LMFDB - Embedded newform 22.4.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [22,4,Mod(1,22)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(22, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("22.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 22 = 2 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	22.a (trivial)

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:	yes
Analytic conductor:	$1.29804202013$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	22.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-2.00000 q^{2} +4.00000 q^{3} +4.00000 q^{4} +14.0000 q^{5} -8.00000 q^{6} -8.00000 q^{7} -8.00000 q^{8} -11.0000 q^{9} +O(q^{10})$

\(q-2.00000 q^{2} +4.00000 q^{3} +4.00000 q^{4} +14.0000 q^{5} -8.00000 q^{6} -8.00000 q^{7} -8.00000 q^{8} -11.0000 q^{9} -28.0000 q^{10} -11.0000 q^{11} +16.0000 q^{12} -50.0000 q^{13} +16.0000 q^{14} +56.0000 q^{15} +16.0000 q^{16} +130.000 q^{17} +22.0000 q^{18} -108.000 q^{19} +56.0000 q^{20} -32.0000 q^{21} +22.0000 q^{22} -96.0000 q^{23} -32.0000 q^{24} +71.0000 q^{25} +100.000 q^{26} -152.000 q^{27} -32.0000 q^{28} +142.000 q^{29} -112.000 q^{30} +40.0000 q^{31} -32.0000 q^{32} -44.0000 q^{33} -260.000 q^{34} -112.000 q^{35} -44.0000 q^{36} +382.000 q^{37} +216.000 q^{38} -200.000 q^{39} -112.000 q^{40} -118.000 q^{41} +64.0000 q^{42} +220.000 q^{43} -44.0000 q^{44} -154.000 q^{45} +192.000 q^{46} +520.000 q^{47} +64.0000 q^{48} -279.000 q^{49} -142.000 q^{50} +520.000 q^{51} -200.000 q^{52} +238.000 q^{53} +304.000 q^{54} -154.000 q^{55} +64.0000 q^{56} -432.000 q^{57} -284.000 q^{58} -852.000 q^{59} +224.000 q^{60} +190.000 q^{61} -80.0000 q^{62} +88.0000 q^{63} +64.0000 q^{64} -700.000 q^{65} +88.0000 q^{66} -12.0000 q^{67} +520.000 q^{68} -384.000 q^{69} +224.000 q^{70} -112.000 q^{71} +88.0000 q^{72} -6.00000 q^{73} -764.000 q^{74} +284.000 q^{75} -432.000 q^{76} +88.0000 q^{77} +400.000 q^{78} +304.000 q^{79} +224.000 q^{80} -311.000 q^{81} +236.000 q^{82} +820.000 q^{83} -128.000 q^{84} +1820.00 q^{85} -440.000 q^{86} +568.000 q^{87} +88.0000 q^{88} +202.000 q^{89} +308.000 q^{90} +400.000 q^{91} -384.000 q^{92} +160.000 q^{93} -1040.00 q^{94} -1512.00 q^{95} -128.000 q^{96} -1406.00 q^{97} +558.000 q^{98} +121.000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

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 $

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	4.00000	0.769800	0.384900	−	0.922958i	$-0.374236\pi$
$3$	4.00000	0.769800	0.384900	+	0.922958i	$0.374236\pi$
$4$	4.00000	0.500000
$5$	14.0000	1.25220	0.626099	−	0.779744i	$-0.284651\pi$
$5$	14.0000	1.25220	0.626099	+	0.779744i	$0.284651\pi$
$6$	−8.00000	−0.544331
$7$	−8.00000	−0.431959	−0.215980	−	0.976398i	$-0.569295\pi$
$7$	−8.00000	−0.431959	−0.215980	+	0.976398i	$0.569295\pi$
$8$	−8.00000	−0.353553
$9$	−11.0000	−0.407407
$10$	−28.0000	−0.885438
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	16.0000	0.384900
$13$	−50.0000	−1.06673	−0.533366	−	0.845885i	$-0.679073\pi$
$13$	−50.0000	−1.06673	−0.533366	+	0.845885i	$0.679073\pi$
$14$	16.0000	0.305441
$15$	56.0000	0.963943
$16$	16.0000	0.250000
$17$	130.000	1.85468	0.927342	−	0.374215i	$-0.122088\pi$
$17$	130.000	1.85468	0.927342	+	0.374215i	$0.122088\pi$
$18$	22.0000	0.288081
$19$	−108.000	−1.30405	−0.652024	−	0.758199i	$-0.726080\pi$
$19$	−108.000	−1.30405	−0.652024	+	0.758199i	$0.726080\pi$
$20$	56.0000	0.626099
$21$	−32.0000	−0.332522
$22$	22.0000	0.213201
$23$	−96.0000	−0.870321	−0.435161	−	0.900353i	$-0.643308\pi$
$23$	−96.0000	−0.870321	−0.435161	+	0.900353i	$0.643308\pi$
$24$	−32.0000	−0.272166
$25$	71.0000	0.568000
$26$	100.000	0.754293
$27$	−152.000	−1.08342
$28$	−32.0000	−0.215980
$29$	142.000	0.909267	0.454633	−	0.890679i	$-0.349770\pi$
$29$	142.000	0.909267	0.454633	+	0.890679i	$0.349770\pi$
$30$	−112.000	−0.681610
$31$	40.0000	0.231749	0.115874	−	0.993264i	$-0.463033\pi$
$31$	40.0000	0.231749	0.115874	+	0.993264i	$0.463033\pi$
$32$	−32.0000	−0.176777
$33$	−44.0000	−0.232104
$34$	−260.000	−1.31146
$35$	−112.000	−0.540899
$36$	−44.0000	−0.203704
$37$	382.000	1.69731	0.848654	−	0.528948i	$-0.177413\pi$
$37$	382.000	1.69731	0.848654	+	0.528948i	$0.177413\pi$
$38$	216.000	0.922101
$39$	−200.000	−0.821170
$40$	−112.000	−0.442719
$41$	−118.000	−0.449476	−0.224738	−	0.974419i	$-0.572153\pi$
$41$	−118.000	−0.449476	−0.224738	+	0.974419i	$0.572153\pi$
$42$	64.0000	0.235129
$43$	220.000	0.780225	0.390113	−	0.920767i	$-0.372436\pi$
$43$	220.000	0.780225	0.390113	+	0.920767i	$0.372436\pi$
$44$	−44.0000	−0.150756
$45$	−154.000	−0.510155
$46$	192.000	0.615410
$47$	520.000	1.61383	0.806913	−	0.590671i	$-0.201137\pi$
$47$	520.000	1.61383	0.806913	+	0.590671i	$0.201137\pi$
$48$	64.0000	0.192450
$49$	−279.000	−0.813411
$50$	−142.000	−0.401637
$51$	520.000	1.42774
$52$	−200.000	−0.533366
$53$	238.000	0.616827	0.308413	−	0.951252i	$-0.400202\pi$
$53$	238.000	0.616827	0.308413	+	0.951252i	$0.400202\pi$
$54$	304.000	0.766096
$55$	−154.000	−0.377552
$56$	64.0000	0.152721
$57$	−432.000	−1.00386
$58$	−284.000	−0.642949
$59$	−852.000	−1.88002	−0.940008	−	0.341152i	$-0.889183\pi$
$59$	−852.000	−1.88002	−0.940008	+	0.341152i	$0.889183\pi$
$60$	224.000	0.481971
$61$	190.000	0.398803	0.199402	−	0.979918i	$-0.436100\pi$
$61$	190.000	0.398803	0.199402	+	0.979918i	$0.436100\pi$
$62$	−80.0000	−0.163871
$63$	88.0000	0.175983
$64$	64.0000	0.125000
$65$	−700.000	−1.33576
$66$	88.0000	0.164122
$67$	−12.0000	−0.0218811	−0.0109405	−	0.999940i	$-0.503483\pi$
$67$	−12.0000	−0.0218811	−0.0109405	+	0.999940i	$0.503483\pi$
$68$	520.000	0.927342
$69$	−384.000	−0.669973
$70$	224.000	0.382473
$71$	−112.000	−0.187211	−0.0936053	−	0.995609i	$-0.529839\pi$
$71$	−112.000	−0.187211	−0.0936053	+	0.995609i	$0.529839\pi$
$72$	88.0000	0.144040
$73$	−6.00000	−0.00961982	−0.00480991	−	0.999988i	$-0.501531\pi$
$73$	−6.00000	−0.00961982	−0.00480991	+	0.999988i	$0.501531\pi$
$74$	−764.000	−1.20018
$75$	284.000	0.437247
$76$	−432.000	−0.652024
$77$	88.0000	0.130241
$78$	400.000	0.580655
$79$	304.000	0.432945	0.216473	−	0.976289i	$-0.430545\pi$
$79$	304.000	0.432945	0.216473	+	0.976289i	$0.430545\pi$
$80$	224.000	0.313050
$81$	−311.000	−0.426612
$82$	236.000	0.317827
$83$	820.000	1.08442	0.542209	−	0.840244i	$-0.317588\pi$
$83$	820.000	1.08442	0.542209	+	0.840244i	$0.317588\pi$
$84$	−128.000	−0.166261
$85$	1820.00	2.32243
$86$	−440.000	−0.551703
$87$	568.000	0.699954
$88$	88.0000	0.106600
$89$	202.000	0.240584	0.120292	−	0.992739i	$-0.461617\pi$
$89$	202.000	0.240584	0.120292	+	0.992739i	$0.461617\pi$
$90$	308.000	0.360734
$91$	400.000	0.460785
$92$	−384.000	−0.435161
$93$	160.000	0.178400
$94$	−1040.00	−1.14115
$95$	−1512.00	−1.63293
$96$	−128.000	−0.136083
$97$	−1406.00	−1.47173	−0.735864	−	0.677129i	$-0.763224\pi$
$97$	−1406.00	−1.47173	−0.735864	+	0.677129i	$0.763224\pi$
$98$	558.000	0.575168
$99$	121.000	0.122838
$100$	284.000	0.284000
$101$	−634.000	−0.624608	−0.312304	−	0.949982i	$-0.601101\pi$
$101$	−634.000	−0.624608	−0.312304	+	0.949982i	$0.601101\pi$
$102$	−1040.00	−1.00956
$103$	304.000	0.290816	0.145408	−	0.989372i	$-0.453551\pi$
$103$	304.000	0.290816	0.145408	+	0.989372i	$0.453551\pi$
$104$	400.000	0.377146
$105$	−448.000	−0.416384
$106$	−476.000	−0.436162
$107$	780.000	0.704724	0.352362	−	0.935864i	$-0.385379\pi$
$107$	780.000	0.704724	0.352362	+	0.935864i	$0.385379\pi$
$108$	−608.000	−0.541711
$109$	−370.000	−0.325134	−0.162567	−	0.986698i	$-0.551977\pi$
$109$	−370.000	−0.325134	−0.162567	+	0.986698i	$0.551977\pi$
$110$	308.000	0.266970
$111$	1528.00	1.30659
$112$	−128.000	−0.107990
$113$	−558.000	−0.464533	−0.232266	−	0.972652i	$-0.574614\pi$
$113$	−558.000	−0.464533	−0.232266	+	0.972652i	$0.574614\pi$
$114$	864.000	0.709833
$115$	−1344.00	−1.08981
$116$	568.000	0.454633
$117$	550.000	0.434594
$118$	1704.00	1.32937
$119$	−1040.00	−0.801148
$120$	−448.000	−0.340805
$121$	121.000	0.0909091
$122$	−380.000	−0.281997
$123$	−472.000	−0.346007
$124$	160.000	0.115874
$125$	−756.000	−0.540950
$126$	−176.000	−0.124439
$127$	−112.000	−0.0782551	−0.0391275	−	0.999234i	$-0.512458\pi$
$127$	−112.000	−0.0782551	−0.0391275	+	0.999234i	$0.512458\pi$
$128$	−128.000	−0.0883883
$129$	880.000	0.600618
$130$	1400.00	0.944524
$131$	1716.00	1.14449	0.572243	−	0.820084i	$-0.306073\pi$
$131$	1716.00	1.14449	0.572243	+	0.820084i	$0.306073\pi$
$132$	−176.000	−0.116052
$133$	864.000	0.563295
$134$	24.0000	0.0154723
$135$	−2128.00	−1.35666
$136$	−1040.00	−0.655730
$137$	−134.000	−0.0835649	−0.0417825	−	0.999127i	$-0.513304\pi$
$137$	−134.000	−0.0835649	−0.0417825	+	0.999127i	$0.513304\pi$
$138$	768.000	0.473743
$139$	−132.000	−0.0805474	−0.0402737	−	0.999189i	$-0.512823\pi$
$139$	−132.000	−0.0805474	−0.0402737	+	0.999189i	$0.512823\pi$
$140$	−448.000	−0.270449
$141$	2080.00	1.24232
$142$	224.000	0.132378
$143$	550.000	0.321632
$144$	−176.000	−0.101852
$145$	1988.00	1.13858
$146$	12.0000	0.00680224
$147$	−1116.00	−0.626164
$148$	1528.00	0.848654
$149$	−394.000	−0.216629	−0.108315	−	0.994117i	$-0.534545\pi$
$149$	−394.000	−0.216629	−0.108315	+	0.994117i	$0.534545\pi$
$150$	−568.000	−0.309180
$151$	−3016.00	−1.62542	−0.812711	−	0.582668i	$-0.802009\pi$
$151$	−3016.00	−1.62542	−0.812711	+	0.582668i	$0.802009\pi$
$152$	864.000	0.461050
$153$	−1430.00	−0.755612
$154$	−176.000	−0.0920941
$155$	560.000	0.290195
$156$	−800.000	−0.410585
$157$	1398.00	0.710653	0.355327	−	0.934742i	$-0.384370\pi$
$157$	1398.00	0.710653	0.355327	+	0.934742i	$0.384370\pi$
$158$	−608.000	−0.306138
$159$	952.000	0.474833
$160$	−448.000	−0.221359
$161$	768.000	0.375943
$162$	622.000	0.301660
$163$	−3788.00	−1.82024	−0.910120	−	0.414345i	$-0.864011\pi$
$163$	−3788.00	−1.82024	−0.910120	+	0.414345i	$0.864011\pi$
$164$	−472.000	−0.224738
$165$	−616.000	−0.290640
$166$	−1640.00	−0.766799
$167$	1368.00	0.633886	0.316943	−	0.948445i	$-0.397344\pi$
$167$	1368.00	0.633886	0.316943	+	0.948445i	$0.397344\pi$
$168$	256.000	0.117564
$169$	303.000	0.137915
$170$	−3640.00	−1.64221
$171$	1188.00	0.531279
$172$	880.000	0.390113
$173$	−882.000	−0.387614	−0.193807	−	0.981040i	$-0.562084\pi$
$173$	−882.000	−0.387614	−0.193807	+	0.981040i	$0.562084\pi$
$174$	−1136.00	−0.494942
$175$	−568.000	−0.245353
$176$	−176.000	−0.0753778
$177$	−3408.00	−1.44724
$178$	−404.000	−0.170118
$179$	2628.00	1.09735	0.548676	−	0.836035i	$-0.315132\pi$
$179$	2628.00	1.09735	0.548676	+	0.836035i	$0.315132\pi$
$180$	−616.000	−0.255077
$181$	3950.00	1.62211	0.811053	−	0.584973i	$-0.198895\pi$
$181$	3950.00	1.62211	0.811053	+	0.584973i	$0.198895\pi$
$182$	−800.000	−0.325824
$183$	760.000	0.306999
$184$	768.000	0.307705
$185$	5348.00	2.12537
$186$	−320.000	−0.126148
$187$	−1430.00	−0.559208
$188$	2080.00	0.806913
$189$	1216.00	0.467995
$190$	3024.00	1.15465
$191$	3400.00	1.28804	0.644019	−	0.765009i	$-0.277266\pi$
$191$	3400.00	1.28804	0.644019	+	0.765009i	$0.277266\pi$
$192$	256.000	0.0962250
$193$	−4942.00	−1.84318	−0.921588	−	0.388170i	$-0.873107\pi$
$193$	−4942.00	−1.84318	−0.921588	+	0.388170i	$0.873107\pi$
$194$	2812.00	1.04067
$195$	−2800.00	−1.02827
$196$	−1116.00	−0.406706
$197$	1446.00	0.522961	0.261480	−	0.965209i	$-0.415789\pi$
$197$	1446.00	0.522961	0.261480	+	0.965209i	$0.415789\pi$
$198$	−242.000	−0.0868596
$199$	−1456.00	−0.518659	−0.259329	−	0.965789i	$-0.583502\pi$
$199$	−1456.00	−0.518659	−0.259329	+	0.965789i	$0.583502\pi$
$200$	−568.000	−0.200818
$201$	−48.0000	−0.0168441
$202$	1268.00	0.441664
$203$	−1136.00	−0.392766
$204$	2080.00	0.713868
$205$	−1652.00	−0.562833
$206$	−608.000	−0.205638
$207$	1056.00	0.354575
$208$	−800.000	−0.266683
$209$	1188.00	0.393185
$210$	896.000	0.294428
$211$	−2124.00	−0.692996	−0.346498	−	0.938051i	$-0.612629\pi$
$211$	−2124.00	−0.692996	−0.346498	+	0.938051i	$0.612629\pi$
$212$	952.000	0.308413
$213$	−448.000	−0.144115
$214$	−1560.00	−0.498315
$215$	3080.00	0.976997
$216$	1216.00	0.383048
$217$	−320.000	−0.100106
$218$	740.000	0.229904
$219$	−24.0000	−0.00740534
$220$	−616.000	−0.188776
$221$	−6500.00	−1.97845
$222$	−3056.00	−0.923898
$223$	−2536.00	−0.761539	−0.380769	−	0.924670i	$-0.624341\pi$
$223$	−2536.00	−0.761539	−0.380769	+	0.924670i	$0.624341\pi$
$224$	256.000	0.0763604
$225$	−781.000	−0.231407
$226$	1116.00	0.328474
$227$	−1356.00	−0.396480	−0.198240	−	0.980154i	$-0.563522\pi$
$227$	−1356.00	−0.396480	−0.198240	+	0.980154i	$0.563522\pi$
$228$	−1728.00	−0.501928
$229$	1438.00	0.414959	0.207480	−	0.978239i	$-0.433474\pi$
$229$	1438.00	0.414959	0.207480	+	0.978239i	$0.433474\pi$
$230$	2688.00	0.770615
$231$	352.000	0.100259
$232$	−1136.00	−0.321474
$233$	−5094.00	−1.43227	−0.716135	−	0.697962i	$-0.754091\pi$
$233$	−5094.00	−1.43227	−0.716135	+	0.697962i	$0.754091\pi$
$234$	−1100.00	−0.307304
$235$	7280.00	2.02083
$236$	−3408.00	−0.940008
$237$	1216.00	0.333281
$238$	2080.00	0.566497
$239$	−5504.00	−1.48964	−0.744820	−	0.667265i	$-0.767465\pi$
$239$	−5504.00	−1.48964	−0.744820	+	0.667265i	$0.767465\pi$
$240$	896.000	0.240986
$241$	−2430.00	−0.649502	−0.324751	−	0.945799i	$-0.605281\pi$
$241$	−2430.00	−0.649502	−0.324751	+	0.945799i	$0.605281\pi$
$242$	−242.000	−0.0642824
$243$	2860.00	0.755017
$244$	760.000	0.199402
$245$	−3906.00	−1.01855
$246$	944.000	0.244664
$247$	5400.00	1.39107
$248$	−320.000	−0.0819356
$249$	3280.00	0.834785
$250$	1512.00	0.382509
$251$	604.000	0.151889	0.0759445	−	0.997112i	$-0.475803\pi$
$251$	604.000	0.151889	0.0759445	+	0.997112i	$0.475803\pi$
$252$	352.000	0.0879917
$253$	1056.00	0.262412
$254$	224.000	0.0553347
$255$	7280.00	1.78781
$256$	256.000	0.0625000
$257$	1602.00	0.388833	0.194416	−	0.980919i	$-0.437719\pi$
$257$	1602.00	0.388833	0.194416	+	0.980919i	$0.437719\pi$
$258$	−1760.00	−0.424701
$259$	−3056.00	−0.733168
$260$	−2800.00	−0.667879
$261$	−1562.00	−0.370442
$262$	−3432.00	−0.809274
$263$	2120.00	0.497052	0.248526	−	0.968625i	$-0.420054\pi$
$263$	2120.00	0.497052	0.248526	+	0.968625i	$0.420054\pi$
$264$	352.000	0.0820610
$265$	3332.00	0.772389
$266$	−1728.00	−0.398310
$267$	808.000	0.185201
$268$	−48.0000	−0.0109405
$269$	790.000	0.179060	0.0895300	−	0.995984i	$-0.471463\pi$
$269$	790.000	0.179060	0.0895300	+	0.995984i	$0.471463\pi$
$270$	4256.00	0.959303
$271$	2032.00	0.455480	0.227740	−	0.973722i	$-0.426866\pi$
$271$	2032.00	0.455480	0.227740	+	0.973722i	$0.426866\pi$
$272$	2080.00	0.463671
$273$	1600.00	0.354712
$274$	268.000	0.0590893
$275$	−781.000	−0.171258
$276$	−1536.00	−0.334987
$277$	−6058.00	−1.31404	−0.657022	−	0.753872i	$-0.728184\pi$
$277$	−6058.00	−1.31404	−0.657022	+	0.753872i	$0.728184\pi$
$278$	264.000	0.0569556
$279$	−440.000	−0.0944162
$280$	896.000	0.191237
$281$	970.000	0.205927	0.102963	−	0.994685i	$-0.467168\pi$
$281$	970.000	0.205927	0.102963	+	0.994685i	$0.467168\pi$
$282$	−4160.00	−0.878455
$283$	5308.00	1.11494	0.557470	−	0.830197i	$-0.311772\pi$
$283$	5308.00	1.11494	0.557470	+	0.830197i	$0.311772\pi$
$284$	−448.000	−0.0936053
$285$	−6048.00	−1.25703
$286$	−1100.00	−0.227428
$287$	944.000	0.194155
$288$	352.000	0.0720201
$289$	11987.0	2.43985
$290$	−3976.00	−0.805099
$291$	−5624.00	−1.13294
$292$	−24.0000	−0.00480991
$293$	−7114.00	−1.41844	−0.709222	−	0.704985i	$-0.750954\pi$
$293$	−7114.00	−1.41844	−0.709222	+	0.704985i	$0.750954\pi$
$294$	2232.00	0.442765
$295$	−11928.0	−2.35415
$296$	−3056.00	−0.600089
$297$	1672.00	0.326664
$298$	788.000	0.153180
$299$	4800.00	0.928399
$300$	1136.00	0.218623
$301$	−1760.00	−0.337026
$302$	6032.00	1.14935
$303$	−2536.00	−0.480823
$304$	−1728.00	−0.326012
$305$	2660.00	0.499381
$306$	2860.00	0.534298
$307$	4388.00	0.815754	0.407877	−	0.913037i	$-0.366269\pi$
$307$	4388.00	0.815754	0.407877	+	0.913037i	$0.366269\pi$
$308$	352.000	0.0651203
$309$	1216.00	0.223870
$310$	−1120.00	−0.205199
$311$	−8752.00	−1.59576	−0.797878	−	0.602818i	$-0.794044\pi$
$311$	−8752.00	−1.59576	−0.797878	+	0.602818i	$0.794044\pi$
$312$	1600.00	0.290327
$313$	4458.00	0.805051	0.402526	−	0.915409i	$-0.368132\pi$
$313$	4458.00	0.805051	0.402526	+	0.915409i	$0.368132\pi$
$314$	−2796.00	−0.502508
$315$	1232.00	0.220366
$316$	1216.00	0.216473
$317$	2614.00	0.463145	0.231572	−	0.972818i	$-0.425613\pi$
$317$	2614.00	0.463145	0.231572	+	0.972818i	$0.425613\pi$
$318$	−1904.00	−0.335758
$319$	−1562.00	−0.274154
$320$	896.000	0.156525
$321$	3120.00	0.542497
$322$	−1536.00	−0.265832
$323$	−14040.0	−2.41860
$324$	−1244.00	−0.213306
$325$	−3550.00	−0.605903
$326$	7576.00	1.28710
$327$	−1480.00	−0.250288
$328$	944.000	0.158914
$329$	−4160.00	−0.697107
$330$	1232.00	0.205513
$331$	4172.00	0.692791	0.346396	−	0.938089i	$-0.387406\pi$
$331$	4172.00	0.692791	0.346396	+	0.938089i	$0.387406\pi$
$332$	3280.00	0.542209
$333$	−4202.00	−0.691496
$334$	−2736.00	−0.448225
$335$	−168.000	−0.0273995
$336$	−512.000	−0.0831306
$337$	−4942.00	−0.798836	−0.399418	−	0.916769i	$-0.630788\pi$
$337$	−4942.00	−0.798836	−0.399418	+	0.916769i	$0.630788\pi$
$338$	−606.000	−0.0975209
$339$	−2232.00	−0.357598
$340$	7280.00	1.16122
$341$	−440.000	−0.0698749
$342$	−2376.00	−0.375671
$343$	4976.00	0.783320
$344$	−1760.00	−0.275851
$345$	−5376.00	−0.838939
$346$	1764.00	0.274084
$347$	11676.0	1.80634	0.903171	−	0.429281i	$-0.141233\pi$
$347$	11676.0	1.80634	0.903171	+	0.429281i	$0.141233\pi$
$348$	2272.00	0.349977
$349$	10894.0	1.67090	0.835448	−	0.549570i	$-0.185208\pi$
$349$	10894.0	1.67090	0.835448	+	0.549570i	$0.185208\pi$
$350$	1136.00	0.173491
$351$	7600.00	1.15572
$352$	352.000	0.0533002
$353$	9378.00	1.41400	0.706998	−	0.707216i	$-0.250049\pi$
$353$	9378.00	1.41400	0.706998	+	0.707216i	$0.250049\pi$
$354$	6816.00	1.02335
$355$	−1568.00	−0.234425
$356$	808.000	0.120292
$357$	−4160.00	−0.616724
$358$	−5256.00	−0.775945
$359$	4264.00	0.626867	0.313434	−	0.949610i	$-0.398521\pi$
$359$	4264.00	0.626867	0.313434	+	0.949610i	$0.398521\pi$
$360$	1232.00	0.180367
$361$	4805.00	0.700539
$362$	−7900.00	−1.14700
$363$	484.000	0.0699819
$364$	1600.00	0.230392
$365$	−84.0000	−0.0120459
$366$	−1520.00	−0.217081
$367$	−2056.00	−0.292431	−0.146216	−	0.989253i	$-0.546709\pi$
$367$	−2056.00	−0.292431	−0.146216	+	0.989253i	$0.546709\pi$
$368$	−1536.00	−0.217580
$369$	1298.00	0.183120
$370$	−10696.0	−1.50286
$371$	−1904.00	−0.266444
$372$	640.000	0.0892001
$373$	8086.00	1.12246	0.561230	−	0.827660i	$-0.310329\pi$
$373$	8086.00	1.12246	0.561230	+	0.827660i	$0.310329\pi$
$374$	2860.00	0.395420
$375$	−3024.00	−0.416423
$376$	−4160.00	−0.570573
$377$	−7100.00	−0.969943
$378$	−2432.00	−0.330922
$379$	−9812.00	−1.32984	−0.664919	−	0.746915i	$-0.731534\pi$
$379$	−9812.00	−1.32984	−0.664919	+	0.746915i	$0.731534\pi$
$380$	−6048.00	−0.816463
$381$	−448.000	−0.0602408
$382$	−6800.00	−0.910781
$383$	−744.000	−0.0992601	−0.0496301	−	0.998768i	$-0.515804\pi$
$383$	−744.000	−0.0992601	−0.0496301	+	0.998768i	$0.515804\pi$
$384$	−512.000	−0.0680414
$385$	1232.00	0.163087
$386$	9884.00	1.30332
$387$	−2420.00	−0.317870
$388$	−5624.00	−0.735864
$389$	1294.00	0.168659	0.0843296	−	0.996438i	$-0.473125\pi$
$389$	1294.00	0.168659	0.0843296	+	0.996438i	$0.473125\pi$
$390$	5600.00	0.727095
$391$	−12480.0	−1.61417
$392$	2232.00	0.287584
$393$	6864.00	0.881025
$394$	−2892.00	−0.369789
$395$	4256.00	0.542133
$396$	484.000	0.0614190
$397$	−10362.0	−1.30996	−0.654980	−	0.755646i	$-0.727323\pi$
$397$	−10362.0	−1.30996	−0.654980	+	0.755646i	$0.727323\pi$
$398$	2912.00	0.366747
$399$	3456.00	0.433625
$400$	1136.00	0.142000
$401$	2706.00	0.336986	0.168493	−	0.985703i	$-0.446110\pi$
$401$	2706.00	0.336986	0.168493	+	0.985703i	$0.446110\pi$
$402$	96.0000	0.0119106
$403$	−2000.00	−0.247214
$404$	−2536.00	−0.312304
$405$	−4354.00	−0.534202
$406$	2272.00	0.277728
$407$	−4202.00	−0.511758
$408$	−4160.00	−0.504781
$409$	−7366.00	−0.890526	−0.445263	−	0.895400i	$-0.646890\pi$
$409$	−7366.00	−0.890526	−0.445263	+	0.895400i	$0.646890\pi$
$410$	3304.00	0.397983
$411$	−536.000	−0.0643283
$412$	1216.00	0.145408
$413$	6816.00	0.812091
$414$	−2112.00	−0.250723
$415$	11480.0	1.35791
$416$	1600.00	0.188573
$417$	−528.000	−0.0620054
$418$	−2376.00	−0.278024
$419$	−1020.00	−0.118927	−0.0594633	−	0.998230i	$-0.518939\pi$
$419$	−1020.00	−0.118927	−0.0594633	+	0.998230i	$0.518939\pi$
$420$	−1792.00	−0.208192
$421$	11886.0	1.37598	0.687991	−	0.725719i	$-0.258493\pi$
$421$	11886.0	1.37598	0.687991	+	0.725719i	$0.258493\pi$
$422$	4248.00	0.490022
$423$	−5720.00	−0.657484
$424$	−1904.00	−0.218081
$425$	9230.00	1.05346
$426$	896.000	0.101905
$427$	−1520.00	−0.172267
$428$	3120.00	0.352362
$429$	2200.00	0.247592
$430$	−6160.00	−0.690841
$431$	−1888.00	−0.211002	−0.105501	−	0.994419i	$-0.533645\pi$
$431$	−1888.00	−0.211002	−0.105501	+	0.994419i	$0.533645\pi$
$432$	−2432.00	−0.270856
$433$	7218.00	0.801097	0.400548	−	0.916276i	$-0.368820\pi$
$433$	7218.00	0.801097	0.400548	+	0.916276i	$0.368820\pi$
$434$	640.000	0.0707857
$435$	7952.00	0.876481
$436$	−1480.00	−0.162567
$437$	10368.0	1.13494
$438$	48.0000	0.00523637
$439$	−16760.0	−1.82212	−0.911061	−	0.412273i	$-0.864735\pi$
$439$	−16760.0	−1.82212	−0.911061	+	0.412273i	$0.864735\pi$
$440$	1232.00	0.133485
$441$	3069.00	0.331390
$442$	13000.0	1.39897
$443$	5148.00	0.552119	0.276060	−	0.961141i	$-0.410971\pi$
$443$	5148.00	0.552119	0.276060	+	0.961141i	$0.410971\pi$
$444$	6112.00	0.653294
$445$	2828.00	0.301259
$446$	5072.00	0.538489
$447$	−1576.00	−0.166761
$448$	−512.000	−0.0539949
$449$	−7998.00	−0.840644	−0.420322	−	0.907375i	$-0.638083\pi$
$449$	−7998.00	−0.840644	−0.420322	+	0.907375i	$0.638083\pi$
$450$	1562.00	0.163630
$451$	1298.00	0.135522
$452$	−2232.00	−0.232266
$453$	−12064.0	−1.25125
$454$	2712.00	0.280353
$455$	5600.00	0.576994
$456$	3456.00	0.354917
$457$	−10166.0	−1.04058	−0.520290	−	0.853989i	$-0.674176\pi$
$457$	−10166.0	−1.04058	−0.520290	+	0.853989i	$0.674176\pi$
$458$	−2876.00	−0.293421
$459$	−19760.0	−2.00941
$460$	−5376.00	−0.544907
$461$	−10626.0	−1.07354	−0.536770	−	0.843728i	$-0.680356\pi$
$461$	−10626.0	−1.07354	−0.536770	+	0.843728i	$0.680356\pi$
$462$	−704.000	−0.0708940
$463$	14776.0	1.48315	0.741576	−	0.670869i	$-0.234079\pi$
$463$	14776.0	1.48315	0.741576	+	0.670869i	$0.234079\pi$
$464$	2272.00	0.227317
$465$	2240.00	0.223393
$466$	10188.0	1.01277
$467$	3076.00	0.304797	0.152399	−	0.988319i	$-0.451300\pi$
$467$	3076.00	0.304797	0.152399	+	0.988319i	$0.451300\pi$
$468$	2200.00	0.217297
$469$	96.0000	0.00945174
$470$	−14560.0	−1.42894
$471$	5592.00	0.547061
$472$	6816.00	0.664686
$473$	−2420.00	−0.235247
$474$	−2432.00	−0.235666
$475$	−7668.00	−0.740699
$476$	−4160.00	−0.400574
$477$	−2618.00	−0.251300
$478$	11008.0	1.05334
$479$	14704.0	1.40259	0.701297	−	0.712869i	$-0.252605\pi$
$479$	14704.0	1.40259	0.701297	+	0.712869i	$0.252605\pi$
$480$	−1792.00	−0.170403
$481$	−19100.0	−1.81057
$482$	4860.00	0.459267
$483$	3072.00	0.289401
$484$	484.000	0.0454545
$485$	−19684.0	−1.84290
$486$	−5720.00	−0.533878
$487$	6688.00	0.622304	0.311152	−	0.950360i	$-0.399285\pi$
$487$	6688.00	0.622304	0.311152	+	0.950360i	$0.399285\pi$
$488$	−1520.00	−0.140998
$489$	−15152.0	−1.40122
$490$	7812.00	0.720225
$491$	−756.000	−0.0694864	−0.0347432	−	0.999396i	$-0.511061\pi$
$491$	−756.000	−0.0694864	−0.0347432	+	0.999396i	$0.511061\pi$
$492$	−1888.00	−0.173003
$493$	18460.0	1.68640
$494$	−10800.0	−0.983634
$495$	1694.00	0.153817
$496$	640.000	0.0579372
$497$	896.000	0.0808674
$498$	−6560.00	−0.590282
$499$	7060.00	0.633365	0.316682	−	0.948532i	$-0.397431\pi$
$499$	7060.00	0.633365	0.316682	+	0.948532i	$0.397431\pi$
$500$	−3024.00	−0.270475
$501$	5472.00	0.487966
$502$	−1208.00	−0.107402
$503$	−488.000	−0.0432581	−0.0216291	−	0.999766i	$-0.506885\pi$
$503$	−488.000	−0.0432581	−0.0216291	+	0.999766i	$0.506885\pi$
$504$	−704.000	−0.0622195
$505$	−8876.00	−0.782132
$506$	−2112.00	−0.185553
$507$	1212.00	0.106167
$508$	−448.000	−0.0391275
$509$	−3386.00	−0.294856	−0.147428	−	0.989073i	$-0.547100\pi$
$509$	−3386.00	−0.294856	−0.147428	+	0.989073i	$0.547100\pi$
$510$	−14560.0	−1.26417
$511$	48.0000	0.00415537
$512$	−512.000	−0.0441942
$513$	16416.0	1.41283
$514$	−3204.00	−0.274946
$515$	4256.00	0.364159
$516$	3520.00	0.300309
$517$	−5720.00	−0.486587
$518$	6112.00	0.518428
$519$	−3528.00	−0.298385
$520$	5600.00	0.472262
$521$	1722.00	0.144803	0.0724013	−	0.997376i	$-0.476934\pi$
$521$	1722.00	0.144803	0.0724013	+	0.997376i	$0.476934\pi$
$522$	3124.00	0.261942
$523$	−2692.00	−0.225073	−0.112536	−	0.993648i	$-0.535897\pi$
$523$	−2692.00	−0.225073	−0.112536	+	0.993648i	$0.535897\pi$
$524$	6864.00	0.572243
$525$	−2272.00	−0.188873
$526$	−4240.00	−0.351469
$527$	5200.00	0.429821
$528$	−704.000	−0.0580259
$529$	−2951.00	−0.242541
$530$	−6664.00	−0.546162
$531$	9372.00	0.765933
$532$	3456.00	0.281648
$533$	5900.00	0.479470
$534$	−1616.00	−0.130957
$535$	10920.0	0.882454
$536$	96.0000	0.00773614
$537$	10512.0	0.844742
$538$	−1580.00	−0.126615
$539$	3069.00	0.245253
$540$	−8512.00	−0.678330
$541$	8574.00	0.681377	0.340689	−	0.940176i	$-0.389340\pi$
$541$	8574.00	0.681377	0.340689	+	0.940176i	$0.389340\pi$
$542$	−4064.00	−0.322073
$543$	15800.0	1.24870
$544$	−4160.00	−0.327865
$545$	−5180.00	−0.407132
$546$	−3200.00	−0.250819
$547$	−2492.00	−0.194790	−0.0973951	−	0.995246i	$-0.531051\pi$
$547$	−2492.00	−0.194790	−0.0973951	+	0.995246i	$0.531051\pi$
$548$	−536.000	−0.0417825
$549$	−2090.00	−0.162475
$550$	1562.00	0.121098
$551$	−15336.0	−1.18573
$552$	3072.00	0.236871
$553$	−2432.00	−0.187015
$554$	12116.0	0.929169
$555$	21392.0	1.63611
$556$	−528.000	−0.0402737
$557$	−226.000	−0.0171920	−0.00859599	−	0.999963i	$-0.502736\pi$
$557$	−226.000	−0.0171920	−0.00859599	+	0.999963i	$0.502736\pi$
$558$	880.000	0.0667623
$559$	−11000.0	−0.832291
$560$	−1792.00	−0.135225
$561$	−5720.00	−0.430479
$562$	−1940.00	−0.145612
$563$	−12924.0	−0.967463	−0.483731	−	0.875216i	$-0.660719\pi$
$563$	−12924.0	−0.967463	−0.483731	+	0.875216i	$0.660719\pi$
$564$	8320.00	0.621162
$565$	−7812.00	−0.581687
$566$	−10616.0	−0.788381
$567$	2488.00	0.184279
$568$	896.000	0.0661890
$569$	8090.00	0.596046	0.298023	−	0.954559i	$-0.403673\pi$
$569$	8090.00	0.596046	0.298023	+	0.954559i	$0.403673\pi$
$570$	12096.0	0.888852
$571$	18188.0	1.33300	0.666501	−	0.745504i	$-0.267791\pi$
$571$	18188.0	1.33300	0.666501	+	0.745504i	$0.267791\pi$
$572$	2200.00	0.160816
$573$	13600.0	0.991533
$574$	−1888.00	−0.137288
$575$	−6816.00	−0.494342
$576$	−704.000	−0.0509259
$577$	−9086.00	−0.655555	−0.327777	−	0.944755i	$-0.606300\pi$
$577$	−9086.00	−0.655555	−0.327777	+	0.944755i	$0.606300\pi$
$578$	−23974.0	−1.72524
$579$	−19768.0	−1.41888
$580$	7952.00	0.569291
$581$	−6560.00	−0.468425
$582$	11248.0	0.801108
$583$	−2618.00	−0.185980
$584$	48.0000	0.00340112
$585$	7700.00	0.544198
$586$	14228.0	1.00299
$587$	−2980.00	−0.209536	−0.104768	−	0.994497i	$-0.533410\pi$
$587$	−2980.00	−0.209536	−0.104768	+	0.994497i	$0.533410\pi$
$588$	−4464.00	−0.313082
$589$	−4320.00	−0.302211
$590$	23856.0	1.66464
$591$	5784.00	0.402575
$592$	6112.00	0.424327
$593$	15122.0	1.04719	0.523597	−	0.851966i	$-0.324590\pi$
$593$	15122.0	1.04719	0.523597	+	0.851966i	$0.324590\pi$
$594$	−3344.00	−0.230987
$595$	−14560.0	−1.00320
$596$	−1576.00	−0.108315
$597$	−5824.00	−0.399264
$598$	−9600.00	−0.656477
$599$	−21504.0	−1.46683	−0.733414	−	0.679783i	$-0.762074\pi$
$599$	−21504.0	−1.46683	−0.733414	+	0.679783i	$0.762074\pi$
$600$	−2272.00	−0.154590
$601$	−7446.00	−0.505372	−0.252686	−	0.967548i	$-0.581314\pi$
$601$	−7446.00	−0.505372	−0.252686	+	0.967548i	$0.581314\pi$
$602$	3520.00	0.238313
$603$	132.000	0.00891452
$604$	−12064.0	−0.812711
$605$	1694.00	0.113836
$606$	5072.00	0.339993
$607$	3728.00	0.249283	0.124642	−	0.992202i	$-0.460222\pi$
$607$	3728.00	0.249283	0.124642	+	0.992202i	$0.460222\pi$
$608$	3456.00	0.230525
$609$	−4544.00	−0.302352
$610$	−5320.00	−0.353116
$611$	−26000.0	−1.72152
$612$	−5720.00	−0.377806
$613$	−19402.0	−1.27837	−0.639184	−	0.769054i	$-0.720728\pi$
$613$	−19402.0	−1.27837	−0.639184	+	0.769054i	$0.720728\pi$
$614$	−8776.00	−0.576825
$615$	−6608.00	−0.433269
$616$	−704.000	−0.0460470
$617$	−14854.0	−0.969205	−0.484603	−	0.874734i	$-0.661036\pi$
$617$	−14854.0	−0.969205	−0.484603	+	0.874734i	$0.661036\pi$
$618$	−2432.00	−0.158300
$619$	17020.0	1.10516	0.552578	−	0.833461i	$-0.313644\pi$
$619$	17020.0	1.10516	0.552578	+	0.833461i	$0.313644\pi$
$620$	2240.00	0.145098
$621$	14592.0	0.942926
$622$	17504.0	1.12837
$623$	−1616.00	−0.103922
$624$	−3200.00	−0.205293
$625$	−19459.0	−1.24538
$626$	−8916.00	−0.569257
$627$	4752.00	0.302674
$628$	5592.00	0.355327
$629$	49660.0	3.14797
$630$	−2464.00	−0.155822
$631$	−9696.00	−0.611714	−0.305857	−	0.952077i	$-0.598943\pi$
$631$	−9696.00	−0.611714	−0.305857	+	0.952077i	$0.598943\pi$
$632$	−2432.00	−0.153069
$633$	−8496.00	−0.533469
$634$	−5228.00	−0.327493
$635$	−1568.00	−0.0979908
$636$	3808.00	0.237417
$637$	13950.0	0.867691
$638$	3124.00	0.193856
$639$	1232.00	0.0762710
$640$	−1792.00	−0.110680
$641$	−24702.0	−1.52211	−0.761053	−	0.648689i	$-0.775317\pi$
$641$	−24702.0	−1.52211	−0.761053	+	0.648689i	$0.775317\pi$
$642$	−6240.00	−0.383603
$643$	6788.00	0.416318	0.208159	−	0.978095i	$-0.433253\pi$
$643$	6788.00	0.416318	0.208159	+	0.978095i	$0.433253\pi$
$644$	3072.00	0.187972
$645$	12320.0	0.752092
$646$	28080.0	1.71021
$647$	14960.0	0.909024	0.454512	−	0.890741i	$-0.349814\pi$
$647$	14960.0	0.909024	0.454512	+	0.890741i	$0.349814\pi$
$648$	2488.00	0.150830
$649$	9372.00	0.566846
$650$	7100.00	0.428438
$651$	−1280.00	−0.0770617
$652$	−15152.0	−0.910120
$653$	11670.0	0.699360	0.349680	−	0.936869i	$-0.386290\pi$
$653$	11670.0	0.699360	0.349680	+	0.936869i	$0.386290\pi$
$654$	2960.00	0.176980
$655$	24024.0	1.43312
$656$	−1888.00	−0.112369
$657$	66.0000	0.00391919
$658$	8320.00	0.492929
$659$	532.000	0.0314473	0.0157237	−	0.999876i	$-0.494995\pi$
$659$	532.000	0.0314473	0.0157237	+	0.999876i	$0.494995\pi$
$660$	−2464.00	−0.145320
$661$	26334.0	1.54958	0.774791	−	0.632217i	$-0.217855\pi$
$661$	26334.0	1.54958	0.774791	+	0.632217i	$0.217855\pi$
$662$	−8344.00	−0.489877
$663$	−26000.0	−1.52301
$664$	−6560.00	−0.383400
$665$	12096.0	0.705358
$666$	8404.00	0.488962
$667$	−13632.0	−0.791354
$668$	5472.00	0.316943
$669$	−10144.0	−0.586233
$670$	336.000	0.0193743
$671$	−2090.00	−0.120244
$672$	1024.00	0.0587822
$673$	−4942.00	−0.283061	−0.141531	−	0.989934i	$-0.545202\pi$
$673$	−4942.00	−0.283061	−0.141531	+	0.989934i	$0.545202\pi$
$674$	9884.00	0.564863
$675$	−10792.0	−0.615384
$676$	1212.00	0.0689577
$677$	23798.0	1.35101	0.675503	−	0.737357i	$-0.263926\pi$
$677$	23798.0	1.35101	0.675503	+	0.737357i	$0.263926\pi$
$678$	4464.00	0.252860
$679$	11248.0	0.635727
$680$	−14560.0	−0.821104
$681$	−5424.00	−0.305210
$682$	880.000	0.0494090
$683$	16940.0	0.949035	0.474518	−	0.880246i	$-0.342623\pi$
$683$	16940.0	0.949035	0.474518	+	0.880246i	$0.342623\pi$
$684$	4752.00	0.265639
$685$	−1876.00	−0.104640
$686$	−9952.00	−0.553891
$687$	5752.00	0.319436
$688$	3520.00	0.195056
$689$	−11900.0	−0.657988
$690$	10752.0	0.593220
$691$	14932.0	0.822055	0.411028	−	0.911623i	$-0.365170\pi$
$691$	14932.0	0.822055	0.411028	+	0.911623i	$0.365170\pi$
$692$	−3528.00	−0.193807
$693$	−968.000	−0.0530610
$694$	−23352.0	−1.27728
$695$	−1848.00	−0.100861
$696$	−4544.00	−0.247471
$697$	−15340.0	−0.833635
$698$	−21788.0	−1.18150
$699$	−20376.0	−1.10256
$700$	−2272.00	−0.122676
$701$	8670.00	0.467135	0.233567	−	0.972341i	$-0.424960\pi$
$701$	8670.00	0.467135	0.233567	+	0.972341i	$0.424960\pi$
$702$	−15200.0	−0.817218
$703$	−41256.0	−2.21337
$704$	−704.000	−0.0376889
$705$	29120.0	1.55563
$706$	−18756.0	−0.999846
$707$	5072.00	0.269805
$708$	−13632.0	−0.723619
$709$	−5954.00	−0.315384	−0.157692	−	0.987488i	$-0.550405\pi$
$709$	−5954.00	−0.315384	−0.157692	+	0.987488i	$0.550405\pi$
$710$	3136.00	0.165763
$711$	−3344.00	−0.176385
$712$	−1616.00	−0.0850592
$713$	−3840.00	−0.201696
$714$	8320.00	0.436090
$715$	7700.00	0.402746
$716$	10512.0	0.548676
$717$	−22016.0	−1.14673
$718$	−8528.00	−0.443262
$719$	−14504.0	−0.752306	−0.376153	−	0.926558i	$-0.622753\pi$
$719$	−14504.0	−0.752306	−0.376153	+	0.926558i	$0.622753\pi$
$720$	−2464.00	−0.127539
$721$	−2432.00	−0.125621
$722$	−9610.00	−0.495356
$723$	−9720.00	−0.499987
$724$	15800.0	0.811053
$725$	10082.0	0.516464
$726$	−968.000	−0.0494846
$727$	19600.0	0.999895	0.499948	−	0.866056i	$-0.333353\pi$
$727$	19600.0	0.999895	0.499948	+	0.866056i	$0.333353\pi$
$728$	−3200.00	−0.162912
$729$	19837.0	1.00782
$730$	168.000	0.00851775
$731$	28600.0	1.44707
$732$	3040.00	0.153499
$733$	−1730.00	−0.0871746	−0.0435873	−	0.999050i	$-0.513879\pi$
$733$	−1730.00	−0.0871746	−0.0435873	+	0.999050i	$0.513879\pi$
$734$	4112.00	0.206780
$735$	−15624.0	−0.784082
$736$	3072.00	0.153852
$737$	132.000	0.00659740
$738$	−2596.00	−0.129485
$739$	38756.0	1.92918	0.964589	−	0.263758i	$-0.0849619\pi$
$739$	38756.0	1.92918	0.964589	+	0.263758i	$0.0849619\pi$
$740$	21392.0	1.06268
$741$	21600.0	1.07084
$742$	3808.00	0.188404
$743$	−15688.0	−0.774612	−0.387306	−	0.921951i	$-0.626594\pi$
$743$	−15688.0	−0.774612	−0.387306	+	0.921951i	$0.626594\pi$
$744$	−1280.00	−0.0630740
$745$	−5516.00	−0.271263
$746$	−16172.0	−0.793698
$747$	−9020.00	−0.441800
$748$	−5720.00	−0.279604
$749$	−6240.00	−0.304412
$750$	6048.00	0.294456
$751$	−30920.0	−1.50238	−0.751190	−	0.660086i	$-0.770520\pi$
$751$	−30920.0	−1.50238	−0.751190	+	0.660086i	$0.770520\pi$
$752$	8320.00	0.403456
$753$	2416.00	0.116924
$754$	14200.0	0.685853
$755$	−42224.0	−2.03535
$756$	4864.00	0.233997
$757$	−30594.0	−1.46890	−0.734450	−	0.678662i	$-0.762560\pi$
$757$	−30594.0	−1.46890	−0.734450	+	0.678662i	$0.762560\pi$
$758$	19624.0	0.940337
$759$	4224.00	0.202005
$760$	12096.0	0.577326
$761$	−8198.00	−0.390509	−0.195254	−	0.980753i	$-0.562553\pi$
$761$	−8198.00	−0.390509	−0.195254	+	0.980753i	$0.562553\pi$
$762$	896.000	0.0425967
$763$	2960.00	0.140445
$764$	13600.0	0.644019
$765$	−20020.0	−0.946176
$766$	1488.00	0.0701875
$767$	42600.0	2.00547
$768$	1024.00	0.0481125
$769$	4402.00	0.206424	0.103212	−	0.994659i	$-0.467088\pi$
$769$	4402.00	0.206424	0.103212	+	0.994659i	$0.467088\pi$
$770$	−2464.00	−0.115320
$771$	6408.00	0.299324
$772$	−19768.0	−0.921588
$773$	−5378.00	−0.250237	−0.125119	−	0.992142i	$-0.539931\pi$
$773$	−5378.00	−0.250237	−0.125119	+	0.992142i	$0.539931\pi$
$774$	4840.00	0.224768
$775$	2840.00	0.131633
$776$	11248.0	0.520335
$777$	−12224.0	−0.564393
$778$	−2588.00	−0.119260
$779$	12744.0	0.586138
$780$	−11200.0	−0.514134
$781$	1232.00	0.0564461
$782$	24960.0	1.14139
$783$	−21584.0	−0.985120
$784$	−4464.00	−0.203353
$785$	19572.0	0.889879
$786$	−13728.0	−0.622979
$787$	−6076.00	−0.275205	−0.137602	−	0.990488i	$-0.543940\pi$
$787$	−6076.00	−0.275205	−0.137602	+	0.990488i	$0.543940\pi$
$788$	5784.00	0.261480
$789$	8480.00	0.382631
$790$	−8512.00	−0.383346
$791$	4464.00	0.200659
$792$	−968.000	−0.0434298
$793$	−9500.00	−0.425416
$794$	20724.0	0.926281
$795$	13328.0	0.594585
$796$	−5824.00	−0.259329
$797$	−5450.00	−0.242219	−0.121110	−	0.992639i	$-0.538645\pi$
$797$	−5450.00	−0.242219	−0.121110	+	0.992639i	$0.538645\pi$
$798$	−6912.00	−0.306619
$799$	67600.0	2.99314
$800$	−2272.00	−0.100409
$801$	−2222.00	−0.0980156
$802$	−5412.00	−0.238285
$803$	66.0000	0.00290048
$804$	−192.000	−0.00842204
$805$	10752.0	0.470756
$806$	4000.00	0.174806
$807$	3160.00	0.137840
$808$	5072.00	0.220832
$809$	30538.0	1.32714	0.663572	−	0.748113i	$-0.269040\pi$
$809$	30538.0	1.32714	0.663572	+	0.748113i	$0.269040\pi$
$810$	8708.00	0.377738
$811$	−16164.0	−0.699870	−0.349935	−	0.936774i	$-0.613796\pi$
$811$	−16164.0	−0.699870	−0.349935	+	0.936774i	$0.613796\pi$
$812$	−4544.00	−0.196383
$813$	8128.00	0.350629
$814$	8404.00	0.361867
$815$	−53032.0	−2.27930
$816$	8320.00	0.356934
$817$	−23760.0	−1.01745
$818$	14732.0	0.629697
$819$	−4400.00	−0.187727
$820$	−6608.00	−0.281416
$821$	−28394.0	−1.20701	−0.603506	−	0.797358i	$-0.706230\pi$
$821$	−28394.0	−1.20701	−0.603506	+	0.797358i	$0.706230\pi$
$822$	1072.00	0.0454870
$823$	28800.0	1.21981	0.609906	−	0.792474i	$-0.291207\pi$
$823$	28800.0	1.21981	0.609906	+	0.792474i	$0.291207\pi$
$824$	−2432.00	−0.102819
$825$	−3124.00	−0.131835
$826$	−13632.0	−0.574235
$827$	−4884.00	−0.205361	−0.102680	−	0.994714i	$-0.532742\pi$
$827$	−4884.00	−0.205361	−0.102680	+	0.994714i	$0.532742\pi$
$828$	4224.00	0.177288
$829$	−5514.00	−0.231012	−0.115506	−	0.993307i	$-0.536849\pi$
$829$	−5514.00	−0.231012	−0.115506	+	0.993307i	$0.536849\pi$
$830$	−22960.0	−0.960185
$831$	−24232.0	−1.01155
$832$	−3200.00	−0.133341
$833$	−36270.0	−1.50862
$834$	1056.00	0.0438445
$835$	19152.0	0.793751
$836$	4752.00	0.196593
$837$	−6080.00	−0.251082
$838$	2040.00	0.0840938
$839$	26592.0	1.09423	0.547114	−	0.837058i	$-0.315726\pi$
$839$	26592.0	1.09423	0.547114	+	0.837058i	$0.315726\pi$
$840$	3584.00	0.147214
$841$	−4225.00	−0.173234
$842$	−23772.0	−0.972966
$843$	3880.00	0.158522
$844$	−8496.00	−0.346498
$845$	4242.00	0.172697
$846$	11440.0	0.464912
$847$	−968.000	−0.0392690
$848$	3808.00	0.154207
$849$	21232.0	0.858281
$850$	−18460.0	−0.744909
$851$	−36672.0	−1.47720
$852$	−1792.00	−0.0720574
$853$	25350.0	1.01755	0.508773	−	0.860900i	$-0.330099\pi$
$853$	25350.0	1.01755	0.508773	+	0.860900i	$0.330099\pi$
$854$	3040.00	0.121811
$855$	16632.0	0.665266
$856$	−6240.00	−0.249157
$857$	−1606.00	−0.0640139	−0.0320070	−	0.999488i	$-0.510190\pi$
$857$	−1606.00	−0.0640139	−0.0320070	+	0.999488i	$0.510190\pi$
$858$	−4400.00	−0.175074
$859$	−47156.0	−1.87304	−0.936520	−	0.350613i	$-0.885973\pi$
$859$	−47156.0	−1.87304	−0.936520	+	0.350613i	$0.885973\pi$
$860$	12320.0	0.488498
$861$	3776.00	0.149461
$862$	3776.00	0.149201
$863$	24168.0	0.953289	0.476644	−	0.879096i	$-0.341853\pi$
$863$	24168.0	0.953289	0.476644	+	0.879096i	$0.341853\pi$
$864$	4864.00	0.191524
$865$	−12348.0	−0.485369
$866$	−14436.0	−0.566461
$867$	47948.0	1.87820
$868$	−1280.00	−0.0500530
$869$	−3344.00	−0.130538
$870$	−15904.0	−0.619766
$871$	600.000	0.0233412
$872$	2960.00	0.114952
$873$	15466.0	0.599593
$874$	−20736.0	−0.802524
$875$	6048.00	0.233668
$876$	−96.0000	−0.00370267
$877$	−29954.0	−1.15333	−0.576667	−	0.816979i	$-0.695647\pi$
$877$	−29954.0	−1.15333	−0.576667	+	0.816979i	$0.695647\pi$
$878$	33520.0	1.28843
$879$	−28456.0	−1.09192
$880$	−2464.00	−0.0943880
$881$	18898.0	0.722690	0.361345	−	0.932432i	$-0.382318\pi$
$881$	18898.0	0.722690	0.361345	+	0.932432i	$0.382318\pi$
$882$	−6138.00	−0.234328
$883$	−17548.0	−0.668785	−0.334393	−	0.942434i	$-0.608531\pi$
$883$	−17548.0	−0.668785	−0.334393	+	0.942434i	$0.608531\pi$
$884$	−26000.0	−0.989225
$885$	−47712.0	−1.81223
$886$	−10296.0	−0.390407
$887$	50952.0	1.92875	0.964375	−	0.264540i	$-0.0852201\pi$
$887$	50952.0	1.92875	0.964375	+	0.264540i	$0.0852201\pi$
$888$	−12224.0	−0.461949
$889$	896.000	0.0338030
$890$	−5656.00	−0.213022
$891$	3421.00	0.128628
$892$	−10144.0	−0.380769
$893$	−56160.0	−2.10450
$894$	3152.00	0.117918
$895$	36792.0	1.37410
$896$	1024.00	0.0381802
$897$	19200.0	0.714682
$898$	15996.0	0.594425
$899$	5680.00	0.210721
$900$	−3124.00	−0.115704
$901$	30940.0	1.14402
$902$	−2596.00	−0.0958285
$903$	−7040.00	−0.259442
$904$	4464.00	0.164237
$905$	55300.0	2.03120
$906$	24128.0	0.884767
$907$	−35028.0	−1.28234	−0.641172	−	0.767397i	$-0.721551\pi$
$907$	−35028.0	−1.28234	−0.641172	+	0.767397i	$0.721551\pi$
$908$	−5424.00	−0.198240
$909$	6974.00	0.254470
$910$	−11200.0	−0.407996
$911$	−12088.0	−0.439619	−0.219810	−	0.975543i	$-0.570544\pi$
$911$	−12088.0	−0.439619	−0.219810	+	0.975543i	$0.570544\pi$
$912$	−6912.00	−0.250964
$913$	−9020.00	−0.326964
$914$	20332.0	0.735802
$915$	10640.0	0.384424
$916$	5752.00	0.207480
$917$	−13728.0	−0.494371
$918$	39520.0	1.42087
$919$	13448.0	0.482708	0.241354	−	0.970437i	$-0.422409\pi$
$919$	13448.0	0.482708	0.241354	+	0.970437i	$0.422409\pi$
$920$	10752.0	0.385308
$921$	17552.0	0.627967
$922$	21252.0	0.759108
$923$	5600.00	0.199703
$924$	1408.00	0.0501297
$925$	27122.0	0.964071
$926$	−29552.0	−1.04875
$927$	−3344.00	−0.118480
$928$	−4544.00	−0.160737
$929$	22338.0	0.788898	0.394449	−	0.918918i	$-0.370936\pi$
$929$	22338.0	0.788898	0.394449	+	0.918918i	$0.370936\pi$
$930$	−4480.00	−0.157962
$931$	30132.0	1.06073
$932$	−20376.0	−0.716135
$933$	−35008.0	−1.22841
$934$	−6152.00	−0.215524
$935$	−20020.0	−0.700240
$936$	−4400.00	−0.153652
$937$	23962.0	0.835437	0.417718	−	0.908577i	$-0.362830\pi$
$937$	23962.0	0.835437	0.417718	+	0.908577i	$0.362830\pi$
$938$	−192.000	−0.00668339
$939$	17832.0	0.619729
$940$	29120.0	1.01041
$941$	−41394.0	−1.43401	−0.717006	−	0.697067i	$-0.754488\pi$
$941$	−41394.0	−1.43401	−0.717006	+	0.697067i	$0.754488\pi$
$942$	−11184.0	−0.386831
$943$	11328.0	0.391188
$944$	−13632.0	−0.470004
$945$	17024.0	0.586022
$946$	4840.00	0.166345
$947$	5620.00	0.192846	0.0964232	−	0.995340i	$-0.469260\pi$
$947$	5620.00	0.192846	0.0964232	+	0.995340i	$0.469260\pi$
$948$	4864.00	0.166641
$949$	300.000	0.0102618
$950$	15336.0	0.523753
$951$	10456.0	0.356529
$952$	8320.00	0.283249
$953$	−3942.00	−0.133992	−0.0669958	−	0.997753i	$-0.521341\pi$
$953$	−3942.00	−0.133992	−0.0669958	+	0.997753i	$0.521341\pi$
$954$	5236.00	0.177696
$955$	47600.0	1.61288
$956$	−22016.0	−0.744820
$957$	−6248.00	−0.211044
$958$	−29408.0	−0.991784
$959$	1072.00	0.0360966
$960$	3584.00	0.120493
$961$	−28191.0	−0.946293
$962$	38200.0	1.28027
$963$	−8580.00	−0.287110
$964$	−9720.00	−0.324751
$965$	−69188.0	−2.30802
$966$	−6144.00	−0.204638
$967$	−36328.0	−1.20810	−0.604048	−	0.796948i	$-0.706447\pi$
$967$	−36328.0	−1.20810	−0.604048	+	0.796948i	$0.706447\pi$
$968$	−968.000	−0.0321412
$969$	−56160.0	−1.86184
$970$	39368.0	1.30312
$971$	30284.0	1.00089	0.500443	−	0.865770i	$-0.333171\pi$
$971$	30284.0	1.00089	0.500443	+	0.865770i	$0.333171\pi$
$972$	11440.0	0.377508
$973$	1056.00	0.0347932
$974$	−13376.0	−0.440036
$975$	−14200.0	−0.466425
$976$	3040.00	0.0997008
$977$	−27694.0	−0.906868	−0.453434	−	0.891290i	$-0.649801\pi$
$977$	−27694.0	−0.906868	−0.453434	+	0.891290i	$0.649801\pi$
$978$	30304.0	0.990813
$979$	−2222.00	−0.0725387
$980$	−15624.0	−0.509276
$981$	4070.00	0.132462
$982$	1512.00	0.0491343
$983$	25232.0	0.818694	0.409347	−	0.912379i	$-0.365757\pi$
$983$	25232.0	0.818694	0.409347	+	0.912379i	$0.365757\pi$
$984$	3776.00	0.122332
$985$	20244.0	0.654850
$986$	−36920.0	−1.19247
$987$	−16640.0	−0.536633
$988$	21600.0	0.695534
$989$	−21120.0	−0.679046
$990$	−3388.00	−0.108765
$991$	30472.0	0.976766	0.488383	−	0.872629i	$-0.337587\pi$
$991$	30472.0	0.976766	0.488383	+	0.872629i	$0.337587\pi$
$992$	−1280.00	−0.0409678
$993$	16688.0	0.533311
$994$	−1792.00	−0.0571819
$995$	−20384.0	−0.649464
$996$	13120.0	0.417393
$997$	44678.0	1.41922	0.709612	−	0.704593i	$-0.248870\pi$
$997$	44678.0	1.41922	0.709612	+	0.704593i	$0.248870\pi$
$998$	−14120.0	−0.447857
$999$	−58064.0	−1.83890

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	22.4.a.b.1.1	✓	1
3.2	odd	2		198.4.a.d.1.1		1
4.3	odd	2		176.4.a.b.1.1		1
5.2	odd	4		550.4.b.b.199.1		2
5.3	odd	4		550.4.b.b.199.2		2
5.4	even	2		550.4.a.k.1.1		1
7.6	odd	2		1078.4.a.a.1.1		1
8.3	odd	2		704.4.a.i.1.1		1
8.5	even	2		704.4.a.d.1.1		1
11.2	odd	10		242.4.c.b.81.1		4
11.3	even	5		242.4.c.h.9.1		4
11.4	even	5		242.4.c.h.27.1		4
11.5	even	5		242.4.c.h.3.1		4
11.6	odd	10		242.4.c.b.3.1		4
11.7	odd	10		242.4.c.b.27.1		4
11.8	odd	10		242.4.c.b.9.1		4
11.9	even	5		242.4.c.h.81.1		4
11.10	odd	2		242.4.a.f.1.1		1
12.11	even	2		1584.4.a.b.1.1		1
33.32	even	2		2178.4.a.a.1.1		1
44.43	even	2		1936.4.a.g.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
22.4.a.b.1.1	✓	1	1.1	even	1	trivial
176.4.a.b.1.1		1	4.3	odd	2
198.4.a.d.1.1		1	3.2	odd	2
242.4.a.f.1.1		1	11.10	odd	2
242.4.c.b.3.1		4	11.6	odd	10
242.4.c.b.9.1		4	11.8	odd	10
242.4.c.b.27.1		4	11.7	odd	10
242.4.c.b.81.1		4	11.2	odd	10
242.4.c.h.3.1		4	11.5	even	5
242.4.c.h.9.1		4	11.3	even	5
242.4.c.h.27.1		4	11.4	even	5
242.4.c.h.81.1		4	11.9	even	5
550.4.a.k.1.1		1	5.4	even	2
550.4.b.b.199.1		2	5.2	odd	4
550.4.b.b.199.2		2	5.3	odd	4
704.4.a.d.1.1		1	8.5	even	2
704.4.a.i.1.1		1	8.3	odd	2
1078.4.a.a.1.1		1	7.6	odd	2
1584.4.a.b.1.1		1	12.11	even	2
1936.4.a.g.1.1		1	44.43	even	2
2178.4.a.a.1.1		1	33.32	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 22 = 2 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	22.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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