LMFDB - Embedded newform 4400.2.b.q.4049.1

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			4049.1
Root			$-0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	4400.4049
Dual form			4400.2.b.q.4049.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.82843i q^{3} -2.00000i q^{7} -5.00000 q^{9} +O(q^{10})$	$q-2.82843i q^{3} -2.00000i q^{7} -5.00000 q^{9} -1.00000 q^{11} -6.82843i q^{13} -1.17157i q^{17} -5.65685 q^{21} -2.82843i q^{23} +5.65685i q^{27} -7.65685 q^{29} +2.82843i q^{33} -3.65685i q^{37} -19.3137 q^{39} +6.00000 q^{41} +6.00000i q^{43} -2.82843i q^{47} +3.00000 q^{49} -3.31371 q^{51} +0.343146i q^{53} -9.65685 q^{59} +13.3137 q^{61} +10.0000i q^{63} -4.48528i q^{67} -8.00000 q^{69} +11.3137 q^{71} -6.82843i q^{73} +2.00000i q^{77} +4.00000 q^{79} +1.00000 q^{81} +6.00000i q^{83} +21.6569i q^{87} -9.31371 q^{89} -13.6569 q^{91} +7.65685i q^{97} +5.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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	$-0.695913\pi$
$3$	− 2.82843i	− 1.63299i	0.577350	−	0.816497i	$-0.304087\pi$
$4$	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
$13$	− 6.82843i	− 1.89386i	−0.321433	−	0.946932i	$-0.604164\pi$
$13$	− 6.82843i	− 1.89386i	0.321433	−	0.946932i	$-0.395836\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 1.17157i	− 0.284148i	−0.989856	−	0.142074i	$-0.954623\pi$
$17$	− 1.17157i	− 0.284148i	0.989856	−	0.142074i	$-0.0453771\pi$
$18$	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	$-0.904719\pi$
$23$	− 2.82843i	− 0.589768i	0.955533	−	0.294884i	$-0.0952810\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	5.65685i	1.08866i
$28$	0	0
$29$	−7.65685	−1.42184	−0.710921	−	0.703272i	$-0.751722\pi$
$29$	−7.65685	−1.42184	−0.710921	+	0.703272i	$0.751722\pi$
$30$	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
$37$	− 3.65685i	− 0.601183i	−0.953753	−	0.300592i	$-0.902816\pi$
$37$	− 3.65685i	− 0.601183i	0.953753	−	0.300592i	$-0.0971841\pi$
$38$	0	0
$39$	−19.3137	−3.09267
$40$	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	$-0.933863\pi$
$47$	− 2.82843i	− 0.412568i	0.978492	−	0.206284i	$-0.0661372\pi$
$48$	0	0
$49$	3.00000	0.428571
$50$	0	0
$51$	−3.31371	−0.464012
$52$	0	0
$53$	0.343146i	0.0471347i	0.999722	+	0.0235673i	$0.00750241\pi$
$53$	0.343146i	0.0471347i	−0.999722	+	0.0235673i	$0.992498\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−9.65685	−1.25722	−0.628608	−	0.777723i	$-0.716375\pi$
$59$	−9.65685	−1.25722	−0.628608	+	0.777723i	$0.716375\pi$
$60$	0	0
$61$	13.3137	1.70465	0.852323	−	0.523016i	$-0.175193\pi$
$61$	13.3137	1.70465	0.852323	+	0.523016i	$0.175193\pi$
$62$	0	0
$63$	10.0000i	1.25988i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 4.48528i	− 0.547964i	−0.961735	−	0.273982i	$-0.911659\pi$
$67$	− 4.48528i	− 0.547964i	0.961735	−	0.273982i	$-0.0883409\pi$
$68$	0	0
$69$	−8.00000	−0.963087
$70$	0	0
$71$	11.3137	1.34269	0.671345	−	0.741145i	$-0.265717\pi$
$71$	11.3137	1.34269	0.671345	+	0.741145i	$0.265717\pi$
$72$	0	0
$73$	− 6.82843i	− 0.799207i	−0.916688	−	0.399603i	$-0.869148\pi$
$73$	− 6.82843i	− 0.799207i	0.916688	−	0.399603i	$-0.130852\pi$
$74$	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
$87$	21.6569i	2.32186i
$88$	0	0
$89$	−9.31371	−0.987251	−0.493626	−	0.869675i	$-0.664329\pi$
$89$	−9.31371	−0.987251	−0.493626	+	0.869675i	$0.664329\pi$
$90$	0	0
$91$	−13.6569	−1.43163
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	7.65685i	0.777436i	0.921357	+	0.388718i	$0.127082\pi$
$97$	7.65685i	0.777436i	−0.921357	+	0.388718i	$0.872918\pi$
$98$	0	0
$99$	5.00000	0.502519
$100$	0	0
$101$	−13.3137	−1.32476	−0.662382	−	0.749166i	$-0.730454\pi$
$101$	−13.3137	−1.32476	−0.662382	+	0.749166i	$0.730454\pi$
$102$	0	0
$103$	− 1.17157i	− 0.115439i	−0.998333	−	0.0577193i	$-0.981617\pi$
$103$	− 1.17157i	− 0.115439i	0.998333	−	0.0577193i	$-0.0183828\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 3.65685i	− 0.353521i	−0.984254	−	0.176761i	$-0.943438\pi$
$107$	− 3.65685i	− 0.353521i	0.984254	−	0.176761i	$-0.0565619\pi$
$108$	0	0
$109$	−3.65685	−0.350263	−0.175132	−	0.984545i	$-0.556035\pi$
$109$	−3.65685	−0.350263	−0.175132	+	0.984545i	$0.556035\pi$
$110$	0	0
$111$	−10.3431	−0.981728
$112$	0	0
$113$	8.34315i	0.784857i	0.919782	+	0.392429i	$0.128365\pi$
$113$	8.34315i	0.784857i	−0.919782	+	0.392429i	$0.871635\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	34.1421i	3.15644i
$118$	0	0
$119$	−2.34315	−0.214796
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	− 16.9706i	− 1.53018i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	15.6569i	1.38932i	0.719338	+	0.694661i	$0.244445\pi$
$127$	15.6569i	1.38932i	−0.719338	+	0.694661i	$0.755555\pi$
$128$	0	0
$129$	16.9706	1.49417
$130$	0	0
$131$	−11.3137	−0.988483	−0.494242	−	0.869325i	$-0.664554\pi$
$131$	−11.3137	−0.988483	−0.494242	+	0.869325i	$0.664554\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 22.9706i	− 1.96251i	−0.192720	−	0.981254i	$-0.561731\pi$
$137$	− 22.9706i	− 1.96251i	0.192720	−	0.981254i	$-0.438269\pi$
$138$	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
$143$	6.82843i	0.571022i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 8.48528i	− 0.699854i
$148$	0	0
$149$	−11.6569	−0.954967	−0.477483	−	0.878641i	$-0.658451\pi$
$149$	−11.6569	−0.954967	−0.477483	+	0.878641i	$0.658451\pi$
$150$	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
$153$	5.85786i	0.473580i
$154$	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
$159$	0.970563	0.0769706
$160$	0	0
$161$	−5.65685	−0.445823
$162$	0	0
$163$	0.485281i	0.0380102i	0.999819	+	0.0190051i	$0.00604987\pi$
$163$	0.485281i	0.0380102i	−0.999819	+	0.0190051i	$0.993950\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	10.9706i	0.848928i	0.905445	+	0.424464i	$0.139537\pi$
$167$	10.9706i	0.848928i	−0.905445	+	0.424464i	$0.860463\pi$
$168$	0	0
$169$	−33.6274	−2.58672
$170$	0	0
$171$	0	0
$172$	0	0
$173$	6.14214i	0.466978i	0.972359	+	0.233489i	$0.0750143\pi$
$173$	6.14214i	0.466978i	−0.972359	+	0.233489i	$0.924986\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	27.3137i	2.05302i
$178$	0	0
$179$	−1.65685	−0.123839	−0.0619196	−	0.998081i	$-0.519722\pi$
$179$	−1.65685	−0.123839	−0.0619196	+	0.998081i	$0.519722\pi$
$180$	0	0
$181$	−1.31371	−0.0976472	−0.0488236	−	0.998807i	$-0.515547\pi$
$181$	−1.31371	−0.0976472	−0.0488236	+	0.998807i	$0.515547\pi$
$182$	0	0
$183$	− 37.6569i	− 2.78367i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	1.17157i	0.0856739i
$188$	0	0
$189$	11.3137	0.822951
$190$	0	0
$191$	19.3137	1.39749	0.698745	−	0.715370i	$-0.253742\pi$
$191$	19.3137	1.39749	0.698745	+	0.715370i	$0.253742\pi$
$192$	0	0
$193$	− 6.82843i	− 0.491521i	−0.969331	−	0.245760i	$-0.920962\pi$
$193$	− 6.82843i	− 0.491521i	0.969331	−	0.245760i	$-0.0790377\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	5.17157i	0.368459i	0.982883	+	0.184230i	$0.0589790\pi$
$197$	5.17157i	0.368459i	−0.982883	+	0.184230i	$0.941021\pi$
$198$	0	0
$199$	21.6569	1.53521	0.767607	−	0.640921i	$-0.221447\pi$
$199$	21.6569	1.53521	0.767607	+	0.640921i	$0.221447\pi$
$200$	0	0
$201$	−12.6863	−0.894822
$202$	0	0
$203$	15.3137i	1.07481i
$204$	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
$219$	−19.3137	−1.30510
$220$	0	0
$221$	−8.00000	−0.538138
$222$	0	0
$223$	5.17157i	0.346314i	0.984894	+	0.173157i	$0.0553968\pi$
$223$	5.17157i	0.346314i	−0.984894	+	0.173157i	$0.944603\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	2.68629i	0.178295i	0.996018	+	0.0891477i	$0.0284143\pi$
$227$	2.68629i	0.178295i	−0.996018	+	0.0891477i	$0.971586\pi$
$228$	0	0
$229$	21.3137	1.40845	0.704225	−	0.709977i	$-0.251295\pi$
$229$	21.3137	1.40845	0.704225	+	0.709977i	$0.251295\pi$
$230$	0	0
$231$	5.65685	0.372194
$232$	0	0
$233$	22.1421i	1.45058i	0.688444	+	0.725290i	$0.258294\pi$
$233$	22.1421i	1.45058i	−0.688444	+	0.725290i	$0.741706\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 11.3137i	− 0.734904i
$238$	0	0
$239$	−0.686292	−0.0443925	−0.0221963	−	0.999754i	$-0.507066\pi$
$239$	−0.686292	−0.0443925	−0.0221963	+	0.999754i	$0.507066\pi$
$240$	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
$257$	− 13.3137i	− 0.830486i	−0.909710	−	0.415243i	$-0.863696\pi$
$257$	− 13.3137i	− 0.830486i	0.909710	−	0.415243i	$-0.136304\pi$
$258$	0	0
$259$	−7.31371	−0.454452
$260$	0	0
$261$	38.2843	2.36974
$262$	0	0
$263$	− 22.9706i	− 1.41643i	−0.705999	−	0.708213i	$-0.749502\pi$
$263$	− 22.9706i	− 1.41643i	0.705999	−	0.708213i	$-0.250498\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	26.3431i	1.61217i
$268$	0	0
$269$	5.31371	0.323983	0.161991	−	0.986792i	$-0.448208\pi$
$269$	5.31371	0.323983	0.161991	+	0.986792i	$0.448208\pi$
$270$	0	0
$271$	15.3137	0.930242	0.465121	−	0.885247i	$-0.346011\pi$
$271$	15.3137	0.930242	0.465121	+	0.885247i	$0.346011\pi$
$272$	0	0
$273$	38.6274i	2.33784i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 1.17157i	− 0.0703930i	−0.999380	−	0.0351965i	$-0.988794\pi$
$277$	− 1.17157i	− 0.0703930i	0.999380	−	0.0351965i	$-0.0112057\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−5.31371	−0.316989	−0.158495	−	0.987360i	$-0.550664\pi$
$281$	−5.31371	−0.316989	−0.158495	+	0.987360i	$0.550664\pi$
$282$	0	0
$283$	12.6274i	0.750622i	0.926899	+	0.375311i	$0.122464\pi$
$283$	12.6274i	0.750622i	−0.926899	+	0.375311i	$0.877536\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 12.0000i	− 0.708338i
$288$	0	0
$289$	15.6274	0.919260
$290$	0	0
$291$	21.6569	1.26955
$292$	0	0
$293$	− 14.8284i	− 0.866286i	−0.901325	−	0.433143i	$-0.857405\pi$
$293$	− 14.8284i	− 0.866286i	0.901325	−	0.433143i	$-0.142595\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 5.65685i	− 0.328244i
$298$	0	0
$299$	−19.3137	−1.11694
$300$	0	0
$301$	12.0000	0.691669
$302$	0	0
$303$	37.6569i	2.16333i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 27.6569i	− 1.57846i	−0.614098	−	0.789230i	$-0.710480\pi$
$307$	− 27.6569i	− 1.57846i	0.614098	−	0.789230i	$-0.289520\pi$
$308$	0	0
$309$	−3.31371	−0.188510
$310$	0	0
$311$	−27.3137	−1.54882	−0.774409	−	0.632685i	$-0.781953\pi$
$311$	−27.3137	−1.54882	−0.774409	+	0.632685i	$0.781953\pi$
$312$	0	0
$313$	21.3137i	1.20472i	0.798224	+	0.602361i	$0.205773\pi$
$313$	21.3137i	1.20472i	−0.798224	+	0.602361i	$0.794227\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 21.3137i	− 1.19710i	−0.801087	−	0.598549i	$-0.795744\pi$
$317$	− 21.3137i	− 1.19710i	0.801087	−	0.598549i	$-0.204256\pi$
$318$	0	0
$319$	7.65685	0.428702
$320$	0	0
$321$	−10.3431	−0.577298
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0	0
$326$	0	0
$327$	10.3431i	0.571977i
$328$	0	0
$329$	−5.65685	−0.311872
$330$	0	0
$331$	−15.3137	−0.841718	−0.420859	−	0.907126i	$-0.638271\pi$
$331$	−15.3137	−0.841718	−0.420859	+	0.907126i	$0.638271\pi$
$332$	0	0
$333$	18.2843i	1.00197i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	3.51472i	0.191459i	0.995407	+	0.0957295i	$0.0305184\pi$
$337$	3.51472i	0.191459i	−0.995407	+	0.0957295i	$0.969482\pi$
$338$	0	0
$339$	23.5980	1.28167
$340$	0	0
$341$	0	0
$342$	0	0
$343$	− 20.0000i	− 1.07990i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 22.9706i	− 1.23312i	−0.787306	−	0.616562i	$-0.788525\pi$
$347$	− 22.9706i	− 1.23312i	0.787306	−	0.616562i	$-0.211475\pi$
$348$	0	0
$349$	6.97056	0.373126	0.186563	−	0.982443i	$-0.440265\pi$
$349$	6.97056	0.373126	0.186563	+	0.982443i	$0.440265\pi$
$350$	0	0
$351$	38.6274	2.06178
$352$	0	0
$353$	− 1.31371i	− 0.0699216i	−0.999389	−	0.0349608i	$-0.988869\pi$
$353$	− 1.31371i	− 0.0699216i	0.999389	−	0.0349608i	$-0.0111306\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	6.62742i	0.350760i
$358$	0	0
$359$	23.3137	1.23045	0.615225	−	0.788351i	$-0.289065\pi$
$359$	23.3137	1.23045	0.615225	+	0.788351i	$0.289065\pi$
$360$	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	$0.0710835\pi$
$367$	8.48528i	0.442928i	−0.975169	+	0.221464i	$0.928916\pi$
$368$	0	0
$369$	−30.0000	−1.56174
$370$	0	0
$371$	0.686292	0.0356305
$372$	0	0
$373$	− 3.79899i	− 0.196704i	−0.995152	−	0.0983521i	$-0.968643\pi$
$373$	− 3.79899i	− 0.196704i	0.995152	−	0.0983521i	$-0.0313571\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	52.2843i	2.69278i
$378$	0	0
$379$	22.3431	1.14769	0.573845	−	0.818964i	$-0.305451\pi$
$379$	22.3431	1.14769	0.573845	+	0.818964i	$0.305451\pi$
$380$	0	0
$381$	44.2843	2.26875
$382$	0	0
$383$	34.1421i	1.74458i	0.488987	+	0.872291i	$0.337366\pi$
$383$	34.1421i	1.74458i	−0.488987	+	0.872291i	$0.662634\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 30.0000i	− 1.52499i
$388$	0	0
$389$	24.6274	1.24866	0.624330	−	0.781161i	$-0.285372\pi$
$389$	24.6274	1.24866	0.624330	+	0.781161i	$0.285372\pi$
$390$	0	0
$391$	−3.31371	−0.167581
$392$	0	0
$393$	32.0000i	1.61419i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 13.3137i	− 0.668196i	−0.942538	−	0.334098i	$-0.891568\pi$
$397$	− 13.3137i	− 0.668196i	0.942538	−	0.334098i	$-0.108432\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	17.3137	0.864605	0.432303	−	0.901729i	$-0.357701\pi$
$401$	17.3137	0.864605	0.432303	+	0.901729i	$0.357701\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	3.65685i	0.181264i
$408$	0	0
$409$	−34.9706	−1.72918	−0.864592	−	0.502475i	$-0.832423\pi$
$409$	−34.9706	−1.72918	−0.864592	+	0.502475i	$0.832423\pi$
$410$	0	0
$411$	−64.9706	−3.20476
$412$	0	0
$413$	19.3137i	0.950365i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	11.3137i	0.554035i
$418$	0	0
$419$	−14.3431	−0.700709	−0.350354	−	0.936617i	$-0.613939\pi$
$419$	−14.3431	−0.700709	−0.350354	+	0.936617i	$0.613939\pi$
$420$	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
$427$	− 26.6274i	− 1.28859i
$428$	0	0
$429$	19.3137	0.932475
$430$	0	0
$431$	−11.3137	−0.544962	−0.272481	−	0.962161i	$-0.587844\pi$
$431$	−11.3137	−0.544962	−0.272481	+	0.962161i	$0.587844\pi$
$432$	0	0
$433$	3.65685i	0.175737i	0.996132	+	0.0878686i	$0.0280056\pi$
$433$	3.65685i	0.175737i	−0.996132	+	0.0878686i	$0.971994\pi$
$434$	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
$443$	21.1716i	1.00589i	0.864318	+	0.502946i	$0.167751\pi$
$443$	21.1716i	1.00589i	−0.864318	+	0.502946i	$0.832249\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	32.9706i	1.55945i
$448$	0	0
$449$	16.6274	0.784696	0.392348	−	0.919817i	$-0.371663\pi$
$449$	16.6274	0.784696	0.392348	+	0.919817i	$0.371663\pi$
$450$	0	0
$451$	−6.00000	−0.282529
$452$	0	0
$453$	− 33.9411i	− 1.59469i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	16.4853i	0.771149i	0.922677	+	0.385574i	$0.125997\pi$
$457$	16.4853i	0.771149i	−0.922677	+	0.385574i	$0.874003\pi$
$458$	0	0
$459$	6.62742	0.309341
$460$	0	0
$461$	−32.6274	−1.51961	−0.759805	−	0.650151i	$-0.774706\pi$
$461$	−32.6274	−1.51961	−0.759805	+	0.650151i	$0.774706\pi$
$462$	0	0
$463$	− 22.1421i	− 1.02903i	−0.857481	−	0.514516i	$-0.827972\pi$
$463$	− 22.1421i	− 1.02903i	0.857481	−	0.514516i	$-0.172028\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 9.17157i	− 0.424410i	−0.977225	−	0.212205i	$-0.931936\pi$
$467$	− 9.17157i	− 0.424410i	0.977225	−	0.212205i	$-0.0680644\pi$
$468$	0	0
$469$	−8.97056	−0.414222
$470$	0	0
$471$	39.5980	1.82458
$472$	0	0
$473$	− 6.00000i	− 0.275880i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 1.71573i	− 0.0785578i
$478$	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
$481$	−24.9706	−1.13856
$482$	0	0
$483$	16.0000i	0.728025i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 7.51472i	− 0.340524i	−0.985399	−	0.170262i	$-0.945539\pi$
$487$	− 7.51472i	− 0.340524i	0.985399	−	0.170262i	$-0.0544615\pi$
$488$	0	0
$489$	1.37258	0.0620703
$490$	0	0
$491$	23.3137	1.05213	0.526066	−	0.850443i	$-0.323666\pi$
$491$	23.3137	1.05213	0.526066	+	0.850443i	$0.323666\pi$
$492$	0	0
$493$	8.97056i	0.404014i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 22.6274i	− 1.01498i
$498$	0	0
$499$	−1.65685	−0.0741710	−0.0370855	−	0.999312i	$-0.511807\pi$
$499$	−1.65685	−0.0741710	−0.0370855	+	0.999312i	$0.511807\pi$
$500$	0	0
$501$	31.0294	1.38629
$502$	0	0
$503$	28.6274i	1.27643i	0.769857	+	0.638217i	$0.220328\pi$
$503$	28.6274i	1.27643i	−0.769857	+	0.638217i	$0.779672\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	95.1127i	4.22410i
$508$	0	0
$509$	−9.31371	−0.412823	−0.206411	−	0.978465i	$-0.566179\pi$
$509$	−9.31371	−0.412823	−0.206411	+	0.978465i	$0.566179\pi$
$510$	0	0
$511$	−13.6569	−0.604144
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	2.82843i	0.124394i
$518$	0	0
$519$	17.3726	0.762572
$520$	0	0
$521$	2.68629	0.117689	0.0588443	−	0.998267i	$-0.481258\pi$
$521$	2.68629	0.117689	0.0588443	+	0.998267i	$0.481258\pi$
$522$	0	0
$523$	− 37.5980i	− 1.64404i	−0.569455	−	0.822022i	$-0.692846\pi$
$523$	− 37.5980i	− 1.64404i	0.569455	−	0.822022i	$-0.307154\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	15.0000	0.652174
$530$	0	0
$531$	48.2843	2.09536
$532$	0	0
$533$	− 40.9706i	− 1.77463i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	4.68629i	0.202228i
$538$	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
$543$	3.71573i	0.159457i
$544$	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
$549$	−66.5685	−2.84108
$550$	0	0
$551$	0	0
$552$	0	0
$553$	− 8.00000i	− 0.340195i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 38.1421i	− 1.61613i	−0.589090	−	0.808067i	$-0.700514\pi$
$557$	− 38.1421i	− 1.61613i	0.589090	−	0.808067i	$-0.299486\pi$
$558$	0	0
$559$	40.9706	1.73287
$560$	0	0
$561$	3.31371	0.139905
$562$	0	0
$563$	− 11.6569i	− 0.491278i	−0.969361	−	0.245639i	$-0.921002\pi$
$563$	− 11.6569i	− 0.491278i	0.969361	−	0.245639i	$-0.0789977\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 2.00000i	− 0.0839921i
$568$	0	0
$569$	−20.3431	−0.852829	−0.426415	−	0.904528i	$-0.640224\pi$
$569$	−20.3431	−0.852829	−0.426415	+	0.904528i	$0.640224\pi$
$570$	0	0
$571$	−45.9411	−1.92258	−0.961288	−	0.275545i	$-0.911142\pi$
$571$	−45.9411	−1.92258	−0.961288	+	0.275545i	$0.911142\pi$
$572$	0	0
$573$	− 54.6274i	− 2.28209i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 6.97056i	− 0.290188i	−0.989418	−	0.145094i	$-0.953651\pi$
$577$	− 6.97056i	− 0.290188i	0.989418	−	0.145094i	$-0.0463485\pi$
$578$	0	0
$579$	−19.3137	−0.802650
$580$	0	0
$581$	12.0000	0.497844
$582$	0	0
$583$	− 0.343146i	− 0.0142116i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	26.1421i	1.07900i	0.841985	+	0.539501i	$0.181387\pi$
$587$	26.1421i	1.07900i	−0.841985	+	0.539501i	$0.818613\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	14.6274	0.601692
$592$	0	0
$593$	20.4853i	0.841230i	0.907239	+	0.420615i	$0.138186\pi$
$593$	20.4853i	0.841230i	−0.907239	+	0.420615i	$0.861814\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 61.2548i	− 2.50699i
$598$	0	0
$599$	5.65685	0.231133	0.115566	−	0.993300i	$-0.463132\pi$
$599$	5.65685	0.231133	0.115566	+	0.993300i	$0.463132\pi$
$600$	0	0
$601$	−43.9411	−1.79240	−0.896198	−	0.443654i	$-0.853682\pi$
$601$	−43.9411	−1.79240	−0.896198	+	0.443654i	$0.853682\pi$
$602$	0	0
$603$	22.4264i	0.913274i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 18.2843i	− 0.742136i	−0.928606	−	0.371068i	$-0.878992\pi$
$607$	− 18.2843i	− 0.742136i	0.928606	−	0.371068i	$-0.121008\pi$
$608$	0	0
$609$	43.3137	1.75516
$610$	0	0
$611$	−19.3137	−0.781349
$612$	0	0
$613$	25.4558i	1.02815i	0.857745	+	0.514076i	$0.171865\pi$
$613$	25.4558i	1.02815i	−0.857745	+	0.514076i	$0.828135\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 11.6569i	− 0.469287i	−0.972081	−	0.234644i	$-0.924608\pi$
$617$	− 11.6569i	− 0.469287i	0.972081	−	0.234644i	$-0.0753923\pi$
$618$	0	0
$619$	−25.6569	−1.03124	−0.515618	−	0.856819i	$-0.672438\pi$
$619$	−25.6569	−1.03124	−0.515618	+	0.856819i	$0.672438\pi$
$620$	0	0
$621$	16.0000	0.642058
$622$	0	0
$623$	18.6274i	0.746292i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−4.28427	−0.170825
$630$	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
$637$	− 20.4853i	− 0.811656i
$638$	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
$643$	− 49.4558i	− 1.95035i	−0.221440	−	0.975174i	$-0.571076\pi$
$643$	− 49.4558i	− 1.95035i	0.221440	−	0.975174i	$-0.428924\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 35.1127i	− 1.38042i	−0.723608	−	0.690211i	$-0.757518\pi$
$647$	− 35.1127i	− 1.38042i	0.723608	−	0.690211i	$-0.242482\pi$
$648$	0	0
$649$	9.65685	0.379065
$650$	0	0
$651$	0	0
$652$	0	0
$653$	0.343146i	0.0134283i	0.999977	+	0.00671417i	$0.00213720\pi$
$653$	0.343146i	0.0134283i	−0.999977	+	0.00671417i	$0.997863\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	34.1421i	1.33201i
$658$	0	0
$659$	−21.9411	−0.854705	−0.427352	−	0.904085i	$-0.640554\pi$
$659$	−21.9411	−0.854705	−0.427352	+	0.904085i	$0.640554\pi$
$660$	0	0
$661$	−0.627417	−0.0244037	−0.0122018	−	0.999926i	$-0.503884\pi$
$661$	−0.627417	−0.0244037	−0.0122018	+	0.999926i	$0.503884\pi$
$662$	0	0
$663$	22.6274i	0.878776i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	21.6569i	0.838557i
$668$	0	0
$669$	14.6274	0.565529
$670$	0	0
$671$	−13.3137	−0.513970
$672$	0	0
$673$	4.48528i	0.172895i	0.996256	+	0.0864474i	$0.0275515\pi$
$673$	4.48528i	0.172895i	−0.996256	+	0.0864474i	$0.972449\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 17.1716i	− 0.659957i	−0.943988	−	0.329979i	$-0.892958\pi$
$677$	− 17.1716i	− 0.659957i	0.943988	−	0.329979i	$-0.107042\pi$
$678$	0	0
$679$	15.3137	0.587686
$680$	0	0
$681$	7.59798	0.291155
$682$	0	0
$683$	− 31.7990i	− 1.21675i	−0.793648	−	0.608377i	$-0.791821\pi$
$683$	− 31.7990i	− 1.21675i	0.793648	−	0.608377i	$-0.208179\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 60.2843i	− 2.29999i
$688$	0	0
$689$	2.34315	0.0892667
$690$	0	0
$691$	16.6863	0.634776	0.317388	−	0.948296i	$-0.397194\pi$
$691$	16.6863	0.634776	0.317388	+	0.948296i	$0.397194\pi$
$692$	0	0
$693$	− 10.0000i	− 0.379869i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 7.02944i	− 0.266259i
$698$	0	0
$699$	62.6274	2.36879
$700$	0	0
$701$	32.6274	1.23232	0.616160	−	0.787621i	$-0.288687\pi$
$701$	32.6274	1.23232	0.616160	+	0.787621i	$0.288687\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	26.6274i	1.00143i
$708$	0	0
$709$	20.6274	0.774679	0.387339	−	0.921937i	$-0.373394\pi$
$709$	20.6274	0.774679	0.387339	+	0.921937i	$0.373394\pi$
$710$	0	0
$711$	−20.0000	−0.750059
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	1.94113i	0.0724927i
$718$	0	0
$719$	29.6569	1.10601	0.553007	−	0.833177i	$-0.313480\pi$
$719$	29.6569	1.10601	0.553007	+	0.833177i	$0.313480\pi$
$720$	0	0
$721$	−2.34315	−0.0872633
$722$	0	0
$723$	− 16.9706i	− 0.631142i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 36.4853i	− 1.35316i	−0.736367	−	0.676582i	$-0.763460\pi$
$727$	− 36.4853i	− 1.35316i	0.736367	−	0.676582i	$-0.236540\pi$
$728$	0	0
$729$	43.0000	1.59259
$730$	0	0
$731$	7.02944	0.259993
$732$	0	0
$733$	33.4558i	1.23572i	0.786288	+	0.617860i	$0.212000\pi$
$733$	33.4558i	1.23572i	−0.786288	+	0.617860i	$0.788000\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	4.48528i	0.165217i
$738$	0	0
$739$	−37.9411	−1.39569	−0.697843	−	0.716250i	$-0.745857\pi$
$739$	−37.9411	−1.39569	−0.697843	+	0.716250i	$0.745857\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 29.5980i	− 1.08584i	−0.839783	−	0.542922i	$-0.817318\pi$
$743$	− 29.5980i	− 1.08584i	0.839783	−	0.542922i	$-0.182682\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 30.0000i	− 1.09764i
$748$	0	0
$749$	−7.31371	−0.267237
$750$	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
$757$	9.31371i	0.338512i	0.985572	+	0.169256i	$0.0541365\pi$
$757$	9.31371i	0.338512i	−0.985572	+	0.169256i	$0.945863\pi$
$758$	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
$763$	7.31371i	0.264774i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	65.9411i	2.38100i
$768$	0	0
$769$	−14.9706	−0.539852	−0.269926	−	0.962881i	$-0.586999\pi$
$769$	−14.9706	−0.539852	−0.269926	+	0.962881i	$0.586999\pi$
$770$	0	0
$771$	−37.6569	−1.35618
$772$	0	0
$773$	− 30.2843i	− 1.08925i	−0.838680	−	0.544625i	$-0.816672\pi$
$773$	− 30.2843i	− 1.08925i	0.838680	−	0.544625i	$-0.183328\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	20.6863i	0.742117i
$778$	0	0
$779$	0	0
$780$	0	0
$781$	−11.3137	−0.404836
$782$	0	0
$783$	− 43.3137i	− 1.54791i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	18.9706i	0.676228i	0.941105	+	0.338114i	$0.109789\pi$
$787$	18.9706i	0.676228i	−0.941105	+	0.338114i	$0.890211\pi$
$788$	0	0
$789$	−64.9706	−2.31301
$790$	0	0
$791$	16.6863	0.593296
$792$	0	0
$793$	− 90.9117i	− 3.22837i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	12.6274i	0.447286i	0.974671	+	0.223643i	$0.0717950\pi$
$797$	12.6274i	0.447286i	−0.974671	+	0.223643i	$0.928205\pi$
$798$	0	0
$799$	−3.31371	−0.117231
$800$	0	0
$801$	46.5685	1.64542
$802$	0	0
$803$	6.82843i	0.240970i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 15.0294i	− 0.529061i
$808$	0	0
$809$	−22.9706	−0.807602	−0.403801	−	0.914847i	$-0.632311\pi$
$809$	−22.9706	−0.807602	−0.403801	+	0.914847i	$0.632311\pi$
$810$	0	0
$811$	13.9411	0.489539	0.244770	−	0.969581i	$-0.421288\pi$
$811$	13.9411	0.489539	0.244770	+	0.969581i	$0.421288\pi$
$812$	0	0
$813$	− 43.3137i	− 1.51908i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	68.2843	2.38605
$820$	0	0
$821$	−18.6863	−0.652156	−0.326078	−	0.945343i	$-0.605727\pi$
$821$	−18.6863	−0.652156	−0.326078	+	0.945343i	$0.605727\pi$
$822$	0	0
$823$	− 36.4853i	− 1.27180i	−0.771773	−	0.635898i	$-0.780630\pi$
$823$	− 36.4853i	− 1.27180i	0.771773	−	0.635898i	$-0.219370\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 34.2843i	− 1.19218i	−0.802917	−	0.596090i	$-0.796720\pi$
$827$	− 34.2843i	− 1.19218i	0.802917	−	0.596090i	$-0.203280\pi$
$828$	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
$831$	−3.31371	−0.114951
$832$	0	0
$833$	− 3.51472i	− 0.121778i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	37.6569	1.30006	0.650029	−	0.759909i	$-0.274757\pi$
$839$	37.6569	1.30006	0.650029	+	0.759909i	$0.274757\pi$
$840$	0	0
$841$	29.6274	1.02164
$842$	0	0
$843$	15.0294i	0.517641i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 2.00000i	− 0.0687208i
$848$	0	0
$849$	35.7157	1.22576
$850$	0	0
$851$	−10.3431	−0.354558
$852$	0	0
$853$	− 32.4853i	− 1.11227i	−0.831090	−	0.556137i	$-0.812283\pi$
$853$	− 32.4853i	− 1.11227i	0.831090	−	0.556137i	$-0.187717\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	48.7696i	1.66594i	0.553321	+	0.832968i	$0.313360\pi$
$857$	48.7696i	1.66594i	−0.553321	+	0.832968i	$0.686640\pi$
$858$	0	0
$859$	−32.2843	−1.10153	−0.550763	−	0.834662i	$-0.685663\pi$
$859$	−32.2843	−1.10153	−0.550763	+	0.834662i	$0.685663\pi$
$860$	0	0
$861$	−33.9411	−1.15671
$862$	0	0
$863$	14.8284i	0.504766i	0.967627	+	0.252383i	$0.0812142\pi$
$863$	14.8284i	0.504766i	−0.967627	+	0.252383i	$0.918786\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 44.2010i	− 1.50115i
$868$	0	0
$869$	−4.00000	−0.135691
$870$	0	0
$871$	−30.6274	−1.03777
$872$	0	0
$873$	− 38.2843i	− 1.29573i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 1.45584i	− 0.0491604i	−0.999698	−	0.0245802i	$-0.992175\pi$
$877$	− 1.45584i	− 0.0491604i	0.999698	−	0.0245802i	$-0.00782490\pi$
$878$	0	0
$879$	−41.9411	−1.41464
$880$	0	0
$881$	−52.6274	−1.77306	−0.886531	−	0.462668i	$-0.846892\pi$
$881$	−52.6274	−1.77306	−0.886531	+	0.462668i	$0.846892\pi$
$882$	0	0
$883$	− 42.8284i	− 1.44129i	−0.693304	−	0.720646i	$-0.743845\pi$
$883$	− 42.8284i	− 1.44129i	0.693304	−	0.720646i	$-0.256155\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 18.2843i	− 0.613926i	−0.951721	−	0.306963i	$-0.900687\pi$
$887$	− 18.2843i	− 0.613926i	0.951721	−	0.306963i	$-0.0993127\pi$
$888$	0	0
$889$	31.3137	1.05023
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	54.6274i	1.82396i
$898$	0	0
$899$	0	0
$900$	0	0
$901$	0.402020	0.0133932
$902$	0	0
$903$	− 33.9411i	− 1.12949i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 44.4853i	− 1.47711i	−0.674193	−	0.738555i	$-0.735509\pi$
$907$	− 44.4853i	− 1.47711i	0.674193	−	0.738555i	$-0.264491\pi$
$908$	0	0
$909$	66.5685	2.20794
$910$	0	0
$911$	−57.9411	−1.91968	−0.959838	−	0.280556i	$-0.909481\pi$
$911$	−57.9411	−1.91968	−0.959838	+	0.280556i	$0.909481\pi$
$912$	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
$921$	−78.2254	−2.57761
$922$	0	0
$923$	− 77.2548i	− 2.54287i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	5.85786i	0.192398i
$928$	0	0
$929$	17.3137	0.568044	0.284022	−	0.958818i	$-0.408331\pi$
$929$	17.3137	0.568044	0.284022	+	0.958818i	$0.408331\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	77.2548i	2.52921i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 49.4558i	− 1.61565i	−0.589421	−	0.807826i	$-0.700644\pi$
$937$	− 49.4558i	− 1.61565i	0.589421	−	0.807826i	$-0.299356\pi$
$938$	0	0
$939$	60.2843	1.96730
$940$	0	0
$941$	−29.3137	−0.955600	−0.477800	−	0.878469i	$-0.658566\pi$
$941$	−29.3137	−0.955600	−0.477800	+	0.878469i	$0.658566\pi$
$942$	0	0
$943$	− 16.9706i	− 0.552638i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	46.8284i	1.52172i	0.648916	+	0.760860i	$0.275222\pi$
$947$	46.8284i	1.52172i	−0.648916	+	0.760860i	$0.724778\pi$
$948$	0	0
$949$	−46.6274	−1.51359
$950$	0	0
$951$	−60.2843	−1.95485
$952$	0	0
$953$	58.8284i	1.90564i	0.303536	+	0.952820i	$0.401833\pi$
$953$	58.8284i	1.90564i	−0.303536	+	0.952820i	$0.598167\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 21.6569i	− 0.700067i
$958$	0	0
$959$	−45.9411	−1.48352
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	18.2843i	0.589202i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	18.9706i	0.610052i	0.952344	+	0.305026i	$0.0986652\pi$
$967$	18.9706i	0.610052i	−0.952344	+	0.305026i	$0.901335\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	31.3137	1.00490	0.502452	−	0.864605i	$-0.332431\pi$
$971$	31.3137	1.00490	0.502452	+	0.864605i	$0.332431\pi$
$972$	0	0
$973$	8.00000i	0.256468i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 43.6569i	− 1.39671i	−0.715753	−	0.698353i	$-0.753916\pi$
$977$	− 43.6569i	− 1.39671i	0.715753	−	0.698353i	$-0.246084\pi$
$978$	0	0
$979$	9.31371	0.297667
$980$	0	0
$981$	18.2843	0.583772
$982$	0	0
$983$	50.1421i	1.59929i	0.600476	+	0.799643i	$0.294978\pi$
$983$	50.1421i	1.59929i	−0.600476	+	0.799643i	$0.705022\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	16.0000i	0.509286i
$988$	0	0
$989$	16.9706	0.539633
$990$	0	0
$991$	−9.94113	−0.315790	−0.157895	−	0.987456i	$-0.550471\pi$
$991$	−9.94113	−0.315790	−0.157895	+	0.987456i	$0.550471\pi$
$992$	0	0
$993$	43.3137i	1.37452i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 9.45584i	− 0.299470i	−0.988726	−	0.149735i	$-0.952158\pi$
$997$	− 9.45584i	− 0.299470i	0.988726	−	0.149735i	$-0.0478420\pi$
$998$	0	0
$999$	20.6863	0.654485

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4400.2.b.q.4049.1		4
4.3	odd	2		275.2.b.d.199.3		4
5.2	odd	4		880.2.a.m.1.1		2
5.3	odd	4		4400.2.a.bn.1.2		2
5.4	even	2	inner	4400.2.b.q.4049.4		4
12.11	even	2		2475.2.c.l.199.2		4
15.2	even	4		7920.2.a.ch.1.1		2
20.3	even	4		275.2.a.c.1.2		2
20.7	even	4		55.2.a.b.1.1	✓	2
20.19	odd	2		275.2.b.d.199.2		4
40.27	even	4		3520.2.a.bn.1.1		2
40.37	odd	4		3520.2.a.bo.1.2		2
55.32	even	4		9680.2.a.bn.1.1		2
60.23	odd	4		2475.2.a.x.1.1		2
60.47	odd	4		495.2.a.b.1.2		2
60.59	even	2		2475.2.c.l.199.3		4
140.27	odd	4		2695.2.a.f.1.1		2
220.7	odd	20		605.2.g.l.511.2		8
220.27	even	20		605.2.g.f.366.2		8
220.43	odd	4		3025.2.a.o.1.1		2
220.47	even	20		605.2.g.f.251.1		8
220.87	odd	4		605.2.a.d.1.2		2
220.107	odd	20		605.2.g.l.251.2		8
220.127	odd	20		605.2.g.l.366.1		8
220.147	even	20		605.2.g.f.511.1		8
220.167	odd	20		605.2.g.l.81.1		8
220.207	even	20		605.2.g.f.81.2		8
260.207	even	4		9295.2.a.g.1.2		2
660.527	even	4		5445.2.a.y.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
55.2.a.b.1.1	✓	2	20.7	even	4
275.2.a.c.1.2		2	20.3	even	4
275.2.b.d.199.2		4	20.19	odd	2
275.2.b.d.199.3		4	4.3	odd	2
495.2.a.b.1.2		2	60.47	odd	4
605.2.a.d.1.2		2	220.87	odd	4
605.2.g.f.81.2		8	220.207	even	20
605.2.g.f.251.1		8	220.47	even	20
605.2.g.f.366.2		8	220.27	even	20
605.2.g.f.511.1		8	220.147	even	20
605.2.g.l.81.1		8	220.167	odd	20
605.2.g.l.251.2		8	220.107	odd	20
605.2.g.l.366.1		8	220.127	odd	20
605.2.g.l.511.2		8	220.7	odd	20
880.2.a.m.1.1		2	5.2	odd	4
2475.2.a.x.1.1		2	60.23	odd	4
2475.2.c.l.199.2		4	12.11	even	2
2475.2.c.l.199.3		4	60.59	even	2
2695.2.a.f.1.1		2	140.27	odd	4
3025.2.a.o.1.1		2	220.43	odd	4
3520.2.a.bn.1.1		2	40.27	even	4
3520.2.a.bo.1.2		2	40.37	odd	4
4400.2.a.bn.1.2		2	5.3	odd	4
4400.2.b.q.4049.1		4	1.1	even	1	trivial
4400.2.b.q.4049.4		4	5.4	even	2	inner
5445.2.a.y.1.1		2	660.527	even	4
7920.2.a.ch.1.1		2	15.2	even	4
9295.2.a.g.1.2		2	260.207	even	4
9680.2.a.bn.1.1		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)

\(f(q)\)	\(=\)	\(q-2.82843i q^{3} -2.00000i q^{7} -5.00000 q^{9} +O(q^{10})\)	\(q-2.82843i q^{3} -2.00000i q^{7} -5.00000 q^{9} -1.00000 q^{11} -6.82843i q^{13} -1.17157i q^{17} -5.65685 q^{21} -2.82843i q^{23} +5.65685i q^{27} -7.65685 q^{29} +2.82843i q^{33} -3.65685i q^{37} -19.3137 q^{39} +6.00000 q^{41} +6.00000i q^{43} -2.82843i q^{47} +3.00000 q^{49} -3.31371 q^{51} +0.343146i q^{53} -9.65685 q^{59} +13.3137 q^{61} +10.0000i q^{63} -4.48528i q^{67} -8.00000 q^{69} +11.3137 q^{71} -6.82843i q^{73} +2.00000i q^{77} +4.00000 q^{79} +1.00000 q^{81} +6.00000i q^{83} +21.6569i q^{87} -9.31371 q^{89} -13.6569 q^{91} +7.65685i q^{97} +5.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 20 q^{9}+O(q^{10}) \) 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

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