LMFDB - Embedded newform 4400.2.b.q.4049.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4400,2,Mod(4049,4400)]

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, 1, 0]))

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

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

chi := DirichletCharacter("4400.4049");

S:= CuspForms(chi, 2);

N := Newforms(S);

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: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$35.1341768894$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 55)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			4049.3
Root			$0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	4400.4049
Dual form			4400.2.b.q.4049.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.82843i q^{3} -2.00000i q^{7} -5.00000 q^{9} +O(q^{10})$	$q+2.82843i q^{3} -2.00000i q^{7} -5.00000 q^{9} -1.00000 q^{11} -1.17157i q^{13} -6.82843i q^{17} +5.65685 q^{21} +2.82843i q^{23} -5.65685i q^{27} +3.65685 q^{29} -2.82843i q^{33} +7.65685i q^{37} +3.31371 q^{39} +6.00000 q^{41} +6.00000i q^{43} +2.82843i q^{47} +3.00000 q^{49} +19.3137 q^{51} +11.6569i q^{53} +1.65685 q^{59} -9.31371 q^{61} +10.0000i q^{63} +12.4853i q^{67} -8.00000 q^{69} -11.3137 q^{71} -1.17157i q^{73} +2.00000i q^{77} +4.00000 q^{79} +1.00000 q^{81} +6.00000i q^{83} +10.3431i q^{87} +13.3137 q^{89} -2.34315 q^{91} -3.65685i q^{97} +5.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 20 q^{9}+O(q^{10}) $ 4 * q - 20 * q^9	$ 4 q - 20 q^{9} - 4 q^{11} - 8 q^{29} - 32 q^{39} + 24 q^{41} + 12 q^{49} + 32 q^{51} - 16 q^{59} + 8 q^{61} - 32 q^{69} + 16 q^{79} + 4 q^{81} + 8 q^{89} - 32 q^{91} + 20 q^{99}+O(q^{100}) $ 4 * q - 20 * q^9 - 4 * q^11 - 8 * q^29 - 32 * q^39 + 24 * q^41 + 12 * q^49 + 32 * q^51 - 16 * q^59 + 8 * q^61 - 32 * q^69 + 16 * q^79 + 4 * q^81 + 8 * q^89 - 32 * q^91 + 20 * q^99

Display 100 coefficients 10 coefficients

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
$3$	2.82843i	1.63299i	0.577350	+	0.816497i	$0.304087\pi$
$3$	2.82843i	1.63299i	−0.577350	+	0.816497i	$0.695913\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 2.00000i	− 0.755929i	−0.925820	−	0.377964i	$-0.876624\pi$
$7$	− 2.00000i	− 0.755929i	0.925820	−	0.377964i	$-0.123376\pi$
$8$	0	0
$9$	−5.00000	−1.66667
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	− 1.17157i	− 0.324936i	−0.986714	−	0.162468i	$-0.948055\pi$
$13$	− 1.17157i	− 0.324936i	0.986714	−	0.162468i	$-0.0519454\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 6.82843i	− 1.65614i	−0.560627	−	0.828068i	$-0.689440\pi$
$17$	− 6.82843i	− 1.65614i	0.560627	−	0.828068i	$-0.310560\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	5.65685	1.23443
$22$	0	0
$23$	2.82843i	0.589768i	0.955533	+	0.294884i	$0.0952810\pi$
$23$	2.82843i	0.589768i	−0.955533	+	0.294884i	$0.904719\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 5.65685i	− 1.08866i
$28$	0	0
$29$	3.65685	0.679061	0.339530	−	0.940595i	$-0.389732\pi$
$29$	3.65685	0.679061	0.339530	+	0.940595i	$0.389732\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	− 2.82843i	− 0.492366i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	7.65685i	1.25878i	0.777090	+	0.629390i	$0.216695\pi$
$37$	7.65685i	1.25878i	−0.777090	+	0.629390i	$0.783305\pi$
$38$	0	0
$39$	3.31371	0.530618
$40$	0	0
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	0	0
$43$	6.00000i	0.914991i	0.889212	+	0.457496i	$0.151253\pi$
$43$	6.00000i	0.914991i	−0.889212	+	0.457496i	$0.848747\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	2.82843i	0.412568i	0.978492	+	0.206284i	$0.0661372\pi$
$47$	2.82843i	0.412568i	−0.978492	+	0.206284i	$0.933863\pi$
$48$	0	0
$49$	3.00000	0.428571
$50$	0	0
$51$	19.3137	2.70446
$52$	0	0
$53$	11.6569i	1.60119i	0.599204	+	0.800596i	$0.295484\pi$
$53$	11.6569i	1.60119i	−0.599204	+	0.800596i	$0.704516\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	1.65685	0.215704	0.107852	−	0.994167i	$-0.465603\pi$
$59$	1.65685	0.215704	0.107852	+	0.994167i	$0.465603\pi$
$60$	0	0
$61$	−9.31371	−1.19250	−0.596249	−	0.802799i	$-0.703343\pi$
$61$	−9.31371	−1.19250	−0.596249	+	0.802799i	$0.703343\pi$
$62$	0	0
$63$	10.0000i	1.25988i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	12.4853i	1.52532i	0.646800	+	0.762660i	$0.276107\pi$
$67$	12.4853i	1.52532i	−0.646800	+	0.762660i	$0.723893\pi$
$68$	0	0
$69$	−8.00000	−0.963087
$70$	0	0
$71$	−11.3137	−1.34269	−0.671345	−	0.741145i	$-0.734283\pi$
$71$	−11.3137	−1.34269	−0.671345	+	0.741145i	$0.734283\pi$
$72$	0	0
$73$	− 1.17157i	− 0.137122i	−0.997647	−	0.0685611i	$-0.978159\pi$
$73$	− 1.17157i	− 0.137122i	0.997647	−	0.0685611i	$-0.0218408\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.00000i	0.227921i
$78$	0	0
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	6.00000i	0.658586i	0.944228	+	0.329293i	$0.106810\pi$
$83$	6.00000i	0.658586i	−0.944228	+	0.329293i	$0.893190\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	10.3431i	1.10890i
$88$	0	0
$89$	13.3137	1.41125	0.705625	−	0.708585i	$-0.250666\pi$
$89$	13.3137	1.41125	0.705625	+	0.708585i	$0.250666\pi$
$90$	0	0
$91$	−2.34315	−0.245628
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 3.65685i	− 0.371297i	−0.982616	−	0.185649i	$-0.940561\pi$
$97$	− 3.65685i	− 0.371297i	0.982616	−	0.185649i	$-0.0594386\pi$
$98$	0	0
$99$	5.00000	0.502519
$100$	0	0
$101$	9.31371	0.926749	0.463374	−	0.886163i	$-0.346639\pi$
$101$	9.31371	0.926749	0.463374	+	0.886163i	$0.346639\pi$
$102$	0	0
$103$	− 6.82843i	− 0.672825i	−0.941715	−	0.336412i	$-0.890786\pi$
$103$	− 6.82843i	− 0.672825i	0.941715	−	0.336412i	$-0.109214\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	7.65685i	0.740216i	0.928989	+	0.370108i	$0.120679\pi$
$107$	7.65685i	0.740216i	−0.928989	+	0.370108i	$0.879321\pi$
$108$	0	0
$109$	7.65685	0.733394	0.366697	−	0.930341i	$-0.380489\pi$
$109$	7.65685	0.733394	0.366697	+	0.930341i	$0.380489\pi$
$110$	0	0
$111$	−21.6569	−2.05558
$112$	0	0
$113$	19.6569i	1.84916i	0.380986	+	0.924581i	$0.375584\pi$
$113$	19.6569i	1.84916i	−0.380986	+	0.924581i	$0.624416\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	5.85786i	0.541560i
$118$	0	0
$119$	−13.6569	−1.25192
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	16.9706i	1.53018i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	4.34315i	0.385392i	0.981259	+	0.192696i	$0.0617231\pi$
$127$	4.34315i	0.385392i	−0.981259	+	0.192696i	$0.938277\pi$
$128$	0	0
$129$	−16.9706	−1.49417
$130$	0	0
$131$	11.3137	0.988483	0.494242	−	0.869325i	$-0.335446\pi$
$131$	11.3137	0.988483	0.494242	+	0.869325i	$0.335446\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	10.9706i	0.937278i	0.883390	+	0.468639i	$0.155256\pi$
$137$	10.9706i	0.937278i	−0.883390	+	0.468639i	$0.844744\pi$
$138$	0	0
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	0	0
$141$	−8.00000	−0.673722
$142$	0	0
$143$	1.17157i	0.0979718i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	8.48528i	0.699854i
$148$	0	0
$149$	−0.343146	−0.0281116	−0.0140558	−	0.999901i	$-0.504474\pi$
$149$	−0.343146	−0.0281116	−0.0140558	+	0.999901i	$0.504474\pi$
$150$	0	0
$151$	12.0000	0.976546	0.488273	−	0.872691i	$-0.337627\pi$
$151$	12.0000	0.976546	0.488273	+	0.872691i	$0.337627\pi$
$152$	0	0
$153$	34.1421i	2.76023i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	14.0000i	1.11732i	0.829396	+	0.558661i	$0.188685\pi$
$157$	14.0000i	1.11732i	−0.829396	+	0.558661i	$0.811315\pi$
$158$	0	0
$159$	−32.9706	−2.61474
$160$	0	0
$161$	5.65685	0.445823
$162$	0	0
$163$	− 16.4853i	− 1.29123i	−0.763664	−	0.645613i	$-0.776602\pi$
$163$	− 16.4853i	− 1.29123i	0.763664	−	0.645613i	$-0.223398\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 22.9706i	− 1.77752i	−0.458377	−	0.888758i	$-0.651569\pi$
$167$	− 22.9706i	− 1.77752i	0.458377	−	0.888758i	$-0.348431\pi$
$168$	0	0
$169$	11.6274	0.894417
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 22.1421i	− 1.68344i	−0.539918	−	0.841718i	$-0.681545\pi$
$173$	− 22.1421i	− 1.68344i	0.539918	−	0.841718i	$-0.318455\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	4.68629i	0.352243i
$178$	0	0
$179$	9.65685	0.721787	0.360894	−	0.932607i	$-0.382472\pi$
$179$	9.65685	0.721787	0.360894	+	0.932607i	$0.382472\pi$
$180$	0	0
$181$	21.3137	1.58424	0.792118	−	0.610368i	$-0.208979\pi$
$181$	21.3137	1.58424	0.792118	+	0.610368i	$0.208979\pi$
$182$	0	0
$183$	− 26.3431i	− 1.94734i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	6.82843i	0.499344i
$188$	0	0
$189$	−11.3137	−0.822951
$190$	0	0
$191$	−3.31371	−0.239772	−0.119886	−	0.992788i	$-0.538253\pi$
$191$	−3.31371	−0.239772	−0.119886	+	0.992788i	$0.538253\pi$
$192$	0	0
$193$	− 1.17157i	− 0.0843317i	−0.999111	−	0.0421658i	$-0.986574\pi$
$193$	− 1.17157i	− 0.0843317i	0.999111	−	0.0421658i	$-0.0134258\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	10.8284i	0.771493i	0.922605	+	0.385747i	$0.126056\pi$
$197$	10.8284i	0.771493i	−0.922605	+	0.385747i	$0.873944\pi$
$198$	0	0
$199$	10.3431	0.733206	0.366603	−	0.930377i	$-0.380521\pi$
$199$	10.3431	0.733206	0.366603	+	0.930377i	$0.380521\pi$
$200$	0	0
$201$	−35.3137	−2.49084
$202$	0	0
$203$	− 7.31371i	− 0.513322i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 14.1421i	− 0.982946i
$208$	0	0
$209$	0	0
$210$	0	0
$211$	16.0000	1.10149	0.550743	−	0.834675i	$-0.314345\pi$
$211$	16.0000	1.10149	0.550743	+	0.834675i	$0.314345\pi$
$212$	0	0
$213$	− 32.0000i	− 2.19260i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	3.31371	0.223920
$220$	0	0
$221$	−8.00000	−0.538138
$222$	0	0
$223$	10.8284i	0.725125i	0.931959	+	0.362563i	$0.118098\pi$
$223$	10.8284i	0.725125i	−0.931959	+	0.362563i	$0.881902\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	25.3137i	1.68013i	0.542486	+	0.840065i	$0.317483\pi$
$227$	25.3137i	1.68013i	−0.542486	+	0.840065i	$0.682517\pi$
$228$	0	0
$229$	−1.31371	−0.0868123	−0.0434062	−	0.999058i	$-0.513821\pi$
$229$	−1.31371	−0.0868123	−0.0434062	+	0.999058i	$0.513821\pi$
$230$	0	0
$231$	−5.65685	−0.372194
$232$	0	0
$233$	− 6.14214i	− 0.402385i	−0.979552	−	0.201192i	$-0.935518\pi$
$233$	− 6.14214i	− 0.402385i	0.979552	−	0.201192i	$-0.0644816\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	11.3137i	0.734904i
$238$	0	0
$239$	−23.3137	−1.50804	−0.754019	−	0.656852i	$-0.771887\pi$
$239$	−23.3137	−1.50804	−0.754019	+	0.656852i	$0.771887\pi$
$240$	0	0
$241$	6.00000	0.386494	0.193247	−	0.981150i	$-0.438098\pi$
$241$	6.00000	0.386494	0.193247	+	0.981150i	$0.438098\pi$
$242$	0	0
$243$	− 14.1421i	− 0.907218i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−16.9706	−1.07547
$250$	0	0
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	0	0
$253$	− 2.82843i	− 0.177822i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	9.31371i	0.580973i	0.956879	+	0.290487i	$0.0938172\pi$
$257$	9.31371i	0.580973i	−0.956879	+	0.290487i	$0.906183\pi$
$258$	0	0
$259$	15.3137	0.951548
$260$	0	0
$261$	−18.2843	−1.13177
$262$	0	0
$263$	10.9706i	0.676474i	0.941061	+	0.338237i	$0.109831\pi$
$263$	10.9706i	0.676474i	−0.941061	+	0.338237i	$0.890169\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	37.6569i	2.30456i
$268$	0	0
$269$	−17.3137	−1.05564	−0.527818	−	0.849358i	$-0.676990\pi$
$269$	−17.3137	−1.05564	−0.527818	+	0.849358i	$0.676990\pi$
$270$	0	0
$271$	−7.31371	−0.444276	−0.222138	−	0.975015i	$-0.571304\pi$
$271$	−7.31371	−0.444276	−0.222138	+	0.975015i	$0.571304\pi$
$272$	0	0
$273$	− 6.62742i	− 0.401110i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 6.82843i	− 0.410280i	−0.978733	−	0.205140i	$-0.934235\pi$
$277$	− 6.82843i	− 0.410280i	0.978733	−	0.205140i	$-0.0657650\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	17.3137	1.03285	0.516425	−	0.856333i	$-0.327263\pi$
$281$	17.3137	1.03285	0.516425	+	0.856333i	$0.327263\pi$
$282$	0	0
$283$	− 32.6274i	− 1.93950i	−0.244103	−	0.969749i	$-0.578493\pi$
$283$	− 32.6274i	− 1.93950i	0.244103	−	0.969749i	$-0.421507\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 12.0000i	− 0.708338i
$288$	0	0
$289$	−29.6274	−1.74279
$290$	0	0
$291$	10.3431	0.606326
$292$	0	0
$293$	− 9.17157i	− 0.535809i	−0.963445	−	0.267905i	$-0.913669\pi$
$293$	− 9.17157i	− 0.535809i	0.963445	−	0.267905i	$-0.0863312\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	5.65685i	0.328244i
$298$	0	0
$299$	3.31371	0.191637
$300$	0	0
$301$	12.0000	0.691669
$302$	0	0
$303$	26.3431i	1.51337i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 16.3431i	− 0.932753i	−0.884586	−	0.466376i	$-0.845559\pi$
$307$	− 16.3431i	− 0.932753i	0.884586	−	0.466376i	$-0.154441\pi$
$308$	0	0
$309$	19.3137	1.09872
$310$	0	0
$311$	−4.68629	−0.265735	−0.132868	−	0.991134i	$-0.542419\pi$
$311$	−4.68629	−0.265735	−0.132868	+	0.991134i	$0.542419\pi$
$312$	0	0
$313$	− 1.31371i	− 0.0742552i	−0.999311	−	0.0371276i	$-0.988179\pi$
$313$	− 1.31371i	− 0.0742552i	0.999311	−	0.0371276i	$-0.0118208\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.31371i	0.0737852i	0.999319	+	0.0368926i	$0.0117459\pi$
$317$	1.31371i	0.0737852i	−0.999319	+	0.0368926i	$0.988254\pi$
$318$	0	0
$319$	−3.65685	−0.204745
$320$	0	0
$321$	−21.6569	−1.20877
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0	0
$326$	0	0
$327$	21.6569i	1.19763i
$328$	0	0
$329$	5.65685	0.311872
$330$	0	0
$331$	7.31371	0.401998	0.200999	−	0.979591i	$-0.435581\pi$
$331$	7.31371	0.401998	0.200999	+	0.979591i	$0.435581\pi$
$332$	0	0
$333$	− 38.2843i	− 2.09797i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	20.4853i	1.11590i	0.829873	+	0.557952i	$0.188413\pi$
$337$	20.4853i	1.11590i	−0.829873	+	0.557952i	$0.811587\pi$
$338$	0	0
$339$	−55.5980	−3.01967
$340$	0	0
$341$	0	0
$342$	0	0
$343$	− 20.0000i	− 1.07990i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	10.9706i	0.588931i	0.955662	+	0.294465i	$0.0951416\pi$
$347$	10.9706i	0.588931i	−0.955662	+	0.294465i	$0.904858\pi$
$348$	0	0
$349$	−26.9706	−1.44370	−0.721851	−	0.692049i	$-0.756708\pi$
$349$	−26.9706	−1.44370	−0.721851	+	0.692049i	$0.756708\pi$
$350$	0	0
$351$	−6.62742	−0.353745
$352$	0	0
$353$	21.3137i	1.13441i	0.823575	+	0.567207i	$0.191976\pi$
$353$	21.3137i	1.13441i	−0.823575	+	0.567207i	$0.808024\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 38.6274i	− 2.04438i
$358$	0	0
$359$	0.686292	0.0362211	0.0181105	−	0.999836i	$-0.494235\pi$
$359$	0.686292	0.0362211	0.0181105	+	0.999836i	$0.494235\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0	0
$363$	2.82843i	0.148454i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 8.48528i	− 0.442928i	−0.975169	−	0.221464i	$-0.928916\pi$
$367$	− 8.48528i	− 0.442928i	0.975169	−	0.221464i	$-0.0710835\pi$
$368$	0	0
$369$	−30.0000	−1.56174
$370$	0	0
$371$	23.3137	1.21039
$372$	0	0
$373$	35.7990i	1.85360i	0.375554	+	0.926801i	$0.377453\pi$
$373$	35.7990i	1.85360i	−0.375554	+	0.926801i	$0.622547\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 4.28427i	− 0.220651i
$378$	0	0
$379$	33.6569	1.72884	0.864418	−	0.502773i	$-0.167687\pi$
$379$	33.6569	1.72884	0.864418	+	0.502773i	$0.167687\pi$
$380$	0	0
$381$	−12.2843	−0.629342
$382$	0	0
$383$	5.85786i	0.299323i	0.988737	+	0.149661i	$0.0478184\pi$
$383$	5.85786i	0.299323i	−0.988737	+	0.149661i	$0.952182\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 30.0000i	− 1.52499i
$388$	0	0
$389$	−20.6274	−1.04585	−0.522926	−	0.852378i	$-0.675160\pi$
$389$	−20.6274	−1.04585	−0.522926	+	0.852378i	$0.675160\pi$
$390$	0	0
$391$	19.3137	0.976736
$392$	0	0
$393$	32.0000i	1.61419i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	9.31371i	0.467442i	0.972304	+	0.233721i	$0.0750902\pi$
$397$	9.31371i	0.467442i	−0.972304	+	0.233721i	$0.924910\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−5.31371	−0.265354	−0.132677	−	0.991159i	$-0.542357\pi$
$401$	−5.31371	−0.265354	−0.132677	+	0.991159i	$0.542357\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 7.65685i	− 0.379536i
$408$	0	0
$409$	−1.02944	−0.0509024	−0.0254512	−	0.999676i	$-0.508102\pi$
$409$	−1.02944	−0.0509024	−0.0254512	+	0.999676i	$0.508102\pi$
$410$	0	0
$411$	−31.0294	−1.53057
$412$	0	0
$413$	− 3.31371i	− 0.163057i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 11.3137i	− 0.554035i
$418$	0	0
$419$	−25.6569	−1.25342	−0.626710	−	0.779253i	$-0.715599\pi$
$419$	−25.6569	−1.25342	−0.626710	+	0.779253i	$0.715599\pi$
$420$	0	0
$421$	−6.00000	−0.292422	−0.146211	−	0.989253i	$-0.546708\pi$
$421$	−6.00000	−0.292422	−0.146211	+	0.989253i	$0.546708\pi$
$422$	0	0
$423$	− 14.1421i	− 0.687614i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	18.6274i	0.901444i
$428$	0	0
$429$	−3.31371	−0.159987
$430$	0	0
$431$	11.3137	0.544962	0.272481	−	0.962161i	$-0.412156\pi$
$431$	11.3137	0.544962	0.272481	+	0.962161i	$0.412156\pi$
$432$	0	0
$433$	− 7.65685i	− 0.367965i	−0.982930	−	0.183982i	$-0.941101\pi$
$433$	− 7.65685i	− 0.367965i	0.982930	−	0.183982i	$-0.0588990\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−16.0000	−0.763638	−0.381819	−	0.924237i	$-0.624702\pi$
$439$	−16.0000	−0.763638	−0.381819	+	0.924237i	$0.624702\pi$
$440$	0	0
$441$	−15.0000	−0.714286
$442$	0	0
$443$	26.8284i	1.27466i	0.770592	+	0.637329i	$0.219961\pi$
$443$	26.8284i	1.27466i	−0.770592	+	0.637329i	$0.780039\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 0.970563i	− 0.0459060i
$448$	0	0
$449$	−28.6274	−1.35101	−0.675506	−	0.737355i	$-0.736075\pi$
$449$	−28.6274	−1.35101	−0.675506	+	0.737355i	$0.736075\pi$
$450$	0	0
$451$	−6.00000	−0.282529
$452$	0	0
$453$	33.9411i	1.59469i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 0.485281i	− 0.0227005i	−0.999936	−	0.0113503i	$-0.996387\pi$
$457$	− 0.485281i	− 0.0227005i	0.999936	−	0.0113503i	$-0.00361298\pi$
$458$	0	0
$459$	−38.6274	−1.80297
$460$	0	0
$461$	12.6274	0.588117	0.294059	−	0.955787i	$-0.404994\pi$
$461$	12.6274	0.588117	0.294059	+	0.955787i	$0.404994\pi$
$462$	0	0
$463$	6.14214i	0.285449i	0.989762	+	0.142725i	$0.0455863\pi$
$463$	6.14214i	0.285449i	−0.989762	+	0.142725i	$0.954414\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 14.8284i	− 0.686178i	−0.939303	−	0.343089i	$-0.888527\pi$
$467$	− 14.8284i	− 0.686178i	0.939303	−	0.343089i	$-0.111473\pi$
$468$	0	0
$469$	24.9706	1.15303
$470$	0	0
$471$	−39.5980	−1.82458
$472$	0	0
$473$	− 6.00000i	− 0.275880i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 58.2843i	− 2.66865i
$478$	0	0
$479$	−36.0000	−1.64488	−0.822441	−	0.568850i	$-0.807388\pi$
$479$	−36.0000	−1.64488	−0.822441	+	0.568850i	$0.807388\pi$
$480$	0	0
$481$	8.97056	0.409022
$482$	0	0
$483$	16.0000i	0.728025i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 24.4853i	− 1.10953i	−0.832006	−	0.554767i	$-0.812807\pi$
$487$	− 24.4853i	− 1.10953i	0.832006	−	0.554767i	$-0.187193\pi$
$488$	0	0
$489$	46.6274	2.10856
$490$	0	0
$491$	0.686292	0.0309719	0.0154860	−	0.999880i	$-0.495070\pi$
$491$	0.686292	0.0309719	0.0154860	+	0.999880i	$0.495070\pi$
$492$	0	0
$493$	− 24.9706i	− 1.12462i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	22.6274i	1.01498i
$498$	0	0
$499$	9.65685	0.432300	0.216150	−	0.976360i	$-0.430650\pi$
$499$	9.65685	0.432300	0.216150	+	0.976360i	$0.430650\pi$
$500$	0	0
$501$	64.9706	2.90267
$502$	0	0
$503$	− 16.6274i	− 0.741380i	−0.928757	−	0.370690i	$-0.879121\pi$
$503$	− 16.6274i	− 0.741380i	0.928757	−	0.370690i	$-0.120879\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	32.8873i	1.46058i
$508$	0	0
$509$	13.3137	0.590120	0.295060	−	0.955479i	$-0.404660\pi$
$509$	13.3137	0.590120	0.295060	+	0.955479i	$0.404660\pi$
$510$	0	0
$511$	−2.34315	−0.103655
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 2.82843i	− 0.124394i
$518$	0	0
$519$	62.6274	2.74904
$520$	0	0
$521$	25.3137	1.10901	0.554507	−	0.832179i	$-0.312907\pi$
$521$	25.3137	1.10901	0.554507	+	0.832179i	$0.312907\pi$
$522$	0	0
$523$	41.5980i	1.81895i	0.415756	+	0.909476i	$0.363517\pi$
$523$	41.5980i	1.81895i	−0.415756	+	0.909476i	$0.636483\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	15.0000	0.652174
$530$	0	0
$531$	−8.28427	−0.359507
$532$	0	0
$533$	− 7.02944i	− 0.304479i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	27.3137i	1.17867i
$538$	0	0
$539$	−3.00000	−0.129219
$540$	0	0
$541$	6.00000	0.257960	0.128980	−	0.991647i	$-0.458830\pi$
$541$	6.00000	0.257960	0.128980	+	0.991647i	$0.458830\pi$
$542$	0	0
$543$	60.2843i	2.58705i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 34.0000i	− 1.45374i	−0.686778	−	0.726868i	$-0.740975\pi$
$547$	− 34.0000i	− 1.45374i	0.686778	−	0.726868i	$-0.259025\pi$
$548$	0	0
$549$	46.5685	1.98750
$550$	0	0
$551$	0	0
$552$	0	0
$553$	− 8.00000i	− 0.340195i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 9.85786i	− 0.417691i	−0.977949	−	0.208846i	$-0.933029\pi$
$557$	− 9.85786i	− 0.417691i	0.977949	−	0.208846i	$-0.0669706\pi$
$558$	0	0
$559$	7.02944	0.297314
$560$	0	0
$561$	−19.3137	−0.815425
$562$	0	0
$563$	− 0.343146i	− 0.0144619i	−0.999974	−	0.00723093i	$-0.997698\pi$
$563$	− 0.343146i	− 0.0144619i	0.999974	−	0.00723093i	$-0.00230170\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 2.00000i	− 0.0839921i
$568$	0	0
$569$	−31.6569	−1.32712	−0.663562	−	0.748121i	$-0.730956\pi$
$569$	−31.6569	−1.32712	−0.663562	+	0.748121i	$0.730956\pi$
$570$	0	0
$571$	21.9411	0.918208	0.459104	−	0.888383i	$-0.348171\pi$
$571$	21.9411	0.918208	0.459104	+	0.888383i	$0.348171\pi$
$572$	0	0
$573$	− 9.37258i	− 0.391545i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	26.9706i	1.12280i	0.827545	+	0.561400i	$0.189737\pi$
$577$	26.9706i	1.12280i	−0.827545	+	0.561400i	$0.810263\pi$
$578$	0	0
$579$	3.31371	0.137713
$580$	0	0
$581$	12.0000	0.497844
$582$	0	0
$583$	− 11.6569i	− 0.482778i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 2.14214i	− 0.0884154i	−0.999022	−	0.0442077i	$-0.985924\pi$
$587$	− 2.14214i	− 0.0884154i	0.999022	−	0.0442077i	$-0.0140763\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−30.6274	−1.25984
$592$	0	0
$593$	3.51472i	0.144332i	0.997393	+	0.0721661i	$0.0229912\pi$
$593$	3.51472i	0.144332i	−0.997393	+	0.0721661i	$0.977009\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	29.2548i	1.19732i
$598$	0	0
$599$	−5.65685	−0.231133	−0.115566	−	0.993300i	$-0.536868\pi$
$599$	−5.65685	−0.231133	−0.115566	+	0.993300i	$0.536868\pi$
$600$	0	0
$601$	23.9411	0.976579	0.488289	−	0.872682i	$-0.337621\pi$
$601$	23.9411	0.976579	0.488289	+	0.872682i	$0.337621\pi$
$602$	0	0
$603$	− 62.4264i	− 2.54220i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	38.2843i	1.55391i	0.629556	+	0.776955i	$0.283237\pi$
$607$	38.2843i	1.55391i	−0.629556	+	0.776955i	$0.716763\pi$
$608$	0	0
$609$	20.6863	0.838251
$610$	0	0
$611$	3.31371	0.134058
$612$	0	0
$613$	− 25.4558i	− 1.02815i	−0.857745	−	0.514076i	$-0.828135\pi$
$613$	− 25.4558i	− 1.02815i	0.857745	−	0.514076i	$-0.171865\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 0.343146i	− 0.0138145i	−0.999976	−	0.00690726i	$-0.997801\pi$
$617$	− 0.343146i	− 0.0138145i	0.999976	−	0.00690726i	$-0.00219867\pi$
$618$	0	0
$619$	−14.3431	−0.576500	−0.288250	−	0.957555i	$-0.593073\pi$
$619$	−14.3431	−0.576500	−0.288250	+	0.957555i	$0.593073\pi$
$620$	0	0
$621$	16.0000	0.642058
$622$	0	0
$623$	− 26.6274i	− 1.06680i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	52.2843	2.08471
$630$	0	0
$631$	0	0		−	1.00000i	$-0.5\pi$
$631$	0	0			1.00000i	$0.5\pi$
$632$	0	0
$633$	45.2548i	1.79872i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 3.51472i	− 0.139258i
$638$	0	0
$639$	56.5685	2.23782
$640$	0	0
$641$	30.0000	1.18493	0.592464	−	0.805597i	$-0.298155\pi$
$641$	30.0000	1.18493	0.592464	+	0.805597i	$0.298155\pi$
$642$	0	0
$643$	1.45584i	0.0574129i	0.999588	+	0.0287064i	$0.00913880\pi$
$643$	1.45584i	0.0574129i	−0.999588	+	0.0287064i	$0.990861\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	27.1127i	1.06591i	0.846144	+	0.532955i	$0.178919\pi$
$647$	27.1127i	1.06591i	−0.846144	+	0.532955i	$0.821081\pi$
$648$	0	0
$649$	−1.65685	−0.0650372
$650$	0	0
$651$	0	0
$652$	0	0
$653$	11.6569i	0.456168i	0.973641	+	0.228084i	$0.0732461\pi$
$653$	11.6569i	0.456168i	−0.973641	+	0.228084i	$0.926754\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	5.85786i	0.228537i
$658$	0	0
$659$	45.9411	1.78961	0.894806	−	0.446455i	$-0.147314\pi$
$659$	45.9411	1.78961	0.894806	+	0.446455i	$0.147314\pi$
$660$	0	0
$661$	44.6274	1.73581	0.867903	−	0.496734i	$-0.165468\pi$
$661$	44.6274	1.73581	0.867903	+	0.496734i	$0.165468\pi$
$662$	0	0
$663$	− 22.6274i	− 0.878776i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	10.3431i	0.400488i
$668$	0	0
$669$	−30.6274	−1.18412
$670$	0	0
$671$	9.31371	0.359552
$672$	0	0
$673$	− 12.4853i	− 0.481272i	−0.970615	−	0.240636i	$-0.922644\pi$
$673$	− 12.4853i	− 0.481272i	0.970615	−	0.240636i	$-0.0773560\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 22.8284i	− 0.877368i	−0.898641	−	0.438684i	$-0.855445\pi$
$677$	− 22.8284i	− 0.877368i	0.898641	−	0.438684i	$-0.144555\pi$
$678$	0	0
$679$	−7.31371	−0.280674
$680$	0	0
$681$	−71.5980	−2.74364
$682$	0	0
$683$	7.79899i	0.298420i	0.988806	+	0.149210i	$0.0476731\pi$
$683$	7.79899i	0.298420i	−0.988806	+	0.149210i	$0.952327\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 3.71573i	− 0.141764i
$688$	0	0
$689$	13.6569	0.520285
$690$	0	0
$691$	39.3137	1.49556	0.747782	−	0.663944i	$-0.231119\pi$
$691$	39.3137	1.49556	0.747782	+	0.663944i	$0.231119\pi$
$692$	0	0
$693$	− 10.0000i	− 0.379869i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 40.9706i	− 1.55187i
$698$	0	0
$699$	17.3726	0.657091
$700$	0	0
$701$	−12.6274	−0.476931	−0.238465	−	0.971151i	$-0.576644\pi$
$701$	−12.6274	−0.476931	−0.238465	+	0.971151i	$0.576644\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 18.6274i	− 0.700556i
$708$	0	0
$709$	−24.6274	−0.924902	−0.462451	−	0.886645i	$-0.653030\pi$
$709$	−24.6274	−0.924902	−0.462451	+	0.886645i	$0.653030\pi$
$710$	0	0
$711$	−20.0000	−0.750059
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 65.9411i	− 2.46262i
$718$	0	0
$719$	18.3431	0.684084	0.342042	−	0.939685i	$-0.388882\pi$
$719$	18.3431	0.684084	0.342042	+	0.939685i	$0.388882\pi$
$720$	0	0
$721$	−13.6569	−0.508608
$722$	0	0
$723$	16.9706i	0.631142i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 19.5147i	− 0.723761i	−0.932225	−	0.361880i	$-0.882135\pi$
$727$	− 19.5147i	− 0.723761i	0.932225	−	0.361880i	$-0.117865\pi$
$728$	0	0
$729$	43.0000	1.59259
$730$	0	0
$731$	40.9706	1.51535
$732$	0	0
$733$	− 17.4558i	− 0.644746i	−0.946613	−	0.322373i	$-0.895519\pi$
$733$	− 17.4558i	− 0.644746i	0.946613	−	0.322373i	$-0.104481\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 12.4853i	− 0.459901i
$738$	0	0
$739$	29.9411	1.10140	0.550701	−	0.834703i	$-0.314360\pi$
$739$	29.9411	1.10140	0.550701	+	0.834703i	$0.314360\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	49.5980i	1.81957i	0.415076	+	0.909787i	$0.363755\pi$
$743$	49.5980i	1.81957i	−0.415076	+	0.909787i	$0.636245\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 30.0000i	− 1.09764i
$748$	0	0
$749$	15.3137	0.559551
$750$	0	0
$751$	16.0000	0.583848	0.291924	−	0.956441i	$-0.405705\pi$
$751$	16.0000	0.583848	0.291924	+	0.956441i	$0.405705\pi$
$752$	0	0
$753$	− 33.9411i	− 1.23688i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 13.3137i	− 0.483895i	−0.970289	−	0.241947i	$-0.922214\pi$
$757$	− 13.3137i	− 0.483895i	0.970289	−	0.241947i	$-0.0777862\pi$
$758$	0	0
$759$	8.00000	0.290382
$760$	0	0
$761$	−30.0000	−1.08750	−0.543750	−	0.839248i	$-0.682996\pi$
$761$	−30.0000	−1.08750	−0.543750	+	0.839248i	$0.682996\pi$
$762$	0	0
$763$	− 15.3137i	− 0.554393i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 1.94113i	− 0.0700900i
$768$	0	0
$769$	18.9706	0.684096	0.342048	−	0.939682i	$-0.388879\pi$
$769$	18.9706	0.684096	0.342048	+	0.939682i	$0.388879\pi$
$770$	0	0
$771$	−26.3431	−0.948725
$772$	0	0
$773$	26.2843i	0.945380i	0.881229	+	0.472690i	$0.156717\pi$
$773$	26.2843i	0.945380i	−0.881229	+	0.472690i	$0.843283\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	43.3137i	1.55387i
$778$	0	0
$779$	0	0
$780$	0	0
$781$	11.3137	0.404836
$782$	0	0
$783$	− 20.6863i	− 0.739268i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 14.9706i	− 0.533643i	−0.963746	−	0.266821i	$-0.914027\pi$
$787$	− 14.9706i	− 0.533643i	0.963746	−	0.266821i	$-0.0859734\pi$
$788$	0	0
$789$	−31.0294	−1.10468
$790$	0	0
$791$	39.3137	1.39783
$792$	0	0
$793$	10.9117i	0.387485i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 32.6274i	− 1.15572i	−0.816135	−	0.577861i	$-0.803887\pi$
$797$	− 32.6274i	− 1.15572i	0.816135	−	0.577861i	$-0.196113\pi$
$798$	0	0
$799$	19.3137	0.683270
$800$	0	0
$801$	−66.5685	−2.35208
$802$	0	0
$803$	1.17157i	0.0413439i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 48.9706i	− 1.72385i
$808$	0	0
$809$	10.9706	0.385704	0.192852	−	0.981228i	$-0.438226\pi$
$809$	10.9706	0.385704	0.192852	+	0.981228i	$0.438226\pi$
$810$	0	0
$811$	−53.9411	−1.89413	−0.947065	−	0.321043i	$-0.895967\pi$
$811$	−53.9411	−1.89413	−0.947065	+	0.321043i	$0.895967\pi$
$812$	0	0
$813$	− 20.6863i	− 0.725500i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	11.7157	0.409381
$820$	0	0
$821$	−41.3137	−1.44186	−0.720929	−	0.693009i	$-0.756285\pi$
$821$	−41.3137	−1.44186	−0.720929	+	0.693009i	$0.756285\pi$
$822$	0	0
$823$	− 19.5147i	− 0.680240i	−0.940382	−	0.340120i	$-0.889532\pi$
$823$	− 19.5147i	− 0.680240i	0.940382	−	0.340120i	$-0.110468\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	22.2843i	0.774900i	0.921891	+	0.387450i	$0.126644\pi$
$827$	22.2843i	0.774900i	−0.921891	+	0.387450i	$0.873356\pi$
$828$	0	0
$829$	−18.0000	−0.625166	−0.312583	−	0.949890i	$-0.601194\pi$
$829$	−18.0000	−0.625166	−0.312583	+	0.949890i	$0.601194\pi$
$830$	0	0
$831$	19.3137	0.669985
$832$	0	0
$833$	− 20.4853i	− 0.709773i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	26.3431	0.909466	0.454733	−	0.890628i	$-0.349735\pi$
$839$	26.3431	0.909466	0.454733	+	0.890628i	$0.349735\pi$
$840$	0	0
$841$	−15.6274	−0.538876
$842$	0	0
$843$	48.9706i	1.68664i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 2.00000i	− 0.0687208i
$848$	0	0
$849$	92.2843	3.16719
$850$	0	0
$851$	−21.6569	−0.742387
$852$	0	0
$853$	− 15.5147i	− 0.531214i	−0.964081	−	0.265607i	$-0.914428\pi$
$853$	− 15.5147i	− 0.531214i	0.964081	−	0.265607i	$-0.0855723\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 24.7696i	− 0.846112i	−0.906104	−	0.423056i	$-0.860957\pi$
$857$	− 24.7696i	− 0.846112i	0.906104	−	0.423056i	$-0.139043\pi$
$858$	0	0
$859$	24.2843	0.828569	0.414284	−	0.910148i	$-0.364032\pi$
$859$	24.2843	0.828569	0.414284	+	0.910148i	$0.364032\pi$
$860$	0	0
$861$	33.9411	1.15671
$862$	0	0
$863$	9.17157i	0.312204i	0.987741	+	0.156102i	$0.0498929\pi$
$863$	9.17157i	0.312204i	−0.987741	+	0.156102i	$0.950107\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 83.7990i	− 2.84596i
$868$	0	0
$869$	−4.00000	−0.135691
$870$	0	0
$871$	14.6274	0.495631
$872$	0	0
$873$	18.2843i	0.618829i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	49.4558i	1.67001i	0.550246	+	0.835003i	$0.314534\pi$
$877$	49.4558i	1.67001i	−0.550246	+	0.835003i	$0.685466\pi$
$878$	0	0
$879$	25.9411	0.874972
$880$	0	0
$881$	−7.37258	−0.248389	−0.124194	−	0.992258i	$-0.539635\pi$
$881$	−7.37258	−0.248389	−0.124194	+	0.992258i	$0.539635\pi$
$882$	0	0
$883$	− 37.1716i	− 1.25092i	−0.780255	−	0.625462i	$-0.784911\pi$
$883$	− 37.1716i	− 1.25092i	0.780255	−	0.625462i	$-0.215089\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	38.2843i	1.28546i	0.766093	+	0.642730i	$0.222198\pi$
$887$	38.2843i	1.28546i	−0.766093	+	0.642730i	$0.777802\pi$
$888$	0	0
$889$	8.68629	0.291329
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	9.37258i	0.312941i
$898$	0	0
$899$	0	0
$900$	0	0
$901$	79.5980	2.65179
$902$	0	0
$903$	33.9411i	1.12949i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 27.5147i	− 0.913611i	−0.889567	−	0.456806i	$-0.848993\pi$
$907$	− 27.5147i	− 0.913611i	0.889567	−	0.456806i	$-0.151007\pi$
$908$	0	0
$909$	−46.5685	−1.54458
$910$	0	0
$911$	9.94113	0.329364	0.164682	−	0.986347i	$-0.447340\pi$
$911$	9.94113	0.329364	0.164682	+	0.986347i	$0.447340\pi$
$912$	0	0
$913$	− 6.00000i	− 0.198571i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 22.6274i	− 0.747223i
$918$	0	0
$919$	−32.0000	−1.05558	−0.527791	−	0.849374i	$-0.676980\pi$
$919$	−32.0000	−1.05558	−0.527791	+	0.849374i	$0.676980\pi$
$920$	0	0
$921$	46.2254	1.52318
$922$	0	0
$923$	13.2548i	0.436288i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	34.1421i	1.12137i
$928$	0	0
$929$	−5.31371	−0.174337	−0.0871686	−	0.996194i	$-0.527782\pi$
$929$	−5.31371	−0.174337	−0.0871686	+	0.996194i	$0.527782\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	− 13.2548i	− 0.433944i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	1.45584i	0.0475604i	0.999717	+	0.0237802i	$0.00757018\pi$
$937$	1.45584i	0.0475604i	−0.999717	+	0.0237802i	$0.992430\pi$
$938$	0	0
$939$	3.71573	0.121258
$940$	0	0
$941$	−6.68629	−0.217967	−0.108983	−	0.994044i	$-0.534760\pi$
$941$	−6.68629	−0.217967	−0.108983	+	0.994044i	$0.534760\pi$
$942$	0	0
$943$	16.9706i	0.552638i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	41.1716i	1.33790i	0.743309	+	0.668948i	$0.233255\pi$
$947$	41.1716i	1.33790i	−0.743309	+	0.668948i	$0.766745\pi$
$948$	0	0
$949$	−1.37258	−0.0445559
$950$	0	0
$951$	−3.71573	−0.120491
$952$	0	0
$953$	53.1716i	1.72240i	0.508269	+	0.861198i	$0.330285\pi$
$953$	53.1716i	1.72240i	−0.508269	+	0.861198i	$0.669715\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 10.3431i	− 0.334346i
$958$	0	0
$959$	21.9411	0.708516
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	− 38.2843i	− 1.23369i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 14.9706i	− 0.481421i	−0.970597	−	0.240710i	$-0.922620\pi$
$967$	− 14.9706i	− 0.481421i	0.970597	−	0.240710i	$-0.0773804\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	8.68629	0.278756	0.139378	−	0.990239i	$-0.455490\pi$
$971$	8.68629	0.278756	0.139378	+	0.990239i	$0.455490\pi$
$972$	0	0
$973$	8.00000i	0.256468i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 32.3431i	− 1.03475i	−0.855759	−	0.517374i	$-0.826909\pi$
$977$	− 32.3431i	− 1.03475i	0.855759	−	0.517374i	$-0.173091\pi$
$978$	0	0
$979$	−13.3137	−0.425508
$980$	0	0
$981$	−38.2843	−1.22232
$982$	0	0
$983$	21.8579i	0.697158i	0.937279	+	0.348579i	$0.113336\pi$
$983$	21.8579i	0.697158i	−0.937279	+	0.348579i	$0.886664\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	16.0000i	0.509286i
$988$	0	0
$989$	−16.9706	−0.539633
$990$	0	0
$991$	57.9411	1.84056	0.920280	−	0.391260i	$-0.127961\pi$
$991$	57.9411	1.84056	0.920280	+	0.391260i	$0.127961\pi$
$992$	0	0
$993$	20.6863i	0.656460i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	41.4558i	1.31292i	0.754361	+	0.656460i	$0.227947\pi$
$997$	41.4558i	1.31292i	−0.754361	+	0.656460i	$0.772053\pi$
$998$	0	0
$999$	43.3137	1.37039

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4400.2.b.q.4049.3		4
4.3	odd	2		275.2.b.d.199.1		4
5.2	odd	4		880.2.a.m.1.2		2
5.3	odd	4		4400.2.a.bn.1.1		2
5.4	even	2	inner	4400.2.b.q.4049.2		4
12.11	even	2		2475.2.c.l.199.4		4
15.2	even	4		7920.2.a.ch.1.2		2
20.3	even	4		275.2.a.c.1.1		2
20.7	even	4		55.2.a.b.1.2	✓	2
20.19	odd	2		275.2.b.d.199.4		4
40.27	even	4		3520.2.a.bn.1.2		2
40.37	odd	4		3520.2.a.bo.1.1		2
55.32	even	4		9680.2.a.bn.1.2		2
60.23	odd	4		2475.2.a.x.1.2		2
60.47	odd	4		495.2.a.b.1.1		2
60.59	even	2		2475.2.c.l.199.1		4
140.27	odd	4		2695.2.a.f.1.2		2
220.7	odd	20		605.2.g.l.511.1		8
220.27	even	20		605.2.g.f.366.1		8
220.43	odd	4		3025.2.a.o.1.2		2
220.47	even	20		605.2.g.f.251.2		8
220.87	odd	4		605.2.a.d.1.1		2
220.107	odd	20		605.2.g.l.251.1		8
220.127	odd	20		605.2.g.l.366.2		8
220.147	even	20		605.2.g.f.511.2		8
220.167	odd	20		605.2.g.l.81.2		8
220.207	even	20		605.2.g.f.81.1		8
260.207	even	4		9295.2.a.g.1.1		2
660.527	even	4		5445.2.a.y.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
55.2.a.b.1.2	✓	2	20.7	even	4
275.2.a.c.1.1		2	20.3	even	4
275.2.b.d.199.1		4	4.3	odd	2
275.2.b.d.199.4		4	20.19	odd	2
495.2.a.b.1.1		2	60.47	odd	4
605.2.a.d.1.1		2	220.87	odd	4
605.2.g.f.81.1		8	220.207	even	20
605.2.g.f.251.2		8	220.47	even	20
605.2.g.f.366.1		8	220.27	even	20
605.2.g.f.511.2		8	220.147	even	20
605.2.g.l.81.2		8	220.167	odd	20
605.2.g.l.251.1		8	220.107	odd	20
605.2.g.l.366.2		8	220.127	odd	20
605.2.g.l.511.1		8	220.7	odd	20
880.2.a.m.1.2		2	5.2	odd	4
2475.2.a.x.1.2		2	60.23	odd	4
2475.2.c.l.199.1		4	60.59	even	2
2475.2.c.l.199.4		4	12.11	even	2
2695.2.a.f.1.2		2	140.27	odd	4
3025.2.a.o.1.2		2	220.43	odd	4
3520.2.a.bn.1.2		2	40.27	even	4
3520.2.a.bo.1.1		2	40.37	odd	4
4400.2.a.bn.1.1		2	5.3	odd	4
4400.2.b.q.4049.2		4	5.4	even	2	inner
4400.2.b.q.4049.3		4	1.1	even	1	trivial
5445.2.a.y.1.2		2	660.527	even	4
7920.2.a.ch.1.2		2	15.2	even	4
9295.2.a.g.1.1		2	260.207	even	4
9680.2.a.bn.1.2		2	55.32	even	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.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.82843i q^{3} -2.00000i q^{7} -5.00000 q^{9} +O(q^{10})\)	\(q+2.82843i q^{3} -2.00000i q^{7} -5.00000 q^{9} -1.00000 q^{11} -1.17157i q^{13} -6.82843i q^{17} +5.65685 q^{21} +2.82843i q^{23} -5.65685i q^{27} +3.65685 q^{29} -2.82843i q^{33} +7.65685i q^{37} +3.31371 q^{39} +6.00000 q^{41} +6.00000i q^{43} +2.82843i q^{47} +3.00000 q^{49} +19.3137 q^{51} +11.6569i q^{53} +1.65685 q^{59} -9.31371 q^{61} +10.0000i q^{63} +12.4853i q^{67} -8.00000 q^{69} -11.3137 q^{71} -1.17157i q^{73} +2.00000i q^{77} +4.00000 q^{79} +1.00000 q^{81} +6.00000i q^{83} +10.3431i q^{87} +13.3137 q^{89} -2.34315 q^{91} -3.65685i q^{97} +5.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 20 q^{9}+O(q^{10}) \) 4 * q - 20 * q^9	\( 4 q - 20 q^{9} - 4 q^{11} - 8 q^{29} - 32 q^{39} + 24 q^{41} + 12 q^{49} + 32 q^{51} - 16 q^{59} + 8 q^{61} - 32 q^{69} + 16 q^{79} + 4 q^{81} + 8 q^{89} - 32 q^{91} + 20 q^{99}+O(q^{100}) \) 4 * q - 20 * q^9 - 4 * q^11 - 8 * q^29 - 32 * q^39 + 24 * q^41 + 12 * q^49 + 32 * q^51 - 16 * q^59 + 8 * q^61 - 32 * q^69 + 16 * q^79 + 4 * q^81 + 8 * q^89 - 32 * q^91 + 20 * q^99

Introduction

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