LMFDB - Embedded newform 4400.2.a.by.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4400,2,Mod(1,4400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4400, 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("4400.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4400 = 2^{4} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4400.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:	$35.1341768894$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.1229.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 7x + 6 $ x^3 - x^2 - 7*x + 6
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2200)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.75153$ of defining polynomial
Character	$\chi$	$=$	4400.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.75153 q^{3} -3.57093 q^{7} +4.57093 q^{9} +O(q^{10})$	$q-2.75153 q^{3} -3.57093 q^{7} +4.57093 q^{9} -1.00000 q^{11} +1.00000 q^{13} +0.751532 q^{17} +2.50306 q^{19} +9.82552 q^{21} -5.75153 q^{23} -4.32246 q^{27} -4.07399 q^{29} +6.14186 q^{31} +2.75153 q^{33} -2.81940 q^{37} -2.75153 q^{39} +1.18060 q^{41} -7.68367 q^{43} +9.82552 q^{47} +5.75153 q^{49} -2.06786 q^{51} +14.2159 q^{53} -6.88726 q^{57} +9.82552 q^{59} +7.07399 q^{61} -16.3225 q^{63} +14.6449 q^{67} +15.8255 q^{69} -5.81940 q^{71} -14.7128 q^{73} +3.57093 q^{77} +3.89339 q^{79} -1.81940 q^{81} +5.57706 q^{83} +11.2097 q^{87} +7.42907 q^{89} -3.57093 q^{91} -16.8995 q^{93} -0.609675 q^{97} -4.57093 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{3} - 3 q^{7} + 6 q^{9}+O(q^{10}) $ 3 * q - q^3 - 3 * q^7 + 6 * q^9	$ 3 q - q^{3} - 3 q^{7} + 6 q^{9} - 3 q^{11} + 3 q^{13} - 5 q^{17} - 7 q^{19} - 10 q^{23} + 2 q^{27} + 10 q^{29} + 3 q^{31} + q^{33} - 8 q^{37} - q^{39} + 4 q^{41} - 9 q^{43} + 10 q^{49} - 13 q^{51} + 5 q^{53} - 27 q^{57} - q^{61} - 34 q^{63} + 14 q^{67} + 18 q^{69} - 17 q^{71} - 21 q^{73} + 3 q^{77} - 11 q^{79} - 5 q^{81} - 20 q^{83} + 25 q^{87} + 30 q^{89} - 3 q^{91} + q^{93} - 10 q^{97} - 6 q^{99}+O(q^{100}) $ 3 * q - q^3 - 3 * q^7 + 6 * q^9 - 3 * q^11 + 3 * q^13 - 5 * q^17 - 7 * q^19 - 10 * q^23 + 2 * q^27 + 10 * q^29 + 3 * q^31 + q^33 - 8 * q^37 - q^39 + 4 * q^41 - 9 * q^43 + 10 * q^49 - 13 * q^51 + 5 * q^53 - 27 * q^57 - q^61 - 34 * q^63 + 14 * q^67 + 18 * q^69 - 17 * q^71 - 21 * q^73 + 3 * q^77 - 11 * q^79 - 5 * q^81 - 20 * q^83 + 25 * q^87 + 30 * q^89 - 3 * q^91 + q^93 - 10 * q^97 - 6 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	−2.75153	−1.58860	−0.794299	−	0.607527i	$-0.792162\pi$
$3$	−2.75153	−1.58860	−0.794299	+	0.607527i	$0.792162\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−3.57093	−1.34968	−0.674842	−	0.737962i	$-0.735788\pi$
$7$	−3.57093	−1.34968	−0.674842	+	0.737962i	$0.735788\pi$
$8$	0	0
$9$	4.57093	1.52364
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	1.00000	0.277350	0.138675	−	0.990338i	$-0.455716\pi$
$13$	1.00000	0.277350	0.138675	+	0.990338i	$0.455716\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0.751532	0.182273	0.0911367	−	0.995838i	$-0.470950\pi$
$17$	0.751532	0.182273	0.0911367	+	0.995838i	$0.470950\pi$
$18$	0	0
$19$	2.50306	0.574242	0.287121	−	0.957894i	$-0.407302\pi$
$19$	2.50306	0.574242	0.287121	+	0.957894i	$0.407302\pi$
$20$	0	0
$21$	9.82552	2.14411
$22$	0	0
$23$	−5.75153	−1.19928	−0.599639	−	0.800271i	$-0.704689\pi$
$23$	−5.75153	−1.19928	−0.599639	+	0.800271i	$0.704689\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−4.32246	−0.831858
$28$	0	0
$29$	−4.07399	−0.756521	−0.378261	−	0.925699i	$-0.623478\pi$
$29$	−4.07399	−0.756521	−0.378261	+	0.925699i	$0.623478\pi$
$30$	0	0
$31$	6.14186	1.10311	0.551555	−	0.834138i	$-0.314035\pi$
$31$	6.14186	1.10311	0.551555	+	0.834138i	$0.314035\pi$
$32$	0	0
$33$	2.75153	0.478980
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−2.81940	−0.463506	−0.231753	−	0.972775i	$-0.574446\pi$
$37$	−2.81940	−0.463506	−0.231753	+	0.972775i	$0.574446\pi$
$38$	0	0
$39$	−2.75153	−0.440598
$40$	0	0
$41$	1.18060	0.184379	0.0921896	−	0.995741i	$-0.470613\pi$
$41$	1.18060	0.184379	0.0921896	+	0.995741i	$0.470613\pi$
$42$	0	0
$43$	−7.68367	−1.17175	−0.585874	−	0.810402i	$-0.699249\pi$
$43$	−7.68367	−1.17175	−0.585874	+	0.810402i	$0.699249\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	9.82552	1.43320	0.716600	−	0.697484i	$-0.245697\pi$
$47$	9.82552	1.43320	0.716600	+	0.697484i	$0.245697\pi$
$48$	0	0
$49$	5.75153	0.821647
$50$	0	0
$51$	−2.06786	−0.289559
$52$	0	0
$53$	14.2159	1.95270	0.976349	−	0.216202i	$-0.0693670\pi$
$53$	14.2159	1.95270	0.976349	+	0.216202i	$0.0693670\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−6.88726	−0.912240
$58$	0	0
$59$	9.82552	1.27917	0.639587	−	0.768719i	$-0.279105\pi$
$59$	9.82552	1.27917	0.639587	+	0.768719i	$0.279105\pi$
$60$	0	0
$61$	7.07399	0.905732	0.452866	−	0.891579i	$-0.350402\pi$
$61$	7.07399	0.905732	0.452866	+	0.891579i	$0.350402\pi$
$62$	0	0
$63$	−16.3225	−2.05644
$64$	0	0
$65$	0	0
$66$	0	0
$67$	14.6449	1.78916	0.894581	−	0.446906i	$-0.147474\pi$
$67$	14.6449	1.78916	0.894581	+	0.446906i	$0.147474\pi$
$68$	0	0
$69$	15.8255	1.90517
$70$	0	0
$71$	−5.81940	−0.690635	−0.345318	−	0.938486i	$-0.612229\pi$
$71$	−5.81940	−0.690635	−0.345318	+	0.938486i	$0.612229\pi$
$72$	0	0
$73$	−14.7128	−1.72200	−0.861001	−	0.508604i	$-0.830162\pi$
$73$	−14.7128	−1.72200	−0.861001	+	0.508604i	$0.830162\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	3.57093	0.406945
$78$	0	0
$79$	3.89339	0.438041	0.219020	−	0.975720i	$-0.429714\pi$
$79$	3.89339	0.438041	0.219020	+	0.975720i	$0.429714\pi$
$80$	0	0
$81$	−1.81940	−0.202155
$82$	0	0
$83$	5.57706	0.612162	0.306081	−	0.952006i	$-0.400982\pi$
$83$	5.57706	0.612162	0.306081	+	0.952006i	$0.400982\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	11.2097	1.20181
$88$	0	0
$89$	7.42907	0.787480	0.393740	−	0.919222i	$-0.371181\pi$
$89$	7.42907	0.787480	0.393740	+	0.919222i	$0.371181\pi$
$90$	0	0
$91$	−3.57093	−0.374335
$92$	0	0
$93$	−16.8995	−1.75240
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−0.609675	−0.0619031	−0.0309515	−	0.999521i	$-0.509854\pi$
$97$	−0.609675	−0.0619031	−0.0309515	+	0.999521i	$0.509854\pi$
$98$	0	0
$99$	−4.57093	−0.459396
$100$	0	0
$101$	0.248468	0.0247235	0.0123617	−	0.999924i	$-0.496065\pi$
$101$	0.248468	0.0247235	0.0123617	+	0.999924i	$0.496065\pi$
$102$	0	0
$103$	−17.5771	−1.73192	−0.865959	−	0.500114i	$-0.833291\pi$
$103$	−17.5771	−1.73192	−0.865959	+	0.500114i	$0.833291\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−14.7868	−1.42949	−0.714746	−	0.699384i	$-0.753458\pi$
$107$	−14.7868	−1.42949	−0.714746	+	0.699384i	$0.753458\pi$
$108$	0	0
$109$	3.42907	0.328445	0.164223	−	0.986423i	$-0.447488\pi$
$109$	3.42907	0.328445	0.164223	+	0.986423i	$0.447488\pi$
$110$	0	0
$111$	7.75766	0.736325
$112$	0	0
$113$	−14.1867	−1.33458	−0.667288	−	0.744800i	$-0.732545\pi$
$113$	−14.1867	−1.33458	−0.667288	+	0.744800i	$0.732545\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	4.57093	0.422583
$118$	0	0
$119$	−2.68367	−0.246011
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−3.24847	−0.292904
$124$	0	0
$125$	0	0
$126$	0	0
$127$	18.8995	1.67706	0.838531	−	0.544855i	$-0.183415\pi$
$127$	18.8995	1.67706	0.838531	+	0.544855i	$0.183415\pi$
$128$	0	0
$129$	21.1419	1.86144
$130$	0	0
$131$	−20.5322	−1.79391	−0.896953	−	0.442127i	$-0.854224\pi$
$131$	−20.5322	−1.79391	−0.896953	+	0.442127i	$0.854224\pi$
$132$	0	0
$133$	−8.93826	−0.775046
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−3.29334	−0.281369	−0.140685	−	0.990054i	$-0.544930\pi$
$137$	−3.29334	−0.281369	−0.140685	+	0.990054i	$0.544930\pi$
$138$	0	0
$139$	−18.4256	−1.56284	−0.781418	−	0.624008i	$-0.785503\pi$
$139$	−18.4256	−1.56284	−0.781418	+	0.624008i	$0.785503\pi$
$140$	0	0
$141$	−27.0352	−2.27678
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−15.8255	−1.30527
$148$	0	0
$149$	8.36121	0.684977	0.342488	−	0.939522i	$-0.388730\pi$
$149$	8.36121	0.684977	0.342488	+	0.939522i	$0.388730\pi$
$150$	0	0
$151$	−5.04487	−0.410546	−0.205273	−	0.978705i	$-0.565808\pi$
$151$	−5.04487	−0.410546	−0.205273	+	0.978705i	$0.565808\pi$
$152$	0	0
$153$	3.43520	0.277719
$154$	0	0
$155$	0	0
$156$	0	0
$157$	12.2837	0.980347	0.490174	−	0.871625i	$-0.336933\pi$
$157$	12.2837	0.980347	0.490174	+	0.871625i	$0.336933\pi$
$158$	0	0
$159$	−39.1154	−3.10205
$160$	0	0
$161$	20.5383	1.61865
$162$	0	0
$163$	12.0801	0.946188	0.473094	−	0.881012i	$-0.343137\pi$
$163$	12.0801	0.946188	0.473094	+	0.881012i	$0.343137\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	20.8317	1.61200	0.806001	−	0.591914i	$-0.201628\pi$
$167$	20.8317	1.61200	0.806001	+	0.591914i	$0.201628\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	0	0
$171$	11.4413	0.874940
$172$	0	0
$173$	2.50306	0.190304	0.0951522	−	0.995463i	$-0.469666\pi$
$173$	2.50306	0.190304	0.0951522	+	0.995463i	$0.469666\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−27.0352	−2.03209
$178$	0	0
$179$	11.5709	0.864852	0.432426	−	0.901669i	$-0.357658\pi$
$179$	11.5709	0.864852	0.432426	+	0.901669i	$0.357658\pi$
$180$	0	0
$181$	−18.4413	−1.37073	−0.685367	−	0.728198i	$-0.740358\pi$
$181$	−18.4413	−1.37073	−0.685367	+	0.728198i	$0.740358\pi$
$182$	0	0
$183$	−19.4643	−1.43884
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−0.751532	−0.0549575
$188$	0	0
$189$	15.4352	1.12275
$190$	0	0
$191$	−3.29334	−0.238298	−0.119149	−	0.992876i	$-0.538017\pi$
$191$	−3.29334	−0.238298	−0.119149	+	0.992876i	$0.538017\pi$
$192$	0	0
$193$	−23.0061	−1.65602	−0.828009	−	0.560715i	$-0.810526\pi$
$193$	−23.0061	−1.65602	−0.828009	+	0.560715i	$0.810526\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	13.2546	0.944351	0.472175	−	0.881505i	$-0.343469\pi$
$197$	13.2546	0.944351	0.472175	+	0.881505i	$0.343469\pi$
$198$	0	0
$199$	−1.65455	−0.117288	−0.0586439	−	0.998279i	$-0.518678\pi$
$199$	−1.65455	−0.117288	−0.0586439	+	0.998279i	$0.518678\pi$
$200$	0	0
$201$	−40.2960	−2.84226
$202$	0	0
$203$	14.5479	1.02107
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−26.2898	−1.82727
$208$	0	0
$209$	−2.50306	−0.173141
$210$	0	0
$211$	20.9674	1.44345	0.721727	−	0.692178i	$-0.243349\pi$
$211$	20.9674	1.44345	0.721727	+	0.692178i	$0.243349\pi$
$212$	0	0
$213$	16.0123	1.09714
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−21.9321	−1.48885
$218$	0	0
$219$	40.4827	2.73557
$220$	0	0
$221$	0.751532	0.0505535
$222$	0	0
$223$	12.8317	0.859271	0.429636	−	0.903002i	$-0.358642\pi$
$223$	12.8317	0.859271	0.429636	+	0.903002i	$0.358642\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−9.42907	−0.625829	−0.312915	−	0.949781i	$-0.601305\pi$
$227$	−9.42907	−0.625829	−0.312915	+	0.949781i	$0.601305\pi$
$228$	0	0
$229$	−4.97701	−0.328890	−0.164445	−	0.986386i	$-0.552583\pi$
$229$	−4.97701	−0.328890	−0.164445	+	0.986386i	$0.552583\pi$
$230$	0	0
$231$	−9.82552	−0.646472
$232$	0	0
$233$	−3.93214	−0.257603	−0.128801	−	0.991670i	$-0.541113\pi$
$233$	−3.93214	−0.257603	−0.128801	+	0.991670i	$0.541113\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−10.7128	−0.695870
$238$	0	0
$239$	−27.7189	−1.79299	−0.896494	−	0.443056i	$-0.853894\pi$
$239$	−27.7189	−1.79299	−0.896494	+	0.443056i	$0.853894\pi$
$240$	0	0
$241$	12.1189	0.780645	0.390322	−	0.920678i	$-0.372364\pi$
$241$	12.1189	0.780645	0.390322	+	0.920678i	$0.372364\pi$
$242$	0	0
$243$	17.9735	1.15300
$244$	0	0
$245$	0	0
$246$	0	0
$247$	2.50306	0.159266
$248$	0	0
$249$	−15.3455	−0.972478
$250$	0	0
$251$	−14.1189	−0.891175	−0.445587	−	0.895238i	$-0.647005\pi$
$251$	−14.1189	−0.891175	−0.445587	+	0.895238i	$0.647005\pi$
$252$	0	0
$253$	5.75153	0.361596
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−13.6062	−0.848730	−0.424365	−	0.905491i	$-0.639503\pi$
$257$	−13.6062	−0.848730	−0.424365	+	0.905491i	$0.639503\pi$
$258$	0	0
$259$	10.0679	0.625587
$260$	0	0
$261$	−18.6219	−1.15267
$262$	0	0
$263$	−28.6959	−1.76947	−0.884733	−	0.466098i	$-0.845660\pi$
$263$	−28.6959	−1.76947	−0.884733	+	0.466098i	$0.845660\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−20.4413	−1.25099
$268$	0	0
$269$	−4.39033	−0.267683	−0.133841	−	0.991003i	$-0.542731\pi$
$269$	−4.39033	−0.267683	−0.133841	+	0.991003i	$0.542731\pi$
$270$	0	0
$271$	−18.8317	−1.14394	−0.571971	−	0.820274i	$-0.693821\pi$
$271$	−18.8317	−1.14394	−0.571971	+	0.820274i	$0.693821\pi$
$272$	0	0
$273$	9.82552	0.594668
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−22.1092	−1.32842	−0.664208	−	0.747548i	$-0.731231\pi$
$277$	−22.1092	−1.32842	−0.664208	+	0.747548i	$0.731231\pi$
$278$	0	0
$279$	28.0740	1.68075
$280$	0	0
$281$	−8.18673	−0.488379	−0.244190	−	0.969727i	$-0.578522\pi$
$281$	−8.18673	−0.488379	−0.244190	+	0.969727i	$0.578522\pi$
$282$	0	0
$283$	−23.8934	−1.42031	−0.710157	−	0.704043i	$-0.751376\pi$
$283$	−23.8934	−1.42031	−0.710157	+	0.704043i	$0.751376\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−4.21585	−0.248854
$288$	0	0
$289$	−16.4352	−0.966776
$290$	0	0
$291$	1.67754	0.0983391
$292$	0	0
$293$	−6.99387	−0.408586	−0.204293	−	0.978910i	$-0.565490\pi$
$293$	−6.99387	−0.408586	−0.204293	+	0.978910i	$0.565490\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	4.32246	0.250815
$298$	0	0
$299$	−5.75153	−0.332620
$300$	0	0
$301$	27.4378	1.58149
$302$	0	0
$303$	−0.683667	−0.0392757
$304$	0	0
$305$	0	0
$306$	0	0
$307$	11.0352	0.629815	0.314907	−	0.949122i	$-0.398027\pi$
$307$	11.0352	0.629815	0.314907	+	0.949122i	$0.398027\pi$
$308$	0	0
$309$	48.3638	2.75132
$310$	0	0
$311$	−23.6449	−1.34078	−0.670390	−	0.742009i	$-0.733873\pi$
$311$	−23.6449	−1.34078	−0.670390	+	0.742009i	$0.733873\pi$
$312$	0	0
$313$	−3.50306	−0.198005	−0.0990024	−	0.995087i	$-0.531565\pi$
$313$	−3.50306	−0.198005	−0.0990024	+	0.995087i	$0.531565\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.10048	0.0618092	0.0309046	−	0.999522i	$-0.490161\pi$
$317$	1.10048	0.0618092	0.0309046	+	0.999522i	$0.490161\pi$
$318$	0	0
$319$	4.07399	0.228100
$320$	0	0
$321$	40.6863	2.27089
$322$	0	0
$323$	1.88113	0.104669
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−9.43520	−0.521768
$328$	0	0
$329$	−35.0862	−1.93437
$330$	0	0
$331$	12.3225	0.677304	0.338652	−	0.940912i	$-0.390029\pi$
$331$	12.3225	0.677304	0.338652	+	0.940912i	$0.390029\pi$
$332$	0	0
$333$	−12.8873	−0.706218
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−10.0000	−0.544735	−0.272367	−	0.962193i	$-0.587807\pi$
$337$	−10.0000	−0.544735	−0.272367	+	0.962193i	$0.587807\pi$
$338$	0	0
$339$	39.0352	2.12010
$340$	0	0
$341$	−6.14186	−0.332600
$342$	0	0
$343$	4.45819	0.240720
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−15.5709	−0.835891	−0.417946	−	0.908472i	$-0.637250\pi$
$347$	−15.5709	−0.835891	−0.417946	+	0.908472i	$0.637250\pi$
$348$	0	0
$349$	−12.1541	−0.650595	−0.325297	−	0.945612i	$-0.605464\pi$
$349$	−12.1541	−0.650595	−0.325297	+	0.945612i	$0.605464\pi$
$350$	0	0
$351$	−4.32246	−0.230716
$352$	0	0
$353$	−18.0061	−0.958370	−0.479185	−	0.877714i	$-0.659068\pi$
$353$	−18.0061	−0.958370	−0.479185	+	0.877714i	$0.659068\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	7.38420	0.390813
$358$	0	0
$359$	7.35508	0.388186	0.194093	−	0.980983i	$-0.437824\pi$
$359$	7.35508	0.388186	0.194093	+	0.980983i	$0.437824\pi$
$360$	0	0
$361$	−12.7347	−0.670246
$362$	0	0
$363$	−2.75153	−0.144418
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−28.5322	−1.48937	−0.744684	−	0.667417i	$-0.767400\pi$
$367$	−28.5322	−1.48937	−0.744684	+	0.667417i	$0.767400\pi$
$368$	0	0
$369$	5.39645	0.280928
$370$	0	0
$371$	−50.7638	−2.63552
$372$	0	0
$373$	18.0510	0.934645	0.467323	−	0.884087i	$-0.345219\pi$
$373$	18.0510	0.934645	0.467323	+	0.884087i	$0.345219\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−4.07399	−0.209821
$378$	0	0
$379$	2.32246	0.119297	0.0596484	−	0.998219i	$-0.481002\pi$
$379$	2.32246	0.119297	0.0596484	+	0.998219i	$0.481002\pi$
$380$	0	0
$381$	−52.0026	−2.66418
$382$	0	0
$383$	5.67141	0.289796	0.144898	−	0.989447i	$-0.453715\pi$
$383$	5.67141	0.289796	0.144898	+	0.989447i	$0.453715\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−35.1215	−1.78533
$388$	0	0
$389$	−26.7347	−1.35550	−0.677751	−	0.735292i	$-0.737045\pi$
$389$	−26.7347	−1.35550	−0.677751	+	0.735292i	$0.737045\pi$
$390$	0	0
$391$	−4.32246	−0.218596
$392$	0	0
$393$	56.4950	2.84979
$394$	0	0
$395$	0	0
$396$	0	0
$397$	9.89339	0.496535	0.248267	−	0.968692i	$-0.420139\pi$
$397$	9.89339	0.496535	0.248267	+	0.968692i	$0.420139\pi$
$398$	0	0
$399$	24.5939	1.23124
$400$	0	0
$401$	−27.2572	−1.36116	−0.680580	−	0.732673i	$-0.738272\pi$
$401$	−27.2572	−1.36116	−0.680580	+	0.732673i	$0.738272\pi$
$402$	0	0
$403$	6.14186	0.305948
$404$	0	0
$405$	0	0
$406$	0	0
$407$	2.81940	0.139752
$408$	0	0
$409$	−2.03875	−0.100809	−0.0504047	−	0.998729i	$-0.516051\pi$
$409$	−2.03875	−0.100809	−0.0504047	+	0.998729i	$0.516051\pi$
$410$	0	0
$411$	9.06174	0.446982
$412$	0	0
$413$	−35.0862	−1.72648
$414$	0	0
$415$	0	0
$416$	0	0
$417$	50.6986	2.48272
$418$	0	0
$419$	−0.310204	−0.0151545	−0.00757723	−	0.999971i	$-0.502412\pi$
$419$	−0.310204	−0.0151545	−0.00757723	+	0.999971i	$0.502412\pi$
$420$	0	0
$421$	−8.28109	−0.403595	−0.201798	−	0.979427i	$-0.564678\pi$
$421$	−8.28109	−0.403595	−0.201798	+	0.979427i	$0.564678\pi$
$422$	0	0
$423$	44.9118	2.18369
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−25.2607	−1.22245
$428$	0	0
$429$	2.75153	0.132845
$430$	0	0
$431$	−4.45819	−0.214743	−0.107372	−	0.994219i	$-0.534243\pi$
$431$	−4.45819	−0.214743	−0.107372	+	0.994219i	$0.534243\pi$
$432$	0	0
$433$	22.9348	1.10217	0.551087	−	0.834448i	$-0.314213\pi$
$433$	22.9348	1.10217	0.551087	+	0.834448i	$0.314213\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−14.3965	−0.688676
$438$	0	0
$439$	0.964753	0.0460451	0.0230226	−	0.999735i	$-0.492671\pi$
$439$	0.964753	0.0460451	0.0230226	+	0.999735i	$0.492671\pi$
$440$	0	0
$441$	26.2898	1.25190
$442$	0	0
$443$	36.5092	1.73460	0.867302	−	0.497782i	$-0.165852\pi$
$443$	36.5092	1.73460	0.867302	+	0.497782i	$0.165852\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−23.0061	−1.08815
$448$	0	0
$449$	8.68017	0.409642	0.204821	−	0.978799i	$-0.434339\pi$
$449$	8.68017	0.409642	0.204821	+	0.978799i	$0.434339\pi$
$450$	0	0
$451$	−1.18060	−0.0555924
$452$	0	0
$453$	13.8811	0.652193
$454$	0	0
$455$	0	0
$456$	0	0
$457$	11.4934	0.537640	0.268820	−	0.963190i	$-0.413366\pi$
$457$	11.4934	0.537640	0.268820	+	0.963190i	$0.413366\pi$
$458$	0	0
$459$	−3.24847	−0.151625
$460$	0	0
$461$	25.0571	1.16703	0.583513	−	0.812103i	$-0.301678\pi$
$461$	25.0571	1.16703	0.583513	+	0.812103i	$0.301678\pi$
$462$	0	0
$463$	−32.7868	−1.52373	−0.761865	−	0.647735i	$-0.775716\pi$
$463$	−32.7868	−1.52373	−0.761865	+	0.647735i	$0.775716\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	41.7603	1.93244	0.966218	−	0.257727i	$-0.0829734\pi$
$467$	41.7603	1.93244	0.966218	+	0.257727i	$0.0829734\pi$
$468$	0	0
$469$	−52.2960	−2.41480
$470$	0	0
$471$	−33.7990	−1.55738
$472$	0	0
$473$	7.68367	0.353295
$474$	0	0
$475$	0	0
$476$	0	0
$477$	64.9796	2.97521
$478$	0	0
$479$	−3.96125	−0.180994	−0.0904972	−	0.995897i	$-0.528846\pi$
$479$	−3.96125	−0.180994	−0.0904972	+	0.995897i	$0.528846\pi$
$480$	0	0
$481$	−2.81940	−0.128553
$482$	0	0
$483$	−56.5118	−2.57138
$484$	0	0
$485$	0	0
$486$	0	0
$487$	22.7990	1.03312	0.516561	−	0.856250i	$-0.327212\pi$
$487$	22.7990	1.03312	0.516561	+	0.856250i	$0.327212\pi$
$488$	0	0
$489$	−33.2388	−1.50311
$490$	0	0
$491$	22.4256	1.01205	0.506026	−	0.862518i	$-0.331114\pi$
$491$	22.4256	1.01205	0.506026	+	0.862518i	$0.331114\pi$
$492$	0	0
$493$	−3.06174	−0.137894
$494$	0	0
$495$	0	0
$496$	0	0
$497$	20.7807	0.932140
$498$	0	0
$499$	23.9321	1.07135	0.535675	−	0.844424i	$-0.320057\pi$
$499$	23.9321	1.07135	0.535675	+	0.844424i	$0.320057\pi$
$500$	0	0
$501$	−57.3190	−2.56082
$502$	0	0
$503$	23.0766	1.02894	0.514468	−	0.857510i	$-0.327989\pi$
$503$	23.0766	1.02894	0.514468	+	0.857510i	$0.327989\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	33.0184	1.46640
$508$	0	0
$509$	21.2220	0.940648	0.470324	−	0.882494i	$-0.344137\pi$
$509$	21.2220	0.940648	0.470324	+	0.882494i	$0.344137\pi$
$510$	0	0
$511$	52.5383	2.32416
$512$	0	0
$513$	−10.8194	−0.477688
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−9.82552	−0.432126
$518$	0	0
$519$	−6.88726	−0.302317
$520$	0	0
$521$	16.1806	0.708885	0.354443	−	0.935078i	$-0.384671\pi$
$521$	16.1806	0.708885	0.354443	+	0.935078i	$0.384671\pi$
$522$	0	0
$523$	−31.8899	−1.39445	−0.697224	−	0.716854i	$-0.745582\pi$
$523$	−31.8899	−1.39445	−0.697224	+	0.716854i	$0.745582\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	4.61580	0.201068
$528$	0	0
$529$	10.0801	0.438266
$530$	0	0
$531$	44.9118	1.94901
$532$	0	0
$533$	1.18060	0.0511376
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−31.8378	−1.37390
$538$	0	0
$539$	−5.75153	−0.247736
$540$	0	0
$541$	12.4739	0.536297	0.268148	−	0.963378i	$-0.413588\pi$
$541$	12.4739	0.536297	0.268148	+	0.963378i	$0.413588\pi$
$542$	0	0
$543$	50.7419	2.17754
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−12.1250	−0.518427	−0.259214	−	0.965820i	$-0.583463\pi$
$547$	−12.1250	−0.518427	−0.259214	+	0.965820i	$0.583463\pi$
$548$	0	0
$549$	32.3347	1.38001
$550$	0	0
$551$	−10.1975	−0.434427
$552$	0	0
$553$	−13.9030	−0.591216
$554$	0	0
$555$	0	0
$556$	0	0
$557$	28.3673	1.20196	0.600981	−	0.799263i	$-0.294777\pi$
$557$	28.3673	1.20196	0.600981	+	0.799263i	$0.294777\pi$
$558$	0	0
$559$	−7.68367	−0.324985
$560$	0	0
$561$	2.06786	0.0873053
$562$	0	0
$563$	−4.08012	−0.171957	−0.0859783	−	0.996297i	$-0.527402\pi$
$563$	−4.08012	−0.171957	−0.0859783	+	0.996297i	$0.527402\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	6.49694	0.272846
$568$	0	0
$569$	−23.0088	−0.964577	−0.482289	−	0.876012i	$-0.660194\pi$
$569$	−23.0088	−0.964577	−0.482289	+	0.876012i	$0.660194\pi$
$570$	0	0
$571$	34.5322	1.44513	0.722563	−	0.691305i	$-0.242964\pi$
$571$	34.5322	1.44513	0.722563	+	0.691305i	$0.242964\pi$
$572$	0	0
$573$	9.06174	0.378559
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−25.1710	−1.04788	−0.523941	−	0.851755i	$-0.675539\pi$
$577$	−25.1710	−1.04788	−0.523941	+	0.851755i	$0.675539\pi$
$578$	0	0
$579$	63.3021	2.63075
$580$	0	0
$581$	−19.9153	−0.826225
$582$	0	0
$583$	−14.2159	−0.588760
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−10.2546	−0.423252	−0.211626	−	0.977351i	$-0.567876\pi$
$587$	−10.2546	−0.423252	−0.211626	+	0.977351i	$0.567876\pi$
$588$	0	0
$589$	15.3735	0.633453
$590$	0	0
$591$	−36.4704	−1.50019
$592$	0	0
$593$	−10.9091	−0.447985	−0.223992	−	0.974591i	$-0.571909\pi$
$593$	−10.9091	−0.447985	−0.223992	+	0.974591i	$0.571909\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	4.55254	0.186323
$598$	0	0
$599$	−29.2607	−1.19556	−0.597780	−	0.801660i	$-0.703951\pi$
$599$	−29.2607	−1.19556	−0.597780	+	0.801660i	$0.703951\pi$
$600$	0	0
$601$	−27.2220	−1.11041	−0.555204	−	0.831714i	$-0.687360\pi$
$601$	−27.2220	−1.11041	−0.555204	+	0.831714i	$0.687360\pi$
$602$	0	0
$603$	66.9409	2.72604
$604$	0	0
$605$	0	0
$606$	0	0
$607$	32.7347	1.32866	0.664330	−	0.747440i	$-0.268717\pi$
$607$	32.7347	1.32866	0.664330	+	0.747440i	$0.268717\pi$
$608$	0	0
$609$	−40.0291	−1.62206
$610$	0	0
$611$	9.82552	0.397498
$612$	0	0
$613$	−14.1894	−0.573103	−0.286551	−	0.958065i	$-0.592509\pi$
$613$	−14.1894	−0.573103	−0.286551	+	0.958065i	$0.592509\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	24.4766	0.985390	0.492695	−	0.870202i	$-0.336012\pi$
$617$	24.4766	0.985390	0.492695	+	0.870202i	$0.336012\pi$
$618$	0	0
$619$	−31.7603	−1.27655	−0.638277	−	0.769807i	$-0.720353\pi$
$619$	−31.7603	−1.27655	−0.638277	+	0.769807i	$0.720353\pi$
$620$	0	0
$621$	24.8608	0.997628
$622$	0	0
$623$	−26.5287	−1.06285
$624$	0	0
$625$	0	0
$626$	0	0
$627$	6.88726	0.275051
$628$	0	0
$629$	−2.11887	−0.0844848
$630$	0	0
$631$	24.0219	0.956296	0.478148	−	0.878279i	$-0.341308\pi$
$631$	24.0219	0.956296	0.478148	+	0.878279i	$0.341308\pi$
$632$	0	0
$633$	−57.6924	−2.29307
$634$	0	0
$635$	0	0
$636$	0	0
$637$	5.75153	0.227884
$638$	0	0
$639$	−26.6000	−1.05228
$640$	0	0
$641$	16.0061	0.632204	0.316102	−	0.948725i	$-0.397626\pi$
$641$	16.0061	0.632204	0.316102	+	0.948725i	$0.397626\pi$
$642$	0	0
$643$	11.0836	0.437095	0.218548	−	0.975826i	$-0.429868\pi$
$643$	11.0836	0.437095	0.218548	+	0.975826i	$0.429868\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−29.0449	−1.14187	−0.570936	−	0.820995i	$-0.693420\pi$
$647$	−29.0449	−1.14187	−0.570936	+	0.820995i	$0.693420\pi$
$648$	0	0
$649$	−9.82552	−0.385686
$650$	0	0
$651$	60.3470	2.36518
$652$	0	0
$653$	−50.3251	−1.96937	−0.984686	−	0.174335i	$-0.944223\pi$
$653$	−50.3251	−1.96937	−0.984686	+	0.174335i	$0.944223\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−67.2511	−2.62372
$658$	0	0
$659$	−26.7189	−1.04082	−0.520411	−	0.853916i	$-0.674221\pi$
$659$	−26.7189	−1.04082	−0.520411	+	0.853916i	$0.674221\pi$
$660$	0	0
$661$	−4.83165	−0.187930	−0.0939648	−	0.995576i	$-0.529954\pi$
$661$	−4.83165	−0.187930	−0.0939648	+	0.995576i	$0.529954\pi$
$662$	0	0
$663$	−2.06786	−0.0803092
$664$	0	0
$665$	0	0
$666$	0	0
$667$	23.4317	0.907279
$668$	0	0
$669$	−35.3067	−1.36504
$670$	0	0
$671$	−7.07399	−0.273088
$672$	0	0
$673$	−20.7542	−0.800014	−0.400007	−	0.916512i	$-0.630992\pi$
$673$	−20.7542	−0.800014	−0.400007	+	0.916512i	$0.630992\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	5.53568	0.212754	0.106377	−	0.994326i	$-0.466075\pi$
$677$	5.53568	0.212754	0.106377	+	0.994326i	$0.466075\pi$
$678$	0	0
$679$	2.17710	0.0835496
$680$	0	0
$681$	25.9444	0.994191
$682$	0	0
$683$	41.0256	1.56980	0.784901	−	0.619621i	$-0.212714\pi$
$683$	41.0256	1.56980	0.784901	+	0.619621i	$0.212714\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	13.6944	0.522474
$688$	0	0
$689$	14.2159	0.541581
$690$	0	0
$691$	−51.5180	−1.95984	−0.979918	−	0.199403i	$-0.936100\pi$
$691$	−51.5180	−1.95984	−0.979918	+	0.199403i	$0.936100\pi$
$692$	0	0
$693$	16.3225	0.620039
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0.887261	0.0336074
$698$	0	0
$699$	10.8194	0.409227
$700$	0	0
$701$	19.2450	0.726872	0.363436	−	0.931619i	$-0.381603\pi$
$701$	19.2450	0.726872	0.363436	+	0.931619i	$0.381603\pi$
$702$	0	0
$703$	−7.05713	−0.266165
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−0.887261	−0.0333689
$708$	0	0
$709$	−8.99387	−0.337772	−0.168886	−	0.985636i	$-0.554017\pi$
$709$	−8.99387	−0.337772	−0.168886	+	0.985636i	$0.554017\pi$
$710$	0	0
$711$	17.7964	0.667417
$712$	0	0
$713$	−35.3251	−1.32294
$714$	0	0
$715$	0	0
$716$	0	0
$717$	76.2695	2.84834
$718$	0	0
$719$	10.6837	0.398434	0.199217	−	0.979955i	$-0.436160\pi$
$719$	10.6837	0.398434	0.199217	+	0.979955i	$0.436160\pi$
$720$	0	0
$721$	62.7664	2.33754
$722$	0	0
$723$	−33.3455	−1.24013
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−1.98424	−0.0735916	−0.0367958	−	0.999323i	$-0.511715\pi$
$727$	−1.98424	−0.0735916	−0.0367958	+	0.999323i	$0.511715\pi$
$728$	0	0
$729$	−43.9965	−1.62950
$730$	0	0
$731$	−5.77452	−0.213578
$732$	0	0
$733$	−32.8765	−1.21432	−0.607161	−	0.794579i	$-0.707692\pi$
$733$	−32.8765	−1.21432	−0.607161	+	0.794579i	$0.707692\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−14.6449	−0.539453
$738$	0	0
$739$	−31.4352	−1.15636	−0.578181	−	0.815908i	$-0.696237\pi$
$739$	−31.4352	−1.15636	−0.578181	+	0.815908i	$0.696237\pi$
$740$	0	0
$741$	−6.88726	−0.253010
$742$	0	0
$743$	28.3638	1.04057	0.520284	−	0.853993i	$-0.325826\pi$
$743$	28.3638	1.04057	0.520284	+	0.853993i	$0.325826\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	25.4923	0.932716
$748$	0	0
$749$	52.8025	1.92936
$750$	0	0
$751$	17.4291	0.635996	0.317998	−	0.948091i	$-0.396989\pi$
$751$	17.4291	0.635996	0.317998	+	0.948091i	$0.396989\pi$
$752$	0	0
$753$	38.8485	1.41572
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−21.8838	−0.795379	−0.397689	−	0.917520i	$-0.630188\pi$
$757$	−21.8838	−0.795379	−0.397689	+	0.917520i	$0.630188\pi$
$758$	0	0
$759$	−15.8255	−0.574430
$760$	0	0
$761$	32.7224	1.18619	0.593093	−	0.805134i	$-0.297907\pi$
$761$	32.7224	1.18619	0.593093	+	0.805134i	$0.297907\pi$
$762$	0	0
$763$	−12.2450	−0.443298
$764$	0	0
$765$	0	0
$766$	0	0
$767$	9.82552	0.354779
$768$	0	0
$769$	−9.54181	−0.344086	−0.172043	−	0.985089i	$-0.555037\pi$
$769$	−9.54181	−0.344086	−0.172043	+	0.985089i	$0.555037\pi$
$770$	0	0
$771$	37.4378	1.34829
$772$	0	0
$773$	−0.0556080	−0.00200008	−0.00100004	−	0.999999i	$-0.500318\pi$
$773$	−0.0556080	−0.00200008	−0.00100004	+	0.999999i	$0.500318\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−27.7021	−0.993806
$778$	0	0
$779$	2.95513	0.105878
$780$	0	0
$781$	5.81940	0.208234
$782$	0	0
$783$	17.6097	0.629318
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−22.0970	−0.787672	−0.393836	−	0.919181i	$-0.628852\pi$
$787$	−22.0970	−0.787672	−0.393836	+	0.919181i	$0.628852\pi$
$788$	0	0
$789$	78.9578	2.81097
$790$	0	0
$791$	50.6598	1.80126
$792$	0	0
$793$	7.07399	0.251205
$794$	0	0
$795$	0	0
$796$	0	0
$797$	35.7577	1.26660	0.633301	−	0.773906i	$-0.281700\pi$
$797$	35.7577	1.26660	0.633301	+	0.773906i	$0.281700\pi$
$798$	0	0
$799$	7.38420	0.261234
$800$	0	0
$801$	33.9578	1.19984
$802$	0	0
$803$	14.7128	0.519203
$804$	0	0
$805$	0	0
$806$	0	0
$807$	12.0801	0.425240
$808$	0	0
$809$	24.9286	0.876444	0.438222	−	0.898867i	$-0.355608\pi$
$809$	24.9286	0.876444	0.438222	+	0.898867i	$0.355608\pi$
$810$	0	0
$811$	−29.6123	−1.03983	−0.519914	−	0.854218i	$-0.674036\pi$
$811$	−29.6123	−1.03983	−0.519914	+	0.854218i	$0.674036\pi$
$812$	0	0
$813$	51.8159	1.81726
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−19.2327	−0.672867
$818$	0	0
$819$	−16.3225	−0.570353
$820$	0	0
$821$	0.328589	0.0114678	0.00573392	−	0.999984i	$-0.498175\pi$
$821$	0.328589	0.0114678	0.00573392	+	0.999984i	$0.498175\pi$
$822$	0	0
$823$	42.5797	1.48423	0.742117	−	0.670270i	$-0.233822\pi$
$823$	42.5797	1.48423	0.742117	+	0.670270i	$0.233822\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	23.1867	0.806282	0.403141	−	0.915138i	$-0.367918\pi$
$827$	23.1867	0.806282	0.403141	+	0.915138i	$0.367918\pi$
$828$	0	0
$829$	−36.2546	−1.25917	−0.629587	−	0.776930i	$-0.716776\pi$
$829$	−36.2546	−1.25917	−0.629587	+	0.776930i	$0.716776\pi$
$830$	0	0
$831$	60.8343	2.11032
$832$	0	0
$833$	4.32246	0.149764
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−26.5479	−0.917631
$838$	0	0
$839$	13.6607	0.471619	0.235809	−	0.971799i	$-0.424226\pi$
$839$	13.6607	0.471619	0.235809	+	0.971799i	$0.424226\pi$
$840$	0	0
$841$	−12.4026	−0.427675
$842$	0	0
$843$	22.5261	0.775839
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−3.57093	−0.122699
$848$	0	0
$849$	65.7434	2.25631
$850$	0	0
$851$	16.2159	0.555872
$852$	0	0
$853$	54.2476	1.85740	0.928701	−	0.370829i	$-0.120926\pi$
$853$	54.2476	1.85740	0.928701	+	0.370829i	$0.120926\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	6.41221	0.219037	0.109518	−	0.993985i	$-0.465069\pi$
$857$	6.41221	0.219037	0.109518	+	0.993985i	$0.465069\pi$
$858$	0	0
$859$	−51.5031	−1.75726	−0.878631	−	0.477501i	$-0.841543\pi$
$859$	−51.5031	−1.75726	−0.878631	+	0.477501i	$0.841543\pi$
$860$	0	0
$861$	11.6000	0.395329
$862$	0	0
$863$	43.2837	1.47339	0.736697	−	0.676223i	$-0.236384\pi$
$863$	43.2837	1.47339	0.736697	+	0.676223i	$0.236384\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	45.2220	1.53582
$868$	0	0
$869$	−3.89339	−0.132074
$870$	0	0
$871$	14.6449	0.496224
$872$	0	0
$873$	−2.78678	−0.0943182
$874$	0	0
$875$	0	0
$876$	0	0
$877$	10.2511	0.346155	0.173077	−	0.984908i	$-0.444629\pi$
$877$	10.2511	0.346155	0.173077	+	0.984908i	$0.444629\pi$
$878$	0	0
$879$	19.2439	0.649079
$880$	0	0
$881$	8.19023	0.275936	0.137968	−	0.990437i	$-0.455943\pi$
$881$	8.19023	0.275936	0.137968	+	0.990437i	$0.455943\pi$
$882$	0	0
$883$	−6.76116	−0.227531	−0.113766	−	0.993508i	$-0.536291\pi$
$883$	−6.76116	−0.227531	−0.113766	+	0.993508i	$0.536291\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	10.3225	0.346594	0.173297	−	0.984870i	$-0.444558\pi$
$887$	10.3225	0.346594	0.173297	+	0.984870i	$0.444558\pi$
$888$	0	0
$889$	−67.4888	−2.26350
$890$	0	0
$891$	1.81940	0.0609521
$892$	0	0
$893$	24.5939	0.823004
$894$	0	0
$895$	0	0
$896$	0	0
$897$	15.8255	0.528399
$898$	0	0
$899$	−25.0219	−0.834527
$900$	0	0
$901$	10.6837	0.355925
$902$	0	0
$903$	−75.4961	−2.51235
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−11.9348	−0.396287	−0.198144	−	0.980173i	$-0.563491\pi$
$907$	−11.9348	−0.396287	−0.198144	+	0.980173i	$0.563491\pi$
$908$	0	0
$909$	1.13573	0.0376698
$910$	0	0
$911$	7.35508	0.243685	0.121842	−	0.992549i	$-0.461120\pi$
$911$	7.35508	0.243685	0.121842	+	0.992549i	$0.461120\pi$
$912$	0	0
$913$	−5.57706	−0.184574
$914$	0	0
$915$	0	0
$916$	0	0
$917$	73.3190	2.42121
$918$	0	0
$919$	25.0184	0.825280	0.412640	−	0.910894i	$-0.364607\pi$
$919$	25.0184	0.825280	0.412640	+	0.910894i	$0.364607\pi$
$920$	0	0
$921$	−30.3638	−1.00052
$922$	0	0
$923$	−5.81940	−0.191548
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−80.3435	−2.63883
$928$	0	0
$929$	4.42557	0.145198	0.0725992	−	0.997361i	$-0.476871\pi$
$929$	4.42557	0.145198	0.0725992	+	0.997361i	$0.476871\pi$
$930$	0	0
$931$	14.3965	0.471825
$932$	0	0
$933$	65.0598	2.12996
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−40.3612	−1.31854	−0.659272	−	0.751905i	$-0.729135\pi$
$937$	−40.3612	−1.31854	−0.659272	+	0.751905i	$0.729135\pi$
$938$	0	0
$939$	9.63879	0.314550
$940$	0	0
$941$	13.2704	0.432601	0.216301	−	0.976327i	$-0.430601\pi$
$941$	13.2704	0.432601	0.216301	+	0.976327i	$0.430601\pi$
$942$	0	0
$943$	−6.79028	−0.221122
$944$	0	0
$945$	0	0
$946$	0	0
$947$	4.81940	0.156609	0.0783047	−	0.996929i	$-0.475049\pi$
$947$	4.81940	0.156609	0.0783047	+	0.996929i	$0.475049\pi$
$948$	0	0
$949$	−14.7128	−0.477597
$950$	0	0
$951$	−3.02801	−0.0981900
$952$	0	0
$953$	16.0123	0.518688	0.259344	−	0.965785i	$-0.416494\pi$
$953$	16.0123	0.518688	0.259344	+	0.965785i	$0.416494\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−11.2097	−0.362359
$958$	0	0
$959$	11.7603	0.379760
$960$	0	0
$961$	6.72241	0.216852
$962$	0	0
$963$	−67.5893	−2.17804
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−50.6211	−1.62786	−0.813932	−	0.580960i	$-0.802677\pi$
$967$	−50.6211	−1.62786	−0.813932	+	0.580960i	$0.802677\pi$
$968$	0	0
$969$	−5.17600	−0.166277
$970$	0	0
$971$	−27.5249	−0.883318	−0.441659	−	0.897183i	$-0.645610\pi$
$971$	−27.5249	−0.883318	−0.441659	+	0.897183i	$0.645610\pi$
$972$	0	0
$973$	65.7964	2.10934
$974$	0	0
$975$	0	0
$976$	0	0
$977$	48.1215	1.53954	0.769772	−	0.638320i	$-0.220370\pi$
$977$	48.1215	1.53954	0.769772	+	0.638320i	$0.220370\pi$
$978$	0	0
$979$	−7.42907	−0.237434
$980$	0	0
$981$	15.6740	0.500434
$982$	0	0
$983$	−12.8960	−0.411319	−0.205660	−	0.978624i	$-0.565934\pi$
$983$	−12.8960	−0.411319	−0.205660	+	0.978624i	$0.565934\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	96.5409	3.07293
$988$	0	0
$989$	44.1929	1.40525
$990$	0	0
$991$	−1.53218	−0.0486714	−0.0243357	−	0.999704i	$-0.507747\pi$
$991$	−1.53218	−0.0486714	−0.0243357	+	0.999704i	$0.507747\pi$
$992$	0	0
$993$	−33.9056	−1.07596
$994$	0	0
$995$	0	0
$996$	0	0
$997$	31.9831	1.01292	0.506458	−	0.862265i	$-0.330954\pi$
$997$	31.9831	1.01292	0.506458	+	0.862265i	$0.330954\pi$
$998$	0	0
$999$	12.1867	0.385571

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4400.2.a.by.1.1		3
4.3	odd	2		2200.2.a.v.1.3	yes	3
5.2	odd	4		4400.2.b.bb.4049.6		6
5.3	odd	4		4400.2.b.bb.4049.1		6
5.4	even	2		4400.2.a.bz.1.3		3
20.3	even	4		2200.2.b.m.1849.6		6
20.7	even	4		2200.2.b.m.1849.1		6
20.19	odd	2		2200.2.a.u.1.1	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2200.2.a.u.1.1	✓	3	20.19	odd	2
2200.2.a.v.1.3	yes	3	4.3	odd	2
2200.2.b.m.1849.1		6	20.7	even	4
2200.2.b.m.1849.6		6	20.3	even	4
4400.2.a.by.1.1		3	1.1	even	1	trivial
4400.2.a.bz.1.3		3	5.4	even	2
4400.2.b.bb.4049.1		6	5.3	odd	4
4400.2.b.bb.4049.6		6	5.2	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:	\( N \)	\(=\)	\( 4400 = 2^{4} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4400.a (trivial)

\(f(q)\)	\(=\)	\(q-2.75153 q^{3} -3.57093 q^{7} +4.57093 q^{9} +O(q^{10})\)	\(q-2.75153 q^{3} -3.57093 q^{7} +4.57093 q^{9} -1.00000 q^{11} +1.00000 q^{13} +0.751532 q^{17} +2.50306 q^{19} +9.82552 q^{21} -5.75153 q^{23} -4.32246 q^{27} -4.07399 q^{29} +6.14186 q^{31} +2.75153 q^{33} -2.81940 q^{37} -2.75153 q^{39} +1.18060 q^{41} -7.68367 q^{43} +9.82552 q^{47} +5.75153 q^{49} -2.06786 q^{51} +14.2159 q^{53} -6.88726 q^{57} +9.82552 q^{59} +7.07399 q^{61} -16.3225 q^{63} +14.6449 q^{67} +15.8255 q^{69} -5.81940 q^{71} -14.7128 q^{73} +3.57093 q^{77} +3.89339 q^{79} -1.81940 q^{81} +5.57706 q^{83} +11.2097 q^{87} +7.42907 q^{89} -3.57093 q^{91} -16.8995 q^{93} -0.609675 q^{97} -4.57093 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{3} - 3 q^{7} + 6 q^{9}+O(q^{10}) \) 3 * q - q^3 - 3 * q^7 + 6 * q^9	\( 3 q - q^{3} - 3 q^{7} + 6 q^{9} - 3 q^{11} + 3 q^{13} - 5 q^{17} - 7 q^{19} - 10 q^{23} + 2 q^{27} + 10 q^{29} + 3 q^{31} + q^{33} - 8 q^{37} - q^{39} + 4 q^{41} - 9 q^{43} + 10 q^{49} - 13 q^{51} + 5 q^{53} - 27 q^{57} - q^{61} - 34 q^{63} + 14 q^{67} + 18 q^{69} - 17 q^{71} - 21 q^{73} + 3 q^{77} - 11 q^{79} - 5 q^{81} - 20 q^{83} + 25 q^{87} + 30 q^{89} - 3 q^{91} + q^{93} - 10 q^{97} - 6 q^{99}+O(q^{100}) \) 3 * q - q^3 - 3 * q^7 + 6 * q^9 - 3 * q^11 + 3 * q^13 - 5 * q^17 - 7 * q^19 - 10 * q^23 + 2 * q^27 + 10 * q^29 + 3 * q^31 + q^33 - 8 * q^37 - q^39 + 4 * q^41 - 9 * q^43 + 10 * q^49 - 13 * q^51 + 5 * q^53 - 27 * q^57 - q^61 - 34 * q^63 + 14 * q^67 + 18 * q^69 - 17 * q^71 - 21 * q^73 + 3 * q^77 - 11 * q^79 - 5 * q^81 - 20 * q^83 + 25 * q^87 + 30 * q^89 - 3 * q^91 + q^93 - 10 * q^97 - 6 * q^99

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