LMFDB - Embedded newform 1200.4.a.bn.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1200,4,Mod(1,1200)]

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, 0]))

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

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

chi := DirichletCharacter("1200.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1200 = 2^{4} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1200.a (trivial)

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:	yes
Analytic conductor:	$70.8022920069$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{41}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 10 $ x^2 - x - 10
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 15)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.70156$ of defining polynomial
Character	$\chi$	$=$	1200.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{3} +16.2094 q^{7} +9.00000 q^{9} +O(q^{10})$	$q-3.00000 q^{3} +16.2094 q^{7} +9.00000 q^{9} +40.2094 q^{11} +19.7906 q^{13} +83.0156 q^{17} +48.8375 q^{19} -48.6281 q^{21} -1.61250 q^{23} -27.0000 q^{27} -24.5344 q^{29} +12.4187 q^{31} -120.628 q^{33} +325.884 q^{37} -59.3719 q^{39} -242.419 q^{41} -367.350 q^{43} +204.544 q^{47} -80.2562 q^{49} -249.047 q^{51} -61.5281 q^{53} -146.512 q^{57} +112.209 q^{59} +477.350 q^{61} +145.884 q^{63} -558.094 q^{67} +4.83749 q^{69} -558.281 q^{71} +1011.77 q^{73} +651.769 q^{77} -1150.47 q^{79} +81.0000 q^{81} +1157.92 q^{83} +73.6032 q^{87} +96.9751 q^{89} +320.794 q^{91} -37.2562 q^{93} -1152.37 q^{97} +361.884 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{3} - 6 q^{7} + 18 q^{9}+O(q^{10}) $ 2 * q - 6 * q^3 - 6 * q^7 + 18 * q^9	$ 2 q - 6 q^{3} - 6 q^{7} + 18 q^{9} + 42 q^{11} + 78 q^{13} + 102 q^{17} - 56 q^{19} + 18 q^{21} + 48 q^{23} - 54 q^{27} - 318 q^{29} - 52 q^{31} - 126 q^{33} + 306 q^{37} - 234 q^{39} - 408 q^{41} - 120 q^{43} - 180 q^{47} + 70 q^{49} - 306 q^{51} + 402 q^{53} + 168 q^{57} + 186 q^{59} + 340 q^{61} - 54 q^{63} - 732 q^{67} - 144 q^{69} + 36 q^{71} + 1332 q^{73} + 612 q^{77} - 380 q^{79} + 162 q^{81} + 984 q^{83} + 954 q^{87} + 1116 q^{89} - 972 q^{91} + 156 q^{93} - 768 q^{97} + 378 q^{99}+O(q^{100}) $ 2 * q - 6 * q^3 - 6 * q^7 + 18 * q^9 + 42 * q^11 + 78 * q^13 + 102 * q^17 - 56 * q^19 + 18 * q^21 + 48 * q^23 - 54 * q^27 - 318 * q^29 - 52 * q^31 - 126 * q^33 + 306 * q^37 - 234 * q^39 - 408 * q^41 - 120 * q^43 - 180 * q^47 + 70 * q^49 - 306 * q^51 + 402 * q^53 + 168 * q^57 + 186 * q^59 + 340 * q^61 - 54 * q^63 - 732 * q^67 - 144 * q^69 + 36 * q^71 + 1332 * q^73 + 612 * q^77 - 380 * q^79 + 162 * q^81 + 984 * q^83 + 954 * q^87 + 1116 * q^89 - 972 * q^91 + 156 * q^93 - 768 * q^97 + 378 * q^99

Display 100 coefficients 10 coefficients

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
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	16.2094	0.875224	0.437612	−	0.899164i	$-0.355824\pi$
$7$	16.2094	0.875224	0.437612	+	0.899164i	$0.355824\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	40.2094	1.10214	0.551072	−	0.834458i	$-0.314219\pi$
$11$	40.2094	1.10214	0.551072	+	0.834458i	$0.314219\pi$
$12$	0	0
$13$	19.7906	0.422226	0.211113	−	0.977462i	$-0.432291\pi$
$13$	19.7906	0.422226	0.211113	+	0.977462i	$0.432291\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	83.0156	1.18437	0.592184	−	0.805803i	$-0.298266\pi$
$17$	83.0156	1.18437	0.592184	+	0.805803i	$0.298266\pi$
$18$	0	0
$19$	48.8375	0.589689	0.294844	−	0.955545i	$-0.404732\pi$
$19$	48.8375	0.589689	0.294844	+	0.955545i	$0.404732\pi$
$20$	0	0
$21$	−48.6281	−0.505311
$22$	0	0
$23$	−1.61250	−0.0146186	−0.00730932	−	0.999973i	$-0.502327\pi$
$23$	−1.61250	−0.0146186	−0.00730932	+	0.999973i	$0.502327\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	−24.5344	−0.157101	−0.0785504	−	0.996910i	$-0.525029\pi$
$29$	−24.5344	−0.157101	−0.0785504	+	0.996910i	$0.525029\pi$
$30$	0	0
$31$	12.4187	0.0719507	0.0359754	−	0.999353i	$-0.488546\pi$
$31$	12.4187	0.0719507	0.0359754	+	0.999353i	$0.488546\pi$
$32$	0	0
$33$	−120.628	−0.636323
$34$	0	0
$35$	0	0
$36$	0	0
$37$	325.884	1.44797	0.723987	−	0.689813i	$-0.242307\pi$
$37$	325.884	1.44797	0.723987	+	0.689813i	$0.242307\pi$
$38$	0	0
$39$	−59.3719	−0.243772
$40$	0	0
$41$	−242.419	−0.923401	−0.461701	−	0.887036i	$-0.652761\pi$
$41$	−242.419	−0.923401	−0.461701	+	0.887036i	$0.652761\pi$
$42$	0	0
$43$	−367.350	−1.30280	−0.651399	−	0.758735i	$-0.725818\pi$
$43$	−367.350	−1.30280	−0.651399	+	0.758735i	$0.725818\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	204.544	0.634804	0.317402	−	0.948291i	$-0.397190\pi$
$47$	204.544	0.634804	0.317402	+	0.948291i	$0.397190\pi$
$48$	0	0
$49$	−80.2562	−0.233983
$50$	0	0
$51$	−249.047	−0.683795
$52$	0	0
$53$	−61.5281	−0.159463	−0.0797314	−	0.996816i	$-0.525406\pi$
$53$	−61.5281	−0.159463	−0.0797314	+	0.996816i	$0.525406\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−146.512	−0.340457
$58$	0	0
$59$	112.209	0.247600	0.123800	−	0.992307i	$-0.460492\pi$
$59$	112.209	0.247600	0.123800	+	0.992307i	$0.460492\pi$
$60$	0	0
$61$	477.350	1.00194	0.500970	−	0.865464i	$-0.332977\pi$
$61$	477.350	1.00194	0.500970	+	0.865464i	$0.332977\pi$
$62$	0	0
$63$	145.884	0.291741
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−558.094	−1.01764	−0.508821	−	0.860872i	$-0.669918\pi$
$67$	−558.094	−1.01764	−0.508821	+	0.860872i	$0.669918\pi$
$68$	0	0
$69$	4.83749	0.00844008
$70$	0	0
$71$	−558.281	−0.933180	−0.466590	−	0.884474i	$-0.654518\pi$
$71$	−558.281	−0.933180	−0.466590	+	0.884474i	$0.654518\pi$
$72$	0	0
$73$	1011.77	1.62217	0.811086	−	0.584927i	$-0.198877\pi$
$73$	1011.77	1.62217	0.811086	+	0.584927i	$0.198877\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	651.769	0.964623
$78$	0	0
$79$	−1150.47	−1.63845	−0.819227	−	0.573470i	$-0.805597\pi$
$79$	−1150.47	−1.63845	−0.819227	+	0.573470i	$0.805597\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	1157.92	1.53131	0.765655	−	0.643251i	$-0.222415\pi$
$83$	1157.92	1.53131	0.765655	+	0.643251i	$0.222415\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	73.6032	0.0907022
$88$	0	0
$89$	96.9751	0.115498	0.0577491	−	0.998331i	$-0.481608\pi$
$89$	96.9751	0.115498	0.0577491	+	0.998331i	$0.481608\pi$
$90$	0	0
$91$	320.794	0.369542
$92$	0	0
$93$	−37.2562	−0.0415408
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−1152.37	−1.20625	−0.603123	−	0.797648i	$-0.706077\pi$
$97$	−1152.37	−1.20625	−0.603123	+	0.797648i	$0.706077\pi$
$98$	0	0
$99$	361.884	0.367381
$100$	0	0
$101$	−1156.49	−1.13936	−0.569679	−	0.821867i	$-0.692932\pi$
$101$	−1156.49	−1.13936	−0.569679	+	0.821867i	$0.692932\pi$
$102$	0	0
$103$	1333.70	1.27585	0.637927	−	0.770096i	$-0.279792\pi$
$103$	1333.70	1.27585	0.637927	+	0.770096i	$0.279792\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	798.263	0.721224	0.360612	−	0.932716i	$-0.382568\pi$
$107$	798.263	0.721224	0.360612	+	0.932716i	$0.382568\pi$
$108$	0	0
$109$	−985.119	−0.865663	−0.432831	−	0.901475i	$-0.642485\pi$
$109$	−985.119	−0.865663	−0.432831	+	0.901475i	$0.642485\pi$
$110$	0	0
$111$	−977.653	−0.835988
$112$	0	0
$113$	1888.25	1.57196	0.785979	−	0.618253i	$-0.212159\pi$
$113$	1888.25	1.57196	0.785979	+	0.618253i	$0.212159\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	178.116	0.140742
$118$	0	0
$119$	1345.63	1.03659
$120$	0	0
$121$	285.794	0.214721
$122$	0	0
$123$	727.256	0.533126
$124$	0	0
$125$	0	0
$126$	0	0
$127$	620.859	0.433798	0.216899	−	0.976194i	$-0.430406\pi$
$127$	620.859	0.433798	0.216899	+	0.976194i	$0.430406\pi$
$128$	0	0
$129$	1102.05	0.752171
$130$	0	0
$131$	2588.35	1.72630	0.863151	−	0.504947i	$-0.168488\pi$
$131$	2588.35	1.72630	0.863151	+	0.504947i	$0.168488\pi$
$132$	0	0
$133$	791.625	0.516110
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−1656.29	−1.03289	−0.516447	−	0.856319i	$-0.672746\pi$
$137$	−1656.29	−1.03289	−0.516447	+	0.856319i	$0.672746\pi$
$138$	0	0
$139$	153.256	0.0935182	0.0467591	−	0.998906i	$-0.485111\pi$
$139$	153.256	0.0935182	0.0467591	+	0.998906i	$0.485111\pi$
$140$	0	0
$141$	−613.631	−0.366504
$142$	0	0
$143$	795.769	0.465353
$144$	0	0
$145$	0	0
$146$	0	0
$147$	240.769	0.135090
$148$	0	0
$149$	1483.38	0.815591	0.407795	−	0.913073i	$-0.366298\pi$
$149$	1483.38	0.815591	0.407795	+	0.913073i	$0.366298\pi$
$150$	0	0
$151$	394.281	0.212491	0.106246	−	0.994340i	$-0.466117\pi$
$151$	394.281	0.212491	0.106246	+	0.994340i	$0.466117\pi$
$152$	0	0
$153$	747.141	0.394789
$154$	0	0
$155$	0	0
$156$	0	0
$157$	1727.05	0.877922	0.438961	−	0.898506i	$-0.355347\pi$
$157$	1727.05	0.877922	0.438961	+	0.898506i	$0.355347\pi$
$158$	0	0
$159$	184.584	0.0920659
$160$	0	0
$161$	−26.1376	−0.0127946
$162$	0	0
$163$	2034.28	0.977529	0.488764	−	0.872416i	$-0.337448\pi$
$163$	2034.28	0.977529	0.488764	+	0.872416i	$0.337448\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−192.900	−0.0893835	−0.0446918	−	0.999001i	$-0.514231\pi$
$167$	−192.900	−0.0893835	−0.0446918	+	0.999001i	$0.514231\pi$
$168$	0	0
$169$	−1805.33	−0.821726
$170$	0	0
$171$	439.537	0.196563
$172$	0	0
$173$	−1239.91	−0.544905	−0.272452	−	0.962169i	$-0.587835\pi$
$173$	−1239.91	−0.544905	−0.272452	+	0.962169i	$0.587835\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−336.628	−0.142952
$178$	0	0
$179$	2636.86	1.10105	0.550525	−	0.834818i	$-0.314427\pi$
$179$	2636.86	1.10105	0.550525	+	0.834818i	$0.314427\pi$
$180$	0	0
$181$	3317.58	1.36240	0.681199	−	0.732099i	$-0.261459\pi$
$181$	3317.58	1.36240	0.681199	+	0.732099i	$0.261459\pi$
$182$	0	0
$183$	−1432.05	−0.578471
$184$	0	0
$185$	0	0
$186$	0	0
$187$	3338.01	1.30534
$188$	0	0
$189$	−437.653	−0.168437
$190$	0	0
$191$	624.506	0.236585	0.118292	−	0.992979i	$-0.462258\pi$
$191$	624.506	0.236585	0.118292	+	0.992979i	$0.462258\pi$
$192$	0	0
$193$	−436.144	−0.162665	−0.0813324	−	0.996687i	$-0.525918\pi$
$193$	−436.144	−0.162665	−0.0813324	+	0.996687i	$0.525918\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	3355.81	1.21366	0.606831	−	0.794831i	$-0.292440\pi$
$197$	3355.81	1.21366	0.606831	+	0.794831i	$0.292440\pi$
$198$	0	0
$199$	−3799.77	−1.35356	−0.676780	−	0.736185i	$-0.736625\pi$
$199$	−3799.77	−1.35356	−0.676780	+	0.736185i	$0.736625\pi$
$200$	0	0
$201$	1674.28	0.587536
$202$	0	0
$203$	−397.687	−0.137498
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−14.5125	−0.00487288
$208$	0	0
$209$	1963.72	0.649922
$210$	0	0
$211$	−2365.27	−0.771715	−0.385857	−	0.922558i	$-0.626094\pi$
$211$	−2365.27	−0.771715	−0.385857	+	0.922558i	$0.626094\pi$
$212$	0	0
$213$	1674.84	0.538772
$214$	0	0
$215$	0	0
$216$	0	0
$217$	201.300	0.0629730
$218$	0	0
$219$	−3035.31	−0.936562
$220$	0	0
$221$	1642.93	0.500070
$222$	0	0
$223$	3328.58	0.999545	0.499772	−	0.866157i	$-0.333417\pi$
$223$	3328.58	0.999545	0.499772	+	0.866157i	$0.333417\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	527.100	0.154118	0.0770592	−	0.997027i	$-0.475447\pi$
$227$	527.100	0.154118	0.0770592	+	0.997027i	$0.475447\pi$
$228$	0	0
$229$	2566.06	0.740479	0.370240	−	0.928936i	$-0.379276\pi$
$229$	2566.06	0.740479	0.370240	+	0.928936i	$0.379276\pi$
$230$	0	0
$231$	−1955.31	−0.556925
$232$	0	0
$233$	5534.99	1.55626	0.778132	−	0.628101i	$-0.216168\pi$
$233$	5534.99	1.55626	0.778132	+	0.628101i	$0.216168\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	3451.41	0.945962
$238$	0	0
$239$	−1010.01	−0.273355	−0.136678	−	0.990616i	$-0.543642\pi$
$239$	−1010.01	−0.273355	−0.136678	+	0.990616i	$0.543642\pi$
$240$	0	0
$241$	−4074.29	−1.08900	−0.544498	−	0.838762i	$-0.683280\pi$
$241$	−4074.29	−1.08900	−0.544498	+	0.838762i	$0.683280\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	966.525	0.248982
$248$	0	0
$249$	−3473.77	−0.884103
$250$	0	0
$251$	−1773.98	−0.446107	−0.223054	−	0.974806i	$-0.571602\pi$
$251$	−1773.98	−0.446107	−0.223054	+	0.974806i	$0.571602\pi$
$252$	0	0
$253$	−64.8375	−0.0161119
$254$	0	0
$255$	0	0
$256$	0	0
$257$	662.784	0.160869	0.0804345	−	0.996760i	$-0.474369\pi$
$257$	662.784	0.160869	0.0804345	+	0.996760i	$0.474369\pi$
$258$	0	0
$259$	5282.38	1.26730
$260$	0	0
$261$	−220.810	−0.0523669
$262$	0	0
$263$	−712.312	−0.167008	−0.0835039	−	0.996507i	$-0.526611\pi$
$263$	−712.312	−0.167008	−0.0835039	+	0.996507i	$0.526611\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−290.925	−0.0666829
$268$	0	0
$269$	−3136.41	−0.710894	−0.355447	−	0.934696i	$-0.615671\pi$
$269$	−3136.41	−0.710894	−0.355447	+	0.934696i	$0.615671\pi$
$270$	0	0
$271$	2275.69	0.510105	0.255053	−	0.966927i	$-0.417907\pi$
$271$	2275.69	0.510105	0.255053	+	0.966927i	$0.417907\pi$
$272$	0	0
$273$	−962.381	−0.213355
$274$	0	0
$275$	0	0
$276$	0	0
$277$	5171.00	1.12164	0.560821	−	0.827937i	$-0.310485\pi$
$277$	5171.00	1.12164	0.560821	+	0.827937i	$0.310485\pi$
$278$	0	0
$279$	111.769	0.0239836
$280$	0	0
$281$	2240.14	0.475571	0.237785	−	0.971318i	$-0.423578\pi$
$281$	2240.14	0.475571	0.237785	+	0.971318i	$0.423578\pi$
$282$	0	0
$283$	−225.244	−0.0473123	−0.0236561	−	0.999720i	$-0.507531\pi$
$283$	−225.244	−0.0473123	−0.0236561	+	0.999720i	$0.507531\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−3929.46	−0.808183
$288$	0	0
$289$	1978.59	0.402726
$290$	0	0
$291$	3457.12	0.696427
$292$	0	0
$293$	1139.86	0.227274	0.113637	−	0.993522i	$-0.463750\pi$
$293$	1139.86	0.227274	0.113637	+	0.993522i	$0.463750\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−1085.65	−0.212108
$298$	0	0
$299$	−31.9123	−0.00617237
$300$	0	0
$301$	−5954.51	−1.14024
$302$	0	0
$303$	3469.47	0.657808
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−5244.86	−0.975049	−0.487525	−	0.873109i	$-0.662100\pi$
$307$	−5244.86	−0.975049	−0.487525	+	0.873109i	$0.662100\pi$
$308$	0	0
$309$	−4001.09	−0.736615
$310$	0	0
$311$	5188.26	0.945977	0.472989	−	0.881068i	$-0.343175\pi$
$311$	5188.26	0.945977	0.472989	+	0.881068i	$0.343175\pi$
$312$	0	0
$313$	486.656	0.0878832	0.0439416	−	0.999034i	$-0.486008\pi$
$313$	486.656	0.0878832	0.0439416	+	0.999034i	$0.486008\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−4218.87	−0.747493	−0.373747	−	0.927531i	$-0.621927\pi$
$317$	−4218.87	−0.747493	−0.373747	+	0.927531i	$0.621927\pi$
$318$	0	0
$319$	−986.512	−0.173148
$320$	0	0
$321$	−2394.79	−0.416399
$322$	0	0
$323$	4054.27	0.698408
$324$	0	0
$325$	0	0
$326$	0	0
$327$	2955.36	0.499791
$328$	0	0
$329$	3315.53	0.555595
$330$	0	0
$331$	−7439.94	−1.23546	−0.617728	−	0.786392i	$-0.711947\pi$
$331$	−7439.94	−1.23546	−0.617728	+	0.786392i	$0.711947\pi$
$332$	0	0
$333$	2932.96	0.482658
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−6555.39	−1.05963	−0.529815	−	0.848113i	$-0.677739\pi$
$337$	−6555.39	−1.05963	−0.529815	+	0.848113i	$0.677739\pi$
$338$	0	0
$339$	−5664.74	−0.907571
$340$	0	0
$341$	499.350	0.0793000
$342$	0	0
$343$	−6860.72	−1.08001
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−1950.56	−0.301763	−0.150881	−	0.988552i	$-0.548211\pi$
$347$	−1950.56	−0.301763	−0.150881	+	0.988552i	$0.548211\pi$
$348$	0	0
$349$	−1426.74	−0.218830	−0.109415	−	0.993996i	$-0.534898\pi$
$349$	−1426.74	−0.218830	−0.109415	+	0.993996i	$0.534898\pi$
$350$	0	0
$351$	−534.347	−0.0812573
$352$	0	0
$353$	−7078.96	−1.06735	−0.533676	−	0.845689i	$-0.679190\pi$
$353$	−7078.96	−1.06735	−0.533676	+	0.845689i	$0.679190\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−4036.89	−0.598474
$358$	0	0
$359$	5409.79	0.795314	0.397657	−	0.917534i	$-0.369823\pi$
$359$	5409.79	0.795314	0.397657	+	0.917534i	$0.369823\pi$
$360$	0	0
$361$	−4473.90	−0.652267
$362$	0	0
$363$	−857.381	−0.123969
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−4940.09	−0.702645	−0.351322	−	0.936255i	$-0.614268\pi$
$367$	−4940.09	−0.702645	−0.351322	+	0.936255i	$0.614268\pi$
$368$	0	0
$369$	−2181.77	−0.307800
$370$	0	0
$371$	−997.332	−0.139566
$372$	0	0
$373$	12891.9	1.78959	0.894797	−	0.446473i	$-0.147320\pi$
$373$	12891.9	1.78959	0.894797	+	0.446473i	$0.147320\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−485.551	−0.0663320
$378$	0	0
$379$	9475.15	1.28418	0.642092	−	0.766627i	$-0.278067\pi$
$379$	9475.15	1.28418	0.642092	+	0.766627i	$0.278067\pi$
$380$	0	0
$381$	−1862.58	−0.250453
$382$	0	0
$383$	5800.97	0.773931	0.386966	−	0.922094i	$-0.373523\pi$
$383$	5800.97	0.773931	0.386966	+	0.922094i	$0.373523\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−3306.15	−0.434266
$388$	0	0
$389$	13779.7	1.79603	0.898016	−	0.439962i	$-0.145008\pi$
$389$	13779.7	1.79603	0.898016	+	0.439962i	$0.145008\pi$
$390$	0	0
$391$	−133.862	−0.0173138
$392$	0	0
$393$	−7765.06	−0.996680
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−2816.46	−0.356056	−0.178028	−	0.984025i	$-0.556972\pi$
$397$	−2816.46	−0.356056	−0.178028	+	0.984025i	$0.556972\pi$
$398$	0	0
$399$	−2374.88	−0.297976
$400$	0	0
$401$	11986.4	1.49270	0.746352	−	0.665551i	$-0.231804\pi$
$401$	11986.4	1.49270	0.746352	+	0.665551i	$0.231804\pi$
$402$	0	0
$403$	245.775	0.0303794
$404$	0	0
$405$	0	0
$406$	0	0
$407$	13103.6	1.59588
$408$	0	0
$409$	−3339.07	−0.403683	−0.201841	−	0.979418i	$-0.564693\pi$
$409$	−3339.07	−0.403683	−0.201841	+	0.979418i	$0.564693\pi$
$410$	0	0
$411$	4968.87	0.596342
$412$	0	0
$413$	1818.84	0.216706
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−459.769	−0.0539927
$418$	0	0
$419$	−1688.52	−0.196873	−0.0984363	−	0.995143i	$-0.531384\pi$
$419$	−1688.52	−0.196873	−0.0984363	+	0.995143i	$0.531384\pi$
$420$	0	0
$421$	−2664.27	−0.308429	−0.154214	−	0.988037i	$-0.549285\pi$
$421$	−2664.27	−0.308429	−0.154214	+	0.988037i	$0.549285\pi$
$422$	0	0
$423$	1840.89	0.211601
$424$	0	0
$425$	0	0
$426$	0	0
$427$	7737.54	0.876923
$428$	0	0
$429$	−2387.31	−0.268672
$430$	0	0
$431$	−12266.0	−1.37084	−0.685420	−	0.728148i	$-0.740381\pi$
$431$	−12266.0	−1.37084	−0.685420	+	0.728148i	$0.740381\pi$
$432$	0	0
$433$	−15647.3	−1.73664	−0.868318	−	0.496008i	$-0.834799\pi$
$433$	−15647.3	−1.73664	−0.868318	+	0.496008i	$0.834799\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−78.7503	−0.00862045
$438$	0	0
$439$	−16131.0	−1.75373	−0.876867	−	0.480733i	$-0.840371\pi$
$439$	−16131.0	−1.75373	−0.876867	+	0.480733i	$0.840371\pi$
$440$	0	0
$441$	−722.306	−0.0779944
$442$	0	0
$443$	−10053.7	−1.07825	−0.539127	−	0.842225i	$-0.681246\pi$
$443$	−10053.7	−1.07825	−0.539127	+	0.842225i	$0.681246\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−4450.13	−0.470882
$448$	0	0
$449$	7477.71	0.785957	0.392979	−	0.919548i	$-0.371445\pi$
$449$	7477.71	0.785957	0.392979	+	0.919548i	$0.371445\pi$
$450$	0	0
$451$	−9747.51	−1.01772
$452$	0	0
$453$	−1182.84	−0.122682
$454$	0	0
$455$	0	0
$456$	0	0
$457$	1363.46	0.139562	0.0697812	−	0.997562i	$-0.477770\pi$
$457$	1363.46	0.139562	0.0697812	+	0.997562i	$0.477770\pi$
$458$	0	0
$459$	−2241.42	−0.227932
$460$	0	0
$461$	5276.77	0.533109	0.266555	−	0.963820i	$-0.414115\pi$
$461$	5276.77	0.533109	0.266555	+	0.963820i	$0.414115\pi$
$462$	0	0
$463$	−5740.02	−0.576159	−0.288079	−	0.957607i	$-0.593017\pi$
$463$	−5740.02	−0.576159	−0.288079	+	0.957607i	$0.593017\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−6233.36	−0.617657	−0.308828	−	0.951118i	$-0.599937\pi$
$467$	−6233.36	−0.617657	−0.308828	+	0.951118i	$0.599937\pi$
$468$	0	0
$469$	−9046.35	−0.890664
$470$	0	0
$471$	−5181.16	−0.506869
$472$	0	0
$473$	−14770.9	−1.43587
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−553.753	−0.0531543
$478$	0	0
$479$	19688.2	1.87803	0.939013	−	0.343881i	$-0.111742\pi$
$479$	19688.2	1.87803	0.939013	+	0.343881i	$0.111742\pi$
$480$	0	0
$481$	6449.46	0.611372
$482$	0	0
$483$	78.4127	0.00738696
$484$	0	0
$485$	0	0
$486$	0	0
$487$	3955.08	0.368012	0.184006	−	0.982925i	$-0.441093\pi$
$487$	3955.08	0.368012	0.184006	+	0.982925i	$0.441093\pi$
$488$	0	0
$489$	−6102.84	−0.564377
$490$	0	0
$491$	13893.5	1.27699	0.638497	−	0.769624i	$-0.279557\pi$
$491$	13893.5	1.27699	0.638497	+	0.769624i	$0.279557\pi$
$492$	0	0
$493$	−2036.74	−0.186065
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−9049.39	−0.816741
$498$	0	0
$499$	−13523.7	−1.21324	−0.606618	−	0.794993i	$-0.707474\pi$
$499$	−13523.7	−1.21324	−0.606618	+	0.794993i	$0.707474\pi$
$500$	0	0
$501$	578.700	0.0516056
$502$	0	0
$503$	−13135.4	−1.16437	−0.582184	−	0.813057i	$-0.697802\pi$
$503$	−13135.4	−1.16437	−0.582184	+	0.813057i	$0.697802\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	5415.99	0.474423
$508$	0	0
$509$	−2222.71	−0.193556	−0.0967778	−	0.995306i	$-0.530854\pi$
$509$	−2222.71	−0.193556	−0.0967778	+	0.995306i	$0.530854\pi$
$510$	0	0
$511$	16400.1	1.41976
$512$	0	0
$513$	−1318.61	−0.113486
$514$	0	0
$515$	0	0
$516$	0	0
$517$	8224.57	0.699645
$518$	0	0
$519$	3719.73	0.314601
$520$	0	0
$521$	−4916.42	−0.413421	−0.206710	−	0.978402i	$-0.566276\pi$
$521$	−4916.42	−0.413421	−0.206710	+	0.978402i	$0.566276\pi$
$522$	0	0
$523$	−17743.4	−1.48349	−0.741746	−	0.670681i	$-0.766002\pi$
$523$	−17743.4	−1.48349	−0.741746	+	0.670681i	$0.766002\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1030.95	0.0852161
$528$	0	0
$529$	−12164.4	−0.999786
$530$	0	0
$531$	1009.88	0.0825334
$532$	0	0
$533$	−4797.62	−0.389884
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−7910.58	−0.635692
$538$	0	0
$539$	−3227.05	−0.257883
$540$	0	0
$541$	−12671.3	−1.00699	−0.503495	−	0.863998i	$-0.667953\pi$
$541$	−12671.3	−1.00699	−0.503495	+	0.863998i	$0.667953\pi$
$542$	0	0
$543$	−9952.74	−0.786580
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−5250.90	−0.410443	−0.205221	−	0.978716i	$-0.565791\pi$
$547$	−5250.90	−0.410443	−0.205221	+	0.978716i	$0.565791\pi$
$548$	0	0
$549$	4296.15	0.333980
$550$	0	0
$551$	−1198.20	−0.0926406
$552$	0	0
$553$	−18648.4	−1.43401
$554$	0	0
$555$	0	0
$556$	0	0
$557$	25830.2	1.96492	0.982462	−	0.186465i	$-0.0597032\pi$
$557$	25830.2	1.96492	0.982462	+	0.186465i	$0.0597032\pi$
$558$	0	0
$559$	−7270.09	−0.550075
$560$	0	0
$561$	−10014.0	−0.753640
$562$	0	0
$563$	−2021.14	−0.151298	−0.0756490	−	0.997135i	$-0.524103\pi$
$563$	−2021.14	−0.151298	−0.0756490	+	0.997135i	$0.524103\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	1312.96	0.0972471
$568$	0	0
$569$	8706.51	0.641469	0.320734	−	0.947169i	$-0.396070\pi$
$569$	8706.51	0.641469	0.320734	+	0.947169i	$0.396070\pi$
$570$	0	0
$571$	12194.5	0.893740	0.446870	−	0.894599i	$-0.352539\pi$
$571$	12194.5	0.893740	0.446870	+	0.894599i	$0.352539\pi$
$572$	0	0
$573$	−1873.52	−0.136592
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−15264.0	−1.10130	−0.550649	−	0.834737i	$-0.685620\pi$
$577$	−15264.0	−1.10130	−0.550649	+	0.834737i	$0.685620\pi$
$578$	0	0
$579$	1308.43	0.0939146
$580$	0	0
$581$	18769.2	1.34024
$582$	0	0
$583$	−2474.01	−0.175751
$584$	0	0
$585$	0	0
$586$	0	0
$587$	8456.89	0.594639	0.297319	−	0.954778i	$-0.403907\pi$
$587$	8456.89	0.594639	0.297319	+	0.954778i	$0.403907\pi$
$588$	0	0
$589$	606.500	0.0424285
$590$	0	0
$591$	−10067.4	−0.700708
$592$	0	0
$593$	−1225.23	−0.0848467	−0.0424234	−	0.999100i	$-0.513508\pi$
$593$	−1225.23	−0.0848467	−0.0424234	+	0.999100i	$0.513508\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	11399.3	0.781478
$598$	0	0
$599$	16060.0	1.09548	0.547741	−	0.836648i	$-0.315488\pi$
$599$	16060.0	1.09548	0.547741	+	0.836648i	$0.315488\pi$
$600$	0	0
$601$	9699.93	0.658350	0.329175	−	0.944269i	$-0.393229\pi$
$601$	9699.93	0.658350	0.329175	+	0.944269i	$0.393229\pi$
$602$	0	0
$603$	−5022.84	−0.339214
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−23661.2	−1.58217	−0.791087	−	0.611703i	$-0.790485\pi$
$607$	−23661.2	−1.58217	−0.791087	+	0.611703i	$0.790485\pi$
$608$	0	0
$609$	1193.06	0.0793847
$610$	0	0
$611$	4048.05	0.268030
$612$	0	0
$613$	−8085.63	−0.532749	−0.266375	−	0.963870i	$-0.585826\pi$
$613$	−8085.63	−0.532749	−0.266375	+	0.963870i	$0.585826\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−11035.1	−0.720029	−0.360014	−	0.932947i	$-0.617228\pi$
$617$	−11035.1	−0.720029	−0.360014	+	0.932947i	$0.617228\pi$
$618$	0	0
$619$	16826.3	1.09258	0.546290	−	0.837596i	$-0.316040\pi$
$619$	16826.3	1.09258	0.546290	+	0.837596i	$0.316040\pi$
$620$	0	0
$621$	43.5374	0.00281336
$622$	0	0
$623$	1571.90	0.101087
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−5891.17	−0.375233
$628$	0	0
$629$	27053.5	1.71493
$630$	0	0
$631$	3705.91	0.233803	0.116902	−	0.993144i	$-0.462704\pi$
$631$	3705.91	0.233803	0.116902	+	0.993144i	$0.462704\pi$
$632$	0	0
$633$	7095.81	0.445550
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−1588.32	−0.0987937
$638$	0	0
$639$	−5024.53	−0.311060
$640$	0	0
$641$	−24597.4	−1.51566	−0.757829	−	0.652453i	$-0.773740\pi$
$641$	−24597.4	−1.51566	−0.757829	+	0.652453i	$0.773740\pi$
$642$	0	0
$643$	21479.5	1.31737	0.658685	−	0.752419i	$-0.271113\pi$
$643$	21479.5	1.31737	0.658685	+	0.752419i	$0.271113\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−27119.7	−1.64789	−0.823946	−	0.566668i	$-0.808232\pi$
$647$	−27119.7	−1.64789	−0.823946	+	0.566668i	$0.808232\pi$
$648$	0	0
$649$	4511.87	0.272891
$650$	0	0
$651$	−603.900	−0.0363575
$652$	0	0
$653$	18476.4	1.10725	0.553627	−	0.832765i	$-0.313243\pi$
$653$	18476.4	1.10725	0.553627	+	0.832765i	$0.313243\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	9105.92	0.540724
$658$	0	0
$659$	19273.5	1.13928	0.569641	−	0.821894i	$-0.307082\pi$
$659$	19273.5	1.13928	0.569641	+	0.821894i	$0.307082\pi$
$660$	0	0
$661$	25605.3	1.50670	0.753352	−	0.657618i	$-0.228436\pi$
$661$	25605.3	1.50670	0.753352	+	0.657618i	$0.228436\pi$
$662$	0	0
$663$	−4928.79	−0.288716
$664$	0	0
$665$	0	0
$666$	0	0
$667$	39.5616	0.00229660
$668$	0	0
$669$	−9985.75	−0.577087
$670$	0	0
$671$	19193.9	1.10428
$672$	0	0
$673$	7855.52	0.449938	0.224969	−	0.974366i	$-0.427772\pi$
$673$	7855.52	0.449938	0.224969	+	0.974366i	$0.427772\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−6763.09	−0.383939	−0.191970	−	0.981401i	$-0.561488\pi$
$677$	−6763.09	−0.383939	−0.191970	+	0.981401i	$0.561488\pi$
$678$	0	0
$679$	−18679.3	−1.05574
$680$	0	0
$681$	−1581.30	−0.0889803
$682$	0	0
$683$	−15608.6	−0.874447	−0.437224	−	0.899353i	$-0.644038\pi$
$683$	−15608.6	−0.874447	−0.437224	+	0.899353i	$0.644038\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−7698.17	−0.427516
$688$	0	0
$689$	−1217.68	−0.0673293
$690$	0	0
$691$	−6203.15	−0.341504	−0.170752	−	0.985314i	$-0.554620\pi$
$691$	−6203.15	−0.341504	−0.170752	+	0.985314i	$0.554620\pi$
$692$	0	0
$693$	5865.92	0.321541
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−20124.5	−1.09365
$698$	0	0
$699$	−16605.0	−0.898509
$700$	0	0
$701$	−16507.9	−0.889435	−0.444718	−	0.895671i	$-0.646696\pi$
$701$	−16507.9	−0.889435	−0.444718	+	0.895671i	$0.646696\pi$
$702$	0	0
$703$	15915.4	0.853854
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−18746.0	−0.997193
$708$	0	0
$709$	−25539.6	−1.35283	−0.676416	−	0.736520i	$-0.736468\pi$
$709$	−25539.6	−1.35283	−0.676416	+	0.736520i	$0.736468\pi$
$710$	0	0
$711$	−10354.2	−0.546151
$712$	0	0
$713$	−20.0252	−0.00105182
$714$	0	0
$715$	0	0
$716$	0	0
$717$	3030.02	0.157822
$718$	0	0
$719$	7353.45	0.381415	0.190708	−	0.981647i	$-0.438922\pi$
$719$	7353.45	0.381415	0.190708	+	0.981647i	$0.438922\pi$
$720$	0	0
$721$	21618.4	1.11666
$722$	0	0
$723$	12222.9	0.628732
$724$	0	0
$725$	0	0
$726$	0	0
$727$	21696.5	1.10685	0.553424	−	0.832900i	$-0.313321\pi$
$727$	21696.5	1.10685	0.553424	+	0.832900i	$0.313321\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−30495.8	−1.54299
$732$	0	0
$733$	90.2714	0.00454877	0.00227439	−	0.999997i	$-0.499276\pi$
$733$	90.2714	0.00454877	0.00227439	+	0.999997i	$0.499276\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−22440.6	−1.12159
$738$	0	0
$739$	−14273.1	−0.710479	−0.355239	−	0.934775i	$-0.615601\pi$
$739$	−14273.1	−0.710479	−0.355239	+	0.934775i	$0.615601\pi$
$740$	0	0
$741$	−2899.57	−0.143750
$742$	0	0
$743$	−15866.6	−0.783429	−0.391715	−	0.920087i	$-0.628118\pi$
$743$	−15866.6	−0.783429	−0.391715	+	0.920087i	$0.628118\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	10421.3	0.510437
$748$	0	0
$749$	12939.3	0.631232
$750$	0	0
$751$	−26776.9	−1.30107	−0.650534	−	0.759477i	$-0.725455\pi$
$751$	−26776.9	−1.30107	−0.650534	+	0.759477i	$0.725455\pi$
$752$	0	0
$753$	5321.95	0.257560
$754$	0	0
$755$	0	0
$756$	0	0
$757$	30478.0	1.46333	0.731666	−	0.681663i	$-0.238743\pi$
$757$	30478.0	1.46333	0.731666	+	0.681663i	$0.238743\pi$
$758$	0	0
$759$	194.512	0.00930218
$760$	0	0
$761$	29104.7	1.38639	0.693195	−	0.720750i	$-0.256202\pi$
$761$	29104.7	1.38639	0.693195	+	0.720750i	$0.256202\pi$
$762$	0	0
$763$	−15968.2	−0.757649
$764$	0	0
$765$	0	0
$766$	0	0
$767$	2220.69	0.104543
$768$	0	0
$769$	4170.65	0.195575	0.0977876	−	0.995207i	$-0.468823\pi$
$769$	4170.65	0.195575	0.0977876	+	0.995207i	$0.468823\pi$
$770$	0	0
$771$	−1988.35	−0.0928778
$772$	0	0
$773$	−17738.5	−0.825367	−0.412684	−	0.910874i	$-0.635409\pi$
$773$	−17738.5	−0.825367	−0.412684	+	0.910874i	$0.635409\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−15847.1	−0.731677
$778$	0	0
$779$	−11839.1	−0.544519
$780$	0	0
$781$	−22448.1	−1.02850
$782$	0	0
$783$	662.429	0.0302341
$784$	0	0
$785$	0	0
$786$	0	0
$787$	3807.92	0.172475	0.0862374	−	0.996275i	$-0.472516\pi$
$787$	3807.92	0.172475	0.0862374	+	0.996275i	$0.472516\pi$
$788$	0	0
$789$	2136.94	0.0964220
$790$	0	0
$791$	30607.3	1.37582
$792$	0	0
$793$	9447.06	0.423045
$794$	0	0
$795$	0	0
$796$	0	0
$797$	23840.3	1.05956	0.529779	−	0.848136i	$-0.322275\pi$
$797$	23840.3	1.05956	0.529779	+	0.848136i	$0.322275\pi$
$798$	0	0
$799$	16980.3	0.751841
$800$	0	0
$801$	872.775	0.0384994
$802$	0	0
$803$	40682.6	1.78787
$804$	0	0
$805$	0	0
$806$	0	0
$807$	9409.24	0.410435
$808$	0	0
$809$	−1984.22	−0.0862316	−0.0431158	−	0.999070i	$-0.513728\pi$
$809$	−1984.22	−0.0862316	−0.0431158	+	0.999070i	$0.513728\pi$
$810$	0	0
$811$	9713.78	0.420588	0.210294	−	0.977638i	$-0.432558\pi$
$811$	9713.78	0.420588	0.210294	+	0.977638i	$0.432558\pi$
$812$	0	0
$813$	−6827.08	−0.294509
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−17940.5	−0.768246
$818$	0	0
$819$	2887.14	0.123181
$820$	0	0
$821$	−19235.4	−0.817686	−0.408843	−	0.912605i	$-0.634068\pi$
$821$	−19235.4	−0.817686	−0.408843	+	0.912605i	$0.634068\pi$
$822$	0	0
$823$	12717.6	0.538650	0.269325	−	0.963049i	$-0.413199\pi$
$823$	12717.6	0.538650	0.269325	+	0.963049i	$0.413199\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	6744.75	0.283601	0.141800	−	0.989895i	$-0.454711\pi$
$827$	6744.75	0.283601	0.141800	+	0.989895i	$0.454711\pi$
$828$	0	0
$829$	3404.22	0.142622	0.0713108	−	0.997454i	$-0.477282\pi$
$829$	3404.22	0.142622	0.0713108	+	0.997454i	$0.477282\pi$
$830$	0	0
$831$	−15513.0	−0.647581
$832$	0	0
$833$	−6662.52	−0.277122
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−335.306	−0.0138469
$838$	0	0
$839$	21361.9	0.879015	0.439508	−	0.898239i	$-0.355153\pi$
$839$	21361.9	0.879015	0.439508	+	0.898239i	$0.355153\pi$
$840$	0	0
$841$	−23787.1	−0.975319
$842$	0	0
$843$	−6720.41	−0.274571
$844$	0	0
$845$	0	0
$846$	0	0
$847$	4632.54	0.187929
$848$	0	0
$849$	675.732	0.0273158
$850$	0	0
$851$	−525.488	−0.0211674
$852$	0	0
$853$	−10728.9	−0.430657	−0.215328	−	0.976542i	$-0.569082\pi$
$853$	−10728.9	−0.430657	−0.215328	+	0.976542i	$0.569082\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−42895.2	−1.70977	−0.854885	−	0.518817i	$-0.826373\pi$
$857$	−42895.2	−1.70977	−0.854885	+	0.518817i	$0.826373\pi$
$858$	0	0
$859$	−35530.5	−1.41127	−0.705637	−	0.708574i	$-0.749339\pi$
$859$	−35530.5	−1.41127	−0.705637	+	0.708574i	$0.749339\pi$
$860$	0	0
$861$	11788.4	0.466605
$862$	0	0
$863$	−5704.35	−0.225004	−0.112502	−	0.993652i	$-0.535886\pi$
$863$	−5704.35	−0.225004	−0.112502	+	0.993652i	$0.535886\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−5935.78	−0.232514
$868$	0	0
$869$	−46259.6	−1.80581
$870$	0	0
$871$	−11045.0	−0.429674
$872$	0	0
$873$	−10371.4	−0.402082
$874$	0	0
$875$	0	0
$876$	0	0
$877$	50249.0	1.93476	0.967382	−	0.253324i	$-0.0815237\pi$
$877$	50249.0	1.93476	0.967382	+	0.253324i	$0.0815237\pi$
$878$	0	0
$879$	−3419.58	−0.131217
$880$	0	0
$881$	−26864.5	−1.02734	−0.513672	−	0.857987i	$-0.671715\pi$
$881$	−26864.5	−1.02734	−0.513672	+	0.857987i	$0.671715\pi$
$882$	0	0
$883$	−18942.1	−0.721918	−0.360959	−	0.932582i	$-0.617551\pi$
$883$	−18942.1	−0.721918	−0.360959	+	0.932582i	$0.617551\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	25344.8	0.959409	0.479705	−	0.877430i	$-0.340744\pi$
$887$	25344.8	0.959409	0.479705	+	0.877430i	$0.340744\pi$
$888$	0	0
$889$	10063.7	0.379670
$890$	0	0
$891$	3256.96	0.122460
$892$	0	0
$893$	9989.40	0.374337
$894$	0	0
$895$	0	0
$896$	0	0
$897$	95.7370	0.00356362
$898$	0	0
$899$	−304.686	−0.0113035
$900$	0	0
$901$	−5107.79	−0.188863
$902$	0	0
$903$	17863.5	0.658318
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−4800.11	−0.175728	−0.0878639	−	0.996132i	$-0.528004\pi$
$907$	−4800.11	−0.175728	−0.0878639	+	0.996132i	$0.528004\pi$
$908$	0	0
$909$	−10408.4	−0.379786
$910$	0	0
$911$	25731.7	0.935819	0.467909	−	0.883776i	$-0.345007\pi$
$911$	25731.7	0.935819	0.467909	+	0.883776i	$0.345007\pi$
$912$	0	0
$913$	46559.4	1.68772
$914$	0	0
$915$	0	0
$916$	0	0
$917$	41955.6	1.51090
$918$	0	0
$919$	12751.9	0.457722	0.228861	−	0.973459i	$-0.426500\pi$
$919$	12751.9	0.457722	0.228861	+	0.973459i	$0.426500\pi$
$920$	0	0
$921$	15734.6	0.562945
$922$	0	0
$923$	−11048.7	−0.394012
$924$	0	0
$925$	0	0
$926$	0	0
$927$	12003.3	0.425285
$928$	0	0
$929$	15557.8	0.549444	0.274722	−	0.961524i	$-0.411414\pi$
$929$	15557.8	0.549444	0.274722	+	0.961524i	$0.411414\pi$
$930$	0	0
$931$	−3919.51	−0.137977
$932$	0	0
$933$	−15564.8	−0.546160
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−23858.0	−0.831811	−0.415905	−	0.909408i	$-0.636535\pi$
$937$	−23858.0	−0.831811	−0.415905	+	0.909408i	$0.636535\pi$
$938$	0	0
$939$	−1459.97	−0.0507394
$940$	0	0
$941$	9748.00	0.337700	0.168850	−	0.985642i	$-0.445995\pi$
$941$	9748.00	0.337700	0.168850	+	0.985642i	$0.445995\pi$
$942$	0	0
$943$	390.899	0.0134989
$944$	0	0
$945$	0	0
$946$	0	0
$947$	51537.0	1.76845	0.884227	−	0.467057i	$-0.154686\pi$
$947$	51537.0	1.76845	0.884227	+	0.467057i	$0.154686\pi$
$948$	0	0
$949$	20023.5	0.684923
$950$	0	0
$951$	12656.6	0.431566
$952$	0	0
$953$	−5631.36	−0.191414	−0.0957071	−	0.995410i	$-0.530511\pi$
$953$	−5631.36	−0.191414	−0.0957071	+	0.995410i	$0.530511\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	2959.54	0.0999668
$958$	0	0
$959$	−26847.4	−0.904013
$960$	0	0
$961$	−29636.8	−0.994823
$962$	0	0
$963$	7184.36	0.240408
$964$	0	0
$965$	0	0
$966$	0	0
$967$	43360.9	1.44198	0.720989	−	0.692946i	$-0.243688\pi$
$967$	43360.9	1.44198	0.720989	+	0.692946i	$0.243688\pi$
$968$	0	0
$969$	−12162.8	−0.403226
$970$	0	0
$971$	−12920.0	−0.427007	−0.213503	−	0.976942i	$-0.568487\pi$
$971$	−12920.0	−0.427007	−0.213503	+	0.976942i	$0.568487\pi$
$972$	0	0
$973$	2484.19	0.0818493
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−10650.4	−0.348759	−0.174379	−	0.984679i	$-0.555792\pi$
$977$	−10650.4	−0.348759	−0.174379	+	0.984679i	$0.555792\pi$
$978$	0	0
$979$	3899.31	0.127296
$980$	0	0
$981$	−8866.07	−0.288554
$982$	0	0
$983$	49450.3	1.60450	0.802248	−	0.596991i	$-0.203637\pi$
$983$	49450.3	1.60450	0.802248	+	0.596991i	$0.203637\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−9946.58	−0.320773
$988$	0	0
$989$	592.351	0.0190452
$990$	0	0
$991$	−9410.47	−0.301648	−0.150824	−	0.988561i	$-0.548193\pi$
$991$	−9410.47	−0.301648	−0.150824	+	0.988561i	$0.548193\pi$
$992$	0	0
$993$	22319.8	0.713291
$994$	0	0
$995$	0	0
$996$	0	0
$997$	532.117	0.0169030	0.00845151	−	0.999964i	$-0.497310\pi$
$997$	532.117	0.0169030	0.00845151	+	0.999964i	$0.497310\pi$
$998$	0	0
$999$	−8798.88	−0.278663

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1200.4.a.bn.1.2		2
4.3	odd	2		75.4.a.f.1.2		2
5.2	odd	4		240.4.f.f.49.4		4
5.3	odd	4		240.4.f.f.49.2		4
5.4	even	2		1200.4.a.bt.1.1		2
12.11	even	2		225.4.a.i.1.1		2
15.2	even	4		720.4.f.j.289.1		4
15.8	even	4		720.4.f.j.289.2		4
20.3	even	4		15.4.b.a.4.1	✓	4
20.7	even	4		15.4.b.a.4.4	yes	4
20.19	odd	2		75.4.a.c.1.1		2
40.3	even	4		960.4.f.q.769.1		4
40.13	odd	4		960.4.f.p.769.3		4
40.27	even	4		960.4.f.q.769.3		4
40.37	odd	4		960.4.f.p.769.1		4
60.23	odd	4		45.4.b.b.19.4		4
60.47	odd	4		45.4.b.b.19.1		4
60.59	even	2		225.4.a.o.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.4.b.a.4.1	✓	4	20.3	even	4
15.4.b.a.4.4	yes	4	20.7	even	4
45.4.b.b.19.1		4	60.47	odd	4
45.4.b.b.19.4		4	60.23	odd	4
75.4.a.c.1.1		2	20.19	odd	2
75.4.a.f.1.2		2	4.3	odd	2
225.4.a.i.1.1		2	12.11	even	2
225.4.a.o.1.2		2	60.59	even	2
240.4.f.f.49.2		4	5.3	odd	4
240.4.f.f.49.4		4	5.2	odd	4
720.4.f.j.289.1		4	15.2	even	4
720.4.f.j.289.2		4	15.8	even	4
960.4.f.p.769.1		4	40.37	odd	4
960.4.f.p.769.3		4	40.13	odd	4
960.4.f.q.769.1		4	40.3	even	4
960.4.f.q.769.3		4	40.27	even	4
1200.4.a.bn.1.2		2	1.1	even	1	trivial
1200.4.a.bt.1.1		2	5.4	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.a (trivial)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} +16.2094 q^{7} +9.00000 q^{9} +O(q^{10})\)	\(q-3.00000 q^{3} +16.2094 q^{7} +9.00000 q^{9} +40.2094 q^{11} +19.7906 q^{13} +83.0156 q^{17} +48.8375 q^{19} -48.6281 q^{21} -1.61250 q^{23} -27.0000 q^{27} -24.5344 q^{29} +12.4187 q^{31} -120.628 q^{33} +325.884 q^{37} -59.3719 q^{39} -242.419 q^{41} -367.350 q^{43} +204.544 q^{47} -80.2562 q^{49} -249.047 q^{51} -61.5281 q^{53} -146.512 q^{57} +112.209 q^{59} +477.350 q^{61} +145.884 q^{63} -558.094 q^{67} +4.83749 q^{69} -558.281 q^{71} +1011.77 q^{73} +651.769 q^{77} -1150.47 q^{79} +81.0000 q^{81} +1157.92 q^{83} +73.6032 q^{87} +96.9751 q^{89} +320.794 q^{91} -37.2562 q^{93} -1152.37 q^{97} +361.884 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{3} - 6 q^{7} + 18 q^{9}+O(q^{10}) \) 2 * q - 6 * q^3 - 6 * q^7 + 18 * q^9	\( 2 q - 6 q^{3} - 6 q^{7} + 18 q^{9} + 42 q^{11} + 78 q^{13} + 102 q^{17} - 56 q^{19} + 18 q^{21} + 48 q^{23} - 54 q^{27} - 318 q^{29} - 52 q^{31} - 126 q^{33} + 306 q^{37} - 234 q^{39} - 408 q^{41} - 120 q^{43} - 180 q^{47} + 70 q^{49} - 306 q^{51} + 402 q^{53} + 168 q^{57} + 186 q^{59} + 340 q^{61} - 54 q^{63} - 732 q^{67} - 144 q^{69} + 36 q^{71} + 1332 q^{73} + 612 q^{77} - 380 q^{79} + 162 q^{81} + 984 q^{83} + 954 q^{87} + 1116 q^{89} - 972 q^{91} + 156 q^{93} - 768 q^{97} + 378 q^{99}+O(q^{100}) \) 2 * q - 6 * q^3 - 6 * q^7 + 18 * q^9 + 42 * q^11 + 78 * q^13 + 102 * q^17 - 56 * q^19 + 18 * q^21 + 48 * q^23 - 54 * q^27 - 318 * q^29 - 52 * q^31 - 126 * q^33 + 306 * q^37 - 234 * q^39 - 408 * q^41 - 120 * q^43 - 180 * q^47 + 70 * q^49 - 306 * q^51 + 402 * q^53 + 168 * q^57 + 186 * q^59 + 340 * q^61 - 54 * q^63 - 732 * q^67 - 144 * q^69 + 36 * q^71 + 1332 * q^73 + 612 * q^77 - 380 * q^79 + 162 * q^81 + 984 * q^83 + 954 * q^87 + 1116 * q^89 - 972 * q^91 + 156 * q^93 - 768 * q^97 + 378 * q^99

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