LMFDB - Embedded newform 4400.2.b.z.4049.3

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 = [4,0,0,0,0,0,0,0,6,0,-4,0,0,0,0,0,0,0,-24] 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:	$4$
Coefficient field:	$\Q(i, \sqrt{5})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2200)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			4049.3
Root			$0.618034i$ of defining polynomial
Character	$\chi$	$=$	4400.4049
Dual form			4400.2.b.z.4049.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.618034i q^{3} -0.381966i q^{7} +2.61803 q^{9} -1.00000 q^{11} +5.47214i q^{13} +5.09017i q^{17} -3.76393 q^{19} +0.236068 q^{21} -0.381966i q^{23} +3.47214i q^{27} -8.32624 q^{29} +8.70820 q^{31} -0.618034i q^{33} -0.236068i q^{37} -3.38197 q^{39} -10.7082 q^{41} -6.94427i q^{43} -2.23607i q^{47} +6.85410 q^{49} -3.14590 q^{51} +3.61803i q^{53} -2.32624i q^{57} -10.7082 q^{59} -1.90983 q^{61} -1.00000i q^{63} -12.0000i q^{67} +0.236068 q^{69} -8.18034 q^{71} +11.6180i q^{73} +0.381966i q^{77} -7.38197 q^{79} +5.70820 q^{81} +13.5623i q^{83} -5.14590i q^{87} +14.3262 q^{89} +2.09017 q^{91} +5.38197i q^{93} -10.3820i q^{97} -2.61803 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 6 q^{9} - 4 q^{11} - 24 q^{19} - 8 q^{21} - 2 q^{29} + 8 q^{31} - 18 q^{39} - 16 q^{41} + 14 q^{49} - 26 q^{51} - 16 q^{59} - 30 q^{61} - 8 q^{69} + 12 q^{71} - 34 q^{79} - 4 q^{81} + 26 q^{89} - 14 q^{91}+ \cdots - 6 q^{99}+O(q^{100}) $ 4 * q + 6 * q^9 - 4 * q^11 - 24 * q^19 - 8 * q^21 - 2 * q^29 + 8 * q^31 - 18 * q^39 - 16 * q^41 + 14 * q^49 - 26 * q^51 - 16 * q^59 - 30 * q^61 - 8 * q^69 + 12 * q^71 - 34 * q^79 - 4 * q^81 + 26 * q^89 - 14 * q^91 - 6 * 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
$3$	0.618034i	0.356822i	0.983956	+	0.178411i	$0.0570957\pi$
$3$	0.618034i	0.356822i	−0.983956	+	0.178411i	$0.942904\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 0.381966i	− 0.144370i	−0.997391	−	0.0721848i	$-0.977003\pi$
$7$	− 0.381966i	− 0.144370i	0.997391	−	0.0721848i	$-0.0229971\pi$
$8$	0	0
$9$	2.61803	0.872678
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	5.47214i	1.51770i	0.651267	+	0.758849i	$0.274238\pi$
$13$	5.47214i	1.51770i	−0.651267	+	0.758849i	$0.725762\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.09017i	1.23455i	0.786748	+	0.617274i	$0.211763\pi$
$17$	5.09017i	1.23455i	−0.786748	+	0.617274i	$0.788237\pi$
$18$	0	0
$19$	−3.76393	−0.863505	−0.431753	−	0.901992i	$-0.642105\pi$
$19$	−3.76393	−0.863505	−0.431753	+	0.901992i	$0.642105\pi$
$20$	0	0
$21$	0.236068	0.0515143
$22$	0	0
$23$	− 0.381966i	− 0.0796454i	−0.999207	−	0.0398227i	$-0.987321\pi$
$23$	− 0.381966i	− 0.0796454i	0.999207	−	0.0398227i	$-0.0126793\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	3.47214i	0.668213i
$28$	0	0
$29$	−8.32624	−1.54614	−0.773072	−	0.634319i	$-0.781281\pi$
$29$	−8.32624	−1.54614	−0.773072	+	0.634319i	$0.781281\pi$
$30$	0	0
$31$	8.70820	1.56404	0.782020	−	0.623254i	$-0.214190\pi$
$31$	8.70820	1.56404	0.782020	+	0.623254i	$0.214190\pi$
$32$	0	0
$33$	− 0.618034i	− 0.107586i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 0.236068i	− 0.0388093i	−0.999812	−	0.0194047i	$-0.993823\pi$
$37$	− 0.236068i	− 0.0388093i	0.999812	−	0.0194047i	$-0.00617709\pi$
$38$	0	0
$39$	−3.38197	−0.541548
$40$	0	0
$41$	−10.7082	−1.67234	−0.836170	−	0.548470i	$-0.815210\pi$
$41$	−10.7082	−1.67234	−0.836170	+	0.548470i	$0.815210\pi$
$42$	0	0
$43$	− 6.94427i	− 1.05899i	−0.848313	−	0.529496i	$-0.822381\pi$
$43$	− 6.94427i	− 1.05899i	0.848313	−	0.529496i	$-0.177619\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 2.23607i	− 0.326164i	−0.986613	−	0.163082i	$-0.947856\pi$
$47$	− 2.23607i	− 0.326164i	0.986613	−	0.163082i	$-0.0521435\pi$
$48$	0	0
$49$	6.85410	0.979157
$50$	0	0
$51$	−3.14590	−0.440514
$52$	0	0
$53$	3.61803i	0.496975i	0.968635	+	0.248488i	$0.0799335\pi$
$53$	3.61803i	0.496975i	−0.968635	+	0.248488i	$0.920066\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 2.32624i	− 0.308118i
$58$	0	0
$59$	−10.7082	−1.39409	−0.697045	−	0.717028i	$-0.745502\pi$
$59$	−10.7082	−1.39409	−0.697045	+	0.717028i	$0.745502\pi$
$60$	0	0
$61$	−1.90983	−0.244529	−0.122264	−	0.992498i	$-0.539016\pi$
$61$	−1.90983	−0.244529	−0.122264	+	0.992498i	$0.539016\pi$
$62$	0	0
$63$	− 1.00000i	− 0.125988i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 12.0000i	− 1.46603i	−0.680211	−	0.733017i	$-0.738112\pi$
$67$	− 12.0000i	− 1.46603i	0.680211	−	0.733017i	$-0.261888\pi$
$68$	0	0
$69$	0.236068	0.0284192
$70$	0	0
$71$	−8.18034	−0.970828	−0.485414	−	0.874284i	$-0.661331\pi$
$71$	−8.18034	−0.970828	−0.485414	+	0.874284i	$0.661331\pi$
$72$	0	0
$73$	11.6180i	1.35979i	0.733310	+	0.679894i	$0.237974\pi$
$73$	11.6180i	1.35979i	−0.733310	+	0.679894i	$0.762026\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0.381966i	0.0435291i
$78$	0	0
$79$	−7.38197	−0.830536	−0.415268	−	0.909699i	$-0.636312\pi$
$79$	−7.38197	−0.830536	−0.415268	+	0.909699i	$0.636312\pi$
$80$	0	0
$81$	5.70820	0.634245
$82$	0	0
$83$	13.5623i	1.48866i	0.667814	+	0.744328i	$0.267230\pi$
$83$	13.5623i	1.48866i	−0.667814	+	0.744328i	$0.732770\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 5.14590i	− 0.551698i
$88$	0	0
$89$	14.3262	1.51858	0.759289	−	0.650753i	$-0.225547\pi$
$89$	14.3262	1.51858	0.759289	+	0.650753i	$0.225547\pi$
$90$	0	0
$91$	2.09017	0.219109
$92$	0	0
$93$	5.38197i	0.558084i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 10.3820i	− 1.05413i	−0.849825	−	0.527064i	$-0.823293\pi$
$97$	− 10.3820i	− 1.05413i	0.849825	−	0.527064i	$-0.176707\pi$
$98$	0	0
$99$	−2.61803	−0.263122
$100$	0	0
$101$	−13.6180	−1.35505	−0.677523	−	0.735502i	$-0.736946\pi$
$101$	−13.6180	−1.35505	−0.677523	+	0.735502i	$0.736946\pi$
$102$	0	0
$103$	12.3262i	1.21454i	0.794495	+	0.607270i	$0.207735\pi$
$103$	12.3262i	1.21454i	−0.794495	+	0.607270i	$0.792265\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 13.1803i	− 1.27419i	−0.770785	−	0.637096i	$-0.780136\pi$
$107$	− 13.1803i	− 1.27419i	0.770785	−	0.637096i	$-0.219864\pi$
$108$	0	0
$109$	−3.56231	−0.341207	−0.170604	−	0.985340i	$-0.554572\pi$
$109$	−3.56231	−0.341207	−0.170604	+	0.985340i	$0.554572\pi$
$110$	0	0
$111$	0.145898	0.0138480
$112$	0	0
$113$	8.23607i	0.774784i	0.921915	+	0.387392i	$0.126624\pi$
$113$	8.23607i	0.774784i	−0.921915	+	0.387392i	$0.873376\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	14.3262i	1.32446i
$118$	0	0
$119$	1.94427	0.178231
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	− 6.61803i	− 0.596728i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 2.14590i	− 0.190418i	−0.995457	−	0.0952088i	$-0.969648\pi$
$127$	− 2.14590i	− 0.190418i	0.995457	−	0.0952088i	$-0.0303519\pi$
$128$	0	0
$129$	4.29180	0.377872
$130$	0	0
$131$	12.5623	1.09757	0.548787	−	0.835962i	$-0.315090\pi$
$131$	12.5623	1.09757	0.548787	+	0.835962i	$0.315090\pi$
$132$	0	0
$133$	1.43769i	0.124664i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	9.56231i	0.816963i	0.912767	+	0.408481i	$0.133942\pi$
$137$	9.56231i	0.816963i	−0.912767	+	0.408481i	$0.866058\pi$
$138$	0	0
$139$	−11.1803	−0.948304	−0.474152	−	0.880443i	$-0.657245\pi$
$139$	−11.1803	−0.948304	−0.474152	+	0.880443i	$0.657245\pi$
$140$	0	0
$141$	1.38197	0.116383
$142$	0	0
$143$	− 5.47214i	− 0.457603i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	4.23607i	0.349385i
$148$	0	0
$149$	7.41641	0.607576	0.303788	−	0.952740i	$-0.401749\pi$
$149$	7.41641	0.607576	0.303788	+	0.952740i	$0.401749\pi$
$150$	0	0
$151$	−15.9443	−1.29753	−0.648763	−	0.760990i	$-0.724713\pi$
$151$	−15.9443	−1.29753	−0.648763	+	0.760990i	$0.724713\pi$
$152$	0	0
$153$	13.3262i	1.07736i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	4.47214i	0.356915i	0.983948	+	0.178458i	$0.0571108\pi$
$157$	4.47214i	0.356915i	−0.983948	+	0.178458i	$0.942889\pi$
$158$	0	0
$159$	−2.23607	−0.177332
$160$	0	0
$161$	−0.145898	−0.0114984
$162$	0	0
$163$	6.38197i	0.499874i	0.968262	+	0.249937i	$0.0804099\pi$
$163$	6.38197i	0.499874i	−0.968262	+	0.249937i	$0.919590\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	6.70820i	0.519096i	0.965730	+	0.259548i	$0.0835736\pi$
$167$	6.70820i	0.519096i	−0.965730	+	0.259548i	$0.916426\pi$
$168$	0	0
$169$	−16.9443	−1.30341
$170$	0	0
$171$	−9.85410	−0.753562
$172$	0	0
$173$	11.2918i	0.858499i	0.903186	+	0.429250i	$0.141222\pi$
$173$	11.2918i	0.858499i	−0.903186	+	0.429250i	$0.858778\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 6.61803i	− 0.497442i
$178$	0	0
$179$	2.09017	0.156227	0.0781133	−	0.996944i	$-0.475110\pi$
$179$	2.09017	0.156227	0.0781133	+	0.996944i	$0.475110\pi$
$180$	0	0
$181$	4.38197	0.325709	0.162854	−	0.986650i	$-0.447930\pi$
$181$	4.38197	0.325709	0.162854	+	0.986650i	$0.447930\pi$
$182$	0	0
$183$	− 1.18034i	− 0.0872532i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 5.09017i	− 0.372230i
$188$	0	0
$189$	1.32624	0.0964696
$190$	0	0
$191$	12.0344	0.870782	0.435391	−	0.900242i	$-0.356610\pi$
$191$	12.0344	0.870782	0.435391	+	0.900242i	$0.356610\pi$
$192$	0	0
$193$	9.52786i	0.685831i	0.939366	+	0.342915i	$0.111414\pi$
$193$	9.52786i	0.685831i	−0.939366	+	0.342915i	$0.888586\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 11.0344i	− 0.786171i	−0.919502	−	0.393086i	$-0.871408\pi$
$197$	− 11.0344i	− 0.786171i	0.919502	−	0.393086i	$-0.128592\pi$
$198$	0	0
$199$	−14.6180	−1.03624	−0.518122	−	0.855306i	$-0.673369\pi$
$199$	−14.6180	−1.03624	−0.518122	+	0.855306i	$0.673369\pi$
$200$	0	0
$201$	7.41641	0.523113
$202$	0	0
$203$	3.18034i	0.223216i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 1.00000i	− 0.0695048i
$208$	0	0
$209$	3.76393	0.260357
$210$	0	0
$211$	1.00000	0.0688428	0.0344214	−	0.999407i	$-0.489041\pi$
$211$	1.00000	0.0688428	0.0344214	+	0.999407i	$0.489041\pi$
$212$	0	0
$213$	− 5.05573i	− 0.346413i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 3.32624i	− 0.225800i
$218$	0	0
$219$	−7.18034	−0.485202
$220$	0	0
$221$	−27.8541	−1.87367
$222$	0	0
$223$	25.6525i	1.71782i	0.512129	+	0.858908i	$0.328857\pi$
$223$	25.6525i	1.71782i	−0.512129	+	0.858908i	$0.671143\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	13.8541i	0.919529i	0.888041	+	0.459765i	$0.152066\pi$
$227$	13.8541i	0.919529i	−0.888041	+	0.459765i	$0.847934\pi$
$228$	0	0
$229$	−12.6180	−0.833823	−0.416912	−	0.908947i	$-0.636888\pi$
$229$	−12.6180	−0.833823	−0.416912	+	0.908947i	$0.636888\pi$
$230$	0	0
$231$	−0.236068	−0.0155321
$232$	0	0
$233$	− 16.5623i	− 1.08503i	−0.840045	−	0.542516i	$-0.817472\pi$
$233$	− 16.5623i	− 1.08503i	0.840045	−	0.542516i	$-0.182528\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 4.56231i	− 0.296354i
$238$	0	0
$239$	−22.3820	−1.44777	−0.723885	−	0.689921i	$-0.757645\pi$
$239$	−22.3820	−1.44777	−0.723885	+	0.689921i	$0.757645\pi$
$240$	0	0
$241$	0.145898	0.00939812	0.00469906	−	0.999989i	$-0.498504\pi$
$241$	0.145898	0.00939812	0.00469906	+	0.999989i	$0.498504\pi$
$242$	0	0
$243$	13.9443i	0.894525i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 20.5967i	− 1.31054i
$248$	0	0
$249$	−8.38197	−0.531186
$250$	0	0
$251$	−18.5066	−1.16812	−0.584062	−	0.811709i	$-0.698538\pi$
$251$	−18.5066	−1.16812	−0.584062	+	0.811709i	$0.698538\pi$
$252$	0	0
$253$	0.381966i	0.0240140i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 5.05573i	− 0.315368i	−0.987490	−	0.157684i	$-0.949597\pi$
$257$	− 5.05573i	− 0.315368i	0.987490	−	0.157684i	$-0.0504027\pi$
$258$	0	0
$259$	−0.0901699	−0.00560289
$260$	0	0
$261$	−21.7984	−1.34929
$262$	0	0
$263$	15.4721i	0.954053i	0.878889	+	0.477026i	$0.158285\pi$
$263$	15.4721i	0.954053i	−0.878889	+	0.477026i	$0.841715\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	8.85410i	0.541862i
$268$	0	0
$269$	2.61803	0.159624	0.0798122	−	0.996810i	$-0.474568\pi$
$269$	2.61803	0.159624	0.0798122	+	0.996810i	$0.474568\pi$
$270$	0	0
$271$	−12.7082	−0.771968	−0.385984	−	0.922505i	$-0.626138\pi$
$271$	−12.7082	−0.771968	−0.385984	+	0.922505i	$0.626138\pi$
$272$	0	0
$273$	1.29180i	0.0781831i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	26.1246i	1.56968i	0.619701	+	0.784838i	$0.287254\pi$
$277$	26.1246i	1.56968i	−0.619701	+	0.784838i	$0.712746\pi$
$278$	0	0
$279$	22.7984	1.36490
$280$	0	0
$281$	4.23607	0.252703	0.126351	−	0.991986i	$-0.459673\pi$
$281$	4.23607	0.252703	0.126351	+	0.991986i	$0.459673\pi$
$282$	0	0
$283$	− 27.4508i	− 1.63178i	−0.578205	−	0.815892i	$-0.696246\pi$
$283$	− 27.4508i	− 1.63178i	0.578205	−	0.815892i	$-0.303754\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	4.09017i	0.241435i
$288$	0	0
$289$	−8.90983	−0.524108
$290$	0	0
$291$	6.41641	0.376136
$292$	0	0
$293$	− 11.5279i	− 0.673465i	−0.941600	−	0.336733i	$-0.890678\pi$
$293$	− 11.5279i	− 0.673465i	0.941600	−	0.336733i	$-0.109322\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 3.47214i	− 0.201474i
$298$	0	0
$299$	2.09017	0.120878
$300$	0	0
$301$	−2.65248	−0.152886
$302$	0	0
$303$	− 8.41641i	− 0.483510i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 23.8541i	− 1.36143i	−0.732550	−	0.680713i	$-0.761670\pi$
$307$	− 23.8541i	− 1.36143i	0.732550	−	0.680713i	$-0.238330\pi$
$308$	0	0
$309$	−7.61803	−0.433375
$310$	0	0
$311$	−5.94427	−0.337069	−0.168534	−	0.985696i	$-0.553903\pi$
$311$	−5.94427	−0.337069	−0.168534	+	0.985696i	$0.553903\pi$
$312$	0	0
$313$	− 10.6525i	− 0.602114i	−0.953606	−	0.301057i	$-0.902661\pi$
$313$	− 10.6525i	− 0.602114i	0.953606	−	0.301057i	$-0.0973394\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 3.38197i	− 0.189950i	−0.995480	−	0.0949751i	$-0.969723\pi$
$317$	− 3.38197i	− 0.189950i	0.995480	−	0.0949751i	$-0.0302771\pi$
$318$	0	0
$319$	8.32624	0.466180
$320$	0	0
$321$	8.14590	0.454660
$322$	0	0
$323$	− 19.1591i	− 1.06604i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 2.20163i	− 0.121750i
$328$	0	0
$329$	−0.854102	−0.0470882
$330$	0	0
$331$	−13.3607	−0.734369	−0.367185	−	0.930148i	$-0.619678\pi$
$331$	−13.3607	−0.734369	−0.367185	+	0.930148i	$0.619678\pi$
$332$	0	0
$333$	− 0.618034i	− 0.0338681i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 35.8885i	− 1.95497i	−0.210997	−	0.977487i	$-0.567671\pi$
$337$	− 35.8885i	− 1.95497i	0.210997	−	0.977487i	$-0.432329\pi$
$338$	0	0
$339$	−5.09017	−0.276460
$340$	0	0
$341$	−8.70820	−0.471576
$342$	0	0
$343$	− 5.29180i	− 0.285730i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	3.61803i	0.194226i	0.995273	+	0.0971131i	$0.0309609\pi$
$347$	3.61803i	0.194226i	−0.995273	+	0.0971131i	$0.969039\pi$
$348$	0	0
$349$	8.12461	0.434900	0.217450	−	0.976071i	$-0.430226\pi$
$349$	8.12461	0.434900	0.217450	+	0.976071i	$0.430226\pi$
$350$	0	0
$351$	−19.0000	−1.01414
$352$	0	0
$353$	19.3607i	1.03047i	0.857050	+	0.515233i	$0.172294\pi$
$353$	19.3607i	1.03047i	−0.857050	+	0.515233i	$0.827706\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	1.20163i	0.0635968i
$358$	0	0
$359$	31.8885	1.68301	0.841506	−	0.540247i	$-0.181669\pi$
$359$	31.8885	1.68301	0.841506	+	0.540247i	$0.181669\pi$
$360$	0	0
$361$	−4.83282	−0.254359
$362$	0	0
$363$	0.618034i	0.0324384i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	35.9787i	1.87807i	0.343817	+	0.939037i	$0.388280\pi$
$367$	35.9787i	1.87807i	−0.343817	+	0.939037i	$0.611720\pi$
$368$	0	0
$369$	−28.0344	−1.45941
$370$	0	0
$371$	1.38197	0.0717481
$372$	0	0
$373$	14.4164i	0.746453i	0.927740	+	0.373227i	$0.121749\pi$
$373$	14.4164i	0.746453i	−0.927740	+	0.373227i	$0.878251\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 45.5623i	− 2.34658i
$378$	0	0
$379$	29.4721	1.51388	0.756941	−	0.653483i	$-0.226693\pi$
$379$	29.4721	1.51388	0.756941	+	0.653483i	$0.226693\pi$
$380$	0	0
$381$	1.32624	0.0679452
$382$	0	0
$383$	18.9443i	0.968007i	0.875066	+	0.484004i	$0.160818\pi$
$383$	18.9443i	0.968007i	−0.875066	+	0.484004i	$0.839182\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 18.1803i	− 0.924159i
$388$	0	0
$389$	−32.9443	−1.67034	−0.835170	−	0.549991i	$-0.814631\pi$
$389$	−32.9443	−1.67034	−0.835170	+	0.549991i	$0.814631\pi$
$390$	0	0
$391$	1.94427	0.0983261
$392$	0	0
$393$	7.76393i	0.391639i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	24.7984i	1.24459i	0.782781	+	0.622297i	$0.213801\pi$
$397$	24.7984i	1.24459i	−0.782781	+	0.622297i	$0.786199\pi$
$398$	0	0
$399$	−0.888544	−0.0444828
$400$	0	0
$401$	0.472136	0.0235773	0.0117887	−	0.999931i	$-0.496247\pi$
$401$	0.472136	0.0235773	0.0117887	+	0.999931i	$0.496247\pi$
$402$	0	0
$403$	47.6525i	2.37374i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0.236068i	0.0117015i
$408$	0	0
$409$	20.5279	1.01504	0.507519	−	0.861641i	$-0.330563\pi$
$409$	20.5279	1.01504	0.507519	+	0.861641i	$0.330563\pi$
$410$	0	0
$411$	−5.90983	−0.291510
$412$	0	0
$413$	4.09017i	0.201264i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 6.90983i	− 0.338376i
$418$	0	0
$419$	−12.5279	−0.612026	−0.306013	−	0.952027i	$-0.598995\pi$
$419$	−12.5279	−0.612026	−0.306013	+	0.952027i	$0.598995\pi$
$420$	0	0
$421$	4.67376	0.227785	0.113893	−	0.993493i	$-0.463668\pi$
$421$	4.67376	0.227785	0.113893	+	0.993493i	$0.463668\pi$
$422$	0	0
$423$	− 5.85410i	− 0.284636i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0.729490i	0.0353025i
$428$	0	0
$429$	3.38197	0.163283
$430$	0	0
$431$	5.76393	0.277639	0.138819	−	0.990318i	$-0.455669\pi$
$431$	5.76393	0.277639	0.138819	+	0.990318i	$0.455669\pi$
$432$	0	0
$433$	10.8885i	0.523270i	0.965167	+	0.261635i	$0.0842617\pi$
$433$	10.8885i	0.523270i	−0.965167	+	0.261635i	$0.915738\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1.43769i	0.0687742i
$438$	0	0
$439$	23.9230	1.14178	0.570891	−	0.821026i	$-0.306598\pi$
$439$	23.9230	1.14178	0.570891	+	0.821026i	$0.306598\pi$
$440$	0	0
$441$	17.9443	0.854489
$442$	0	0
$443$	24.2918i	1.15414i	0.816695	+	0.577069i	$0.195804\pi$
$443$	24.2918i	1.15414i	−0.816695	+	0.577069i	$0.804196\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	4.58359i	0.216796i
$448$	0	0
$449$	−7.72949	−0.364777	−0.182389	−	0.983227i	$-0.558383\pi$
$449$	−7.72949	−0.364777	−0.182389	+	0.983227i	$0.558383\pi$
$450$	0	0
$451$	10.7082	0.504230
$452$	0	0
$453$	− 9.85410i	− 0.462986i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 19.1459i	− 0.895607i	−0.894132	−	0.447804i	$-0.852206\pi$
$457$	− 19.1459i	− 0.895607i	0.894132	−	0.447804i	$-0.147794\pi$
$458$	0	0
$459$	−17.6738	−0.824941
$460$	0	0
$461$	6.88854	0.320831	0.160416	−	0.987050i	$-0.448717\pi$
$461$	6.88854	0.320831	0.160416	+	0.987050i	$0.448717\pi$
$462$	0	0
$463$	− 36.2361i	− 1.68403i	−0.539452	−	0.842016i	$-0.681369\pi$
$463$	− 36.2361i	− 1.68403i	0.539452	−	0.842016i	$-0.318631\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	34.7082i	1.60610i	0.595909	+	0.803052i	$0.296792\pi$
$467$	34.7082i	1.60610i	−0.595909	+	0.803052i	$0.703208\pi$
$468$	0	0
$469$	−4.58359	−0.211651
$470$	0	0
$471$	−2.76393	−0.127355
$472$	0	0
$473$	6.94427i	0.319298i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	9.47214i	0.433699i
$478$	0	0
$479$	−4.41641	−0.201791	−0.100895	−	0.994897i	$-0.532171\pi$
$479$	−4.41641	−0.201791	−0.100895	+	0.994897i	$0.532171\pi$
$480$	0	0
$481$	1.29180	0.0589008
$482$	0	0
$483$	− 0.0901699i	− 0.00410287i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 25.1803i	− 1.14103i	−0.821287	−	0.570515i	$-0.806744\pi$
$487$	− 25.1803i	− 1.14103i	0.821287	−	0.570515i	$-0.193256\pi$
$488$	0	0
$489$	−3.94427	−0.178366
$490$	0	0
$491$	7.65248	0.345351	0.172676	−	0.984979i	$-0.444759\pi$
$491$	7.65248	0.345351	0.172676	+	0.984979i	$0.444759\pi$
$492$	0	0
$493$	− 42.3820i	− 1.90879i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	3.12461i	0.140158i
$498$	0	0
$499$	40.5623	1.81582	0.907909	−	0.419167i	$-0.137678\pi$
$499$	40.5623	1.81582	0.907909	+	0.419167i	$0.137678\pi$
$500$	0	0
$501$	−4.14590	−0.185225
$502$	0	0
$503$	28.1803i	1.25650i	0.778012	+	0.628250i	$0.216228\pi$
$503$	28.1803i	1.25650i	−0.778012	+	0.628250i	$0.783772\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 10.4721i	− 0.465084i
$508$	0	0
$509$	−26.8541	−1.19029	−0.595144	−	0.803619i	$-0.702905\pi$
$509$	−26.8541	−1.19029	−0.595144	+	0.803619i	$0.702905\pi$
$510$	0	0
$511$	4.43769	0.196312
$512$	0	0
$513$	− 13.0689i	− 0.577005i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	2.23607i	0.0983422i
$518$	0	0
$519$	−6.97871	−0.306332
$520$	0	0
$521$	2.76393	0.121090	0.0605450	−	0.998165i	$-0.480716\pi$
$521$	2.76393	0.121090	0.0605450	+	0.998165i	$0.480716\pi$
$522$	0	0
$523$	17.4164i	0.761566i	0.924664	+	0.380783i	$0.124346\pi$
$523$	17.4164i	0.761566i	−0.924664	+	0.380783i	$0.875654\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	44.3262i	1.93088i
$528$	0	0
$529$	22.8541	0.993657
$530$	0	0
$531$	−28.0344	−1.21659
$532$	0	0
$533$	− 58.5967i	− 2.53811i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	1.29180i	0.0557451i
$538$	0	0
$539$	−6.85410	−0.295227
$540$	0	0
$541$	40.4508	1.73912	0.869559	−	0.493829i	$-0.164403\pi$
$541$	40.4508	1.73912	0.869559	+	0.493829i	$0.164403\pi$
$542$	0	0
$543$	2.70820i	0.116220i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 2.67376i	− 0.114322i	−0.998365	−	0.0571609i	$-0.981795\pi$
$547$	− 2.67376i	− 0.114322i	0.998365	−	0.0571609i	$-0.0182048\pi$
$548$	0	0
$549$	−5.00000	−0.213395
$550$	0	0
$551$	31.3394	1.33510
$552$	0	0
$553$	2.81966i	0.119904i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	7.47214i	0.316605i	0.987391	+	0.158302i	$0.0506020\pi$
$557$	7.47214i	0.316605i	−0.987391	+	0.158302i	$0.949398\pi$
$558$	0	0
$559$	38.0000	1.60723
$560$	0	0
$561$	3.14590	0.132820
$562$	0	0
$563$	1.03444i	0.0435965i	0.999762	+	0.0217983i	$0.00693915\pi$
$563$	1.03444i	0.0435965i	−0.999762	+	0.0217983i	$0.993061\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 2.18034i	− 0.0915657i
$568$	0	0
$569$	20.2016	0.846896	0.423448	−	0.905920i	$-0.360820\pi$
$569$	20.2016	0.846896	0.423448	+	0.905920i	$0.360820\pi$
$570$	0	0
$571$	34.1591	1.42951	0.714756	−	0.699374i	$-0.246538\pi$
$571$	34.1591	1.42951	0.714756	+	0.699374i	$0.246538\pi$
$572$	0	0
$573$	7.43769i	0.310714i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 29.7426i	− 1.23820i	−0.785311	−	0.619101i	$-0.787497\pi$
$577$	− 29.7426i	− 1.23820i	0.785311	−	0.619101i	$-0.212503\pi$
$578$	0	0
$579$	−5.88854	−0.244720
$580$	0	0
$581$	5.18034	0.214917
$582$	0	0
$583$	− 3.61803i	− 0.149844i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	7.20163i	0.297243i	0.988894	+	0.148621i	$0.0474836\pi$
$587$	7.20163i	0.297243i	−0.988894	+	0.148621i	$0.952516\pi$
$588$	0	0
$589$	−32.7771	−1.35056
$590$	0	0
$591$	6.81966	0.280523
$592$	0	0
$593$	− 16.3475i	− 0.671312i	−0.941985	−	0.335656i	$-0.891042\pi$
$593$	− 16.3475i	− 0.671312i	0.941985	−	0.335656i	$-0.108958\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 9.03444i	− 0.369755i
$598$	0	0
$599$	−34.3951	−1.40535	−0.702673	−	0.711513i	$-0.748010\pi$
$599$	−34.3951	−1.40535	−0.702673	+	0.711513i	$0.748010\pi$
$600$	0	0
$601$	4.67376	0.190647	0.0953234	−	0.995446i	$-0.469611\pi$
$601$	4.67376	0.190647	0.0953234	+	0.995446i	$0.469611\pi$
$602$	0	0
$603$	− 31.4164i	− 1.27938i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	8.83282i	0.358513i	0.983802	+	0.179256i	$0.0573692\pi$
$607$	8.83282i	0.358513i	−0.983802	+	0.179256i	$0.942631\pi$
$608$	0	0
$609$	−1.96556	−0.0796484
$610$	0	0
$611$	12.2361	0.495018
$612$	0	0
$613$	46.3262i	1.87110i	0.353195	+	0.935550i	$0.385095\pi$
$613$	46.3262i	1.87110i	−0.353195	+	0.935550i	$0.614905\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	13.2361i	0.532864i	0.963854	+	0.266432i	$0.0858448\pi$
$617$	13.2361i	0.532864i	−0.963854	+	0.266432i	$0.914155\pi$
$618$	0	0
$619$	40.7082	1.63620	0.818100	−	0.575075i	$-0.195027\pi$
$619$	40.7082	1.63620	0.818100	+	0.575075i	$0.195027\pi$
$620$	0	0
$621$	1.32624	0.0532201
$622$	0	0
$623$	− 5.47214i	− 0.219236i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	2.32624i	0.0929010i
$628$	0	0
$629$	1.20163	0.0479120
$630$	0	0
$631$	−38.4508	−1.53070	−0.765352	−	0.643612i	$-0.777435\pi$
$631$	−38.4508	−1.53070	−0.765352	+	0.643612i	$0.777435\pi$
$632$	0	0
$633$	0.618034i	0.0245646i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	37.5066i	1.48606i
$638$	0	0
$639$	−21.4164	−0.847220
$640$	0	0
$641$	−10.4164	−0.411423	−0.205712	−	0.978613i	$-0.565951\pi$
$641$	−10.4164	−0.411423	−0.205712	+	0.978613i	$0.565951\pi$
$642$	0	0
$643$	− 25.4164i	− 1.00233i	−0.865353	−	0.501163i	$-0.832906\pi$
$643$	− 25.4164i	− 1.00233i	0.865353	−	0.501163i	$-0.167094\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	16.4164i	0.645396i	0.946502	+	0.322698i	$0.104590\pi$
$647$	16.4164i	0.645396i	−0.946502	+	0.322698i	$0.895410\pi$
$648$	0	0
$649$	10.7082	0.420334
$650$	0	0
$651$	2.05573	0.0805703
$652$	0	0
$653$	− 47.3820i	− 1.85420i	−0.374815	−	0.927100i	$-0.622294\pi$
$653$	− 47.3820i	− 1.85420i	0.374815	−	0.927100i	$-0.377706\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	30.4164i	1.18666i
$658$	0	0
$659$	−19.7426	−0.769064	−0.384532	−	0.923112i	$-0.625637\pi$
$659$	−19.7426	−0.769064	−0.384532	+	0.923112i	$0.625637\pi$
$660$	0	0
$661$	26.5967	1.03449	0.517247	−	0.855836i	$-0.326957\pi$
$661$	26.5967	1.03449	0.517247	+	0.855836i	$0.326957\pi$
$662$	0	0
$663$	− 17.2148i	− 0.668567i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	3.18034i	0.123143i
$668$	0	0
$669$	−15.8541	−0.612955
$670$	0	0
$671$	1.90983	0.0737282
$672$	0	0
$673$	18.5967i	0.716852i	0.933558	+	0.358426i	$0.116686\pi$
$673$	18.5967i	0.716852i	−0.933558	+	0.358426i	$0.883314\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	1.59675i	0.0613680i	0.999529	+	0.0306840i	$0.00976855\pi$
$677$	1.59675i	0.0613680i	−0.999529	+	0.0306840i	$0.990231\pi$
$678$	0	0
$679$	−3.96556	−0.152184
$680$	0	0
$681$	−8.56231	−0.328108
$682$	0	0
$683$	1.18034i	0.0451645i	0.999745	+	0.0225822i	$0.00718876\pi$
$683$	1.18034i	0.0451645i	−0.999745	+	0.0225822i	$0.992811\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 7.79837i	− 0.297527i
$688$	0	0
$689$	−19.7984	−0.754258
$690$	0	0
$691$	−40.6312	−1.54568	−0.772842	−	0.634599i	$-0.781165\pi$
$691$	−40.6312	−1.54568	−0.772842	+	0.634599i	$0.781165\pi$
$692$	0	0
$693$	1.00000i	0.0379869i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 54.5066i	− 2.06458i
$698$	0	0
$699$	10.2361	0.387164
$700$	0	0
$701$	−37.5279	−1.41741	−0.708704	−	0.705506i	$-0.750720\pi$
$701$	−37.5279	−1.41741	−0.708704	+	0.705506i	$0.750720\pi$
$702$	0	0
$703$	0.888544i	0.0335121i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	5.20163i	0.195627i
$708$	0	0
$709$	24.3607	0.914885	0.457442	−	0.889239i	$-0.348766\pi$
$709$	24.3607	0.914885	0.457442	+	0.889239i	$0.348766\pi$
$710$	0	0
$711$	−19.3262	−0.724791
$712$	0	0
$713$	− 3.32624i	− 0.124569i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 13.8328i	− 0.516596i
$718$	0	0
$719$	49.8328	1.85845	0.929225	−	0.369514i	$-0.120476\pi$
$719$	49.8328	1.85845	0.929225	+	0.369514i	$0.120476\pi$
$720$	0	0
$721$	4.70820	0.175343
$722$	0	0
$723$	0.0901699i	0.00335346i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 13.0902i	− 0.485488i	−0.970090	−	0.242744i	$-0.921953\pi$
$727$	− 13.0902i	− 0.485488i	0.970090	−	0.242744i	$-0.0780474\pi$
$728$	0	0
$729$	8.50658	0.315058
$730$	0	0
$731$	35.3475	1.30738
$732$	0	0
$733$	4.12461i	0.152346i	0.997095	+	0.0761730i	$0.0242701\pi$
$733$	4.12461i	0.152346i	−0.997095	+	0.0761730i	$0.975730\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	12.0000i	0.442026i
$738$	0	0
$739$	−2.85410	−0.104990	−0.0524949	−	0.998621i	$-0.516717\pi$
$739$	−2.85410	−0.104990	−0.0524949	+	0.998621i	$0.516717\pi$
$740$	0	0
$741$	12.7295	0.467630
$742$	0	0
$743$	47.2148i	1.73214i	0.499921	+	0.866071i	$0.333362\pi$
$743$	47.2148i	1.73214i	−0.499921	+	0.866071i	$0.666638\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	35.5066i	1.29912i
$748$	0	0
$749$	−5.03444	−0.183955
$750$	0	0
$751$	0.0344419	0.00125680	0.000628401	−	1.00000i	$-0.499800\pi$
$751$	0.0344419	0.00125680	0.000628401		1.00000i	$0.499800\pi$
$752$	0	0
$753$	− 11.4377i	− 0.416813i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 4.88854i	− 0.177677i	−0.996046	−	0.0888386i	$-0.971684\pi$
$757$	− 4.88854i	− 0.177677i	0.996046	−	0.0888386i	$-0.0283155\pi$
$758$	0	0
$759$	−0.236068	−0.00856872
$760$	0	0
$761$	8.00000	0.290000	0.145000	−	0.989432i	$-0.453682\pi$
$761$	8.00000	0.290000	0.145000	+	0.989432i	$0.453682\pi$
$762$	0	0
$763$	1.36068i	0.0492599i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 58.5967i	− 2.11581i
$768$	0	0
$769$	41.6525	1.50203	0.751013	−	0.660287i	$-0.229565\pi$
$769$	41.6525	1.50203	0.751013	+	0.660287i	$0.229565\pi$
$770$	0	0
$771$	3.12461	0.112530
$772$	0	0
$773$	− 48.6312i	− 1.74914i	−0.484897	−	0.874571i	$-0.661143\pi$
$773$	− 48.6312i	− 1.74914i	0.484897	−	0.874571i	$-0.338857\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 0.0557281i	− 0.00199923i
$778$	0	0
$779$	40.3050	1.44407
$780$	0	0
$781$	8.18034	0.292716
$782$	0	0
$783$	− 28.9098i	− 1.03315i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 1.76393i	− 0.0628774i	−0.999506	−	0.0314387i	$-0.989991\pi$
$787$	− 1.76393i	− 0.0628774i	0.999506	−	0.0314387i	$-0.0100089\pi$
$788$	0	0
$789$	−9.56231	−0.340427
$790$	0	0
$791$	3.14590	0.111855
$792$	0	0
$793$	− 10.4508i	− 0.371121i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	35.5623i	1.25968i	0.776724	+	0.629841i	$0.216880\pi$
$797$	35.5623i	1.25968i	−0.776724	+	0.629841i	$0.783120\pi$
$798$	0	0
$799$	11.3820	0.402665
$800$	0	0
$801$	37.5066	1.32523
$802$	0	0
$803$	− 11.6180i	− 0.409992i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	1.61803i	0.0569575i
$808$	0	0
$809$	50.2492	1.76667	0.883334	−	0.468744i	$-0.155293\pi$
$809$	50.2492	1.76667	0.883334	+	0.468744i	$0.155293\pi$
$810$	0	0
$811$	13.9443	0.489650	0.244825	−	0.969567i	$-0.421270\pi$
$811$	13.9443	0.489650	0.244825	+	0.969567i	$0.421270\pi$
$812$	0	0
$813$	− 7.85410i	− 0.275455i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	26.1378i	0.914445i
$818$	0	0
$819$	5.47214	0.191212
$820$	0	0
$821$	−52.7214	−1.83999	−0.919994	−	0.391932i	$-0.871807\pi$
$821$	−52.7214	−1.83999	−0.919994	+	0.391932i	$0.871807\pi$
$822$	0	0
$823$	− 14.8328i	− 0.517039i	−0.966006	−	0.258520i	$-0.916765\pi$
$823$	− 14.8328i	− 0.517039i	0.966006	−	0.258520i	$-0.0832347\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	9.23607i	0.321170i	0.987022	+	0.160585i	$0.0513380\pi$
$827$	9.23607i	0.321170i	−0.987022	+	0.160585i	$0.948662\pi$
$828$	0	0
$829$	23.1591	0.804347	0.402174	−	0.915563i	$-0.368255\pi$
$829$	23.1591	0.804347	0.402174	+	0.915563i	$0.368255\pi$
$830$	0	0
$831$	−16.1459	−0.560095
$832$	0	0
$833$	34.8885i	1.20882i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	30.2361i	1.04511i
$838$	0	0
$839$	−8.85410	−0.305678	−0.152839	−	0.988251i	$-0.548842\pi$
$839$	−8.85410	−0.305678	−0.152839	+	0.988251i	$0.548842\pi$
$840$	0	0
$841$	40.3262	1.39056
$842$	0	0
$843$	2.61803i	0.0901699i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 0.381966i	− 0.0131245i
$848$	0	0
$849$	16.9656	0.582256
$850$	0	0
$851$	−0.0901699	−0.00309099
$852$	0	0
$853$	− 11.5623i	− 0.395886i	−0.980214	−	0.197943i	$-0.936574\pi$
$853$	− 11.5623i	− 0.395886i	0.980214	−	0.197943i	$-0.0634261\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	11.8328i	0.404201i	0.979365	+	0.202101i	$0.0647768\pi$
$857$	11.8328i	0.404201i	−0.979365	+	0.202101i	$0.935223\pi$
$858$	0	0
$859$	−21.7082	−0.740674	−0.370337	−	0.928897i	$-0.620758\pi$
$859$	−21.7082	−0.740674	−0.370337	+	0.928897i	$0.620758\pi$
$860$	0	0
$861$	−2.52786	−0.0861494
$862$	0	0
$863$	− 52.4164i	− 1.78428i	−0.451764	−	0.892138i	$-0.649205\pi$
$863$	− 52.4164i	− 1.78428i	0.451764	−	0.892138i	$-0.350795\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 5.50658i	− 0.187013i
$868$	0	0
$869$	7.38197	0.250416
$870$	0	0
$871$	65.6656	2.22500
$872$	0	0
$873$	− 27.1803i	− 0.919915i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	45.8885i	1.54955i	0.632239	+	0.774773i	$0.282136\pi$
$877$	45.8885i	1.54955i	−0.632239	+	0.774773i	$0.717864\pi$
$878$	0	0
$879$	7.12461	0.240307
$880$	0	0
$881$	14.4508	0.486861	0.243431	−	0.969918i	$-0.421727\pi$
$881$	14.4508	0.486861	0.243431	+	0.969918i	$0.421727\pi$
$882$	0	0
$883$	25.8328i	0.869343i	0.900589	+	0.434672i	$0.143136\pi$
$883$	25.8328i	0.869343i	−0.900589	+	0.434672i	$0.856864\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	17.0000i	0.570804i	0.958408	+	0.285402i	$0.0921271\pi$
$887$	17.0000i	0.570804i	−0.958408	+	0.285402i	$0.907873\pi$
$888$	0	0
$889$	−0.819660	−0.0274905
$890$	0	0
$891$	−5.70820	−0.191232
$892$	0	0
$893$	8.41641i	0.281644i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	1.29180i	0.0431318i
$898$	0	0
$899$	−72.5066	−2.41823
$900$	0	0
$901$	−18.4164	−0.613540
$902$	0	0
$903$	− 1.63932i	− 0.0545532i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	23.7771i	0.789505i	0.918787	+	0.394753i	$0.129170\pi$
$907$	23.7771i	0.789505i	−0.918787	+	0.394753i	$0.870830\pi$
$908$	0	0
$909$	−35.6525	−1.18252
$910$	0	0
$911$	13.0557	0.432556	0.216278	−	0.976332i	$-0.430608\pi$
$911$	13.0557	0.432556	0.216278	+	0.976332i	$0.430608\pi$
$912$	0	0
$913$	− 13.5623i	− 0.448847i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 4.79837i	− 0.158456i
$918$	0	0
$919$	−52.3607	−1.72722	−0.863610	−	0.504161i	$-0.831802\pi$
$919$	−52.3607	−1.72722	−0.863610	+	0.504161i	$0.831802\pi$
$920$	0	0
$921$	14.7426	0.485787
$922$	0	0
$923$	− 44.7639i	− 1.47342i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	32.2705i	1.05990i
$928$	0	0
$929$	36.8197	1.20801	0.604007	−	0.796979i	$-0.293570\pi$
$929$	36.8197	1.20801	0.604007	+	0.796979i	$0.293570\pi$
$930$	0	0
$931$	−25.7984	−0.845508
$932$	0	0
$933$	− 3.67376i	− 0.120274i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	0.583592i	0.0190651i	0.999955	+	0.00953256i	$0.00303435\pi$
$937$	0.583592i	0.0190651i	−0.999955	+	0.00953256i	$0.996966\pi$
$938$	0	0
$939$	6.58359	0.214847
$940$	0	0
$941$	−8.81966	−0.287513	−0.143756	−	0.989613i	$-0.545918\pi$
$941$	−8.81966	−0.287513	−0.143756	+	0.989613i	$0.545918\pi$
$942$	0	0
$943$	4.09017i	0.133194i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	8.12461i	0.264014i	0.991249	+	0.132007i	$0.0421422\pi$
$947$	8.12461i	0.264014i	−0.991249	+	0.132007i	$0.957858\pi$
$948$	0	0
$949$	−63.5755	−2.06375
$950$	0	0
$951$	2.09017	0.0677784
$952$	0	0
$953$	− 0.111456i	− 0.00361042i	−0.999998	−	0.00180521i	$-0.999425\pi$
$953$	− 0.111456i	− 0.00361042i	0.999998	−	0.00180521i	$-0.000574616\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	5.14590i	0.166343i
$958$	0	0
$959$	3.65248	0.117945
$960$	0	0
$961$	44.8328	1.44622
$962$	0	0
$963$	− 34.5066i	− 1.11196i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 2.72949i	− 0.0877745i	−0.999036	−	0.0438872i	$-0.986026\pi$
$967$	− 2.72949i	− 0.0877745i	0.999036	−	0.0438872i	$-0.0139742\pi$
$968$	0	0
$969$	11.8409	0.380386
$970$	0	0
$971$	0.381966	0.0122579	0.00612894	−	0.999981i	$-0.498049\pi$
$971$	0.381966	0.0122579	0.00612894	+	0.999981i	$0.498049\pi$
$972$	0	0
$973$	4.27051i	0.136906i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 33.5410i	− 1.07307i	−0.843877	−	0.536536i	$-0.819733\pi$
$977$	− 33.5410i	− 1.07307i	0.843877	−	0.536536i	$-0.180267\pi$
$978$	0	0
$979$	−14.3262	−0.457869
$980$	0	0
$981$	−9.32624	−0.297764
$982$	0	0
$983$	40.9443i	1.30592i	0.757393	+	0.652960i	$0.226473\pi$
$983$	40.9443i	1.30592i	−0.757393	+	0.652960i	$0.773527\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 0.527864i	− 0.0168021i
$988$	0	0
$989$	−2.65248	−0.0843438
$990$	0	0
$991$	49.0902	1.55940	0.779700	−	0.626153i	$-0.215371\pi$
$991$	49.0902	1.55940	0.779700	+	0.626153i	$0.215371\pi$
$992$	0	0
$993$	− 8.25735i	− 0.262039i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 56.0344i	− 1.77463i	−0.461165	−	0.887314i	$-0.652568\pi$
$997$	− 56.0344i	− 1.77463i	0.461165	−	0.887314i	$-0.347432\pi$
$998$	0	0
$999$	0.819660	0.0259329

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4400.2.b.z.4049.3		4
4.3	odd	2		2200.2.b.k.1849.2		4
5.2	odd	4		4400.2.a.bm.1.2		2
5.3	odd	4		4400.2.a.bo.1.1		2
5.4	even	2	inner	4400.2.b.z.4049.2		4
20.3	even	4		2200.2.a.p.1.2	✓	2
20.7	even	4		2200.2.a.q.1.1	yes	2
20.19	odd	2		2200.2.b.k.1849.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2200.2.a.p.1.2	✓	2	20.3	even	4
2200.2.a.q.1.1	yes	2	20.7	even	4
2200.2.b.k.1849.2		4	4.3	odd	2
2200.2.b.k.1849.3		4	20.19	odd	2
4400.2.a.bm.1.2		2	5.2	odd	4
4400.2.a.bo.1.1		2	5.3	odd	4
4400.2.b.z.4049.2		4	5.4	even	2	inner
4400.2.b.z.4049.3		4	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+0.618034i q^{3} -0.381966i q^{7} +2.61803 q^{9} -1.00000 q^{11} +5.47214i q^{13} +5.09017i q^{17} -3.76393 q^{19} +0.236068 q^{21} -0.381966i q^{23} +3.47214i q^{27} -8.32624 q^{29} +8.70820 q^{31} -0.618034i q^{33} -0.236068i q^{37} -3.38197 q^{39} -10.7082 q^{41} -6.94427i q^{43} -2.23607i q^{47} +6.85410 q^{49} -3.14590 q^{51} +3.61803i q^{53} -2.32624i q^{57} -10.7082 q^{59} -1.90983 q^{61} -1.00000i q^{63} -12.0000i q^{67} +0.236068 q^{69} -8.18034 q^{71} +11.6180i q^{73} +0.381966i q^{77} -7.38197 q^{79} +5.70820 q^{81} +13.5623i q^{83} -5.14590i q^{87} +14.3262 q^{89} +2.09017 q^{91} +5.38197i q^{93} -10.3820i q^{97} -2.61803 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 6 q^{9} - 4 q^{11} - 24 q^{19} - 8 q^{21} - 2 q^{29} + 8 q^{31} - 18 q^{39} - 16 q^{41} + 14 q^{49} - 26 q^{51} - 16 q^{59} - 30 q^{61} - 8 q^{69} + 12 q^{71} - 34 q^{79} - 4 q^{81} + 26 q^{89} - 14 q^{91}+ \cdots - 6 q^{99}+O(q^{100}) \) 4 * q + 6 * q^9 - 4 * q^11 - 24 * q^19 - 8 * q^21 - 2 * q^29 + 8 * q^31 - 18 * q^39 - 16 * q^41 + 14 * q^49 - 26 * q^51 - 16 * q^59 - 30 * q^61 - 8 * q^69 + 12 * q^71 - 34 * q^79 - 4 * q^81 + 26 * q^89 - 14 * q^91 - 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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