LMFDB - Embedded newform 2200.2.a.y.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2200,2,Mod(1,2200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2200, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("2200.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2200 = 2^{3} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2200.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:	$17.5670884447$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.54764.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 9x^{2} + 3x + 2 $ x^4 - x^3 - 9x^2 + 3x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 440)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.67673$ of defining polynomial
Character	$\chi$	$=$	2200.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.67673 q^{3} +4.38559 q^{7} +4.16490 q^{9} +1.00000 q^{11} +4.00000 q^{13} +5.87995 q^{17} -0.526486 q^{19} -11.7391 q^{21} +6.15025 q^{23} -3.11812 q^{27} -0.967873 q^{29} -9.60629 q^{31} -2.67673 q^{33} +1.76466 q^{37} -10.7069 q^{39} +2.50564 q^{41} -3.85911 q^{43} -4.91208 q^{47} +12.2334 q^{49} -15.7391 q^{51} -5.29767 q^{53} +1.40926 q^{57} -2.63841 q^{59} +8.96787 q^{61} +18.2655 q^{63} -7.97440 q^{67} -16.4626 q^{69} +13.8718 q^{71} +12.7712 q^{73} +4.38559 q^{77} -13.2768 q^{79} -4.14832 q^{81} -7.61901 q^{83} +2.59074 q^{87} +2.83510 q^{89} +17.5424 q^{91} +25.7135 q^{93} -10.4800 q^{97} +4.16490 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + q^{3} + q^{7} + 7 q^{9} + 4 q^{11} + 16 q^{13} + 7 q^{17} - 9 q^{19} - 7 q^{21} + 6 q^{23} + 13 q^{27} + 3 q^{29} - 15 q^{31} + q^{33} + 5 q^{37} + 4 q^{39} + 10 q^{41} + 8 q^{43} - 10 q^{47} + 9 q^{49}+ \cdots + 7 q^{99}+O(q^{100}) $ 4 * q + q^3 + q^7 + 7 * q^9 + 4 * q^11 + 16 * q^13 + 7 * q^17 - 9 * q^19 - 7 * q^21 + 6 * q^23 + 13 * q^27 + 3 * q^29 - 15 * q^31 + q^33 + 5 * q^37 + 4 * q^39 + 10 * q^41 + 8 * q^43 - 10 * q^47 + 9 * q^49 - 23 * q^51 + 5 * q^53 - 15 * q^57 + 6 * q^59 + 29 * q^61 + 40 * q^63 + 6 * q^67 - 30 * q^69 - q^71 + 18 * q^73 + q^77 - 20 * q^79 + 44 * q^81 + 26 * q^83 + 31 * q^87 + 21 * q^89 + 4 * q^91 + 25 * q^93 - 4 * q^97 + 7 * q^99

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
$3$	−2.67673	−1.54541	−0.772706	−	0.634764i	$-0.781097\pi$
$3$	−2.67673	−1.54541	−0.772706	+	0.634764i	$0.781097\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.38559	1.65760	0.828799	−	0.559546i	$-0.189025\pi$
$7$	4.38559	1.65760	0.828799	+	0.559546i	$0.189025\pi$
$8$	0	0
$9$	4.16490	1.38830
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	4.00000	1.10940	0.554700	−	0.832050i	$-0.312833\pi$
$13$	4.00000	1.10940	0.554700	+	0.832050i	$0.312833\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.87995	1.42610	0.713049	−	0.701114i	$-0.247314\pi$
$17$	5.87995	1.42610	0.713049	+	0.701114i	$0.247314\pi$
$18$	0	0
$19$	−0.526486	−0.120784	−0.0603920	−	0.998175i	$-0.519235\pi$
$19$	−0.526486	−0.120784	−0.0603920	+	0.998175i	$0.519235\pi$
$20$	0	0
$21$	−11.7391	−2.56167
$22$	0	0
$23$	6.15025	1.28242	0.641208	−	0.767367i	$-0.278434\pi$
$23$	6.15025	1.28242	0.641208	+	0.767367i	$0.278434\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−3.11812	−0.600083
$28$	0	0
$29$	−0.967873	−0.179730	−0.0898648	−	0.995954i	$-0.528643\pi$
$29$	−0.967873	−0.179730	−0.0898648	+	0.995954i	$0.528643\pi$
$30$	0	0
$31$	−9.60629	−1.72534	−0.862670	−	0.505767i	$-0.831209\pi$
$31$	−9.60629	−1.72534	−0.862670	+	0.505767i	$0.831209\pi$
$32$	0	0
$33$	−2.67673	−0.465959
$34$	0	0
$35$	0	0
$36$	0	0
$37$	1.76466	0.290108	0.145054	−	0.989424i	$-0.453664\pi$
$37$	1.76466	0.290108	0.145054	+	0.989424i	$0.453664\pi$
$38$	0	0
$39$	−10.7069	−1.71448
$40$	0	0
$41$	2.50564	0.391315	0.195658	−	0.980672i	$-0.437316\pi$
$41$	2.50564	0.391315	0.195658	+	0.980672i	$0.437316\pi$
$42$	0	0
$43$	−3.85911	−0.588508	−0.294254	−	0.955727i	$-0.595071\pi$
$43$	−3.85911	−0.588508	−0.294254	+	0.955727i	$0.595071\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−4.91208	−0.716500	−0.358250	−	0.933626i	$-0.616626\pi$
$47$	−4.91208	−0.716500	−0.358250	+	0.933626i	$0.616626\pi$
$48$	0	0
$49$	12.2334	1.74763
$50$	0	0
$51$	−15.7391	−2.20391
$52$	0	0
$53$	−5.29767	−0.727691	−0.363845	−	0.931459i	$-0.618536\pi$
$53$	−5.29767	−0.727691	−0.363845	+	0.931459i	$0.618536\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	1.40926	0.186661
$58$	0	0
$59$	−2.63841	−0.343492	−0.171746	−	0.985141i	$-0.554941\pi$
$59$	−2.63841	−0.343492	−0.171746	+	0.985141i	$0.554941\pi$
$60$	0	0
$61$	8.96787	1.14822	0.574109	−	0.818779i	$-0.305348\pi$
$61$	8.96787	1.14822	0.574109	+	0.818779i	$0.305348\pi$
$62$	0	0
$63$	18.2655	2.30124
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−7.97440	−0.974228	−0.487114	−	0.873338i	$-0.661950\pi$
$67$	−7.97440	−0.974228	−0.487114	+	0.873338i	$0.661950\pi$
$68$	0	0
$69$	−16.4626	−1.98186
$70$	0	0
$71$	13.8718	1.64628	0.823142	−	0.567836i	$-0.192219\pi$
$71$	13.8718	1.64628	0.823142	+	0.567836i	$0.192219\pi$
$72$	0	0
$73$	12.7712	1.49475	0.747377	−	0.664400i	$-0.231313\pi$
$73$	12.7712	1.49475	0.747377	+	0.664400i	$0.231313\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	4.38559	0.499785
$78$	0	0
$79$	−13.2768	−1.49376	−0.746880	−	0.664959i	$-0.768449\pi$
$79$	−13.2768	−1.49376	−0.746880	+	0.664959i	$0.768449\pi$
$80$	0	0
$81$	−4.14832	−0.460924
$82$	0	0
$83$	−7.61901	−0.836295	−0.418147	−	0.908379i	$-0.637320\pi$
$83$	−7.61901	−0.836295	−0.418147	+	0.908379i	$0.637320\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	2.59074	0.277756
$88$	0	0
$89$	2.83510	0.300520	0.150260	−	0.988646i	$-0.451989\pi$
$89$	2.83510	0.300520	0.150260	+	0.988646i	$0.451989\pi$
$90$	0	0
$91$	17.5424	1.83894
$92$	0	0
$93$	25.7135	2.66636
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−10.4800	−1.06409	−0.532044	−	0.846717i	$-0.678576\pi$
$97$	−10.4800	−1.06409	−0.532044	+	0.846717i	$0.678576\pi$
$98$	0	0
$99$	4.16490	0.418588
$100$	0	0
$101$	8.70693	0.866372	0.433186	−	0.901305i	$-0.357389\pi$
$101$	8.70693	0.866372	0.433186	+	0.901305i	$0.357389\pi$
$102$	0	0
$103$	15.6190	1.53899	0.769493	−	0.638655i	$-0.220509\pi$
$103$	15.6190	1.53899	0.769493	+	0.638655i	$0.220509\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	7.08792	0.685215	0.342608	−	0.939479i	$-0.388690\pi$
$107$	7.08792	0.685215	0.342608	+	0.939479i	$0.388690\pi$
$108$	0	0
$109$	−15.5424	−1.48869	−0.744344	−	0.667796i	$-0.767238\pi$
$109$	−15.5424	−1.48869	−0.744344	+	0.667796i	$0.767238\pi$
$110$	0	0
$111$	−4.72351	−0.448336
$112$	0	0
$113$	8.11530	0.763423	0.381711	−	0.924282i	$-0.375335\pi$
$113$	8.11530	0.763423	0.381711	+	0.924282i	$0.375335\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	16.6596	1.54018
$118$	0	0
$119$	25.7871	2.36390
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−6.70693	−0.604744
$124$	0	0
$125$	0	0
$126$	0	0
$127$	4.32980	0.384207	0.192104	−	0.981375i	$-0.438469\pi$
$127$	4.32980	0.384207	0.192104	+	0.981375i	$0.438469\pi$
$128$	0	0
$129$	10.3298	0.909488
$130$	0	0
$131$	−17.6748	−1.54425	−0.772127	−	0.635468i	$-0.780807\pi$
$131$	−17.6748	−1.54425	−0.772127	+	0.635468i	$0.780807\pi$
$132$	0	0
$133$	−2.30895	−0.200211
$134$	0	0
$135$	0	0
$136$	0	0
$137$	18.4800	1.57886	0.789428	−	0.613843i	$-0.210377\pi$
$137$	18.4800	1.57886	0.789428	+	0.613843i	$0.210377\pi$
$138$	0	0
$139$	−5.55861	−0.471475	−0.235738	−	0.971817i	$-0.575751\pi$
$139$	−5.55861	−0.471475	−0.235738	+	0.971817i	$0.575751\pi$
$140$	0	0
$141$	13.1483	1.10729
$142$	0	0
$143$	4.00000	0.334497
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−32.7456	−2.70081
$148$	0	0
$149$	20.9516	1.71642	0.858212	−	0.513295i	$-0.171575\pi$
$149$	20.9516	1.71642	0.858212	+	0.513295i	$0.171575\pi$
$150$	0	0
$151$	−1.67020	−0.135919	−0.0679596	−	0.997688i	$-0.521649\pi$
$151$	−1.67020	−0.135919	−0.0679596	+	0.997688i	$0.521649\pi$
$152$	0	0
$153$	24.4894	1.97985
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−8.94228	−0.713671	−0.356836	−	0.934167i	$-0.616144\pi$
$157$	−8.94228	−0.713671	−0.356836	+	0.934167i	$0.616144\pi$
$158$	0	0
$159$	14.1804	1.12458
$160$	0	0
$161$	26.9725	2.12573
$162$	0	0
$163$	−9.43856	−0.739285	−0.369643	−	0.929174i	$-0.620520\pi$
$163$	−9.43856	−0.739285	−0.369643	+	0.929174i	$0.620520\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	8.96787	0.693955	0.346977	−	0.937873i	$-0.387208\pi$
$167$	8.96787	0.693955	0.346977	+	0.937873i	$0.387208\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	0	0
$171$	−2.19276	−0.167684
$172$	0	0
$173$	2.64653	0.201212	0.100606	−	0.994926i	$-0.467922\pi$
$173$	2.64653	0.201212	0.100606	+	0.994926i	$0.467922\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	7.06232	0.530837
$178$	0	0
$179$	20.2225	1.51150	0.755749	−	0.654861i	$-0.227273\pi$
$179$	20.2225	1.51150	0.755749	+	0.654861i	$0.227273\pi$
$180$	0	0
$181$	15.3453	1.14061	0.570305	−	0.821433i	$-0.306825\pi$
$181$	15.3453	1.14061	0.570305	+	0.821433i	$0.306825\pi$
$182$	0	0
$183$	−24.0046	−1.77447
$184$	0	0
$185$	0	0
$186$	0	0
$187$	5.87995	0.429985
$188$	0	0
$189$	−13.6748	−0.994696
$190$	0	0
$191$	−5.62713	−0.407165	−0.203582	−	0.979058i	$-0.565258\pi$
$191$	−5.62713	−0.407165	−0.203582	+	0.979058i	$0.565258\pi$
$192$	0	0
$193$	−7.23342	−0.520673	−0.260336	−	0.965518i	$-0.583833\pi$
$193$	−7.23342	−0.520673	−0.260336	+	0.965518i	$0.583833\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	5.35347	0.381419	0.190709	−	0.981647i	$-0.438921\pi$
$197$	5.35347	0.381419	0.190709	+	0.981647i	$0.438921\pi$
$198$	0	0
$199$	7.23342	0.512763	0.256382	−	0.966576i	$-0.417470\pi$
$199$	7.23342	0.512763	0.256382	+	0.966576i	$0.417470\pi$
$200$	0	0
$201$	21.3453	1.50558
$202$	0	0
$203$	−4.24470	−0.297919
$204$	0	0
$205$	0	0
$206$	0	0
$207$	25.6152	1.78038
$208$	0	0
$209$	−0.526486	−0.0364178
$210$	0	0
$211$	−12.2447	−0.842960	−0.421480	−	0.906838i	$-0.638489\pi$
$211$	−12.2447	−0.842960	−0.421480	+	0.906838i	$0.638489\pi$
$212$	0	0
$213$	−37.1312	−2.54419
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−42.1292	−2.85992
$218$	0	0
$219$	−34.1850	−2.31001
$220$	0	0
$221$	23.5198	1.58211
$222$	0	0
$223$	19.4525	1.30264	0.651318	−	0.758805i	$-0.274216\pi$
$223$	19.4525	1.30264	0.651318	+	0.758805i	$0.274216\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	3.90080	0.258905	0.129452	−	0.991586i	$-0.458678\pi$
$227$	3.90080	0.258905	0.129452	+	0.991586i	$0.458678\pi$
$228$	0	0
$229$	13.4096	0.886131	0.443066	−	0.896489i	$-0.353891\pi$
$229$	13.4096	0.886131	0.443066	+	0.896489i	$0.353891\pi$
$230$	0	0
$231$	−11.7391	−0.772373
$232$	0	0
$233$	−3.36192	−0.220247	−0.110123	−	0.993918i	$-0.535125\pi$
$233$	−3.36192	−0.220247	−0.110123	+	0.993918i	$0.535125\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	35.5385	2.30847
$238$	0	0
$239$	−13.2768	−0.858806	−0.429403	−	0.903113i	$-0.641276\pi$
$239$	−13.2768	−0.858806	−0.429403	+	0.903113i	$0.641276\pi$
$240$	0	0
$241$	−17.7437	−1.14297	−0.571485	−	0.820613i	$-0.693632\pi$
$241$	−17.7437	−1.14297	−0.571485	+	0.820613i	$0.693632\pi$
$242$	0	0
$243$	20.4583	1.31240
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−2.10594	−0.133998
$248$	0	0
$249$	20.3940	1.29242
$250$	0	0
$251$	8.57416	0.541196	0.270598	−	0.962692i	$-0.412779\pi$
$251$	8.57416	0.541196	0.270598	+	0.962692i	$0.412779\pi$
$252$	0	0
$253$	6.15025	0.386663
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−2.70693	−0.168854	−0.0844269	−	0.996430i	$-0.526906\pi$
$257$	−2.70693	−0.168854	−0.0844269	+	0.996430i	$0.526906\pi$
$258$	0	0
$259$	7.73906	0.480882
$260$	0	0
$261$	−4.03109	−0.249518
$262$	0	0
$263$	14.3213	0.883092	0.441546	−	0.897239i	$-0.354430\pi$
$263$	14.3213	0.883092	0.441546	+	0.897239i	$0.354430\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−7.58881	−0.464428
$268$	0	0
$269$	−11.9357	−0.727735	−0.363868	−	0.931451i	$-0.618544\pi$
$269$	−11.9357	−0.727735	−0.363868	+	0.931451i	$0.618544\pi$
$270$	0	0
$271$	−24.4195	−1.48338	−0.741689	−	0.670744i	$-0.765975\pi$
$271$	−24.4195	−1.48338	−0.741689	+	0.670744i	$0.765975\pi$
$272$	0	0
$273$	−46.9562	−2.84192
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−23.0074	−1.38238	−0.691191	−	0.722672i	$-0.742914\pi$
$277$	−23.0074	−1.38238	−0.691191	+	0.722672i	$0.742914\pi$
$278$	0	0
$279$	−40.0092	−2.39529
$280$	0	0
$281$	−10.6596	−0.635898	−0.317949	−	0.948108i	$-0.602994\pi$
$281$	−10.6596	−0.635898	−0.317949	+	0.948108i	$0.602994\pi$
$282$	0	0
$283$	10.3771	0.616857	0.308428	−	0.951248i	$-0.400197\pi$
$283$	10.3771	0.616857	0.308428	+	0.951248i	$0.400197\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	10.9887	0.648644
$288$	0	0
$289$	17.5738	1.03375
$290$	0	0
$291$	28.0523	1.64445
$292$	0	0
$293$	−22.7069	−1.32655	−0.663277	−	0.748374i	$-0.730835\pi$
$293$	−22.7069	−1.32655	−0.663277	+	0.748374i	$0.730835\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−3.11812	−0.180932
$298$	0	0
$299$	24.6010	1.42271
$300$	0	0
$301$	−16.9245	−0.975510
$302$	0	0
$303$	−23.3061	−1.33890
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−5.49436	−0.313580	−0.156790	−	0.987632i	$-0.550115\pi$
$307$	−5.49436	−0.313580	−0.156790	+	0.987632i	$0.550115\pi$
$308$	0	0
$309$	−41.8079	−2.37837
$310$	0	0
$311$	−24.2447	−1.37479	−0.687395	−	0.726283i	$-0.741246\pi$
$311$	−24.2447	−1.37479	−0.687395	+	0.726283i	$0.741246\pi$
$312$	0	0
$313$	−11.7562	−0.664500	−0.332250	−	0.943191i	$-0.607808\pi$
$313$	−11.7562	−0.664500	−0.332250	+	0.943191i	$0.607808\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−9.05387	−0.508516	−0.254258	−	0.967136i	$-0.581831\pi$
$317$	−9.05387	−0.508516	−0.254258	+	0.967136i	$0.581831\pi$
$318$	0	0
$319$	−0.967873	−0.0541905
$320$	0	0
$321$	−18.9725	−1.05894
$322$	0	0
$323$	−3.09571	−0.172250
$324$	0	0
$325$	0	0
$326$	0	0
$327$	41.6028	2.30064
$328$	0	0
$329$	−21.5424	−1.18767
$330$	0	0
$331$	−3.29733	−0.181238	−0.0906190	−	0.995886i	$-0.528885\pi$
$331$	−3.29733	−0.181238	−0.0906190	+	0.995886i	$0.528885\pi$
$332$	0	0
$333$	7.34961	0.402756
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−3.17302	−0.172845	−0.0864227	−	0.996259i	$-0.527544\pi$
$337$	−3.17302	−0.172845	−0.0864227	+	0.996259i	$0.527544\pi$
$338$	0	0
$339$	−21.7225	−1.17980
$340$	0	0
$341$	−9.60629	−0.520210
$342$	0	0
$343$	22.9516	1.23927
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−4.44139	−0.238426	−0.119213	−	0.992869i	$-0.538037\pi$
$347$	−4.44139	−0.238426	−0.119213	+	0.992869i	$0.538037\pi$
$348$	0	0
$349$	29.2380	1.56508	0.782538	−	0.622603i	$-0.213925\pi$
$349$	29.2380	1.56508	0.782538	+	0.622603i	$0.213925\pi$
$350$	0	0
$351$	−12.4725	−0.665732
$352$	0	0
$353$	1.04927	0.0558469	0.0279234	−	0.999610i	$-0.491111\pi$
$353$	1.04927	0.0558469	0.0279234	+	0.999610i	$0.491111\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−69.0251	−3.65319
$358$	0	0
$359$	9.69565	0.511717	0.255858	−	0.966714i	$-0.417642\pi$
$359$	9.69565	0.511717	0.255858	+	0.966714i	$0.417642\pi$
$360$	0	0
$361$	−18.7228	−0.985411
$362$	0	0
$363$	−2.67673	−0.140492
$364$	0	0
$365$	0	0
$366$	0	0
$367$	11.3921	0.594664	0.297332	−	0.954774i	$-0.403903\pi$
$367$	11.3921	0.594664	0.297332	+	0.954774i	$0.403903\pi$
$368$	0	0
$369$	10.4357	0.543263
$370$	0	0
$371$	−23.2334	−1.20622
$372$	0	0
$373$	−6.47634	−0.335332	−0.167666	−	0.985844i	$-0.553623\pi$
$373$	−6.47634	−0.335332	−0.167666	+	0.985844i	$0.553623\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−3.87149	−0.199392
$378$	0	0
$379$	−33.4993	−1.72074	−0.860372	−	0.509667i	$-0.829768\pi$
$379$	−33.4993	−1.72074	−0.860372	+	0.509667i	$0.829768\pi$
$380$	0	0
$381$	−11.5897	−0.593759
$382$	0	0
$383$	30.8702	1.57740	0.788698	−	0.614781i	$-0.210756\pi$
$383$	30.8702	1.57740	0.788698	+	0.614781i	$0.210756\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−16.0728	−0.817026
$388$	0	0
$389$	−23.3453	−1.18366	−0.591828	−	0.806064i	$-0.701594\pi$
$389$	−23.3453	−1.18366	−0.591828	+	0.806064i	$0.701594\pi$
$390$	0	0
$391$	36.1632	1.82885
$392$	0	0
$393$	47.3107	2.38651
$394$	0	0
$395$	0	0
$396$	0	0
$397$	19.4781	0.977579	0.488789	−	0.872402i	$-0.337439\pi$
$397$	19.4781	0.977579	0.488789	+	0.872402i	$0.337439\pi$
$398$	0	0
$399$	6.18045	0.309409
$400$	0	0
$401$	2.24470	0.112095	0.0560474	−	0.998428i	$-0.482150\pi$
$401$	2.24470	0.112095	0.0560474	+	0.998428i	$0.482150\pi$
$402$	0	0
$403$	−38.4251	−1.91409
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1.76466	0.0874707
$408$	0	0
$409$	29.4781	1.45760	0.728799	−	0.684727i	$-0.240079\pi$
$409$	29.4781	1.45760	0.728799	+	0.684727i	$0.240079\pi$
$410$	0	0
$411$	−49.4661	−2.43998
$412$	0	0
$413$	−11.5710	−0.569372
$414$	0	0
$415$	0	0
$416$	0	0
$417$	14.8789	0.728624
$418$	0	0
$419$	7.64831	0.373644	0.186822	−	0.982394i	$-0.440181\pi$
$419$	7.64831	0.373644	0.186822	+	0.982394i	$0.440181\pi$
$420$	0	0
$421$	29.4781	1.43668	0.718338	−	0.695695i	$-0.244903\pi$
$421$	29.4781	1.43668	0.718338	+	0.695695i	$0.244903\pi$
$422$	0	0
$423$	−20.4583	−0.994717
$424$	0	0
$425$	0	0
$426$	0	0
$427$	39.3294	1.90328
$428$	0	0
$429$	−10.7069	−0.516935
$430$	0	0
$431$	−2.08970	−0.100657	−0.0503286	−	0.998733i	$-0.516027\pi$
$431$	−2.08970	−0.100657	−0.0503286	+	0.998733i	$0.516027\pi$
$432$	0	0
$433$	−5.17777	−0.248828	−0.124414	−	0.992230i	$-0.539705\pi$
$433$	−5.17777	−0.248828	−0.124414	+	0.992230i	$0.539705\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−3.23802	−0.154895
$438$	0	0
$439$	27.6066	1.31759	0.658796	−	0.752322i	$-0.271066\pi$
$439$	27.6066	1.31759	0.658796	+	0.752322i	$0.271066\pi$
$440$	0	0
$441$	50.9509	2.42623
$442$	0	0
$443$	−14.1502	−0.672299	−0.336149	−	0.941809i	$-0.609125\pi$
$443$	−14.1502	−0.672299	−0.336149	+	0.941809i	$0.609125\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−56.0819	−2.65258
$448$	0	0
$449$	−6.46257	−0.304987	−0.152494	−	0.988304i	$-0.548730\pi$
$449$	−6.46257	−0.304987	−0.152494	+	0.988304i	$0.548730\pi$
$450$	0	0
$451$	2.50564	0.117986
$452$	0	0
$453$	4.47069	0.210051
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−38.5582	−1.80368	−0.901839	−	0.432071i	$-0.857783\pi$
$457$	−38.5582	−1.80368	−0.901839	+	0.432071i	$0.857783\pi$
$458$	0	0
$459$	−18.3344	−0.855776
$460$	0	0
$461$	−18.4234	−0.858064	−0.429032	−	0.903289i	$-0.641145\pi$
$461$	−18.4234	−0.858064	−0.429032	+	0.903289i	$0.641145\pi$
$462$	0	0
$463$	7.09163	0.329576	0.164788	−	0.986329i	$-0.447306\pi$
$463$	7.09163	0.329576	0.164788	+	0.986329i	$0.447306\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	40.4328	1.87101	0.935503	−	0.353319i	$-0.114947\pi$
$467$	40.4328	1.87101	0.935503	+	0.353319i	$0.114947\pi$
$468$	0	0
$469$	−34.9725	−1.61488
$470$	0	0
$471$	23.9361	1.10292
$472$	0	0
$473$	−3.85911	−0.177442
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−22.0643	−1.01025
$478$	0	0
$479$	−10.7486	−0.491117	−0.245559	−	0.969382i	$-0.578971\pi$
$479$	−10.7486	−0.491117	−0.245559	+	0.969382i	$0.578971\pi$
$480$	0	0
$481$	7.05862	0.321845
$482$	0	0
$483$	−72.1981	−3.28513
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−38.6396	−1.75093	−0.875465	−	0.483282i	$-0.839445\pi$
$487$	−38.6396	−1.75093	−0.875465	+	0.483282i	$0.839445\pi$
$488$	0	0
$489$	25.2645	1.14250
$490$	0	0
$491$	32.9460	1.48683	0.743416	−	0.668830i	$-0.233204\pi$
$491$	32.9460	1.48683	0.743416	+	0.668830i	$0.233204\pi$
$492$	0	0
$493$	−5.69105	−0.256312
$494$	0	0
$495$	0	0
$496$	0	0
$497$	60.8362	2.72888
$498$	0	0
$499$	26.9562	1.20673	0.603363	−	0.797466i	$-0.293827\pi$
$499$	26.9562	1.20673	0.603363	+	0.797466i	$0.293827\pi$
$500$	0	0
$501$	−24.0046	−1.07245
$502$	0	0
$503$	14.0766	0.627646	0.313823	−	0.949481i	$-0.398390\pi$
$503$	14.0766	0.627646	0.313823	+	0.949481i	$0.398390\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−8.03020	−0.356634
$508$	0	0
$509$	1.28109	0.0567833	0.0283917	−	0.999597i	$-0.490961\pi$
$509$	1.28109	0.0567833	0.0283917	+	0.999597i	$0.490961\pi$
$510$	0	0
$511$	56.0092	2.47770
$512$	0	0
$513$	1.64165	0.0724804
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−4.91208	−0.216033
$518$	0	0
$519$	−7.08407	−0.310956
$520$	0	0
$521$	−29.2394	−1.28100	−0.640501	−	0.767958i	$-0.721273\pi$
$521$	−29.2394	−1.28100	−0.640501	+	0.767958i	$0.721273\pi$
$522$	0	0
$523$	−2.50564	−0.109564	−0.0547820	−	0.998498i	$-0.517446\pi$
$523$	−2.50564	−0.109564	−0.0547820	+	0.998498i	$0.517446\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−56.4845	−2.46050
$528$	0	0
$529$	14.8255	0.644589
$530$	0	0
$531$	−10.9887	−0.476870
$532$	0	0
$533$	10.0226	0.434125
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−54.1301	−2.33589
$538$	0	0
$539$	12.2334	0.526931
$540$	0	0
$541$	−3.27222	−0.140684	−0.0703420	−	0.997523i	$-0.522409\pi$
$541$	−3.27222	−0.140684	−0.0703420	+	0.997523i	$0.522409\pi$
$542$	0	0
$543$	−41.0754	−1.76271
$544$	0	0
$545$	0	0
$546$	0	0
$547$	3.03673	0.129841	0.0649205	−	0.997890i	$-0.479321\pi$
$547$	3.03673	0.129841	0.0649205	+	0.997890i	$0.479321\pi$
$548$	0	0
$549$	37.3503	1.59407
$550$	0	0
$551$	0.509572	0.0217085
$552$	0	0
$553$	−58.2267	−2.47605
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−1.52366	−0.0645596	−0.0322798	−	0.999479i	$-0.510277\pi$
$557$	−1.52366	−0.0645596	−0.0322798	+	0.999479i	$0.510277\pi$
$558$	0	0
$559$	−15.4364	−0.652891
$560$	0	0
$561$	−15.7391	−0.664504
$562$	0	0
$563$	27.9195	1.17667	0.588333	−	0.808618i	$-0.299784\pi$
$563$	27.9195	1.17667	0.588333	+	0.808618i	$0.299784\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−18.1928	−0.764027
$568$	0	0
$569$	−13.7437	−0.576164	−0.288082	−	0.957606i	$-0.593018\pi$
$569$	−13.7437	−0.576164	−0.288082	+	0.957606i	$0.593018\pi$
$570$	0	0
$571$	−1.42618	−0.0596836	−0.0298418	−	0.999555i	$-0.509500\pi$
$571$	−1.42618	−0.0596836	−0.0298418	+	0.999555i	$0.509500\pi$
$572$	0	0
$573$	15.0623	0.629238
$574$	0	0
$575$	0	0
$576$	0	0
$577$	6.35154	0.264418	0.132209	−	0.991222i	$-0.457793\pi$
$577$	6.35154	0.264418	0.132209	+	0.991222i	$0.457793\pi$
$578$	0	0
$579$	19.3619	0.804654
$580$	0	0
$581$	−33.4139	−1.38624
$582$	0	0
$583$	−5.29767	−0.219407
$584$	0	0
$585$	0	0
$586$	0	0
$587$	22.1681	0.914974	0.457487	−	0.889216i	$-0.348750\pi$
$587$	22.1681	0.914974	0.457487	+	0.889216i	$0.348750\pi$
$588$	0	0
$589$	5.05757	0.208394
$590$	0	0
$591$	−14.3298	−0.589449
$592$	0	0
$593$	1.05297	0.0432403	0.0216202	−	0.999766i	$-0.493118\pi$
$593$	1.05297	0.0432403	0.0216202	+	0.999766i	$0.493118\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−19.3619	−0.792431
$598$	0	0
$599$	−5.12747	−0.209503	−0.104751	−	0.994498i	$-0.533405\pi$
$599$	−5.12747	−0.209503	−0.104751	+	0.994498i	$0.533405\pi$
$600$	0	0
$601$	−1.71821	−0.0700874	−0.0350437	−	0.999386i	$-0.511157\pi$
$601$	−1.71821	−0.0700874	−0.0350437	+	0.999386i	$0.511157\pi$
$602$	0	0
$603$	−33.2126	−1.35252
$604$	0	0
$605$	0	0
$606$	0	0
$607$	4.42728	0.179698	0.0898489	−	0.995955i	$-0.471362\pi$
$607$	4.42728	0.179698	0.0898489	+	0.995955i	$0.471362\pi$
$608$	0	0
$609$	11.3619	0.460408
$610$	0	0
$611$	−19.6483	−0.794886
$612$	0	0
$613$	−33.8334	−1.36652	−0.683258	−	0.730177i	$-0.739438\pi$
$613$	−33.8334	−1.36652	−0.683258	+	0.730177i	$0.739438\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	40.3778	1.62555	0.812775	−	0.582578i	$-0.197956\pi$
$617$	40.3778	1.62555	0.812775	+	0.582578i	$0.197956\pi$
$618$	0	0
$619$	−1.62713	−0.0653999	−0.0327000	−	0.999465i	$-0.510411\pi$
$619$	−1.62713	−0.0653999	−0.0327000	+	0.999465i	$0.510411\pi$
$620$	0	0
$621$	−19.1772	−0.769555
$622$	0	0
$623$	12.4336	0.498142
$624$	0	0
$625$	0	0
$626$	0	0
$627$	1.40926	0.0562805
$628$	0	0
$629$	10.3761	0.413722
$630$	0	0
$631$	3.71223	0.147781	0.0738907	−	0.997266i	$-0.476458\pi$
$631$	3.71223	0.147781	0.0738907	+	0.997266i	$0.476458\pi$
$632$	0	0
$633$	32.7758	1.30272
$634$	0	0
$635$	0	0
$636$	0	0
$637$	48.9337	1.93882
$638$	0	0
$639$	57.7748	2.28553
$640$	0	0
$641$	32.5562	1.28589	0.642946	−	0.765911i	$-0.277712\pi$
$641$	32.5562	1.28589	0.642946	+	0.765911i	$0.277712\pi$
$642$	0	0
$643$	27.9017	1.10034	0.550168	−	0.835054i	$-0.314564\pi$
$643$	27.9017	1.10034	0.550168	+	0.835054i	$0.314564\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	28.7873	1.13174	0.565872	−	0.824493i	$-0.308540\pi$
$647$	28.7873	1.13174	0.565872	+	0.824493i	$0.308540\pi$
$648$	0	0
$649$	−2.63841	−0.103567
$650$	0	0
$651$	112.769	4.41976
$652$	0	0
$653$	34.7152	1.35851	0.679256	−	0.733901i	$-0.262303\pi$
$653$	34.7152	1.35851	0.679256	+	0.733901i	$0.262303\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	53.1907	2.07517
$658$	0	0
$659$	−25.5632	−0.995801	−0.497901	−	0.867234i	$-0.665896\pi$
$659$	−25.5632	−0.995801	−0.497901	+	0.867234i	$0.665896\pi$
$660$	0	0
$661$	1.40960	0.0548269	0.0274135	−	0.999624i	$-0.491273\pi$
$661$	1.40960	0.0548269	0.0274135	+	0.999624i	$0.491273\pi$
$662$	0	0
$663$	−62.9562	−2.44502
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−5.95266	−0.230488
$668$	0	0
$669$	−52.0692	−2.01311
$670$	0	0
$671$	8.96787	0.346201
$672$	0	0
$673$	0.0557959	0.00215078	0.00107539	−	0.999999i	$-0.499658\pi$
$673$	0.0557959	0.00215078	0.00107539	+	0.999999i	$0.499658\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	20.8316	0.800623	0.400311	−	0.916379i	$-0.368902\pi$
$677$	20.8316	0.800623	0.400311	+	0.916379i	$0.368902\pi$
$678$	0	0
$679$	−45.9612	−1.76383
$680$	0	0
$681$	−10.4414	−0.400115
$682$	0	0
$683$	21.5085	0.822999	0.411499	−	0.911410i	$-0.365005\pi$
$683$	21.5085	0.822999	0.411499	+	0.911410i	$0.365005\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−35.8939	−1.36944
$688$	0	0
$689$	−21.1907	−0.807301
$690$	0	0
$691$	5.49862	0.209178	0.104589	−	0.994516i	$-0.466647\pi$
$691$	5.49862	0.209178	0.104589	+	0.994516i	$0.466647\pi$
$692$	0	0
$693$	18.2655	0.693851
$694$	0	0
$695$	0	0
$696$	0	0
$697$	14.7330	0.558054
$698$	0	0
$699$	8.99897	0.340372
$700$	0	0
$701$	−6.00460	−0.226791	−0.113395	−	0.993550i	$-0.536173\pi$
$701$	−6.00460	−0.226791	−0.113395	+	0.993550i	$0.536173\pi$
$702$	0	0
$703$	−0.929066	−0.0350404
$704$	0	0
$705$	0	0
$706$	0	0
$707$	38.1850	1.43610
$708$	0	0
$709$	17.7168	0.665369	0.332685	−	0.943038i	$-0.392046\pi$
$709$	17.7168	0.665369	0.332685	+	0.943038i	$0.392046\pi$
$710$	0	0
$711$	−55.2966	−2.07379
$712$	0	0
$713$	−59.0810	−2.21260
$714$	0	0
$715$	0	0
$716$	0	0
$717$	35.5385	1.32721
$718$	0	0
$719$	30.5308	1.13860	0.569302	−	0.822128i	$-0.307213\pi$
$719$	30.5308	1.13860	0.569302	+	0.822128i	$0.307213\pi$
$720$	0	0
$721$	68.4986	2.55102
$722$	0	0
$723$	47.4950	1.76636
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−43.9063	−1.62839	−0.814197	−	0.580589i	$-0.802823\pi$
$727$	−43.9063	−1.62839	−0.814197	+	0.580589i	$0.802823\pi$
$728$	0	0
$729$	−42.3165	−1.56728
$730$	0	0
$731$	−22.6914	−0.839270
$732$	0	0
$733$	−20.6427	−0.762455	−0.381227	−	0.924481i	$-0.624498\pi$
$733$	−20.6427	−0.762455	−0.381227	+	0.924481i	$0.624498\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−7.97440	−0.293741
$738$	0	0
$739$	−7.73446	−0.284517	−0.142258	−	0.989830i	$-0.545436\pi$
$739$	−7.73446	−0.284517	−0.142258	+	0.989830i	$0.545436\pi$
$740$	0	0
$741$	5.63705	0.207082
$742$	0	0
$743$	−29.1755	−1.07034	−0.535172	−	0.844743i	$-0.679753\pi$
$743$	−29.1755	−1.07034	−0.535172	+	0.844743i	$0.679753\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−31.7324	−1.16103
$748$	0	0
$749$	31.0847	1.13581
$750$	0	0
$751$	−9.38310	−0.342394	−0.171197	−	0.985237i	$-0.554763\pi$
$751$	−9.38310	−0.342394	−0.171197	+	0.985237i	$0.554763\pi$
$752$	0	0
$753$	−22.9507	−0.836371
$754$	0	0
$755$	0	0
$756$	0	0
$757$	12.8997	0.468847	0.234424	−	0.972135i	$-0.424680\pi$
$757$	12.8997	0.468847	0.234424	+	0.972135i	$0.424680\pi$
$758$	0	0
$759$	−16.4626	−0.597553
$760$	0	0
$761$	15.6963	0.568991	0.284496	−	0.958677i	$-0.408174\pi$
$761$	15.6963	0.568991	0.284496	+	0.958677i	$0.408174\pi$
$762$	0	0
$763$	−68.1625	−2.46765
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−10.5536	−0.381070
$768$	0	0
$769$	34.1123	1.23012	0.615060	−	0.788480i	$-0.289132\pi$
$769$	34.1123	1.23012	0.615060	+	0.788480i	$0.289132\pi$
$770$	0	0
$771$	7.24573	0.260949
$772$	0	0
$773$	−52.2539	−1.87944	−0.939721	−	0.341942i	$-0.888915\pi$
$773$	−52.2539	−1.87944	−0.939721	+	0.341942i	$0.888915\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−20.7154	−0.743160
$778$	0	0
$779$	−1.31918	−0.0472647
$780$	0	0
$781$	13.8718	0.496373
$782$	0	0
$783$	3.01795	0.107853
$784$	0	0
$785$	0	0
$786$	0	0
$787$	6.89902	0.245924	0.122962	−	0.992411i	$-0.460761\pi$
$787$	6.89902	0.245924	0.122962	+	0.992411i	$0.460761\pi$
$788$	0	0
$789$	−38.3344	−1.36474
$790$	0	0
$791$	35.5904	1.26545
$792$	0	0
$793$	35.8715	1.27383
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−0.544296	−0.0192800	−0.00963998	−	0.999954i	$-0.503069\pi$
$797$	−0.544296	−0.0192800	−0.00963998	+	0.999954i	$0.503069\pi$
$798$	0	0
$799$	−28.8828	−1.02180
$800$	0	0
$801$	11.8079	0.417212
$802$	0	0
$803$	12.7712	0.450685
$804$	0	0
$805$	0	0
$806$	0	0
$807$	31.9488	1.12465
$808$	0	0
$809$	26.3354	0.925905	0.462952	−	0.886383i	$-0.346790\pi$
$809$	26.3354	0.925905	0.462952	+	0.886383i	$0.346790\pi$
$810$	0	0
$811$	−24.4686	−0.859207	−0.429604	−	0.903018i	$-0.641347\pi$
$811$	−24.4686	−0.859207	−0.429604	+	0.903018i	$0.641347\pi$
$812$	0	0
$813$	65.3645	2.29243
$814$	0	0
$815$	0	0
$816$	0	0
$817$	2.03176	0.0710824
$818$	0	0
$819$	73.0622	2.55300
$820$	0	0
$821$	42.8503	1.49549	0.747743	−	0.663989i	$-0.231138\pi$
$821$	42.8503	1.49549	0.747743	+	0.663989i	$0.231138\pi$
$822$	0	0
$823$	18.7417	0.653296	0.326648	−	0.945146i	$-0.394081\pi$
$823$	18.7417	0.653296	0.326648	+	0.945146i	$0.394081\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−30.6681	−1.06644	−0.533218	−	0.845978i	$-0.679017\pi$
$827$	−30.6681	−1.06644	−0.533218	+	0.845978i	$0.679017\pi$
$828$	0	0
$829$	−19.8758	−0.690314	−0.345157	−	0.938545i	$-0.612174\pi$
$829$	−19.8758	−0.690314	−0.345157	+	0.938545i	$0.612174\pi$
$830$	0	0
$831$	61.5847	2.13635
$832$	0	0
$833$	71.9319	2.49229
$834$	0	0
$835$	0	0
$836$	0	0
$837$	29.9536	1.03535
$838$	0	0
$839$	−48.3178	−1.66812	−0.834058	−	0.551677i	$-0.813988\pi$
$839$	−48.3178	−1.66812	−0.834058	+	0.551677i	$0.813988\pi$
$840$	0	0
$841$	−28.0632	−0.967697
$842$	0	0
$843$	28.5329	0.982725
$844$	0	0
$845$	0	0
$846$	0	0
$847$	4.38559	0.150691
$848$	0	0
$849$	−27.7768	−0.953298
$850$	0	0
$851$	10.8531	0.372038
$852$	0	0
$853$	39.4686	1.35138	0.675690	−	0.737186i	$-0.263846\pi$
$853$	39.4686	1.35138	0.675690	+	0.737186i	$0.263846\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	16.7666	0.572736	0.286368	−	0.958120i	$-0.407552\pi$
$857$	16.7666	0.572736	0.286368	+	0.958120i	$0.407552\pi$
$858$	0	0
$859$	0.350976	0.0119751	0.00598757	−	0.999982i	$-0.498094\pi$
$859$	0.350976	0.0119751	0.00598757	+	0.999982i	$0.498094\pi$
$860$	0	0
$861$	−29.4139	−1.00242
$862$	0	0
$863$	−35.2087	−1.19852	−0.599259	−	0.800555i	$-0.704538\pi$
$863$	−35.2087	−1.19852	−0.599259	+	0.800555i	$0.704538\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−47.0404	−1.59758
$868$	0	0
$869$	−13.2768	−0.450385
$870$	0	0
$871$	−31.8976	−1.08081
$872$	0	0
$873$	−43.6483	−1.47727
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−8.61790	−0.291006	−0.145503	−	0.989358i	$-0.546480\pi$
$877$	−8.61790	−0.291006	−0.145503	+	0.989358i	$0.546480\pi$
$878$	0	0
$879$	60.7804	2.05007
$880$	0	0
$881$	−42.5162	−1.43241	−0.716204	−	0.697891i	$-0.754122\pi$
$881$	−42.5162	−1.43241	−0.716204	+	0.697891i	$0.754122\pi$
$882$	0	0
$883$	25.1568	0.846593	0.423296	−	0.905991i	$-0.360873\pi$
$883$	25.1568	0.846593	0.423296	+	0.905991i	$0.360873\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	6.27505	0.210696	0.105348	−	0.994435i	$-0.466404\pi$
$887$	6.27505	0.210696	0.105348	+	0.994435i	$0.466404\pi$
$888$	0	0
$889$	18.9887	0.636861
$890$	0	0
$891$	−4.14832	−0.138974
$892$	0	0
$893$	2.58614	0.0865418
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−65.8503	−2.19868
$898$	0	0
$899$	9.29767	0.310095
$900$	0	0
$901$	−31.1500	−1.03776
$902$	0	0
$903$	45.3023	1.50757
$904$	0	0
$905$	0	0
$906$	0	0
$907$	11.0925	0.368321	0.184161	−	0.982896i	$-0.441043\pi$
$907$	11.0925	0.368321	0.184161	+	0.982896i	$0.441043\pi$
$908$	0	0
$909$	36.2635	1.20278
$910$	0	0
$911$	−32.8175	−1.08729	−0.543646	−	0.839315i	$-0.682956\pi$
$911$	−32.8175	−1.08729	−0.543646	+	0.839315i	$0.682956\pi$
$912$	0	0
$913$	−7.61901	−0.252152
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−77.5145	−2.55975
$918$	0	0
$919$	0.658922	0.0217358	0.0108679	−	0.999941i	$-0.496541\pi$
$919$	0.658922	0.0217358	0.0108679	+	0.999941i	$0.496541\pi$
$920$	0	0
$921$	14.7069	0.484610
$922$	0	0
$923$	55.4873	1.82639
$924$	0	0
$925$	0	0
$926$	0	0
$927$	65.0516	2.13657
$928$	0	0
$929$	−28.0519	−0.920354	−0.460177	−	0.887827i	$-0.652214\pi$
$929$	−28.0519	−0.920354	−0.460177	+	0.887827i	$0.652214\pi$
$930$	0	0
$931$	−6.44072	−0.211086
$932$	0	0
$933$	64.8966	2.12462
$934$	0	0
$935$	0	0
$936$	0	0
$937$	26.6652	0.871115	0.435558	−	0.900161i	$-0.356551\pi$
$937$	26.6652	0.871115	0.435558	+	0.900161i	$0.356551\pi$
$938$	0	0
$939$	31.4682	1.02693
$940$	0	0
$941$	−0.222135	−0.00724139	−0.00362069	−	0.999993i	$-0.501153\pi$
$941$	−0.222135	−0.00724139	−0.00362069	+	0.999993i	$0.501153\pi$
$942$	0	0
$943$	15.4103	0.501829
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−4.99028	−0.162162	−0.0810812	−	0.996707i	$-0.525837\pi$
$947$	−4.99028	−0.162162	−0.0810812	+	0.996707i	$0.525837\pi$
$948$	0	0
$949$	51.0847	1.65828
$950$	0	0
$951$	24.2348	0.785867
$952$	0	0
$953$	−9.66807	−0.313179	−0.156590	−	0.987664i	$-0.550050\pi$
$953$	−9.66807	−0.313179	−0.156590	+	0.987664i	$0.550050\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	2.59074	0.0837467
$958$	0	0
$959$	81.0459	2.61711
$960$	0	0
$961$	61.2807	1.97680
$962$	0	0
$963$	29.5205	0.951284
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−19.8033	−0.636832	−0.318416	−	0.947951i	$-0.603151\pi$
$967$	−19.8033	−0.636832	−0.318416	+	0.947951i	$0.603151\pi$
$968$	0	0
$969$	8.28639	0.266197
$970$	0	0
$971$	39.0410	1.25289	0.626443	−	0.779468i	$-0.284510\pi$
$971$	39.0410	1.25289	0.626443	+	0.779468i	$0.284510\pi$
$972$	0	0
$973$	−24.3778	−0.781517
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−50.8314	−1.62624	−0.813121	−	0.582095i	$-0.802233\pi$
$977$	−50.8314	−1.62624	−0.813121	+	0.582095i	$0.802233\pi$
$978$	0	0
$979$	2.83510	0.0906103
$980$	0	0
$981$	−64.7324	−2.06675
$982$	0	0
$983$	−21.7569	−0.693936	−0.346968	−	0.937877i	$-0.612789\pi$
$983$	−21.7569	−0.693936	−0.346968	+	0.937877i	$0.612789\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	57.6632	1.83544
$988$	0	0
$989$	−23.7345	−0.754712
$990$	0	0
$991$	−33.0622	−1.05025	−0.525127	−	0.851024i	$-0.675982\pi$
$991$	−33.0622	−1.05025	−0.525127	+	0.851024i	$0.675982\pi$
$992$	0	0
$993$	8.82608	0.280087
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−19.4008	−0.614430	−0.307215	−	0.951640i	$-0.599397\pi$
$997$	−19.4008	−0.614430	−0.307215	+	0.951640i	$0.599397\pi$
$998$	0	0
$999$	−5.50241	−0.174088

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2200.2.a.y.1.1		4
4.3	odd	2		4400.2.a.cb.1.4		4
5.2	odd	4		440.2.b.d.89.7	yes	8
5.3	odd	4		440.2.b.d.89.2	✓	8
5.4	even	2		2200.2.a.x.1.4		4
15.2	even	4		3960.2.d.f.3169.2		8
15.8	even	4		3960.2.d.f.3169.1		8
20.3	even	4		880.2.b.j.529.7		8
20.7	even	4		880.2.b.j.529.2		8
20.19	odd	2		4400.2.a.ce.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
440.2.b.d.89.2	✓	8	5.3	odd	4
440.2.b.d.89.7	yes	8	5.2	odd	4
880.2.b.j.529.2		8	20.7	even	4
880.2.b.j.529.7		8	20.3	even	4
2200.2.a.x.1.4		4	5.4	even	2
2200.2.a.y.1.1		4	1.1	even	1	trivial
3960.2.d.f.3169.1		8	15.8	even	4
3960.2.d.f.3169.2		8	15.2	even	4
4400.2.a.cb.1.4		4	4.3	odd	2
4400.2.a.ce.1.1		4	20.19	odd	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:	\( N \)	\(=\)	\( 2200 = 2^{3} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2200.a (trivial)

\(f(q)\)	\(=\)	\(q-2.67673 q^{3} +4.38559 q^{7} +4.16490 q^{9} +1.00000 q^{11} +4.00000 q^{13} +5.87995 q^{17} -0.526486 q^{19} -11.7391 q^{21} +6.15025 q^{23} -3.11812 q^{27} -0.967873 q^{29} -9.60629 q^{31} -2.67673 q^{33} +1.76466 q^{37} -10.7069 q^{39} +2.50564 q^{41} -3.85911 q^{43} -4.91208 q^{47} +12.2334 q^{49} -15.7391 q^{51} -5.29767 q^{53} +1.40926 q^{57} -2.63841 q^{59} +8.96787 q^{61} +18.2655 q^{63} -7.97440 q^{67} -16.4626 q^{69} +13.8718 q^{71} +12.7712 q^{73} +4.38559 q^{77} -13.2768 q^{79} -4.14832 q^{81} -7.61901 q^{83} +2.59074 q^{87} +2.83510 q^{89} +17.5424 q^{91} +25.7135 q^{93} -10.4800 q^{97} +4.16490 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{3} + q^{7} + 7 q^{9} + 4 q^{11} + 16 q^{13} + 7 q^{17} - 9 q^{19} - 7 q^{21} + 6 q^{23} + 13 q^{27} + 3 q^{29} - 15 q^{31} + q^{33} + 5 q^{37} + 4 q^{39} + 10 q^{41} + 8 q^{43} - 10 q^{47} + 9 q^{49}+ \cdots + 7 q^{99}+O(q^{100}) \) 4 * q + q^3 + q^7 + 7 * q^9 + 4 * q^11 + 16 * q^13 + 7 * q^17 - 9 * q^19 - 7 * q^21 + 6 * q^23 + 13 * q^27 + 3 * q^29 - 15 * q^31 + q^33 + 5 * q^37 + 4 * q^39 + 10 * q^41 + 8 * q^43 - 10 * q^47 + 9 * q^49 - 23 * q^51 + 5 * q^53 - 15 * q^57 + 6 * q^59 + 29 * q^61 + 40 * q^63 + 6 * q^67 - 30 * q^69 - q^71 + 18 * q^73 + q^77 - 20 * q^79 + 44 * q^81 + 26 * q^83 + 31 * q^87 + 21 * q^89 + 4 * q^91 + 25 * q^93 - 4 * q^97 + 7 * q^99

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