	+	0.908600i	$0.637152\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		−	836.038i		−	1.64899i
$508$		0			0
$509$		−	251.727i		−	0.494553i	−0.968945	−	0.247276i	$-0.920464\pi$
$509$		−	251.727i		−	0.494553i	0.968945	−	0.247276i	$-0.0795356\pi$
$510$		0			0
$511$			244.630i			0.478728i
$512$		0			0
$513$		−	685.359i		−	1.33598i
$514$		0			0
$515$		0			0
$516$		0			0
$517$			1057.06i			2.04460i
$518$		0			0
$519$			1105.44i			2.12994i
$520$		0			0
$521$	−737.118			−1.41481			−0.707407	−	0.706806i	$-0.750135\pi$
$521$	−737.118			−1.41481			−0.707407	+	0.706806i	$0.750135\pi$
$522$		0			0
$523$			532.630i			1.01841i	0.860645	+	0.509206i	$0.170061\pi$
$523$			532.630i			1.01841i	−0.860645	+	0.509206i	$0.829939\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	359.790	+	272.589i	0.682714	+	0.517246i
$528$		0			0
$529$	−1265.43			−2.39212
$530$		0			0
$531$	705.885			1.32935
$532$		0			0
$533$			88.3220i			0.165707i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	180.152			0.335479
$538$		0			0
$539$		−	260.531i		−	0.483360i
$540$		0			0
$541$	442.330			0.817616			0.408808	−	0.912621i	$-0.365945\pi$
$541$	442.330			0.817616			0.408808	+	0.912621i	$0.365945\pi$
$542$		0			0
$543$	−761.282			−1.40199
$544$		0			0
$545$		0			0
$546$		0			0
$547$	488.873			0.893736			0.446868	−	0.894600i	$-0.352539\pi$
$547$	488.873			0.893736			0.446868	+	0.894600i	$0.352539\pi$
$548$		0			0
$549$			1304.25i			2.37568i
$550$		0			0
$551$			285.684i			0.518484i
$552$		0			0
$553$		−	131.451i		−	0.237705i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	85.4882i		−	0.153480i	−0.997051	−	0.0767398i	$-0.975549\pi$
$557$		−	85.4882i		−	0.153480i	0.997051	−	0.0767398i	$-0.0244511\pi$
$558$		0			0
$559$	−331.162			−0.592419
$560$		0			0
$561$			1116.11i			1.98949i
$562$		0			0
$563$	−691.132			−1.22759			−0.613794	−	0.789466i	$-0.710358\pi$
$563$	−691.132			−1.22759			−0.613794	+	0.789466i	$0.710358\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−1551.90			−2.73703
$568$		0			0
$569$			663.688i			1.16641i	0.812325	+	0.583206i	$0.198202\pi$
$569$			663.688i			1.16641i	−0.812325	+	0.583206i	$0.801798\pi$
$570$		0			0
$571$		−	467.534i		−	0.818799i	−0.912355	−	0.409399i	$-0.865738\pi$
$571$		−	467.534i		−	0.818799i	0.912355	−	0.409399i	$-0.134262\pi$
$572$		0			0
$573$		−	1224.58i		−	2.13713i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−120.318			−0.208524			−0.104262	−	0.994550i	$-0.533248\pi$
$577$	−120.318			−0.208524			−0.104262	+	0.994550i	$0.533248\pi$
$578$		0			0
$579$		−	1841.50i		−	3.18049i
$580$		0			0
$581$			673.151i			1.15861i
$582$		0			0
$583$	866.654			1.48654
$584$		0			0
$585$		0			0
$586$		0			0
$587$			172.997i			0.294714i	0.989083	+	0.147357i	$0.0470767\pi$
$587$			172.997i			0.294714i	−0.989083	+	0.147357i	$0.952923\pi$
$588$		0			0
$589$	185.459	−	244.788i	0.314872	−	0.415600i
$590$		0			0
$591$	−743.393			−1.25786
$592$		0			0
$593$	908.051			1.53128			0.765641	−	0.643268i	$-0.222422\pi$
$593$	908.051			1.53128			0.765641	+	0.643268i	$0.222422\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−1156.60			−1.93735
$598$		0			0
$599$	−332.817			−0.555621			−0.277810	−	0.960636i	$-0.589609\pi$
$599$	−332.817			−0.555621			−0.277810	+	0.960636i	$0.589609\pi$
$600$		0			0
$601$		−	264.210i		−	0.439618i	−0.975543	−	0.219809i	$-0.929457\pi$
$601$		−	264.210i		−	0.439618i	0.975543	−	0.219809i	$-0.0705434\pi$
$602$		0			0
$603$	−932.826			−1.54697
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−602.496			−0.992580			−0.496290	−	0.868157i	$-0.665305\pi$
$607$	−602.496			−0.992580			−0.496290	+	0.868157i	$0.665305\pi$
$608$		0			0
$609$	1311.62			2.15372
$610$		0			0
$611$			320.287i			0.524202i
$612$		0			0
$613$		−	361.758i		−	0.590143i	−0.955475	−	0.295072i	$-0.904657\pi$
$613$		−	361.758i		−	0.590143i	0.955475	−	0.295072i	$-0.0953435\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−193.847			−0.314177			−0.157088	−	0.987585i	$-0.550211\pi$
$617$	−193.847			−0.314177			−0.157088	+	0.987585i	$0.550211\pi$
$618$		0			0
$619$			636.092i			1.02761i	0.857906	+	0.513806i	$0.171765\pi$
$619$			636.092i			1.02761i	−0.857906	+	0.513806i	$0.828235\pi$
$620$		0			0
$621$	2930.55			4.71908
$622$		0			0
$623$		−	655.146i		−	1.05160i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	759.358			1.21110
$628$		0			0
$629$	−110.552			−0.175758
$630$		0			0
$631$			10.4108i			0.0164989i	0.999966	+	0.00824946i	$0.00262591\pi$
$631$			10.4108i			0.0164989i	−0.999966	+	0.00824946i	$0.997374\pi$
$632$		0			0
$633$			337.389i			0.532999i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		−	78.9404i		−	0.123925i
$638$		0			0
$639$	−1443.27			−2.25864
$640$		0			0
$641$			708.503i			1.10531i	0.833410	+	0.552654i	$0.186385\pi$
$641$			708.503i			1.10531i	−0.833410	+	0.552654i	$0.813615\pi$
$642$		0			0
$643$		−	966.662i		−	1.50336i	−0.659527	−	0.751681i	$-0.729243\pi$
$643$		−	966.662i		−	1.50336i	0.659527	−	0.751681i	$-0.270757\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	494.108i		−	0.763691i	−0.924226	−	0.381845i	$-0.875289\pi$
$647$		−	494.108i		−	0.763691i	0.924226	−	0.381845i	$-0.124711\pi$
$648$		0			0
$649$			455.046i			0.701150i
$650$		0			0
$651$	−1123.86	−	851.469i	−1.72635	−	1.30794i
$652$		0			0
$653$	−968.247			−1.48277			−0.741384	−	0.671081i	$-0.765830\pi$
$653$	−968.247			−1.48277			−0.741384	+	0.671081i	$0.765830\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$			639.514i			0.973386i
$658$		0			0
$659$	460.240			0.698392			0.349196	−	0.937050i	$-0.386455\pi$
$659$	460.240			0.698392			0.349196	+	0.937050i	$0.386455\pi$
$660$		0			0
$661$	−123.009			−0.186096			−0.0930479	−	0.995662i	$-0.529661\pi$
$661$	−123.009			−0.186096			−0.0930479	+	0.995662i	$0.529661\pi$
$662$		0			0
$663$			338.178i			0.510072i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−1221.57			−1.83144
$668$		0			0
$669$	−183.328			−0.274033
$670$		0			0
$671$	−840.780			−1.25303
$672$		0			0
$673$		−	412.521i		−	0.612959i	−0.951877	−	0.306479i	$-0.900849\pi$
$673$		−	412.521i		−	0.612959i	0.951877	−	0.306479i	$-0.0991510\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	591.622i		−	0.873888i	−0.899489	−	0.436944i	$-0.856061\pi$
$677$		−	591.622i		−	0.873888i	0.899489	−	0.436944i	$-0.143939\pi$
$678$		0			0
$679$	−1243.23			−1.83098
$680$		0			0
$681$			599.247i			0.879952i
$682$		0			0
$683$	−948.264			−1.38838			−0.694190	−	0.719792i	$-0.744237\pi$
$683$	−948.264			−1.38838			−0.694190	+	0.719792i	$0.744237\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−1182.35			−1.72103
$688$		0			0
$689$	262.594			0.381124
$690$		0			0
$691$	342.214			0.495245			0.247622	−	0.968857i	$-0.420351\pi$
$691$	342.214			0.495245			0.247622	+	0.968857i	$0.420351\pi$
$692$		0			0
$693$		−	2458.31i		−	3.54735i
$694$		0			0
$695$		0			0
$696$		0			0
$697$			305.924i			0.438915i
$698$		0			0
$699$		−	14.5266i		−	0.0207819i
$700$		0			0
$701$	−705.285			−1.00611			−0.503056	−	0.864254i	$-0.667791\pi$
$701$	−705.285			−1.00611			−0.503056	+	0.864254i	$0.667791\pi$
$702$		0			0
$703$			75.2153i			0.106992i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	1182.87			1.67309
$708$		0			0
$709$			238.378i			0.336218i	0.985768	+	0.168109i	$0.0537660\pi$
$709$			238.378i			0.336218i	−0.985768	+	0.168109i	$0.946234\pi$
$710$		0			0
$711$		−	343.641i		−	0.483320i
$712$		0			0
$713$	1046.70	+	793.013i	1.46802	+	1.11222i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−21.2305			−0.0296102
$718$		0			0
$719$			569.195i			0.791648i	0.918326	+	0.395824i	$0.129541\pi$
$719$			569.195i			0.791648i	−0.918326	+	0.395824i	$0.870459\pi$
$720$		0			0
$721$	−1131.55			−1.56942
$722$		0			0
$723$	1680.87			2.32485
$724$		0			0
$725$		0			0
$726$		0			0
$727$	151.968			0.209034			0.104517	−	0.994523i	$-0.466670\pi$
$727$	151.968			0.209034			0.104517	+	0.994523i	$0.466670\pi$
$728$		0			0
$729$	−617.166			−0.846592
$730$		0			0
$731$	−1147.06			−1.56916
$732$		0			0
$733$	90.8457			0.123937			0.0619684	−	0.998078i	$-0.480262\pi$
$733$	90.8457			0.123937			0.0619684	+	0.998078i	$0.480262\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	601.343i		−	0.815933i
$738$		0			0
$739$			218.954i			0.296284i	0.988966	+	0.148142i	$0.0473293\pi$
$739$			218.954i			0.296284i	−0.988966	+	0.148142i	$0.952671\pi$
$740$		0			0
$741$	230.084			0.310505
$742$		0			0
$743$			622.440i			0.837739i	0.908046	+	0.418870i	$0.137574\pi$
$743$			622.440i			0.837739i	−0.908046	+	0.418870i	$0.862426\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$			1759.76i			2.35577i
$748$		0			0
$749$	−1657.19			−2.21253
$750$		0			0
$751$	−885.884			−1.17961			−0.589803	−	0.807547i	$-0.700795\pi$
$751$	−885.884			−1.17961			−0.589803	+	0.807547i	$0.700795\pi$
$752$		0			0
$753$	1133.13			1.50482
$754$		0			0
$755$		0			0
$756$		0			0
$757$		−	503.464i		−	0.665079i	−0.943089	−	0.332539i	$-0.892095\pi$
$757$		−	503.464i		−	0.665079i	0.943089	−	0.332539i	$-0.107905\pi$
$758$		0			0
$759$			3246.97i			4.27796i
$760$		0			0
$761$		−	164.495i		−	0.216156i	−0.994142	−	0.108078i	$-0.965530\pi$
$761$		−	164.495i		−	0.216156i	0.994142	−	0.108078i	$-0.0344696\pi$
$762$		0			0
$763$	−224.992			−0.294878
$764$		0			0
$765$		0			0
$766$		0			0
$767$			137.878i			0.179763i
$768$		0			0
$769$	7.25285			0.00943154			0.00471577	−	0.999989i	$-0.498499\pi$
$769$	7.25285			0.00943154			0.00471577	+	0.999989i	$0.498499\pi$
$770$		0			0
$771$		−	1943.09i		−	2.52022i
$772$		0			0
$773$		−	1148.70i		−	1.48603i	−0.669275	−	0.743015i	$-0.733395\pi$
$773$		−	1148.70i		−	1.48603i	0.669275	−	0.743015i	$-0.266605\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	345.324			0.444432
$778$		0			0
$779$	208.140			0.267188
$780$		0			0
$781$		−	930.401i		−	1.19130i
$782$		0			0
$783$	1994.99			2.54788
$784$		0			0
$785$		0			0
$786$		0			0
$787$		−	1085.59i		−	1.37940i	−0.724096	−	0.689699i	$-0.757743\pi$
$787$		−	1085.59i		−	1.37940i	0.724096	−	0.689699i	$-0.242257\pi$
$788$		0			0
$789$	−2215.31			−2.80775
$790$		0			0
$791$	−843.372			−1.06621
$792$		0			0
$793$	−254.755			−0.321254
$794$		0			0
$795$		0			0
$796$		0			0
$797$			510.042i			0.639953i	0.947426	+	0.319976i	$0.103675\pi$
$797$			510.042i			0.639953i	−0.947426	+	0.319976i	$0.896325\pi$
$798$		0			0
$799$			1109.39i			1.38847i
$800$		0			0
$801$		−	1712.69i		−	2.13819i
$802$		0			0
$803$	−412.261			−0.513401
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−755.242			−0.935864
$808$		0			0
$809$			470.140i			0.581137i	0.956854	+	0.290569i	$0.0938445\pi$
$809$			470.140i			0.581137i	−0.956854	+	0.290569i	$0.906156\pi$
$810$		0			0
$811$	1485.18			1.83129			0.915646	−	0.401985i	$-0.131679\pi$
$811$	1485.18			1.83129			0.915646	+	0.401985i	$0.131679\pi$
$812$		0			0
$813$	−279.564			−0.343868
$814$		0			0
$815$		0			0
$816$		0			0
$817$			780.417i			0.955223i
$818$		0			0
$819$		−	744.863i		−	0.909479i
$820$		0			0
$821$			1200.10i			1.46176i	0.682508	+	0.730878i	$0.260889\pi$
$821$			1200.10i			1.46176i	−0.682508	+	0.730878i	$0.739111\pi$
$822$		0			0
$823$			492.390i			0.598287i	0.954208	+	0.299144i	$0.0967010\pi$
$823$			492.390i			0.598287i	−0.954208	+	0.299144i	$0.903299\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	1400.13i		−	1.69302i	−0.532374	−	0.846509i	$-0.678700\pi$
$827$		−	1400.13i		−	1.69302i	0.532374	−	0.846509i	$-0.321300\pi$
$828$		0			0
$829$		−	1581.78i		−	1.90806i	−0.299719	−	0.954028i	$-0.596893\pi$
$829$		−	1581.78i		−	1.90806i	0.299719	−	0.954028i	$-0.403107\pi$
$830$		0			0
$831$	−1537.96			−1.85074
$832$		0			0
$833$		−	273.428i		−	0.328245i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−1709.40	−	1295.10i	−2.04230	−	1.54731i
$838$		0			0
$839$	290.663			0.346440			0.173220	−	0.984883i	$-0.444583\pi$
$839$	290.663			0.346440			0.173220	+	0.984883i	$0.444583\pi$
$840$		0			0
$841$	9.41065			0.0111898
$842$		0			0
$843$			25.1723i			0.0298603i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	588.580			0.694899
$848$		0			0
$849$			803.426i			0.946321i
$850$		0			0
$851$	−321.616			−0.377927
$852$		0			0
$853$	322.666			0.378272			0.189136	−	0.981951i	$-0.439431\pi$
$853$	322.666			0.378272			0.189136	+	0.981951i	$0.439431\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−172.075			−0.200788			−0.100394	−	0.994948i	$-0.532010\pi$
$857$	−172.075			−0.200788			−0.100394	+	0.994948i	$0.532010\pi$
$858$		0			0
$859$			256.271i			0.298337i	0.988812	+	0.149168i	$0.0476596\pi$
$859$			256.271i			0.298337i	−0.988812	+	0.149168i	$0.952340\pi$
$860$		0			0
$861$		−	955.597i		−	1.10987i
$862$		0			0
$863$		−	880.247i		−	1.01998i	−0.860179	−	0.509992i	$-0.829648\pi$
$863$		−	880.247i		−	1.01998i	0.860179	−	0.509992i	$-0.170352\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		−	425.277i		−	0.490515i
$868$		0			0
$869$	221.527			0.254921
$870$		0			0
$871$		−	182.206i		−	0.209191i
$872$		0			0
$873$	−3250.07			−3.72288
$874$		0			0
$875$		0			0
$876$		0			0
$877$	1244.58			1.41913			0.709567	−	0.704638i	$-0.248890\pi$
$877$	1244.58			1.41913			0.709567	+	0.704638i	$0.248890\pi$
$878$		0			0
$879$			2687.08i			3.05697i
$880$		0			0
$881$			1339.98i			1.52098i	0.649350	+	0.760489i	$0.275041\pi$
$881$			1339.98i			1.52098i	−0.649350	+	0.760489i	$0.724959\pi$
$882$		0			0
$883$			1231.61i			1.39480i	0.716684	+	0.697398i	$0.245659\pi$
$883$			1231.61i			1.39480i	−0.716684	+	0.697398i	$0.754341\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	634.586			0.715429			0.357715	−	0.933831i	$-0.383556\pi$
$887$	634.586			0.715429			0.357715	+	0.933831i	$0.383556\pi$
$888$		0			0
$889$			98.3893i			0.110674i
$890$		0			0
$891$		−	2615.32i		−	2.93526i
$892$		0			0
$893$	754.788			0.845228
$894$		0			0
$895$		0			0
$896$		0			0
$897$			983.825i			1.09679i
$898$		0			0
$899$	712.546	+	539.848i	0.792599	+	0.600498i
$900$		0			0
$901$	909.557			1.00950
$902$		0			0
$903$	3583.00			3.96788
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−955.958			−1.05398			−0.526989	−	0.849872i	$-0.676679\pi$
$907$	−955.958			−1.05398			−0.526989	+	0.849872i	$0.676679\pi$
$908$		0			0
$909$	3092.28			3.40185
$910$		0			0
$911$			516.824i			0.567315i	0.958926	+	0.283657i	$0.0915479\pi$
$911$			516.824i			0.567315i	−0.958926	+	0.283657i	$0.908452\pi$
$912$		0			0
$913$	−1134.42			−1.24252
$914$		0			0
$915$		0			0
$916$		0			0
$917$	529.416			0.577335
$918$		0			0
$919$	1773.59			1.92991			0.964954	−	0.262419i	$-0.0845202\pi$
$919$	1773.59			1.92991			0.964954	+	0.262419i	$0.0845202\pi$
$920$		0			0
$921$		−	1789.74i		−	1.94326i
$922$		0			0
$923$		−	281.910i		−	0.305428i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−2958.12			−3.19106
$928$		0			0
$929$		−	1035.10i		−	1.11421i	−0.830441	−	0.557106i	$-0.811912\pi$
$929$		−	1035.10i		−	1.11421i	0.830441	−	0.557106i	$-0.188088\pi$
$930$		0			0
$931$	−186.031			−0.199818
$932$		0			0
$933$			90.7430i			0.0972594i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	199.308			0.212709			0.106354	−	0.994328i	$-0.466082\pi$
$937$	199.308			0.212709			0.106354	+	0.994328i	$0.466082\pi$
$938$		0			0
$939$	3260.49			3.47230
$940$		0			0
$941$			921.737i			0.979530i	0.871855	+	0.489765i	$0.162917\pi$
$941$			921.737i			0.979530i	−0.871855	+	0.489765i	$0.837083\pi$
$942$		0			0
$943$			889.992i			0.943788i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			498.328i			0.526218i	0.964766	+	0.263109i	$0.0847479\pi$
$947$			498.328i			0.526218i	−0.964766	+	0.263109i	$0.915252\pi$
$948$		0			0
$949$	−124.914			−0.131627
$950$		0			0
$951$			404.124i			0.424946i
$952$		0			0
$953$			470.381i			0.493579i	0.969069	+	0.246790i	$0.0793757\pi$
$953$			470.381i			0.493579i	−0.969069	+	0.246790i	$0.920624\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$			2210.39i			2.30971i
$958$		0			0
$959$		−	2109.57i		−	2.19976i
$960$		0			0
$961$	−260.088	−	925.135i	−0.270643	−	0.962680i
$962$		0			0
$963$	−4332.23			−4.49868
$964$		0			0
$965$		0			0
$966$		0			0
$967$			1157.79i			1.19730i	0.801011	+	0.598649i	$0.204296\pi$
$967$			1157.79i			1.19730i	−0.801011	+	0.598649i	$0.795704\pi$
$968$		0			0
$969$	796.949			0.822445
$970$		0			0
$971$	−1560.93			−1.60755			−0.803774	−	0.594935i	$-0.797178\pi$
$971$	−1560.93			−1.60755			−0.803774	+	0.594935i	$0.797178\pi$
$972$		0			0
$973$			1451.46i			1.49174i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−840.627			−0.860417			−0.430208	−	0.902730i	$-0.641560\pi$
$977$	−840.627			−0.860417			−0.430208	+	0.902730i	$0.641560\pi$
$978$		0			0
$979$	1104.08			1.12776
$980$		0			0
$981$	−588.176			−0.599568
$982$		0			0
$983$		−	1379.41i		−	1.40326i	−0.712539	−	0.701632i	$-0.752455\pi$
$983$		−	1379.41i		−	1.40326i	0.712539	−	0.701632i	$-0.247545\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		−	3465.33i		−	3.51098i
$988$		0			0
$989$	−3337.01			−3.37413
$990$		0			0
$991$			1898.42i			1.91566i	0.287331	+	0.957831i	$0.407232\pi$
$991$			1898.42i			1.91566i	−0.287331	+	0.957831i	$0.592768\pi$
$992$		0			0
$993$	−460.857			−0.464105
$994$		0			0
$995$		0			0
$996$		0			0
$997$	948.866			0.951722			0.475861	−	0.879521i	$-0.342137\pi$
$997$	948.866			0.951722			0.475861	+	0.879521i	$0.342137\pi$
$998$		0			0
$999$	525.242			0.525768

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3100.3.d.g.1301.1	yes	22
5.2	odd	4		3100.3.f.d.1549.1		44
5.3	odd	4		3100.3.f.d.1549.44		44
5.4	even	2		3100.3.d.f.1301.22	yes	22
31.30	odd	2	inner	3100.3.d.g.1301.22	yes	22
155.92	even	4		3100.3.f.d.1549.43		44
155.123	even	4		3100.3.f.d.1549.2		44
155.154	odd	2		3100.3.d.f.1301.1	✓	22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3100.3.d.f.1301.1	✓	22	155.154	odd	2
3100.3.d.f.1301.22	yes	22	5.4	even	2
3100.3.d.g.1301.1	yes	22	1.1	even	1	trivial
3100.3.d.g.1301.22	yes	22	31.30	odd	2	inner
3100.3.f.d.1549.1		44	5.2	odd	4
3100.3.f.d.1549.2		44	155.123	even	4
3100.3.f.d.1549.43		44	155.92	even	4
3100.3.f.d.1549.44		44	5.3	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-5.52468i q^{3} -8.23275 q^{7} -21.5221 q^{9} +O(q^{10})\)	\(q-5.52468i q^{3} -8.23275 q^{7} -21.5221 q^{9} -13.8742i q^{11} -4.20384i q^{13} -14.5610i q^{17} -9.90677 q^{19} +45.4833i q^{21} -42.3608i q^{23} +69.1808i q^{27} -28.8373i q^{29} +(-18.7205 + 24.7092i) q^{31} -76.6504 q^{33} -7.59231i q^{37} -23.2249 q^{39} -21.0098 q^{41} -78.7761i q^{43} -76.1891 q^{47} +18.7781 q^{49} -80.4449 q^{51} +62.4653i q^{53} +54.7318i q^{57} -32.7981 q^{59} -60.6004i q^{61} +177.186 q^{63} +43.3426 q^{67} -234.030 q^{69} +67.0600 q^{71} -29.7143i q^{73} +114.223i q^{77} +15.9669i q^{79} +188.503 q^{81} -81.7651i q^{83} -159.317 q^{87} +79.5780i q^{89} +34.6092i q^{91} +(136.510 + 103.425i) q^{93} +151.011 q^{97} +298.602i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q + 4 q^{7} - 72 q^{9}+O(q^{10}) \) 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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