LMFDB - Embedded newform 4400.2.b.bb.4049.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4400,2,Mod(4049,4400)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4400.4049"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4400, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4400 = 2^{4} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4400.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [6,0,0,0,0,0,0,0,-12,0,-6,0,0,0,0,0,0,0,14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$35.1341768894$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.96668224.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 15x^{4} + 61x^{2} + 36 $ x^6 + 15x^4 + 61x^2 + 36
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2200)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			4049.1
Root			$-2.75153i$ of defining polynomial
Character	$\chi$	$=$	4400.4049
Dual form			4400.2.b.bb.4049.6

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.75153i q^{3} +3.57093i q^{7} -4.57093 q^{9} -1.00000 q^{11} +1.00000i q^{13} -0.751532i q^{17} -2.50306 q^{19} +9.82552 q^{21} -5.75153i q^{23} +4.32246i q^{27} +4.07399 q^{29} +6.14186 q^{31} +2.75153i q^{33} +2.81940i q^{37} +2.75153 q^{39} +1.18060 q^{41} -7.68367i q^{43} -9.82552i q^{47} -5.75153 q^{49} -2.06786 q^{51} +14.2159i q^{53} +6.88726i q^{57} -9.82552 q^{59} +7.07399 q^{61} -16.3225i q^{63} -14.6449i q^{67} -15.8255 q^{69} -5.81940 q^{71} -14.7128i q^{73} -3.57093i q^{77} -3.89339 q^{79} -1.81940 q^{81} +5.57706i q^{83} -11.2097i q^{87} -7.42907 q^{89} -3.57093 q^{91} -16.8995i q^{93} +0.609675i q^{97} +4.57093 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 12 q^{9} - 6 q^{11} + 14 q^{19} - 20 q^{29} + 6 q^{31} + 2 q^{39} + 8 q^{41} - 20 q^{49} - 26 q^{51} - 2 q^{61} - 36 q^{69} - 34 q^{71} + 22 q^{79} - 10 q^{81} - 60 q^{89} - 6 q^{91} + 12 q^{99}+O(q^{100}) $ 6 * q - 12 * q^9 - 6 * q^11 + 14 * q^19 - 20 * q^29 + 6 * q^31 + 2 * q^39 + 8 * q^41 - 20 * q^49 - 26 * q^51 - 2 * q^61 - 36 * q^69 - 34 * q^71 + 22 * q^79 - 10 * q^81 - 60 * q^89 - 6 * q^91 + 12 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4400\mathbb{Z}\right)^\times$.

$n$	$177$	$1201$	$2751$	$3301$
$\chi(n)$	$-1$	$1$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	− 2.75153i	− 1.58860i	−0.607527	−	0.794299i	$-0.707838\pi$
$3$	− 2.75153i	− 1.58860i	0.607527	−	0.794299i	$-0.292162\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3.57093i	1.34968i	0.737962	+	0.674842i	$0.235788\pi$
$7$	3.57093i	1.34968i	−0.737962	+	0.674842i	$0.764212\pi$
$8$	0	0
$9$	−4.57093	−1.52364
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	1.00000i	0.277350i	0.990338	+	0.138675i	$0.0442844\pi$
$13$	1.00000i	0.277350i	−0.990338	+	0.138675i	$0.955716\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 0.751532i	− 0.182273i	−0.995838	−	0.0911367i	$-0.970950\pi$
$17$	− 0.751532i	− 0.182273i	0.995838	−	0.0911367i	$-0.0290500\pi$
$18$	0	0
$19$	−2.50306	−0.574242	−0.287121	−	0.957894i	$-0.592698\pi$
$19$	−2.50306	−0.574242	−0.287121	+	0.957894i	$0.592698\pi$
$20$	0	0
$21$	9.82552	2.14411
$22$	0	0
$23$	− 5.75153i	− 1.19928i	−0.800271	−	0.599639i	$-0.795311\pi$
$23$	− 5.75153i	− 1.19928i	0.800271	−	0.599639i	$-0.204689\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	4.32246i	0.831858i
$28$	0	0
$29$	4.07399	0.756521	0.378261	−	0.925699i	$-0.376522\pi$
$29$	4.07399	0.756521	0.378261	+	0.925699i	$0.376522\pi$
$30$	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
$33$	2.75153i	0.478980i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	2.81940i	0.463506i	0.972775	+	0.231753i	$0.0744461\pi$
$37$	2.81940i	0.463506i	−0.972775	+	0.231753i	$0.925554\pi$
$38$	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
$43$	− 7.68367i	− 1.17175i	−0.810402	−	0.585874i	$-0.800751\pi$
$43$	− 7.68367i	− 1.17175i	0.810402	−	0.585874i	$-0.199249\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 9.82552i	− 1.43320i	−0.697484	−	0.716600i	$-0.745697\pi$
$47$	− 9.82552i	− 1.43320i	0.697484	−	0.716600i	$-0.254303\pi$
$48$	0	0
$49$	−5.75153	−0.821647
$50$	0	0
$51$	−2.06786	−0.289559
$52$	0	0
$53$	14.2159i	1.95270i	0.216202	+	0.976349i	$0.430633\pi$
$53$	14.2159i	1.95270i	−0.216202	+	0.976349i	$0.569367\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	6.88726i	0.912240i
$58$	0	0
$59$	−9.82552	−1.27917	−0.639587	−	0.768719i	$-0.720895\pi$
$59$	−9.82552	−1.27917	−0.639587	+	0.768719i	$0.720895\pi$
$60$	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
$63$	− 16.3225i	− 2.05644i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 14.6449i	− 1.78916i	−0.446906	−	0.894581i	$-0.647474\pi$
$67$	− 14.6449i	− 1.78916i	0.446906	−	0.894581i	$-0.352526\pi$
$68$	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
$73$	− 14.7128i	− 1.72200i	−0.508604	−	0.861001i	$-0.669838\pi$
$73$	− 14.7128i	− 1.72200i	0.508604	−	0.861001i	$-0.330162\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 3.57093i	− 0.406945i
$78$	0	0
$79$	−3.89339	−0.438041	−0.219020	−	0.975720i	$-0.570286\pi$
$79$	−3.89339	−0.438041	−0.219020	+	0.975720i	$0.570286\pi$
$80$	0	0
$81$	−1.81940	−0.202155
$82$	0	0
$83$	5.57706i	0.612162i	0.952006	+	0.306081i	$0.0990178\pi$
$83$	5.57706i	0.612162i	−0.952006	+	0.306081i	$0.900982\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 11.2097i	− 1.20181i
$88$	0	0
$89$	−7.42907	−0.787480	−0.393740	−	0.919222i	$-0.628819\pi$
$89$	−7.42907	−0.787480	−0.393740	+	0.919222i	$0.628819\pi$
$90$	0	0
$91$	−3.57093	−0.374335
$92$	0	0
$93$	− 16.8995i	− 1.75240i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	0.609675i	0.0619031i	0.999521	+	0.0309515i	$0.00985375\pi$
$97$	0.609675i	0.0619031i	−0.999521	+	0.0309515i	$0.990146\pi$
$98$	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
$103$	− 17.5771i	− 1.73192i	−0.500114	−	0.865959i	$-0.666709\pi$
$103$	− 17.5771i	− 1.73192i	0.500114	−	0.865959i	$-0.333291\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	14.7868i	1.42949i	0.699384	+	0.714746i	$0.253458\pi$
$107$	14.7868i	1.42949i	−0.699384	+	0.714746i	$0.746542\pi$
$108$	0	0
$109$	−3.42907	−0.328445	−0.164223	−	0.986423i	$-0.552512\pi$
$109$	−3.42907	−0.328445	−0.164223	+	0.986423i	$0.552512\pi$
$110$	0	0
$111$	7.75766	0.736325
$112$	0	0
$113$	− 14.1867i	− 1.33458i	−0.744800	−	0.667288i	$-0.767455\pi$
$113$	− 14.1867i	− 1.33458i	0.744800	−	0.667288i	$-0.232545\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 4.57093i	− 0.422583i
$118$	0	0
$119$	2.68367	0.246011
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	− 3.24847i	− 0.292904i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 18.8995i	− 1.67706i	−0.544855	−	0.838531i	$-0.683415\pi$
$127$	− 18.8995i	− 1.67706i	0.544855	−	0.838531i	$-0.316585\pi$
$128$	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
$133$	− 8.93826i	− 0.775046i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	3.29334i	0.281369i	0.990054	+	0.140685i	$0.0449304\pi$
$137$	3.29334i	0.281369i	−0.990054	+	0.140685i	$0.955070\pi$
$138$	0	0
$139$	18.4256	1.56284	0.781418	−	0.624008i	$-0.214497\pi$
$139$	18.4256	1.56284	0.781418	+	0.624008i	$0.214497\pi$
$140$	0	0
$141$	−27.0352	−2.27678
$142$	0	0
$143$	− 1.00000i	− 0.0836242i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	15.8255i	1.30527i
$148$	0	0
$149$	−8.36121	−0.684977	−0.342488	−	0.939522i	$-0.611270\pi$
$149$	−8.36121	−0.684977	−0.342488	+	0.939522i	$0.611270\pi$
$150$	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
$153$	3.43520i	0.277719i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 12.2837i	− 0.980347i	−0.871625	−	0.490174i	$-0.836933\pi$
$157$	− 12.2837i	− 0.980347i	0.871625	−	0.490174i	$-0.163067\pi$
$158$	0	0
$159$	39.1154	3.10205
$160$	0	0
$161$	20.5383	1.61865
$162$	0	0
$163$	12.0801i	0.946188i	0.881012	+	0.473094i	$0.156863\pi$
$163$	12.0801i	0.946188i	−0.881012	+	0.473094i	$0.843137\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 20.8317i	− 1.61200i	−0.591914	−	0.806001i	$-0.701628\pi$
$167$	− 20.8317i	− 1.61200i	0.591914	−	0.806001i	$-0.298372\pi$
$168$	0	0
$169$	12.0000	0.923077
$170$	0	0
$171$	11.4413	0.874940
$172$	0	0
$173$	2.50306i	0.190304i	0.995463	+	0.0951522i	$0.0303338\pi$
$173$	2.50306i	0.190304i	−0.995463	+	0.0951522i	$0.969666\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	27.0352i	2.03209i
$178$	0	0
$179$	−11.5709	−0.864852	−0.432426	−	0.901669i	$-0.642342\pi$
$179$	−11.5709	−0.864852	−0.432426	+	0.901669i	$0.642342\pi$
$180$	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
$183$	− 19.4643i	− 1.43884i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0.751532i	0.0549575i
$188$	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
$193$	− 23.0061i	− 1.65602i	−0.560715	−	0.828009i	$-0.689474\pi$
$193$	− 23.0061i	− 1.65602i	0.560715	−	0.828009i	$-0.310526\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 13.2546i	− 0.944351i	−0.881505	−	0.472175i	$-0.843469\pi$
$197$	− 13.2546i	− 0.944351i	0.881505	−	0.472175i	$-0.156531\pi$
$198$	0	0
$199$	1.65455	0.117288	0.0586439	−	0.998279i	$-0.481322\pi$
$199$	1.65455	0.117288	0.0586439	+	0.998279i	$0.481322\pi$
$200$	0	0
$201$	−40.2960	−2.84226
$202$	0	0
$203$	14.5479i	1.02107i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	26.2898i	1.82727i
$208$	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
$213$	16.0123i	1.09714i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	21.9321i	1.48885i
$218$	0	0
$219$	−40.4827	−2.73557
$220$	0	0
$221$	0.751532	0.0505535
$222$	0	0
$223$	12.8317i	0.859271i	0.903002	+	0.429636i	$0.141358\pi$
$223$	12.8317i	0.859271i	−0.903002	+	0.429636i	$0.858642\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	9.42907i	0.625829i	0.949781	+	0.312915i	$0.101305\pi$
$227$	9.42907i	0.625829i	−0.949781	+	0.312915i	$0.898695\pi$
$228$	0	0
$229$	4.97701	0.328890	0.164445	−	0.986386i	$-0.447417\pi$
$229$	4.97701	0.328890	0.164445	+	0.986386i	$0.447417\pi$
$230$	0	0
$231$	−9.82552	−0.646472
$232$	0	0
$233$	− 3.93214i	− 0.257603i	−0.991670	−	0.128801i	$-0.958887\pi$
$233$	− 3.93214i	− 0.257603i	0.991670	−	0.128801i	$-0.0411130\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	10.7128i	0.695870i
$238$	0	0
$239$	27.7189	1.79299	0.896494	−	0.443056i	$-0.146106\pi$
$239$	27.7189	1.79299	0.896494	+	0.443056i	$0.146106\pi$
$240$	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
$243$	17.9735i	1.15300i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 2.50306i	− 0.159266i
$248$	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
$253$	5.75153i	0.361596i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	13.6062i	0.848730i	0.905491	+	0.424365i	$0.139503\pi$
$257$	13.6062i	0.848730i	−0.905491	+	0.424365i	$0.860497\pi$
$258$	0	0
$259$	−10.0679	−0.625587
$260$	0	0
$261$	−18.6219	−1.15267
$262$	0	0
$263$	− 28.6959i	− 1.76947i	−0.466098	−	0.884733i	$-0.654340\pi$
$263$	− 28.6959i	− 1.76947i	0.466098	−	0.884733i	$-0.345660\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	20.4413i	1.25099i
$268$	0	0
$269$	4.39033	0.267683	0.133841	−	0.991003i	$-0.457269\pi$
$269$	4.39033	0.267683	0.133841	+	0.991003i	$0.457269\pi$
$270$	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
$273$	9.82552i	0.594668i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	22.1092i	1.32842i	0.747548	+	0.664208i	$0.231231\pi$
$277$	22.1092i	1.32842i	−0.747548	+	0.664208i	$0.768769\pi$
$278$	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
$283$	− 23.8934i	− 1.42031i	−0.704043	−	0.710157i	$-0.748624\pi$
$283$	− 23.8934i	− 1.42031i	0.704043	−	0.710157i	$-0.251376\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	4.21585i	0.248854i
$288$	0	0
$289$	16.4352	0.966776
$290$	0	0
$291$	1.67754	0.0983391
$292$	0	0
$293$	− 6.99387i	− 0.408586i	−0.978910	−	0.204293i	$-0.934510\pi$
$293$	− 6.99387i	− 0.408586i	0.978910	−	0.204293i	$-0.0654896\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 4.32246i	− 0.250815i
$298$	0	0
$299$	5.75153	0.332620
$300$	0	0
$301$	27.4378	1.58149
$302$	0	0
$303$	− 0.683667i	− 0.0392757i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 11.0352i	− 0.629815i	−0.949122	−	0.314907i	$-0.898027\pi$
$307$	− 11.0352i	− 0.629815i	0.949122	−	0.314907i	$-0.101973\pi$
$308$	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
$313$	− 3.50306i	− 0.198005i	−0.995087	−	0.0990024i	$-0.968435\pi$
$313$	− 3.50306i	− 0.198005i	0.995087	−	0.0990024i	$-0.0315652\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 1.10048i	− 0.0618092i	−0.999522	−	0.0309046i	$-0.990161\pi$
$317$	− 1.10048i	− 0.0618092i	0.999522	−	0.0309046i	$-0.00983881\pi$
$318$	0	0
$319$	−4.07399	−0.228100
$320$	0	0
$321$	40.6863	2.27089
$322$	0	0
$323$	1.88113i	0.104669i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	9.43520i	0.521768i
$328$	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
$333$	− 12.8873i	− 0.706218i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	10.0000i	0.544735i	0.962193	+	0.272367i	$0.0878066\pi$
$337$	10.0000i	0.544735i	−0.962193	+	0.272367i	$0.912193\pi$
$338$	0	0
$339$	−39.0352	−2.12010
$340$	0	0
$341$	−6.14186	−0.332600
$342$	0	0
$343$	4.45819i	0.240720i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	15.5709i	0.835891i	0.908472	+	0.417946i	$0.137250\pi$
$347$	15.5709i	0.835891i	−0.908472	+	0.417946i	$0.862750\pi$
$348$	0	0
$349$	12.1541	0.650595	0.325297	−	0.945612i	$-0.394536\pi$
$349$	12.1541	0.650595	0.325297	+	0.945612i	$0.394536\pi$
$350$	0	0
$351$	−4.32246	−0.230716
$352$	0	0
$353$	− 18.0061i	− 0.958370i	−0.877714	−	0.479185i	$-0.840932\pi$
$353$	− 18.0061i	− 0.958370i	0.877714	−	0.479185i	$-0.159068\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 7.38420i	− 0.390813i
$358$	0	0
$359$	−7.35508	−0.388186	−0.194093	−	0.980983i	$-0.562176\pi$
$359$	−7.35508	−0.388186	−0.194093	+	0.980983i	$0.562176\pi$
$360$	0	0
$361$	−12.7347	−0.670246
$362$	0	0
$363$	− 2.75153i	− 0.144418i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	28.5322i	1.48937i	0.667417	+	0.744684i	$0.267400\pi$
$367$	28.5322i	1.48937i	−0.667417	+	0.744684i	$0.732600\pi$
$368$	0	0
$369$	−5.39645	−0.280928
$370$	0	0
$371$	−50.7638	−2.63552
$372$	0	0
$373$	18.0510i	0.934645i	0.884087	+	0.467323i	$0.154781\pi$
$373$	18.0510i	0.934645i	−0.884087	+	0.467323i	$0.845219\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	4.07399i	0.209821i
$378$	0	0
$379$	−2.32246	−0.119297	−0.0596484	−	0.998219i	$-0.518998\pi$
$379$	−2.32246	−0.119297	−0.0596484	+	0.998219i	$0.518998\pi$
$380$	0	0
$381$	−52.0026	−2.66418
$382$	0	0
$383$	5.67141i	0.289796i	0.989447	+	0.144898i	$0.0462853\pi$
$383$	5.67141i	0.289796i	−0.989447	+	0.144898i	$0.953715\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	35.1215i	1.78533i
$388$	0	0
$389$	26.7347	1.35550	0.677751	−	0.735292i	$-0.262955\pi$
$389$	26.7347	1.35550	0.677751	+	0.735292i	$0.262955\pi$
$390$	0	0
$391$	−4.32246	−0.218596
$392$	0	0
$393$	56.4950i	2.84979i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 9.89339i	− 0.496535i	−0.968692	−	0.248267i	$-0.920139\pi$
$397$	− 9.89339i	− 0.496535i	0.968692	−	0.248267i	$-0.0798612\pi$
$398$	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
$403$	6.14186i	0.305948i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 2.81940i	− 0.139752i
$408$	0	0
$409$	2.03875	0.100809	0.0504047	−	0.998729i	$-0.483949\pi$
$409$	2.03875	0.100809	0.0504047	+	0.998729i	$0.483949\pi$
$410$	0	0
$411$	9.06174	0.446982
$412$	0	0
$413$	− 35.0862i	− 1.72648i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 50.6986i	− 2.48272i
$418$	0	0
$419$	0.310204	0.0151545	0.00757723	−	0.999971i	$-0.497588\pi$
$419$	0.310204	0.0151545	0.00757723	+	0.999971i	$0.497588\pi$
$420$	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
$423$	44.9118i	2.18369i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	25.2607i	1.22245i
$428$	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
$433$	22.9348i	1.10217i	0.834448	+	0.551087i	$0.185787\pi$
$433$	22.9348i	1.10217i	−0.834448	+	0.551087i	$0.814213\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	14.3965i	0.688676i
$438$	0	0
$439$	−0.964753	−0.0460451	−0.0230226	−	0.999735i	$-0.507329\pi$
$439$	−0.964753	−0.0460451	−0.0230226	+	0.999735i	$0.507329\pi$
$440$	0	0
$441$	26.2898	1.25190
$442$	0	0
$443$	36.5092i	1.73460i	0.497782	+	0.867302i	$0.334148\pi$
$443$	36.5092i	1.73460i	−0.497782	+	0.867302i	$0.665852\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	23.0061i	1.08815i
$448$	0	0
$449$	−8.68017	−0.409642	−0.204821	−	0.978799i	$-0.565661\pi$
$449$	−8.68017	−0.409642	−0.204821	+	0.978799i	$0.565661\pi$
$450$	0	0
$451$	−1.18060	−0.0555924
$452$	0	0
$453$	13.8811i	0.652193i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 11.4934i	− 0.537640i	−0.963190	−	0.268820i	$-0.913366\pi$
$457$	− 11.4934i	− 0.537640i	0.963190	−	0.268820i	$-0.0866337\pi$
$458$	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
$463$	− 32.7868i	− 1.52373i	−0.647735	−	0.761865i	$-0.724284\pi$
$463$	− 32.7868i	− 1.52373i	0.647735	−	0.761865i	$-0.275716\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 41.7603i	− 1.93244i	−0.257727	−	0.966218i	$-0.582973\pi$
$467$	− 41.7603i	− 1.93244i	0.257727	−	0.966218i	$-0.417027\pi$
$468$	0	0
$469$	52.2960	2.41480
$470$	0	0
$471$	−33.7990	−1.55738
$472$	0	0
$473$	7.68367i	0.353295i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 64.9796i	− 2.97521i
$478$	0	0
$479$	3.96125	0.180994	0.0904972	−	0.995897i	$-0.471154\pi$
$479$	3.96125	0.180994	0.0904972	+	0.995897i	$0.471154\pi$
$480$	0	0
$481$	−2.81940	−0.128553
$482$	0	0
$483$	− 56.5118i	− 2.57138i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 22.7990i	− 1.03312i	−0.856250	−	0.516561i	$-0.827212\pi$
$487$	− 22.7990i	− 1.03312i	0.856250	−	0.516561i	$-0.172788\pi$
$488$	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
$493$	− 3.06174i	− 0.137894i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 20.7807i	− 0.932140i
$498$	0	0
$499$	−23.9321	−1.07135	−0.535675	−	0.844424i	$-0.679943\pi$
$499$	−23.9321	−1.07135	−0.535675	+	0.844424i	$0.679943\pi$
$500$	0	0
$501$	−57.3190	−2.56082
$502$	0	0
$503$	23.0766i	1.02894i	0.857510	+	0.514468i	$0.172011\pi$
$503$	23.0766i	1.02894i	−0.857510	+	0.514468i	$0.827989\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 33.0184i	− 1.46640i
$508$	0	0
$509$	−21.2220	−0.940648	−0.470324	−	0.882494i	$-0.655863\pi$
$509$	−21.2220	−0.940648	−0.470324	+	0.882494i	$0.655863\pi$
$510$	0	0
$511$	52.5383	2.32416
$512$	0	0
$513$	− 10.8194i	− 0.477688i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	9.82552i	0.432126i
$518$	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
$523$	− 31.8899i	− 1.39445i	−0.716854	−	0.697224i	$-0.754418\pi$
$523$	− 31.8899i	− 1.39445i	0.716854	−	0.697224i	$-0.245582\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 4.61580i	− 0.201068i
$528$	0	0
$529$	−10.0801	−0.438266
$530$	0	0
$531$	44.9118	1.94901
$532$	0	0
$533$	1.18060i	0.0511376i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	31.8378i	1.37390i
$538$	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
$543$	50.7419i	2.17754i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	12.1250i	0.518427i	0.965820	+	0.259214i	$0.0834634\pi$
$547$	12.1250i	0.518427i	−0.965820	+	0.259214i	$0.916537\pi$
$548$	0	0
$549$	−32.3347	−1.38001
$550$	0	0
$551$	−10.1975	−0.434427
$552$	0	0
$553$	− 13.9030i	− 0.591216i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 28.3673i	− 1.20196i	−0.799263	−	0.600981i	$-0.794777\pi$
$557$	− 28.3673i	− 1.20196i	0.799263	−	0.600981i	$-0.205223\pi$
$558$	0	0
$559$	7.68367	0.324985
$560$	0	0
$561$	2.06786	0.0873053
$562$	0	0
$563$	− 4.08012i	− 0.171957i	−0.996297	−	0.0859783i	$-0.972598\pi$
$563$	− 4.08012i	− 0.171957i	0.996297	−	0.0859783i	$-0.0274016\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 6.49694i	− 0.272846i
$568$	0	0
$569$	23.0088	0.964577	0.482289	−	0.876012i	$-0.339806\pi$
$569$	23.0088	0.964577	0.482289	+	0.876012i	$0.339806\pi$
$570$	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
$573$	9.06174i	0.378559i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	25.1710i	1.04788i	0.851755	+	0.523941i	$0.175539\pi$
$577$	25.1710i	1.04788i	−0.851755	+	0.523941i	$0.824461\pi$
$578$	0	0
$579$	−63.3021	−2.63075
$580$	0	0
$581$	−19.9153	−0.826225
$582$	0	0
$583$	− 14.2159i	− 0.588760i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	10.2546i	0.423252i	0.977351	+	0.211626i	$0.0678760\pi$
$587$	10.2546i	0.423252i	−0.977351	+	0.211626i	$0.932124\pi$
$588$	0	0
$589$	−15.3735	−0.633453
$590$	0	0
$591$	−36.4704	−1.50019
$592$	0	0
$593$	− 10.9091i	− 0.447985i	−0.974591	−	0.223992i	$-0.928091\pi$
$593$	− 10.9091i	− 0.447985i	0.974591	−	0.223992i	$-0.0719091\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 4.55254i	− 0.186323i
$598$	0	0
$599$	29.2607	1.19556	0.597780	−	0.801660i	$-0.296049\pi$
$599$	29.2607	1.19556	0.597780	+	0.801660i	$0.296049\pi$
$600$	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
$603$	66.9409i	2.72604i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 32.7347i	− 1.32866i	−0.747440	−	0.664330i	$-0.768717\pi$
$607$	− 32.7347i	− 1.32866i	0.747440	−	0.664330i	$-0.231283\pi$
$608$	0	0
$609$	40.0291	1.62206
$610$	0	0
$611$	9.82552	0.397498
$612$	0	0
$613$	− 14.1894i	− 0.573103i	−0.958065	−	0.286551i	$-0.907491\pi$
$613$	− 14.1894i	− 0.573103i	0.958065	−	0.286551i	$-0.0925089\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 24.4766i	− 0.985390i	−0.870202	−	0.492695i	$-0.836012\pi$
$617$	− 24.4766i	− 0.985390i	0.870202	−	0.492695i	$-0.163988\pi$
$618$	0	0
$619$	31.7603	1.27655	0.638277	−	0.769807i	$-0.279647\pi$
$619$	31.7603	1.27655	0.638277	+	0.769807i	$0.279647\pi$
$620$	0	0
$621$	24.8608	0.997628
$622$	0	0
$623$	− 26.5287i	− 1.06285i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 6.88726i	− 0.275051i
$628$	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
$633$	− 57.6924i	− 2.29307i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 5.75153i	− 0.227884i
$638$	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
$643$	11.0836i	0.437095i	0.975826	+	0.218548i	$0.0701319\pi$
$643$	11.0836i	0.437095i	−0.975826	+	0.218548i	$0.929868\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	29.0449i	1.14187i	0.820995	+	0.570936i	$0.193420\pi$
$647$	29.0449i	1.14187i	−0.820995	+	0.570936i	$0.806580\pi$
$648$	0	0
$649$	9.82552	0.385686
$650$	0	0
$651$	60.3470	2.36518
$652$	0	0
$653$	− 50.3251i	− 1.96937i	−0.174335	−	0.984686i	$-0.555777\pi$
$653$	− 50.3251i	− 1.96937i	0.174335	−	0.984686i	$-0.444223\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	67.2511i	2.62372i
$658$	0	0
$659$	26.7189	1.04082	0.520411	−	0.853916i	$-0.325779\pi$
$659$	26.7189	1.04082	0.520411	+	0.853916i	$0.325779\pi$
$660$	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
$663$	− 2.06786i	− 0.0803092i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 23.4317i	− 0.907279i
$668$	0	0
$669$	35.3067	1.36504
$670$	0	0
$671$	−7.07399	−0.273088
$672$	0	0
$673$	− 20.7542i	− 0.800014i	−0.916512	−	0.400007i	$-0.869008\pi$
$673$	− 20.7542i	− 0.800014i	0.916512	−	0.400007i	$-0.130992\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 5.53568i	− 0.212754i	−0.994326	−	0.106377i	$-0.966075\pi$
$677$	− 5.53568i	− 0.212754i	0.994326	−	0.106377i	$-0.0339250\pi$
$678$	0	0
$679$	−2.17710	−0.0835496
$680$	0	0
$681$	25.9444	0.994191
$682$	0	0
$683$	41.0256i	1.56980i	0.619621	+	0.784901i	$0.287286\pi$
$683$	41.0256i	1.56980i	−0.619621	+	0.784901i	$0.712714\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 13.6944i	− 0.522474i
$688$	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
$693$	16.3225i	0.620039i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 0.887261i	− 0.0336074i
$698$	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
$703$	− 7.05713i	− 0.266165i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0.887261i	0.0333689i
$708$	0	0
$709$	8.99387	0.337772	0.168886	−	0.985636i	$-0.445983\pi$
$709$	8.99387	0.337772	0.168886	+	0.985636i	$0.445983\pi$
$710$	0	0
$711$	17.7964	0.667417
$712$	0	0
$713$	− 35.3251i	− 1.32294i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 76.2695i	− 2.84834i
$718$	0	0
$719$	−10.6837	−0.398434	−0.199217	−	0.979955i	$-0.563840\pi$
$719$	−10.6837	−0.398434	−0.199217	+	0.979955i	$0.563840\pi$
$720$	0	0
$721$	62.7664	2.33754
$722$	0	0
$723$	− 33.3455i	− 1.24013i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	1.98424i	0.0735916i	0.999323	+	0.0367958i	$0.0117151\pi$
$727$	1.98424i	0.0735916i	−0.999323	+	0.0367958i	$0.988285\pi$
$728$	0	0
$729$	43.9965	1.62950
$730$	0	0
$731$	−5.77452	−0.213578
$732$	0	0
$733$	− 32.8765i	− 1.21432i	−0.794579	−	0.607161i	$-0.792308\pi$
$733$	− 32.8765i	− 1.21432i	0.794579	−	0.607161i	$-0.207692\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	14.6449i	0.539453i
$738$	0	0
$739$	31.4352	1.15636	0.578181	−	0.815908i	$-0.303763\pi$
$739$	31.4352	1.15636	0.578181	+	0.815908i	$0.303763\pi$
$740$	0	0
$741$	−6.88726	−0.253010
$742$	0	0
$743$	28.3638i	1.04057i	0.853993	+	0.520284i	$0.174174\pi$
$743$	28.3638i	1.04057i	−0.853993	+	0.520284i	$0.825826\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 25.4923i	− 0.932716i
$748$	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
$753$	38.8485i	1.41572i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	21.8838i	0.795379i	0.917520	+	0.397689i	$0.130188\pi$
$757$	21.8838i	0.795379i	−0.917520	+	0.397689i	$0.869812\pi$
$758$	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
$763$	− 12.2450i	− 0.443298i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 9.82552i	− 0.354779i
$768$	0	0
$769$	9.54181	0.344086	0.172043	−	0.985089i	$-0.444963\pi$
$769$	9.54181	0.344086	0.172043	+	0.985089i	$0.444963\pi$
$770$	0	0
$771$	37.4378	1.34829
$772$	0	0
$773$	− 0.0556080i	− 0.00200008i	−0.999999	−	0.00100004i	$-0.999682\pi$
$773$	− 0.0556080i	− 0.00200008i	0.999999	−	0.00100004i	$-0.000318323\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	27.7021i	0.993806i
$778$	0	0
$779$	−2.95513	−0.105878
$780$	0	0
$781$	5.81940	0.208234
$782$	0	0
$783$	17.6097i	0.629318i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	22.0970i	0.787672i	0.919181	+	0.393836i	$0.128852\pi$
$787$	22.0970i	0.787672i	−0.919181	+	0.393836i	$0.871148\pi$
$788$	0	0
$789$	−78.9578	−2.81097
$790$	0	0
$791$	50.6598	1.80126
$792$	0	0
$793$	7.07399i	0.251205i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 35.7577i	− 1.26660i	−0.773906	−	0.633301i	$-0.781700\pi$
$797$	− 35.7577i	− 1.26660i	0.773906	−	0.633301i	$-0.218300\pi$
$798$	0	0
$799$	−7.38420	−0.261234
$800$	0	0
$801$	33.9578	1.19984
$802$	0	0
$803$	14.7128i	0.519203i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 12.0801i	− 0.425240i
$808$	0	0
$809$	−24.9286	−0.876444	−0.438222	−	0.898867i	$-0.644392\pi$
$809$	−24.9286	−0.876444	−0.438222	+	0.898867i	$0.644392\pi$
$810$	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
$813$	51.8159i	1.81726i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	19.2327i	0.672867i
$818$	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
$823$	42.5797i	1.48423i	0.670270	+	0.742117i	$0.266178\pi$
$823$	42.5797i	1.48423i	−0.670270	+	0.742117i	$0.733822\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 23.1867i	− 0.806282i	−0.915138	−	0.403141i	$-0.867918\pi$
$827$	− 23.1867i	− 0.806282i	0.915138	−	0.403141i	$-0.132082\pi$
$828$	0	0
$829$	36.2546	1.25917	0.629587	−	0.776930i	$-0.283224\pi$
$829$	36.2546	1.25917	0.629587	+	0.776930i	$0.283224\pi$
$830$	0	0
$831$	60.8343	2.11032
$832$	0	0
$833$	4.32246i	0.149764i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	26.5479i	0.917631i
$838$	0	0
$839$	−13.6607	−0.471619	−0.235809	−	0.971799i	$-0.575774\pi$
$839$	−13.6607	−0.471619	−0.235809	+	0.971799i	$0.575774\pi$
$840$	0	0
$841$	−12.4026	−0.427675
$842$	0	0
$843$	22.5261i	0.775839i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	3.57093i	0.122699i
$848$	0	0
$849$	−65.7434	−2.25631
$850$	0	0
$851$	16.2159	0.555872
$852$	0	0
$853$	54.2476i	1.85740i	0.370829	+	0.928701i	$0.379074\pi$
$853$	54.2476i	1.85740i	−0.370829	+	0.928701i	$0.620926\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 6.41221i	− 0.219037i	−0.993985	−	0.109518i	$-0.965069\pi$
$857$	− 6.41221i	− 0.219037i	0.993985	−	0.109518i	$-0.0349309\pi$
$858$	0	0
$859$	51.5031	1.75726	0.878631	−	0.477501i	$-0.158457\pi$
$859$	51.5031	1.75726	0.878631	+	0.477501i	$0.158457\pi$
$860$	0	0
$861$	11.6000	0.395329
$862$	0	0
$863$	43.2837i	1.47339i	0.676223	+	0.736697i	$0.263616\pi$
$863$	43.2837i	1.47339i	−0.676223	+	0.736697i	$0.736384\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 45.2220i	− 1.53582i
$868$	0	0
$869$	3.89339	0.132074
$870$	0	0
$871$	14.6449	0.496224
$872$	0	0
$873$	− 2.78678i	− 0.0943182i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 10.2511i	− 0.346155i	−0.984908	−	0.173077i	$-0.944629\pi$
$877$	− 10.2511i	− 0.346155i	0.984908	−	0.173077i	$-0.0553711\pi$
$878$	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
$883$	− 6.76116i	− 0.227531i	−0.993508	−	0.113766i	$-0.963709\pi$
$883$	− 6.76116i	− 0.227531i	0.993508	−	0.113766i	$-0.0362913\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 10.3225i	− 0.346594i	−0.984870	−	0.173297i	$-0.944558\pi$
$887$	− 10.3225i	− 0.346594i	0.984870	−	0.173297i	$-0.0554421\pi$
$888$	0	0
$889$	67.4888	2.26350
$890$	0	0
$891$	1.81940	0.0609521
$892$	0	0
$893$	24.5939i	0.823004i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 15.8255i	− 0.528399i
$898$	0	0
$899$	25.0219	0.834527
$900$	0	0
$901$	10.6837	0.355925
$902$	0	0
$903$	− 75.4961i	− 2.51235i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	11.9348i	0.396287i	0.980173	+	0.198144i	$0.0634913\pi$
$907$	11.9348i	0.396287i	−0.980173	+	0.198144i	$0.936509\pi$
$908$	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
$913$	− 5.57706i	− 0.184574i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 73.3190i	− 2.42121i
$918$	0	0
$919$	−25.0184	−0.825280	−0.412640	−	0.910894i	$-0.635393\pi$
$919$	−25.0184	−0.825280	−0.412640	+	0.910894i	$0.635393\pi$
$920$	0	0
$921$	−30.3638	−1.00052
$922$	0	0
$923$	− 5.81940i	− 0.191548i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	80.3435i	2.63883i
$928$	0	0
$929$	−4.42557	−0.145198	−0.0725992	−	0.997361i	$-0.523129\pi$
$929$	−4.42557	−0.145198	−0.0725992	+	0.997361i	$0.523129\pi$
$930$	0	0
$931$	14.3965	0.471825
$932$	0	0
$933$	65.0598i	2.12996i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	40.3612i	1.31854i	0.751905	+	0.659272i	$0.229135\pi$
$937$	40.3612i	1.31854i	−0.751905	+	0.659272i	$0.770865\pi$
$938$	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
$943$	− 6.79028i	− 0.221122i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 4.81940i	− 0.156609i	−0.996929	−	0.0783047i	$-0.975049\pi$
$947$	− 4.81940i	− 0.156609i	0.996929	−	0.0783047i	$-0.0249507\pi$
$948$	0	0
$949$	14.7128	0.477597
$950$	0	0
$951$	−3.02801	−0.0981900
$952$	0	0
$953$	16.0123i	0.518688i	0.965785	+	0.259344i	$0.0835063\pi$
$953$	16.0123i	0.518688i	−0.965785	+	0.259344i	$0.916494\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	11.2097i	0.362359i
$958$	0	0
$959$	−11.7603	−0.379760
$960$	0	0
$961$	6.72241	0.216852
$962$	0	0
$963$	− 67.5893i	− 2.17804i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	50.6211i	1.62786i	0.580960	+	0.813932i	$0.302677\pi$
$967$	50.6211i	1.62786i	−0.580960	+	0.813932i	$0.697323\pi$
$968$	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
$973$	65.7964i	2.10934i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 48.1215i	− 1.53954i	−0.638320	−	0.769772i	$-0.720370\pi$
$977$	− 48.1215i	− 1.53954i	0.638320	−	0.769772i	$-0.279630\pi$
$978$	0	0
$979$	7.42907	0.237434
$980$	0	0
$981$	15.6740	0.500434
$982$	0	0
$983$	− 12.8960i	− 0.411319i	−0.978624	−	0.205660i	$-0.934066\pi$
$983$	− 12.8960i	− 0.411319i	0.978624	−	0.205660i	$-0.0659340\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 96.5409i	− 3.07293i
$988$	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
$993$	− 33.9056i	− 1.07596i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 31.9831i	− 1.01292i	−0.862265	−	0.506458i	$-0.830954\pi$
$997$	− 31.9831i	− 1.01292i	0.862265	−	0.506458i	$-0.169046\pi$
$998$	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.b.bb.4049.1		6
4.3	odd	2		2200.2.b.m.1849.6		6
5.2	odd	4		4400.2.a.by.1.1		3
5.3	odd	4		4400.2.a.bz.1.3		3
5.4	even	2	inner	4400.2.b.bb.4049.6		6
20.3	even	4		2200.2.a.u.1.1	✓	3
20.7	even	4		2200.2.a.v.1.3	yes	3
20.19	odd	2		2200.2.b.m.1849.1		6

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

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 4400 = 2^{4} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4400.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.75153i q^{3} +3.57093i q^{7} -4.57093 q^{9} -1.00000 q^{11} +1.00000i q^{13} -0.751532i q^{17} -2.50306 q^{19} +9.82552 q^{21} -5.75153i q^{23} +4.32246i q^{27} +4.07399 q^{29} +6.14186 q^{31} +2.75153i q^{33} +2.81940i q^{37} +2.75153 q^{39} +1.18060 q^{41} -7.68367i q^{43} -9.82552i q^{47} -5.75153 q^{49} -2.06786 q^{51} +14.2159i q^{53} +6.88726i q^{57} -9.82552 q^{59} +7.07399 q^{61} -16.3225i q^{63} -14.6449i q^{67} -15.8255 q^{69} -5.81940 q^{71} -14.7128i q^{73} -3.57093i q^{77} -3.89339 q^{79} -1.81940 q^{81} +5.57706i q^{83} -11.2097i q^{87} -7.42907 q^{89} -3.57093 q^{91} -16.8995i q^{93} +0.609675i q^{97} +4.57093 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 12 q^{9} - 6 q^{11} + 14 q^{19} - 20 q^{29} + 6 q^{31} + 2 q^{39} + 8 q^{41} - 20 q^{49} - 26 q^{51} - 2 q^{61} - 36 q^{69} - 34 q^{71} + 22 q^{79} - 10 q^{81} - 60 q^{89} - 6 q^{91} + 12 q^{99}+O(q^{100}) \) 6 * q - 12 * q^9 - 6 * q^11 + 14 * q^19 - 20 * q^29 + 6 * q^31 + 2 * q^39 + 8 * q^41 - 20 * q^49 - 26 * q^51 - 2 * q^61 - 36 * q^69 - 34 * q^71 + 22 * q^79 - 10 * q^81 - 60 * q^89 - 6 * q^91 + 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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