LMFDB - Embedded newform 1200.4.f.i.49.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1200,4,Mod(49,1200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1200, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1200.49");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1200 = 2^{4} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1200.f (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:	$70.8022920069$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 600)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	1200.49
Dual form			1200.4.f.i.49.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000i q^{3} +5.00000i q^{7} -9.00000 q^{9} -14.0000 q^{11} +1.00000i q^{13} -46.0000i q^{17} +19.0000 q^{19} +15.0000 q^{21} +46.0000i q^{23} +27.0000i q^{27} -14.0000 q^{29} -133.000 q^{31} +42.0000i q^{33} -258.000i q^{37} +3.00000 q^{39} +84.0000 q^{41} +167.000i q^{43} +410.000i q^{47} +318.000 q^{49} -138.000 q^{51} +456.000i q^{53} -57.0000i q^{57} -194.000 q^{59} -17.0000 q^{61} -45.0000i q^{63} +653.000i q^{67} +138.000 q^{69} -828.000 q^{71} +570.000i q^{73} -70.0000i q^{77} -552.000 q^{79} +81.0000 q^{81} -142.000i q^{83} +42.0000i q^{87} +1104.00 q^{89} -5.00000 q^{91} +399.000i q^{93} -841.000i q^{97} +126.000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 18 q^{9} - 28 q^{11} + 38 q^{19} + 30 q^{21} - 28 q^{29} - 266 q^{31} + 6 q^{39} + 168 q^{41} + 636 q^{49} - 276 q^{51} - 388 q^{59} - 34 q^{61} + 276 q^{69} - 1656 q^{71} - 1104 q^{79} + 162 q^{81}+ \cdots + 252 q^{99}+O(q^{100}) $ 2 * q - 18 * q^9 - 28 * q^11 + 38 * q^19 + 30 * q^21 - 28 * q^29 - 266 * q^31 + 6 * q^39 + 168 * q^41 + 636 * q^49 - 276 * q^51 - 388 * q^59 - 34 * q^61 + 276 * q^69 - 1656 * q^71 - 1104 * q^79 + 162 * q^81 + 2208 * q^89 - 10 * q^91 + 252 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times$.

$n$	$401$	$577$	$751$	$901$
$\chi(n)$	$1$	$-1$	$1$	$1$

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
$3$	− 3.00000i	− 0.577350i
$3$	− 3.00000i	− 0.577350i
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	5.00000i	0.269975i	0.990847	+	0.134987i	$0.0430994\pi$
$7$	5.00000i	0.269975i	−0.990847	+	0.134987i	$0.956901\pi$
$8$	0	0
$9$	−9.00000	−0.333333
$10$	0	0
$11$	−14.0000	−0.383742	−0.191871	−	0.981420i	$-0.561455\pi$
$11$	−14.0000	−0.383742	−0.191871	+	0.981420i	$0.561455\pi$
$12$	0	0
$13$	1.00000i	0.0213346i	0.999943	+	0.0106673i	$0.00339558\pi$
$13$	1.00000i	0.0213346i	−0.999943	+	0.0106673i	$0.996604\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 46.0000i	− 0.656273i	−0.944630	−	0.328136i	$-0.893579\pi$
$17$	− 46.0000i	− 0.656273i	0.944630	−	0.328136i	$-0.106421\pi$
$18$	0	0
$19$	19.0000	0.229416	0.114708	−	0.993399i	$-0.463407\pi$
$19$	19.0000	0.229416	0.114708	+	0.993399i	$0.463407\pi$
$20$	0	0
$21$	15.0000	0.155870
$22$	0	0
$23$	46.0000i	0.417029i	0.978019	+	0.208514i	$0.0668628\pi$
$23$	46.0000i	0.417029i	−0.978019	+	0.208514i	$0.933137\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	27.0000i	0.192450i
$28$	0	0
$29$	−14.0000	−0.0896460	−0.0448230	−	0.998995i	$-0.514272\pi$
$29$	−14.0000	−0.0896460	−0.0448230	+	0.998995i	$0.514272\pi$
$30$	0	0
$31$	−133.000	−0.770565	−0.385282	−	0.922799i	$-0.625896\pi$
$31$	−133.000	−0.770565	−0.385282	+	0.922799i	$0.625896\pi$
$32$	0	0
$33$	42.0000i	0.221553i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 258.000i	− 1.14635i	−0.819433	−	0.573175i	$-0.805712\pi$
$37$	− 258.000i	− 1.14635i	0.819433	−	0.573175i	$-0.194288\pi$
$38$	0	0
$39$	3.00000	0.0123176
$40$	0	0
$41$	84.0000	0.319966	0.159983	−	0.987120i	$-0.448856\pi$
$41$	84.0000	0.319966	0.159983	+	0.987120i	$0.448856\pi$
$42$	0	0
$43$	167.000i	0.592262i	0.955147	+	0.296131i	$0.0956965\pi$
$43$	167.000i	0.592262i	−0.955147	+	0.296131i	$0.904304\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	410.000i	1.27244i	0.771508	+	0.636220i	$0.219503\pi$
$47$	410.000i	1.27244i	−0.771508	+	0.636220i	$0.780497\pi$
$48$	0	0
$49$	318.000	0.927114
$50$	0	0
$51$	−138.000	−0.378899
$52$	0	0
$53$	456.000i	1.18182i	0.806738	+	0.590910i	$0.201231\pi$
$53$	456.000i	1.18182i	−0.806738	+	0.590910i	$0.798769\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 57.0000i	− 0.132453i
$58$	0	0
$59$	−194.000	−0.428079	−0.214039	−	0.976825i	$-0.568662\pi$
$59$	−194.000	−0.428079	−0.214039	+	0.976825i	$0.568662\pi$
$60$	0	0
$61$	−17.0000	−0.0356824	−0.0178412	−	0.999841i	$-0.505679\pi$
$61$	−17.0000	−0.0356824	−0.0178412	+	0.999841i	$0.505679\pi$
$62$	0	0
$63$	− 45.0000i	− 0.0899915i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	653.000i	1.19070i	0.803468	+	0.595348i	$0.202986\pi$
$67$	653.000i	1.19070i	−0.803468	+	0.595348i	$0.797014\pi$
$68$	0	0
$69$	138.000	0.240772
$70$	0	0
$71$	−828.000	−1.38402	−0.692011	−	0.721887i	$-0.743275\pi$
$71$	−828.000	−1.38402	−0.692011	+	0.721887i	$0.743275\pi$
$72$	0	0
$73$	570.000i	0.913883i	0.889497	+	0.456941i	$0.151055\pi$
$73$	570.000i	0.913883i	−0.889497	+	0.456941i	$0.848945\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 70.0000i	− 0.103601i
$78$	0	0
$79$	−552.000	−0.786137	−0.393069	−	0.919509i	$-0.628587\pi$
$79$	−552.000	−0.786137	−0.393069	+	0.919509i	$0.628587\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	− 142.000i	− 0.187789i	−0.995582	−	0.0938947i	$-0.970068\pi$
$83$	− 142.000i	− 0.187789i	0.995582	−	0.0938947i	$-0.0299317\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	42.0000i	0.0517572i
$88$	0	0
$89$	1104.00	1.31487	0.657437	−	0.753510i	$-0.271641\pi$
$89$	1104.00	1.31487	0.657437	+	0.753510i	$0.271641\pi$
$90$	0	0
$91$	−5.00000	−0.00575981
$92$	0	0
$93$	399.000i	0.444886i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 841.000i	− 0.880316i	−0.897920	−	0.440158i	$-0.854923\pi$
$97$	− 841.000i	− 0.880316i	0.897920	−	0.440158i	$-0.145077\pi$
$98$	0	0
$99$	126.000	0.127914
$100$	0	0
$101$	552.000	0.543822	0.271911	−	0.962322i	$-0.412344\pi$
$101$	552.000	0.543822	0.271911	+	0.962322i	$0.412344\pi$
$102$	0	0
$103$	308.000i	0.294642i	0.989089	+	0.147321i	$0.0470651\pi$
$103$	308.000i	0.294642i	−0.989089	+	0.147321i	$0.952935\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	984.000i	0.889036i	0.895770	+	0.444518i	$0.146625\pi$
$107$	984.000i	0.889036i	−0.895770	+	0.444518i	$0.853375\pi$
$108$	0	0
$109$	1843.00	1.61952	0.809759	−	0.586763i	$-0.199598\pi$
$109$	1843.00	1.61952	0.809759	+	0.586763i	$0.199598\pi$
$110$	0	0
$111$	−774.000	−0.661845
$112$	0	0
$113$	876.000i	0.729267i	0.931151	+	0.364633i	$0.118806\pi$
$113$	876.000i	0.729267i	−0.931151	+	0.364633i	$0.881194\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 9.00000i	− 0.00711154i
$118$	0	0
$119$	230.000	0.177177
$120$	0	0
$121$	−1135.00	−0.852742
$122$	0	0
$123$	− 252.000i	− 0.184732i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	2376.00i	1.66013i	0.557670	+	0.830063i	$0.311695\pi$
$127$	2376.00i	1.66013i	−0.557670	+	0.830063i	$0.688305\pi$
$128$	0	0
$129$	501.000	0.341943
$130$	0	0
$131$	−1056.00	−0.704299	−0.352149	−	0.935944i	$-0.614549\pi$
$131$	−1056.00	−0.704299	−0.352149	+	0.935944i	$0.614549\pi$
$132$	0	0
$133$	95.0000i	0.0619364i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	778.000i	0.485175i	0.970129	+	0.242588i	$0.0779962\pi$
$137$	778.000i	0.485175i	−0.970129	+	0.242588i	$0.922004\pi$
$138$	0	0
$139$	1692.00	1.03247	0.516236	−	0.856446i	$-0.327333\pi$
$139$	1692.00	1.03247	0.516236	+	0.856446i	$0.327333\pi$
$140$	0	0
$141$	1230.00	0.734643
$142$	0	0
$143$	− 14.0000i	− 0.00818698i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 954.000i	− 0.535269i
$148$	0	0
$149$	494.000	0.271611	0.135806	−	0.990736i	$-0.456638\pi$
$149$	494.000	0.271611	0.135806	+	0.990736i	$0.456638\pi$
$150$	0	0
$151$	841.000	0.453242	0.226621	−	0.973983i	$-0.427232\pi$
$151$	841.000	0.453242	0.226621	+	0.973983i	$0.427232\pi$
$152$	0	0
$153$	414.000i	0.218758i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 19.0000i	− 0.00965838i	−0.999988	−	0.00482919i	$-0.998463\pi$
$157$	− 19.0000i	− 0.00965838i	0.999988	−	0.00482919i	$-0.00153718\pi$
$158$	0	0
$159$	1368.00	0.682324
$160$	0	0
$161$	−230.000	−0.112587
$162$	0	0
$163$	2261.00i	1.08647i	0.839580	+	0.543237i	$0.182801\pi$
$163$	2261.00i	1.08647i	−0.839580	+	0.543237i	$0.817199\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	2112.00i	0.978632i	0.872107	+	0.489316i	$0.162753\pi$
$167$	2112.00i	0.978632i	−0.872107	+	0.489316i	$0.837247\pi$
$168$	0	0
$169$	2196.00	0.999545
$170$	0	0
$171$	−171.000	−0.0764719
$172$	0	0
$173$	562.000i	0.246983i	0.992346	+	0.123492i	$0.0394092\pi$
$173$	562.000i	0.246983i	−0.992346	+	0.123492i	$0.960591\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	582.000i	0.247151i
$178$	0	0
$179$	3718.00	1.55249	0.776247	−	0.630429i	$-0.217121\pi$
$179$	3718.00	1.55249	0.776247	+	0.630429i	$0.217121\pi$
$180$	0	0
$181$	−1639.00	−0.673071	−0.336536	−	0.941671i	$-0.609255\pi$
$181$	−1639.00	−0.673071	−0.336536	+	0.941671i	$0.609255\pi$
$182$	0	0
$183$	51.0000i	0.0206012i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	644.000i	0.251839i
$188$	0	0
$189$	−135.000	−0.0519566
$190$	0	0
$191$	−2410.00	−0.912992	−0.456496	−	0.889725i	$-0.650896\pi$
$191$	−2410.00	−0.912992	−0.456496	+	0.889725i	$0.650896\pi$
$192$	0	0
$193$	− 2621.00i	− 0.977532i	−0.872415	−	0.488766i	$-0.837447\pi$
$193$	− 2621.00i	− 0.977532i	0.872415	−	0.488766i	$-0.162553\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	4954.00i	1.79166i	0.444392	+	0.895832i	$0.353420\pi$
$197$	4954.00i	1.79166i	−0.444392	+	0.895832i	$0.646580\pi$
$198$	0	0
$199$	1739.00	0.619470	0.309735	−	0.950823i	$-0.399760\pi$
$199$	1739.00	0.619470	0.309735	+	0.950823i	$0.399760\pi$
$200$	0	0
$201$	1959.00	0.687449
$202$	0	0
$203$	− 70.0000i	− 0.0242022i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 414.000i	− 0.139010i
$208$	0	0
$209$	−266.000	−0.0880364
$210$	0	0
$211$	4525.00	1.47637	0.738184	−	0.674599i	$-0.235683\pi$
$211$	4525.00	1.47637	0.738184	+	0.674599i	$0.235683\pi$
$212$	0	0
$213$	2484.00i	0.799065i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 665.000i	− 0.208033i
$218$	0	0
$219$	1710.00	0.527631
$220$	0	0
$221$	46.0000	0.0140013
$222$	0	0
$223$	3211.00i	0.964235i	0.876106	+	0.482118i	$0.160132\pi$
$223$	3211.00i	0.964235i	−0.876106	+	0.482118i	$0.839868\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 2484.00i	− 0.726295i	−0.931732	−	0.363147i	$-0.881702\pi$
$227$	− 2484.00i	− 0.726295i	0.931732	−	0.363147i	$-0.118298\pi$
$228$	0	0
$229$	1847.00	0.532983	0.266492	−	0.963837i	$-0.414136\pi$
$229$	1847.00	0.532983	0.266492	+	0.963837i	$0.414136\pi$
$230$	0	0
$231$	−210.000	−0.0598138
$232$	0	0
$233$	− 1020.00i	− 0.286792i	−0.989665	−	0.143396i	$-0.954198\pi$
$233$	− 1020.00i	− 0.286792i	0.989665	−	0.143396i	$-0.0458022\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	1656.00i	0.453877i
$238$	0	0
$239$	1176.00	0.318281	0.159140	−	0.987256i	$-0.449128\pi$
$239$	1176.00	0.318281	0.159140	+	0.987256i	$0.449128\pi$
$240$	0	0
$241$	−6967.00	−1.86217	−0.931087	−	0.364797i	$-0.881138\pi$
$241$	−6967.00	−1.86217	−0.931087	+	0.364797i	$0.881138\pi$
$242$	0	0
$243$	− 243.000i	− 0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	19.0000i	0.00489450i
$248$	0	0
$249$	−426.000	−0.108420
$250$	0	0
$251$	−1380.00	−0.347031	−0.173516	−	0.984831i	$-0.555513\pi$
$251$	−1380.00	−0.347031	−0.173516	+	0.984831i	$0.555513\pi$
$252$	0	0
$253$	− 644.000i	− 0.160031i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	6924.00i	1.68057i	0.542143	+	0.840286i	$0.317613\pi$
$257$	6924.00i	1.68057i	−0.542143	+	0.840286i	$0.682387\pi$
$258$	0	0
$259$	1290.00	0.309485
$260$	0	0
$261$	126.000	0.0298820
$262$	0	0
$263$	− 1884.00i	− 0.441720i	−0.975305	−	0.220860i	$-0.929114\pi$
$263$	− 1884.00i	− 0.441720i	0.975305	−	0.220860i	$-0.0708864\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 3312.00i	− 0.759143i
$268$	0	0
$269$	−3610.00	−0.818236	−0.409118	−	0.912481i	$-0.634164\pi$
$269$	−3610.00	−0.818236	−0.409118	+	0.912481i	$0.634164\pi$
$270$	0	0
$271$	6072.00	1.36106	0.680531	−	0.732719i	$-0.261749\pi$
$271$	6072.00	1.36106	0.680531	+	0.732719i	$0.261749\pi$
$272$	0	0
$273$	15.0000i	0.00332543i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	2803.00i	0.608000i	0.952672	+	0.304000i	$0.0983223\pi$
$277$	2803.00i	0.608000i	−0.952672	+	0.304000i	$0.901678\pi$
$278$	0	0
$279$	1197.00	0.256855
$280$	0	0
$281$	−6694.00	−1.42111	−0.710553	−	0.703644i	$-0.751555\pi$
$281$	−6694.00	−1.42111	−0.710553	+	0.703644i	$0.751555\pi$
$282$	0	0
$283$	6481.00i	1.36133i	0.732596	+	0.680663i	$0.238308\pi$
$283$	6481.00i	1.36133i	−0.732596	+	0.680663i	$0.761692\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	420.000i	0.0863826i
$288$	0	0
$289$	2797.00	0.569306
$290$	0	0
$291$	−2523.00	−0.508250
$292$	0	0
$293$	3014.00i	0.600955i	0.953789	+	0.300477i	$0.0971460\pi$
$293$	3014.00i	0.600955i	−0.953789	+	0.300477i	$0.902854\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 378.000i	− 0.0738511i
$298$	0	0
$299$	−46.0000	−0.00889715
$300$	0	0
$301$	−835.000	−0.159896
$302$	0	0
$303$	− 1656.00i	− 0.313976i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 5369.00i	− 0.998127i	−0.866565	−	0.499064i	$-0.833677\pi$
$307$	− 5369.00i	− 0.998127i	0.866565	−	0.499064i	$-0.166323\pi$
$308$	0	0
$309$	924.000	0.170112
$310$	0	0
$311$	−4846.00	−0.883574	−0.441787	−	0.897120i	$-0.645655\pi$
$311$	−4846.00	−0.883574	−0.441787	+	0.897120i	$0.645655\pi$
$312$	0	0
$313$	− 757.000i	− 0.136703i	−0.997661	−	0.0683517i	$-0.978226\pi$
$313$	− 757.000i	− 0.136703i	0.997661	−	0.0683517i	$-0.0217740\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	7632.00i	1.35223i	0.736798	+	0.676113i	$0.236337\pi$
$317$	7632.00i	1.35223i	−0.736798	+	0.676113i	$0.763663\pi$
$318$	0	0
$319$	196.000	0.0344009
$320$	0	0
$321$	2952.00	0.513285
$322$	0	0
$323$	− 874.000i	− 0.150559i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 5529.00i	− 0.935029i
$328$	0	0
$329$	−2050.00	−0.343526
$330$	0	0
$331$	6780.00	1.12587	0.562934	−	0.826502i	$-0.309672\pi$
$331$	6780.00	1.12587	0.562934	+	0.826502i	$0.309672\pi$
$332$	0	0
$333$	2322.00i	0.382117i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 7849.00i	− 1.26873i	−0.773033	−	0.634365i	$-0.781261\pi$
$337$	− 7849.00i	− 1.26873i	0.773033	−	0.634365i	$-0.218739\pi$
$338$	0	0
$339$	2628.00	0.421042
$340$	0	0
$341$	1862.00	0.295698
$342$	0	0
$343$	3305.00i	0.520272i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	634.000i	0.0980833i	0.998797	+	0.0490416i	$0.0156167\pi$
$347$	634.000i	0.0980833i	−0.998797	+	0.0490416i	$0.984383\pi$
$348$	0	0
$349$	−930.000	−0.142641	−0.0713206	−	0.997453i	$-0.522721\pi$
$349$	−930.000	−0.142641	−0.0713206	+	0.997453i	$0.522721\pi$
$350$	0	0
$351$	−27.0000	−0.00410585
$352$	0	0
$353$	− 4286.00i	− 0.646234i	−0.946359	−	0.323117i	$-0.895269\pi$
$353$	− 4286.00i	− 0.646234i	0.946359	−	0.323117i	$-0.104731\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 690.000i	− 0.102293i
$358$	0	0
$359$	−4236.00	−0.622751	−0.311375	−	0.950287i	$-0.600790\pi$
$359$	−4236.00	−0.622751	−0.311375	+	0.950287i	$0.600790\pi$
$360$	0	0
$361$	−6498.00	−0.947368
$362$	0	0
$363$	3405.00i	0.492331i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	1451.00i	0.206380i	0.994662	+	0.103190i	$0.0329050\pi$
$367$	1451.00i	0.206380i	−0.994662	+	0.103190i	$0.967095\pi$
$368$	0	0
$369$	−756.000	−0.106655
$370$	0	0
$371$	−2280.00	−0.319061
$372$	0	0
$373$	3115.00i	0.432409i	0.976348	+	0.216205i	$0.0693678\pi$
$373$	3115.00i	0.432409i	−0.976348	+	0.216205i	$0.930632\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 14.0000i	− 0.00191256i
$378$	0	0
$379$	−1415.00	−0.191777	−0.0958887	−	0.995392i	$-0.530569\pi$
$379$	−1415.00	−0.191777	−0.0958887	+	0.995392i	$0.530569\pi$
$380$	0	0
$381$	7128.00	0.958474
$382$	0	0
$383$	180.000i	0.0240145i	0.999928	+	0.0120073i	$0.00382213\pi$
$383$	180.000i	0.0240145i	−0.999928	+	0.0120073i	$0.996178\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 1503.00i	− 0.197421i
$388$	0	0
$389$	−12372.0	−1.61256	−0.806279	−	0.591535i	$-0.798522\pi$
$389$	−12372.0	−1.61256	−0.806279	+	0.591535i	$0.798522\pi$
$390$	0	0
$391$	2116.00	0.273685
$392$	0	0
$393$	3168.00i	0.406627i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 5767.00i	− 0.729062i	−0.931191	−	0.364531i	$-0.881229\pi$
$397$	− 5767.00i	− 0.729062i	0.931191	−	0.364531i	$-0.118771\pi$
$398$	0	0
$399$	285.000	0.0357590
$400$	0	0
$401$	−3120.00	−0.388542	−0.194271	−	0.980948i	$-0.562234\pi$
$401$	−3120.00	−0.388542	−0.194271	+	0.980948i	$0.562234\pi$
$402$	0	0
$403$	− 133.000i	− 0.0164397i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	3612.00i	0.439902i
$408$	0	0
$409$	−1501.00	−0.181466	−0.0907331	−	0.995875i	$-0.528921\pi$
$409$	−1501.00	−0.181466	−0.0907331	+	0.995875i	$0.528921\pi$
$410$	0	0
$411$	2334.00	0.280116
$412$	0	0
$413$	− 970.000i	− 0.115570i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 5076.00i	− 0.596098i
$418$	0	0
$419$	−9072.00	−1.05775	−0.528874	−	0.848701i	$-0.677386\pi$
$419$	−9072.00	−1.05775	−0.528874	+	0.848701i	$0.677386\pi$
$420$	0	0
$421$	7350.00	0.850872	0.425436	−	0.904989i	$-0.360121\pi$
$421$	7350.00	0.850872	0.425436	+	0.904989i	$0.360121\pi$
$422$	0	0
$423$	− 3690.00i	− 0.424146i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 85.0000i	− 0.00963334i
$428$	0	0
$429$	−42.0000	−0.00472676
$430$	0	0
$431$	−5962.00	−0.666310	−0.333155	−	0.942872i	$-0.608113\pi$
$431$	−5962.00	−0.666310	−0.333155	+	0.942872i	$0.608113\pi$
$432$	0	0
$433$	− 10093.0i	− 1.12018i	−0.828431	−	0.560091i	$-0.810766\pi$
$433$	− 10093.0i	− 1.12018i	0.828431	−	0.560091i	$-0.189234\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	874.000i	0.0956730i
$438$	0	0
$439$	−2555.00	−0.277776	−0.138888	−	0.990308i	$-0.544353\pi$
$439$	−2555.00	−0.277776	−0.138888	+	0.990308i	$0.544353\pi$
$440$	0	0
$441$	−2862.00	−0.309038
$442$	0	0
$443$	− 6240.00i	− 0.669236i	−0.942354	−	0.334618i	$-0.891393\pi$
$443$	− 6240.00i	− 0.669236i	0.942354	−	0.334618i	$-0.108607\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 1482.00i	− 0.156815i
$448$	0	0
$449$	−3324.00	−0.349375	−0.174687	−	0.984624i	$-0.555891\pi$
$449$	−3324.00	−0.349375	−0.174687	+	0.984624i	$0.555891\pi$
$450$	0	0
$451$	−1176.00	−0.122784
$452$	0	0
$453$	− 2523.00i	− 0.261680i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 16774.0i	− 1.71697i	−0.512840	−	0.858484i	$-0.671407\pi$
$457$	− 16774.0i	− 1.71697i	0.512840	−	0.858484i	$-0.328593\pi$
$458$	0	0
$459$	1242.00	0.126300
$460$	0	0
$461$	14304.0	1.44513	0.722564	−	0.691304i	$-0.242964\pi$
$461$	14304.0	1.44513	0.722564	+	0.691304i	$0.242964\pi$
$462$	0	0
$463$	− 6936.00i	− 0.696206i	−0.937456	−	0.348103i	$-0.886826\pi$
$463$	− 6936.00i	− 0.696206i	0.937456	−	0.348103i	$-0.113174\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	15622.0i	1.54797i	0.633207	+	0.773983i	$0.281738\pi$
$467$	15622.0i	1.54797i	−0.633207	+	0.773983i	$0.718262\pi$
$468$	0	0
$469$	−3265.00	−0.321458
$470$	0	0
$471$	−57.0000	−0.00557627
$472$	0	0
$473$	− 2338.00i	− 0.227276i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 4104.00i	− 0.393940i
$478$	0	0
$479$	−13354.0	−1.27382	−0.636910	−	0.770938i	$-0.719788\pi$
$479$	−13354.0	−1.27382	−0.636910	+	0.770938i	$0.719788\pi$
$480$	0	0
$481$	258.000	0.0244569
$482$	0	0
$483$	690.000i	0.0650023i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 461.000i	− 0.0428951i	−0.999770	−	0.0214475i	$-0.993173\pi$
$487$	− 461.000i	− 0.0428951i	0.999770	−	0.0214475i	$-0.00682749\pi$
$488$	0	0
$489$	6783.00	0.627276
$490$	0	0
$491$	−3768.00	−0.346329	−0.173164	−	0.984893i	$-0.555399\pi$
$491$	−3768.00	−0.346329	−0.173164	+	0.984893i	$0.555399\pi$
$492$	0	0
$493$	644.000i	0.0588323i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 4140.00i	− 0.373651i
$498$	0	0
$499$	−14317.0	−1.28440	−0.642201	−	0.766536i	$-0.721979\pi$
$499$	−14317.0	−1.28440	−0.642201	+	0.766536i	$0.721979\pi$
$500$	0	0
$501$	6336.00	0.565013
$502$	0	0
$503$	− 9228.00i	− 0.818004i	−0.912533	−	0.409002i	$-0.865877\pi$
$503$	− 9228.00i	− 0.818004i	0.912533	−	0.409002i	$-0.134123\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 6588.00i	− 0.577087i
$508$	0	0
$509$	−4574.00	−0.398308	−0.199154	−	0.979968i	$-0.563819\pi$
$509$	−4574.00	−0.398308	−0.199154	+	0.979968i	$0.563819\pi$
$510$	0	0
$511$	−2850.00	−0.246725
$512$	0	0
$513$	513.000i	0.0441511i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 5740.00i	− 0.488288i
$518$	0	0
$519$	1686.00	0.142596
$520$	0	0
$521$	−8494.00	−0.714259	−0.357129	−	0.934055i	$-0.616245\pi$
$521$	−8494.00	−0.714259	−0.357129	+	0.934055i	$0.616245\pi$
$522$	0	0
$523$	− 8263.00i	− 0.690852i	−0.938446	−	0.345426i	$-0.887734\pi$
$523$	− 8263.00i	− 0.690852i	0.938446	−	0.345426i	$-0.112266\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	6118.00i	0.505701i
$528$	0	0
$529$	10051.0	0.826087
$530$	0	0
$531$	1746.00	0.142693
$532$	0	0
$533$	84.0000i	0.00682635i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 11154.0i	− 0.896333i
$538$	0	0
$539$	−4452.00	−0.355772
$540$	0	0
$541$	21157.0	1.68135	0.840675	−	0.541540i	$-0.182158\pi$
$541$	21157.0	1.68135	0.840675	+	0.541540i	$0.182158\pi$
$542$	0	0
$543$	4917.00i	0.388598i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 4048.00i	− 0.316417i	−0.987406	−	0.158208i	$-0.949428\pi$
$547$	− 4048.00i	− 0.316417i	0.987406	−	0.158208i	$-0.0505718\pi$
$548$	0	0
$549$	153.000	0.0118941
$550$	0	0
$551$	−266.000	−0.0205662
$552$	0	0
$553$	− 2760.00i	− 0.212237i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	6758.00i	0.514086i	0.966400	+	0.257043i	$0.0827481\pi$
$557$	6758.00i	0.514086i	−0.966400	+	0.257043i	$0.917252\pi$
$558$	0	0
$559$	−167.000	−0.0126357
$560$	0	0
$561$	1932.00	0.145399
$562$	0	0
$563$	− 24506.0i	− 1.83447i	−0.398350	−	0.917233i	$-0.630417\pi$
$563$	− 24506.0i	− 1.83447i	0.398350	−	0.917233i	$-0.369583\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	405.000i	0.0299972i
$568$	0	0
$569$	1430.00	0.105358	0.0526790	−	0.998611i	$-0.483224\pi$
$569$	1430.00	0.105358	0.0526790	+	0.998611i	$0.483224\pi$
$570$	0	0
$571$	−3691.00	−0.270514	−0.135257	−	0.990811i	$-0.543186\pi$
$571$	−3691.00	−0.270514	−0.135257	+	0.990811i	$0.543186\pi$
$572$	0	0
$573$	7230.00i	0.527116i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 4571.00i	− 0.329798i	−0.986310	−	0.164899i	$-0.947270\pi$
$577$	− 4571.00i	− 0.329798i	0.986310	−	0.164899i	$-0.0527298\pi$
$578$	0	0
$579$	−7863.00	−0.564378
$580$	0	0
$581$	710.000	0.0506984
$582$	0	0
$583$	− 6384.00i	− 0.453513i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	14808.0i	1.04121i	0.853797	+	0.520606i	$0.174294\pi$
$587$	14808.0i	1.04121i	−0.853797	+	0.520606i	$0.825706\pi$
$588$	0	0
$589$	−2527.00	−0.176780
$590$	0	0
$591$	14862.0	1.03442
$592$	0	0
$593$	− 24588.0i	− 1.70271i	−0.524588	−	0.851356i	$-0.675781\pi$
$593$	− 24588.0i	− 1.70271i	0.524588	−	0.851356i	$-0.324219\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 5217.00i	− 0.357651i
$598$	0	0
$599$	−27564.0	−1.88019	−0.940096	−	0.340911i	$-0.889265\pi$
$599$	−27564.0	−1.88019	−0.940096	+	0.340911i	$0.889265\pi$
$600$	0	0
$601$	10987.0	0.745706	0.372853	−	0.927891i	$-0.378380\pi$
$601$	10987.0	0.745706	0.372853	+	0.927891i	$0.378380\pi$
$602$	0	0
$603$	− 5877.00i	− 0.396899i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 13200.0i	− 0.882655i	−0.897346	−	0.441327i	$-0.854508\pi$
$607$	− 13200.0i	− 0.882655i	0.897346	−	0.441327i	$-0.145492\pi$
$608$	0	0
$609$	−210.000	−0.0139731
$610$	0	0
$611$	−410.000	−0.0271470
$612$	0	0
$613$	21066.0i	1.38801i	0.719972	+	0.694003i	$0.244155\pi$
$613$	21066.0i	1.38801i	−0.719972	+	0.694003i	$0.755845\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 12336.0i	− 0.804909i	−0.915440	−	0.402454i	$-0.868157\pi$
$617$	− 12336.0i	− 0.804909i	0.915440	−	0.402454i	$-0.131843\pi$
$618$	0	0
$619$	−1441.00	−0.0935681	−0.0467841	−	0.998905i	$-0.514897\pi$
$619$	−1441.00	−0.0935681	−0.0467841	+	0.998905i	$0.514897\pi$
$620$	0	0
$621$	−1242.00	−0.0802572
$622$	0	0
$623$	5520.00i	0.354983i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	798.000i	0.0508278i
$628$	0	0
$629$	−11868.0	−0.752318
$630$	0	0
$631$	9839.00	0.620736	0.310368	−	0.950616i	$-0.399548\pi$
$631$	9839.00	0.620736	0.310368	+	0.950616i	$0.399548\pi$
$632$	0	0
$633$	− 13575.0i	− 0.852382i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	318.000i	0.0197796i
$638$	0	0
$639$	7452.00	0.461340
$640$	0	0
$641$	−21564.0	−1.32875	−0.664373	−	0.747401i	$-0.731302\pi$
$641$	−21564.0	−1.32875	−0.664373	+	0.747401i	$0.731302\pi$
$642$	0	0
$643$	− 8604.00i	− 0.527696i	−0.964564	−	0.263848i	$-0.915008\pi$
$643$	− 8604.00i	− 0.527696i	0.964564	−	0.263848i	$-0.0849918\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	3444.00i	0.209270i	0.994511	+	0.104635i	$0.0333674\pi$
$647$	3444.00i	0.209270i	−0.994511	+	0.104635i	$0.966633\pi$
$648$	0	0
$649$	2716.00	0.164272
$650$	0	0
$651$	−1995.00	−0.120108
$652$	0	0
$653$	− 3518.00i	− 0.210827i	−0.994428	−	0.105413i	$-0.966383\pi$
$653$	− 3518.00i	− 0.210827i	0.994428	−	0.105413i	$-0.0336166\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 5130.00i	− 0.304628i
$658$	0	0
$659$	−12612.0	−0.745514	−0.372757	−	0.927929i	$-0.621588\pi$
$659$	−12612.0	−0.745514	−0.372757	+	0.927929i	$0.621588\pi$
$660$	0	0
$661$	27090.0	1.59407	0.797034	−	0.603935i	$-0.206401\pi$
$661$	27090.0	1.59407	0.797034	+	0.603935i	$0.206401\pi$
$662$	0	0
$663$	− 138.000i	− 0.00808367i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 644.000i	− 0.0373850i
$668$	0	0
$669$	9633.00	0.556701
$670$	0	0
$671$	238.000	0.0136928
$672$	0	0
$673$	− 5442.00i	− 0.311699i	−0.987781	−	0.155850i	$-0.950188\pi$
$673$	− 5442.00i	− 0.311699i	0.987781	−	0.155850i	$-0.0498115\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	15226.0i	0.864376i	0.901783	+	0.432188i	$0.142258\pi$
$677$	15226.0i	0.864376i	−0.901783	+	0.432188i	$0.857742\pi$
$678$	0	0
$679$	4205.00	0.237663
$680$	0	0
$681$	−7452.00	−0.419326
$682$	0	0
$683$	552.000i	0.0309249i	0.999880	+	0.0154624i	$0.00492204\pi$
$683$	552.000i	0.0309249i	−0.999880	+	0.0154624i	$0.995078\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 5541.00i	− 0.307718i
$688$	0	0
$689$	−456.000	−0.0252137
$690$	0	0
$691$	−9776.00	−0.538201	−0.269100	−	0.963112i	$-0.586726\pi$
$691$	−9776.00	−0.538201	−0.269100	+	0.963112i	$0.586726\pi$
$692$	0	0
$693$	630.000i	0.0345335i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 3864.00i	− 0.209985i
$698$	0	0
$699$	−3060.00	−0.165579
$700$	0	0
$701$	13066.0	0.703989	0.351994	−	0.936002i	$-0.385504\pi$
$701$	13066.0	0.703989	0.351994	+	0.936002i	$0.385504\pi$
$702$	0	0
$703$	− 4902.00i	− 0.262991i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	2760.00i	0.146818i
$708$	0	0
$709$	−28985.0	−1.53534	−0.767669	−	0.640847i	$-0.778583\pi$
$709$	−28985.0	−1.53534	−0.767669	+	0.640847i	$0.778583\pi$
$710$	0	0
$711$	4968.00	0.262046
$712$	0	0
$713$	− 6118.00i	− 0.321348i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 3528.00i	− 0.183760i
$718$	0	0
$719$	15722.0	0.815482	0.407741	−	0.913098i	$-0.366317\pi$
$719$	15722.0	0.815482	0.407741	+	0.913098i	$0.366317\pi$
$720$	0	0
$721$	−1540.00	−0.0795459
$722$	0	0
$723$	20901.0i	1.07513i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 32611.0i	− 1.66365i	−0.555036	−	0.831826i	$-0.687296\pi$
$727$	− 32611.0i	− 1.66365i	0.555036	−	0.831826i	$-0.312704\pi$
$728$	0	0
$729$	−729.000	−0.0370370
$730$	0	0
$731$	7682.00	0.388685
$732$	0	0
$733$	8358.00i	0.421159i	0.977577	+	0.210580i	$0.0675351\pi$
$733$	8358.00i	0.421159i	−0.977577	+	0.210580i	$0.932465\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 9142.00i	− 0.456920i
$738$	0	0
$739$	20604.0	1.02562	0.512808	−	0.858503i	$-0.328605\pi$
$739$	20604.0	1.02562	0.512808	+	0.858503i	$0.328605\pi$
$740$	0	0
$741$	57.0000	0.00282584
$742$	0	0
$743$	19476.0i	0.961649i	0.876817	+	0.480824i	$0.159663\pi$
$743$	19476.0i	0.961649i	−0.876817	+	0.480824i	$0.840337\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	1278.00i	0.0625965i
$748$	0	0
$749$	−4920.00	−0.240017
$750$	0	0
$751$	−3864.00	−0.187749	−0.0938744	−	0.995584i	$-0.529925\pi$
$751$	−3864.00	−0.187749	−0.0938744	+	0.995584i	$0.529925\pi$
$752$	0	0
$753$	4140.00i	0.200359i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	18871.0i	0.906048i	0.891498	+	0.453024i	$0.149655\pi$
$757$	18871.0i	0.906048i	−0.891498	+	0.453024i	$0.850345\pi$
$758$	0	0
$759$	−1932.00	−0.0923941
$760$	0	0
$761$	−36372.0	−1.73257	−0.866284	−	0.499552i	$-0.833498\pi$
$761$	−36372.0	−1.73257	−0.866284	+	0.499552i	$0.833498\pi$
$762$	0	0
$763$	9215.00i	0.437229i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 194.000i	− 0.00913290i
$768$	0	0
$769$	−4603.00	−0.215850	−0.107925	−	0.994159i	$-0.534421\pi$
$769$	−4603.00	−0.215850	−0.107925	+	0.994159i	$0.534421\pi$
$770$	0	0
$771$	20772.0	0.970279
$772$	0	0
$773$	− 36.0000i	− 0.00167507i	−1.00000		0.000837536i	$-0.999733\pi$
$773$	− 36.0000i	− 0.00167507i	1.00000		0.000837536i	$-0.000266596\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 3870.00i	− 0.178681i
$778$	0	0
$779$	1596.00	0.0734052
$780$	0	0
$781$	11592.0	0.531107
$782$	0	0
$783$	− 378.000i	− 0.0172524i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 20281.0i	− 0.918602i	−0.888281	−	0.459301i	$-0.848100\pi$
$787$	− 20281.0i	− 0.918602i	0.888281	−	0.459301i	$-0.151900\pi$
$788$	0	0
$789$	−5652.00	−0.255027
$790$	0	0
$791$	−4380.00	−0.196884
$792$	0	0
$793$	− 17.0000i	0 0.000761271i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	37524.0i	1.66771i	0.551980	+	0.833857i	$0.313872\pi$
$797$	37524.0i	1.66771i	−0.551980	+	0.833857i	$0.686128\pi$
$798$	0	0
$799$	18860.0	0.835067
$800$	0	0
$801$	−9936.00	−0.438291
$802$	0	0
$803$	− 7980.00i	− 0.350695i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	10830.0i	0.472409i
$808$	0	0
$809$	−31224.0	−1.35696	−0.678478	−	0.734621i	$-0.737360\pi$
$809$	−31224.0	−1.35696	−0.678478	+	0.734621i	$0.737360\pi$
$810$	0	0
$811$	−32579.0	−1.41061	−0.705304	−	0.708905i	$-0.749190\pi$
$811$	−32579.0	−1.41061	−0.705304	+	0.708905i	$0.749190\pi$
$812$	0	0
$813$	− 18216.0i	− 0.785809i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	3173.00i	0.135874i
$818$	0	0
$819$	45.0000	0.00191994
$820$	0	0
$821$	−19810.0	−0.842112	−0.421056	−	0.907035i	$-0.638340\pi$
$821$	−19810.0	−0.842112	−0.421056	+	0.907035i	$0.638340\pi$
$822$	0	0
$823$	− 10273.0i	− 0.435108i	−0.976048	−	0.217554i	$-0.930192\pi$
$823$	− 10273.0i	− 0.435108i	0.976048	−	0.217554i	$-0.0698079\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	16656.0i	0.700346i	0.936685	+	0.350173i	$0.113877\pi$
$827$	16656.0i	0.700346i	−0.936685	+	0.350173i	$0.886123\pi$
$828$	0	0
$829$	4790.00	0.200680	0.100340	−	0.994953i	$-0.468007\pi$
$829$	4790.00	0.200680	0.100340	+	0.994953i	$0.468007\pi$
$830$	0	0
$831$	8409.00	0.351029
$832$	0	0
$833$	− 14628.0i	− 0.608440i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 3591.00i	− 0.148295i
$838$	0	0
$839$	7414.00	0.305077	0.152539	−	0.988298i	$-0.451255\pi$
$839$	7414.00	0.305077	0.152539	+	0.988298i	$0.451255\pi$
$840$	0	0
$841$	−24193.0	−0.991964
$842$	0	0
$843$	20082.0i	0.820475i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 5675.00i	− 0.230219i
$848$	0	0
$849$	19443.0	0.785962
$850$	0	0
$851$	11868.0	0.478061
$852$	0	0
$853$	30155.0i	1.21042i	0.796066	+	0.605210i	$0.206911\pi$
$853$	30155.0i	1.21042i	−0.796066	+	0.605210i	$0.793089\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	8244.00i	0.328599i	0.986410	+	0.164300i	$0.0525364\pi$
$857$	8244.00i	0.328599i	−0.986410	+	0.164300i	$0.947464\pi$
$858$	0	0
$859$	17552.0	0.697167	0.348584	−	0.937278i	$-0.386663\pi$
$859$	17552.0	0.697167	0.348584	+	0.937278i	$0.386663\pi$
$860$	0	0
$861$	1260.00	0.0498730
$862$	0	0
$863$	34104.0i	1.34521i	0.740003	+	0.672604i	$0.234824\pi$
$863$	34104.0i	1.34521i	−0.740003	+	0.672604i	$0.765176\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 8391.00i	− 0.328689i
$868$	0	0
$869$	7728.00	0.301674
$870$	0	0
$871$	−653.000	−0.0254031
$872$	0	0
$873$	7569.00i	0.293439i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	46229.0i	1.77998i	0.455980	+	0.889990i	$0.349289\pi$
$877$	46229.0i	1.77998i	−0.455980	+	0.889990i	$0.650711\pi$
$878$	0	0
$879$	9042.00	0.346961
$880$	0	0
$881$	−22440.0	−0.858142	−0.429071	−	0.903271i	$-0.641159\pi$
$881$	−22440.0	−0.858142	−0.429071	+	0.903271i	$0.641159\pi$
$882$	0	0
$883$	17143.0i	0.653350i	0.945137	+	0.326675i	$0.105928\pi$
$883$	17143.0i	0.653350i	−0.945137	+	0.326675i	$0.894072\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 23626.0i	− 0.894344i	−0.894448	−	0.447172i	$-0.852431\pi$
$887$	− 23626.0i	− 0.894344i	0.894448	−	0.447172i	$-0.147569\pi$
$888$	0	0
$889$	−11880.0	−0.448192
$890$	0	0
$891$	−1134.00	−0.0426380
$892$	0	0
$893$	7790.00i	0.291918i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	138.000i	0.00513677i
$898$	0	0
$899$	1862.00	0.0690781
$900$	0	0
$901$	20976.0	0.775596
$902$	0	0
$903$	2505.00i	0.0923158i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 11268.0i	− 0.412511i	−0.978498	−	0.206256i	$-0.933872\pi$
$907$	− 11268.0i	− 0.412511i	0.978498	−	0.206256i	$-0.0661279\pi$
$908$	0	0
$909$	−4968.00	−0.181274
$910$	0	0
$911$	−10046.0	−0.365355	−0.182678	−	0.983173i	$-0.558476\pi$
$911$	−10046.0	−0.365355	−0.182678	+	0.983173i	$0.558476\pi$
$912$	0	0
$913$	1988.00i	0.0720626i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 5280.00i	− 0.190143i
$918$	0	0
$919$	15359.0	0.551302	0.275651	−	0.961258i	$-0.411107\pi$
$919$	15359.0	0.551302	0.275651	+	0.961258i	$0.411107\pi$
$920$	0	0
$921$	−16107.0	−0.576269
$922$	0	0
$923$	− 828.000i	− 0.0295276i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 2772.00i	− 0.0982141i
$928$	0	0
$929$	39790.0	1.40524	0.702620	−	0.711565i	$-0.252014\pi$
$929$	39790.0	1.40524	0.702620	+	0.711565i	$0.252014\pi$
$930$	0	0
$931$	6042.00	0.212694
$932$	0	0
$933$	14538.0i	0.510132i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	17009.0i	0.593020i	0.955030	+	0.296510i	$0.0958228\pi$
$937$	17009.0i	0.593020i	−0.955030	+	0.296510i	$0.904177\pi$
$938$	0	0
$939$	−2271.00	−0.0789258
$940$	0	0
$941$	−11674.0	−0.404422	−0.202211	−	0.979342i	$-0.564813\pi$
$941$	−11674.0	−0.404422	−0.202211	+	0.979342i	$0.564813\pi$
$942$	0	0
$943$	3864.00i	0.133435i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	6026.00i	0.206778i	0.994641	+	0.103389i	$0.0329686\pi$
$947$	6026.00i	0.206778i	−0.994641	+	0.103389i	$0.967031\pi$
$948$	0	0
$949$	−570.000	−0.0194973
$950$	0	0
$951$	22896.0	0.780708
$952$	0	0
$953$	− 14088.0i	− 0.478862i	−0.970913	−	0.239431i	$-0.923039\pi$
$953$	− 14088.0i	− 0.478862i	0.970913	−	0.239431i	$-0.0769608\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 588.000i	− 0.0198614i
$958$	0	0
$959$	−3890.00	−0.130985
$960$	0	0
$961$	−12102.0	−0.406230
$962$	0	0
$963$	− 8856.00i	− 0.296345i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 11208.0i	− 0.372725i	−0.982481	−	0.186362i	$-0.940330\pi$
$967$	− 11208.0i	− 0.372725i	0.982481	−	0.186362i	$-0.0596699\pi$
$968$	0	0
$969$	−2622.00	−0.0869255
$970$	0	0
$971$	26054.0	0.861084	0.430542	−	0.902571i	$-0.358322\pi$
$971$	26054.0	0.861084	0.430542	+	0.902571i	$0.358322\pi$
$972$	0	0
$973$	8460.00i	0.278741i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	26870.0i	0.879885i	0.898026	+	0.439942i	$0.145001\pi$
$977$	26870.0i	0.879885i	−0.898026	+	0.439942i	$0.854999\pi$
$978$	0	0
$979$	−15456.0	−0.504572
$980$	0	0
$981$	−16587.0	−0.539839
$982$	0	0
$983$	23388.0i	0.758862i	0.925220	+	0.379431i	$0.123880\pi$
$983$	23388.0i	0.758862i	−0.925220	+	0.379431i	$0.876120\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	6150.00i	0.198335i
$988$	0	0
$989$	−7682.00	−0.246990
$990$	0	0
$991$	−17345.0	−0.555986	−0.277993	−	0.960583i	$-0.589669\pi$
$991$	−17345.0	−0.555986	−0.277993	+	0.960583i	$0.589669\pi$
$992$	0	0
$993$	− 20340.0i	− 0.650021i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	25998.0i	0.825842i	0.910767	+	0.412921i	$0.135492\pi$
$997$	25998.0i	0.825842i	−0.910767	+	0.412921i	$0.864508\pi$
$998$	0	0
$999$	6966.00	0.220615

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1200.4.f.i.49.1		2
4.3	odd	2		600.4.f.e.49.2		2
5.2	odd	4		1200.4.a.g.1.1		1
5.3	odd	4		1200.4.a.bd.1.1		1
5.4	even	2	inner	1200.4.f.i.49.2		2
12.11	even	2		1800.4.f.l.649.1		2
20.3	even	4		600.4.a.e.1.1	✓	1
20.7	even	4		600.4.a.n.1.1	yes	1
20.19	odd	2		600.4.f.e.49.1		2
60.23	odd	4		1800.4.a.m.1.1		1
60.47	odd	4		1800.4.a.v.1.1		1
60.59	even	2		1800.4.f.l.649.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
600.4.a.e.1.1	✓	1	20.3	even	4
600.4.a.n.1.1	yes	1	20.7	even	4
600.4.f.e.49.1		2	20.19	odd	2
600.4.f.e.49.2		2	4.3	odd	2
1200.4.a.g.1.1		1	5.2	odd	4
1200.4.a.bd.1.1		1	5.3	odd	4
1200.4.f.i.49.1		2	1.1	even	1	trivial
1200.4.f.i.49.2		2	5.4	even	2	inner
1800.4.a.m.1.1		1	60.23	odd	4
1800.4.a.v.1.1		1	60.47	odd	4
1800.4.f.l.649.1		2	12.11	even	2
1800.4.f.l.649.2		2	60.59	even	2

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:	\( N \)	\(=\)	\( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1200.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.00000i q^{3} +5.00000i q^{7} -9.00000 q^{9} -14.0000 q^{11} +1.00000i q^{13} -46.0000i q^{17} +19.0000 q^{19} +15.0000 q^{21} +46.0000i q^{23} +27.0000i q^{27} -14.0000 q^{29} -133.000 q^{31} +42.0000i q^{33} -258.000i q^{37} +3.00000 q^{39} +84.0000 q^{41} +167.000i q^{43} +410.000i q^{47} +318.000 q^{49} -138.000 q^{51} +456.000i q^{53} -57.0000i q^{57} -194.000 q^{59} -17.0000 q^{61} -45.0000i q^{63} +653.000i q^{67} +138.000 q^{69} -828.000 q^{71} +570.000i q^{73} -70.0000i q^{77} -552.000 q^{79} +81.0000 q^{81} -142.000i q^{83} +42.0000i q^{87} +1104.00 q^{89} -5.00000 q^{91} +399.000i q^{93} -841.000i q^{97} +126.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 18 q^{9} - 28 q^{11} + 38 q^{19} + 30 q^{21} - 28 q^{29} - 266 q^{31} + 6 q^{39} + 168 q^{41} + 636 q^{49} - 276 q^{51} - 388 q^{59} - 34 q^{61} + 276 q^{69} - 1656 q^{71} - 1104 q^{79} + 162 q^{81}+ \cdots + 252 q^{99}+O(q^{100}) \) 2 * q - 18 * q^9 - 28 * q^11 + 38 * q^19 + 30 * q^21 - 28 * q^29 - 266 * q^31 + 6 * q^39 + 168 * q^41 + 636 * q^49 - 276 * q^51 - 388 * q^59 - 34 * q^61 + 276 * q^69 - 1656 * q^71 - 1104 * q^79 + 162 * q^81 + 2208 * q^89 - 10 * q^91 + 252 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more