LMFDB - Embedded newform 1200.4.f.f.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.f.49.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000i q^{3} -19.0000i q^{7} -9.00000 q^{9} +O(q^{10})$	$q-3.00000i q^{3} -19.0000i q^{7} -9.00000 q^{9} -22.0000 q^{11} +1.00000i q^{13} +58.0000i q^{17} -53.0000 q^{19} -57.0000 q^{21} -58.0000i q^{23} +27.0000i q^{27} -22.0000 q^{29} +35.0000 q^{31} +66.0000i q^{33} +270.000i q^{37} +3.00000 q^{39} -468.000 q^{41} +431.000i q^{43} -230.000i q^{47} -18.0000 q^{49} +174.000 q^{51} +159.000i q^{57} +446.000 q^{59} +127.000 q^{61} +171.000i q^{63} -811.000i q^{67} -174.000 q^{69} -36.0000 q^{71} +522.000i q^{73} +418.000i q^{77} +1368.00 q^{79} +81.0000 q^{81} +1138.00i q^{83} +66.0000i q^{87} -144.000 q^{89} +19.0000 q^{91} -105.000i q^{93} +1079.00i q^{97} +198.000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 18 q^{9}+O(q^{10}) $ 2 * q - 18 * q^9	$ 2 q - 18 q^{9} - 44 q^{11} - 106 q^{19} - 114 q^{21} - 44 q^{29} + 70 q^{31} + 6 q^{39} - 936 q^{41} - 36 q^{49} + 348 q^{51} + 892 q^{59} + 254 q^{61} - 348 q^{69} - 72 q^{71} + 2736 q^{79} + 162 q^{81} - 288 q^{89} + 38 q^{91} + 396 q^{99}+O(q^{100}) $ 2 * q - 18 * q^9 - 44 * q^11 - 106 * q^19 - 114 * q^21 - 44 * q^29 + 70 * q^31 + 6 * q^39 - 936 * q^41 - 36 * q^49 + 348 * q^51 + 892 * q^59 + 254 * q^61 - 348 * q^69 - 72 * q^71 + 2736 * q^79 + 162 * q^81 - 288 * q^89 + 38 * q^91 + 396 * q^99

Display 100 coefficients 10 coefficients

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
$7$	− 19.0000i	− 1.02590i	−0.858417	−	0.512952i	$-0.828552\pi$
$7$	− 19.0000i	− 1.02590i	0.858417	−	0.512952i	$-0.171448\pi$
$8$	0	0
$9$	−9.00000	−0.333333
$10$	0	0
$11$	−22.0000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−22.0000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	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
$17$	58.0000i	0.827474i	0.910396	+	0.413737i	$0.135777\pi$
$17$	58.0000i	0.827474i	−0.910396	+	0.413737i	$0.864223\pi$
$18$	0	0
$19$	−53.0000	−0.639949	−0.319975	−	0.947426i	$-0.603674\pi$
$19$	−53.0000	−0.639949	−0.319975	+	0.947426i	$0.603674\pi$
$20$	0	0
$21$	−57.0000	−0.592306
$22$	0	0
$23$	− 58.0000i	− 0.525819i	−0.964821	−	0.262909i	$-0.915318\pi$
$23$	− 58.0000i	− 0.525819i	0.964821	−	0.262909i	$-0.0846821\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	27.0000i	0.192450i
$28$	0	0
$29$	−22.0000	−0.140872	−0.0704362	−	0.997516i	$-0.522439\pi$
$29$	−22.0000	−0.140872	−0.0704362	+	0.997516i	$0.522439\pi$
$30$	0	0
$31$	35.0000	0.202780	0.101390	−	0.994847i	$-0.467671\pi$
$31$	35.0000	0.202780	0.101390	+	0.994847i	$0.467671\pi$
$32$	0	0
$33$	66.0000i	0.348155i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	270.000i	1.19967i	0.800124	+	0.599834i	$0.204767\pi$
$37$	270.000i	1.19967i	−0.800124	+	0.599834i	$0.795233\pi$
$38$	0	0
$39$	3.00000	0.0123176
$40$	0	0
$41$	−468.000	−1.78267	−0.891333	−	0.453349i	$-0.850229\pi$
$41$	−468.000	−1.78267	−0.891333	+	0.453349i	$0.850229\pi$
$42$	0	0
$43$	431.000i	1.52853i	0.644901	+	0.764266i	$0.276899\pi$
$43$	431.000i	1.52853i	−0.644901	+	0.764266i	$0.723101\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 230.000i	− 0.713807i	−0.934141	−	0.356904i	$-0.883832\pi$
$47$	− 230.000i	− 0.713807i	0.934141	−	0.356904i	$-0.116168\pi$
$48$	0	0
$49$	−18.0000	−0.0524781
$50$	0	0
$51$	174.000	0.477743
$52$	0	0
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	159.000i	0.369475i
$58$	0	0
$59$	446.000	0.984140	0.492070	−	0.870556i	$-0.336240\pi$
$59$	446.000	0.984140	0.492070	+	0.870556i	$0.336240\pi$
$60$	0	0
$61$	127.000	0.266569	0.133284	−	0.991078i	$-0.457448\pi$
$61$	127.000	0.266569	0.133284	+	0.991078i	$0.457448\pi$
$62$	0	0
$63$	171.000i	0.341968i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 811.000i	− 1.47880i	−0.673268	−	0.739399i	$-0.735110\pi$
$67$	− 811.000i	− 1.47880i	0.673268	−	0.739399i	$-0.264890\pi$
$68$	0	0
$69$	−174.000	−0.303582
$70$	0	0
$71$	−36.0000	−0.0601748	−0.0300874	−	0.999547i	$-0.509579\pi$
$71$	−36.0000	−0.0601748	−0.0300874	+	0.999547i	$0.509579\pi$
$72$	0	0
$73$	522.000i	0.836924i	0.908234	+	0.418462i	$0.137431\pi$
$73$	522.000i	0.836924i	−0.908234	+	0.418462i	$0.862569\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	418.000i	0.618643i
$78$	0	0
$79$	1368.00	1.94825	0.974127	−	0.226002i	$-0.0725657\pi$
$79$	1368.00	1.94825	0.974127	+	0.226002i	$0.0725657\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	1138.00i	1.50496i	0.658615	+	0.752480i	$0.271143\pi$
$83$	1138.00i	1.50496i	−0.658615	+	0.752480i	$0.728857\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	66.0000i	0.0813327i
$88$	0	0
$89$	−144.000	−0.171505	−0.0857526	−	0.996316i	$-0.527329\pi$
$89$	−144.000	−0.171505	−0.0857526	+	0.996316i	$0.527329\pi$
$90$	0	0
$91$	19.0000	0.0218873
$92$	0	0
$93$	− 105.000i	− 0.117075i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	1079.00i	1.12944i	0.825282	+	0.564721i	$0.191016\pi$
$97$	1079.00i	1.12944i	−0.825282	+	0.564721i	$0.808984\pi$
$98$	0	0
$99$	198.000	0.201008
$100$	0	0
$101$	−1440.00	−1.41867	−0.709333	−	0.704873i	$-0.751004\pi$
$101$	−1440.00	−1.41867	−0.709333	+	0.704873i	$0.751004\pi$
$102$	0	0
$103$	− 124.000i	− 0.118622i	−0.998240	−	0.0593111i	$-0.981110\pi$
$103$	− 124.000i	− 0.118622i	0.998240	−	0.0593111i	$-0.0188904\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 432.000i	− 0.390309i	−0.980773	−	0.195154i	$-0.937479\pi$
$107$	− 432.000i	− 0.390309i	0.980773	−	0.195154i	$-0.0625208\pi$
$108$	0	0
$109$	−701.000	−0.615997	−0.307998	−	0.951387i	$-0.599659\pi$
$109$	−701.000	−0.615997	−0.307998	+	0.951387i	$0.599659\pi$
$110$	0	0
$111$	810.000	0.692629
$112$	0	0
$113$	− 1044.00i	− 0.869126i	−0.900641	−	0.434563i	$-0.856903\pi$
$113$	− 1044.00i	− 0.869126i	0.900641	−	0.434563i	$-0.143097\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 9.00000i	− 0.00711154i
$118$	0	0
$119$	1102.00	0.848909
$120$	0	0
$121$	−847.000	−0.636364
$122$	0	0
$123$	1404.00i	1.02922i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	504.000i	0.352148i	0.984377	+	0.176074i	$0.0563398\pi$
$127$	504.000i	0.352148i	−0.984377	+	0.176074i	$0.943660\pi$
$128$	0	0
$129$	1293.00	0.882498
$130$	0	0
$131$	−72.0000	−0.0480204	−0.0240102	−	0.999712i	$-0.507643\pi$
$131$	−72.0000	−0.0480204	−0.0240102	+	0.999712i	$0.507643\pi$
$132$	0	0
$133$	1007.00i	0.656526i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1850.00i	1.15369i	0.816852	+	0.576847i	$0.195717\pi$
$137$	1850.00i	1.15369i	−0.816852	+	0.576847i	$0.804283\pi$
$138$	0	0
$139$	1836.00	1.12034	0.560171	−	0.828377i	$-0.310735\pi$
$139$	1836.00	1.12034	0.560171	+	0.828377i	$0.310735\pi$
$140$	0	0
$141$	−690.000	−0.412117
$142$	0	0
$143$	− 22.0000i	− 0.0128653i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	54.0000i	0.0302983i
$148$	0	0
$149$	2398.00	1.31847	0.659234	−	0.751938i	$-0.270881\pi$
$149$	2398.00	1.31847	0.659234	+	0.751938i	$0.270881\pi$
$150$	0	0
$151$	−1871.00	−1.00834	−0.504172	−	0.863604i	$-0.668202\pi$
$151$	−1871.00	−1.00834	−0.504172	+	0.863604i	$0.668202\pi$
$152$	0	0
$153$	− 522.000i	− 0.275825i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	3293.00i	1.67395i	0.547242	+	0.836975i	$0.315678\pi$
$157$	3293.00i	1.67395i	−0.547242	+	0.836975i	$0.684322\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−1102.00	−0.539440
$162$	0	0
$163$	− 883.000i	− 0.424306i	−0.977236	−	0.212153i	$-0.931952\pi$
$163$	− 883.000i	− 0.424306i	0.977236	−	0.212153i	$-0.0680475\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	4104.00i	1.90166i	0.309715	+	0.950830i	$0.399766\pi$
$167$	4104.00i	1.90166i	−0.309715	+	0.950830i	$0.600234\pi$
$168$	0	0
$169$	2196.00	0.999545
$170$	0	0
$171$	477.000	0.213316
$172$	0	0
$173$	1706.00i	0.749739i	0.927078	+	0.374869i	$0.122312\pi$
$173$	1706.00i	0.749739i	−0.927078	+	0.374869i	$0.877688\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 1338.00i	− 0.568193i
$178$	0	0
$179$	662.000	0.276426	0.138213	−	0.990403i	$-0.455864\pi$
$179$	662.000	0.276426	0.138213	+	0.990403i	$0.455864\pi$
$180$	0	0
$181$	4121.00	1.69233	0.846164	−	0.532922i	$-0.178906\pi$
$181$	4121.00	1.69233	0.846164	+	0.532922i	$0.178906\pi$
$182$	0	0
$183$	− 381.000i	− 0.153903i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 1276.00i	− 0.498986i
$188$	0	0
$189$	513.000	0.197435
$190$	0	0
$191$	958.000	0.362924	0.181462	−	0.983398i	$-0.441917\pi$
$191$	958.000	0.362924	0.181462	+	0.983398i	$0.441917\pi$
$192$	0	0
$193$	3187.00i	1.18863i	0.804233	+	0.594314i	$0.202576\pi$
$193$	3187.00i	1.18863i	−0.804233	+	0.594314i	$0.797424\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2282.00i	0.825308i	0.910888	+	0.412654i	$0.135398\pi$
$197$	2282.00i	0.825308i	−0.910888	+	0.412654i	$0.864602\pi$
$198$	0	0
$199$	1043.00	0.371539	0.185770	−	0.982593i	$-0.440522\pi$
$199$	1043.00	0.371539	0.185770	+	0.982593i	$0.440522\pi$
$200$	0	0
$201$	−2433.00	−0.853784
$202$	0	0
$203$	418.000i	0.144521i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	522.000i	0.175273i
$208$	0	0
$209$	1166.00	0.385904
$210$	0	0
$211$	−4139.00	−1.35043	−0.675214	−	0.737621i	$-0.735949\pi$
$211$	−4139.00	−1.35043	−0.675214	+	0.737621i	$0.735949\pi$
$212$	0	0
$213$	108.000i	0.0347420i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 665.000i	− 0.208033i
$218$	0	0
$219$	1566.00	0.483199
$220$	0	0
$221$	−58.0000	−0.0176539
$222$	0	0
$223$	− 413.000i	− 0.124020i	−0.998076	−	0.0620101i	$-0.980249\pi$
$223$	− 413.000i	− 0.124020i	0.998076	−	0.0620101i	$-0.0197511\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	5652.00i	1.65258i	0.563242	+	0.826292i	$0.309554\pi$
$227$	5652.00i	1.65258i	−0.563242	+	0.826292i	$0.690446\pi$
$228$	0	0
$229$	4391.00	1.26710	0.633549	−	0.773703i	$-0.281597\pi$
$229$	4391.00	1.26710	0.633549	+	0.773703i	$0.281597\pi$
$230$	0	0
$231$	1254.00	0.357174
$232$	0	0
$233$	− 2052.00i	− 0.576957i	−0.957486	−	0.288479i	$-0.906851\pi$
$233$	− 2052.00i	− 0.576957i	0.957486	−	0.288479i	$-0.0931494\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 4104.00i	− 1.12482i
$238$	0	0
$239$	−4320.00	−1.16919	−0.584597	−	0.811324i	$-0.698748\pi$
$239$	−4320.00	−1.16919	−0.584597	+	0.811324i	$0.698748\pi$
$240$	0	0
$241$	4265.00	1.13997	0.569985	−	0.821655i	$-0.306949\pi$
$241$	4265.00	1.13997	0.569985	+	0.821655i	$0.306949\pi$
$242$	0	0
$243$	− 243.000i	− 0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 53.0000i	− 0.0136531i
$248$	0	0
$249$	3414.00	0.868889
$250$	0	0
$251$	2412.00	0.606550	0.303275	−	0.952903i	$-0.401920\pi$
$251$	2412.00	0.606550	0.303275	+	0.952903i	$0.401920\pi$
$252$	0	0
$253$	1276.00i	0.317081i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 6948.00i	− 1.68640i	−0.537601	−	0.843199i	$-0.680669\pi$
$257$	− 6948.00i	− 1.68640i	0.537601	−	0.843199i	$-0.319331\pi$
$258$	0	0
$259$	5130.00	1.23074
$260$	0	0
$261$	198.000	0.0469574
$262$	0	0
$263$	3564.00i	0.835611i	0.908537	+	0.417805i	$0.137201\pi$
$263$	3564.00i	0.835611i	−0.908537	+	0.417805i	$0.862799\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	432.000i	0.0990186i
$268$	0	0
$269$	1534.00	0.347694	0.173847	−	0.984773i	$-0.444380\pi$
$269$	1534.00	0.347694	0.173847	+	0.984773i	$0.444380\pi$
$270$	0	0
$271$	−7704.00	−1.72688	−0.863440	−	0.504451i	$-0.831695\pi$
$271$	−7704.00	−1.72688	−0.863440	+	0.504451i	$0.831695\pi$
$272$	0	0
$273$	− 57.0000i	− 0.0126366i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 4877.00i	− 1.05787i	−0.848662	−	0.528936i	$-0.822591\pi$
$277$	− 4877.00i	− 1.05787i	0.848662	−	0.528936i	$-0.177409\pi$
$278$	0	0
$279$	−315.000	−0.0675934
$280$	0	0
$281$	−3758.00	−0.797806	−0.398903	−	0.916993i	$-0.630609\pi$
$281$	−3758.00	−0.797806	−0.398903	+	0.916993i	$0.630609\pi$
$282$	0	0
$283$	− 935.000i	− 0.196396i	−0.995167	−	0.0981978i	$-0.968692\pi$
$283$	− 935.000i	− 0.196396i	0.995167	−	0.0981978i	$-0.0313078\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	8892.00i	1.82884i
$288$	0	0
$289$	1549.00	0.315286
$290$	0	0
$291$	3237.00	0.652084
$292$	0	0
$293$	6214.00i	1.23900i	0.784998	+	0.619498i	$0.212664\pi$
$293$	6214.00i	1.23900i	−0.784998	+	0.619498i	$0.787336\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 594.000i	− 0.116052i
$298$	0	0
$299$	58.0000	0.0112181
$300$	0	0
$301$	8189.00	1.56813
$302$	0	0
$303$	4320.00i	0.819068i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 3905.00i	− 0.725961i	−0.931797	−	0.362981i	$-0.881759\pi$
$307$	− 3905.00i	− 0.725961i	0.931797	−	0.362981i	$-0.118241\pi$
$308$	0	0
$309$	−372.000	−0.0684865
$310$	0	0
$311$	−9662.00	−1.76168	−0.880839	−	0.473416i	$-0.843021\pi$
$311$	−9662.00	−1.76168	−0.880839	+	0.473416i	$0.843021\pi$
$312$	0	0
$313$	5147.00i	0.929475i	0.885449	+	0.464737i	$0.153851\pi$
$313$	5147.00i	0.929475i	−0.885449	+	0.464737i	$0.846149\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	9216.00i	1.63288i	0.577432	+	0.816439i	$0.304055\pi$
$317$	9216.00i	1.63288i	−0.577432	+	0.816439i	$0.695945\pi$
$318$	0	0
$319$	484.000	0.0849492
$320$	0	0
$321$	−1296.00	−0.225345
$322$	0	0
$323$	− 3074.00i	− 0.529542i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	2103.00i	0.355646i
$328$	0	0
$329$	−4370.00	−0.732298
$330$	0	0
$331$	−2196.00	−0.364662	−0.182331	−	0.983237i	$-0.558364\pi$
$331$	−2196.00	−0.364662	−0.182331	+	0.983237i	$0.558364\pi$
$332$	0	0
$333$	− 2430.00i	− 0.399889i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 2521.00i	− 0.407500i	−0.979023	−	0.203750i	$-0.934687\pi$
$337$	− 2521.00i	− 0.407500i	0.979023	−	0.203750i	$-0.0653130\pi$
$338$	0	0
$339$	−3132.00	−0.501790
$340$	0	0
$341$	−770.000	−0.122281
$342$	0	0
$343$	− 6175.00i	− 0.972066i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	7034.00i	1.08820i	0.839021	+	0.544099i	$0.183129\pi$
$347$	7034.00i	1.08820i	−0.839021	+	0.544099i	$0.816871\pi$
$348$	0	0
$349$	−7362.00	−1.12917	−0.564583	−	0.825376i	$-0.690963\pi$
$349$	−7362.00	−1.12917	−0.564583	+	0.825376i	$0.690963\pi$
$350$	0	0
$351$	−27.0000	−0.00410585
$352$	0	0
$353$	− 9382.00i	− 1.41460i	−0.706914	−	0.707300i	$-0.749913\pi$
$353$	− 9382.00i	− 1.41460i	0.706914	−	0.707300i	$-0.250087\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 3306.00i	− 0.490118i
$358$	0	0
$359$	−10116.0	−1.48719	−0.743596	−	0.668629i	$-0.766881\pi$
$359$	−10116.0	−1.48719	−0.743596	+	0.668629i	$0.766881\pi$
$360$	0	0
$361$	−4050.00	−0.590465
$362$	0	0
$363$	2541.00i	0.367405i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 3277.00i	− 0.466098i	−0.972465	−	0.233049i	$-0.925130\pi$
$367$	− 3277.00i	− 0.466098i	0.972465	−	0.233049i	$-0.0748703\pi$
$368$	0	0
$369$	4212.00	0.594222
$370$	0	0
$371$	0	0
$372$	0	0
$373$	10891.0i	1.51184i	0.654667	+	0.755918i	$0.272809\pi$
$373$	10891.0i	1.51184i	−0.654667	+	0.755918i	$0.727191\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 22.0000i	− 0.00300546i
$378$	0	0
$379$	−2591.00	−0.351163	−0.175581	−	0.984465i	$-0.556181\pi$
$379$	−2591.00	−0.351163	−0.175581	+	0.984465i	$0.556181\pi$
$380$	0	0
$381$	1512.00	0.203313
$382$	0	0
$383$	612.000i	0.0816494i	0.999166	+	0.0408247i	$0.0129985\pi$
$383$	612.000i	0.0816494i	−0.999166	+	0.0408247i	$0.987001\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 3879.00i	− 0.509511i
$388$	0	0
$389$	−3708.00	−0.483298	−0.241649	−	0.970364i	$-0.577688\pi$
$389$	−3708.00	−0.483298	−0.241649	+	0.970364i	$0.577688\pi$
$390$	0	0
$391$	3364.00	0.435102
$392$	0	0
$393$	216.000i	0.0277246i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	3833.00i	0.484566i	0.970206	+	0.242283i	$0.0778963\pi$
$397$	3833.00i	0.484566i	−0.970206	+	0.242283i	$0.922104\pi$
$398$	0	0
$399$	3021.00	0.379046
$400$	0	0
$401$	−9288.00	−1.15666	−0.578330	−	0.815803i	$-0.696295\pi$
$401$	−9288.00	−1.15666	−0.578330	+	0.815803i	$0.696295\pi$
$402$	0	0
$403$	35.0000i	0.00432624i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 5940.00i	− 0.723427i
$408$	0	0
$409$	755.000	0.0912771	0.0456386	−	0.998958i	$-0.485468\pi$
$409$	755.000	0.0912771	0.0456386	+	0.998958i	$0.485468\pi$
$410$	0	0
$411$	5550.00	0.666086
$412$	0	0
$413$	− 8474.00i	− 1.00963i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 5508.00i	− 0.646830i
$418$	0	0
$419$	−9576.00	−1.11651	−0.558256	−	0.829669i	$-0.688529\pi$
$419$	−9576.00	−1.11651	−0.558256	+	0.829669i	$0.688529\pi$
$420$	0	0
$421$	9414.00	1.08981	0.544905	−	0.838498i	$-0.316566\pi$
$421$	9414.00	1.08981	0.544905	+	0.838498i	$0.316566\pi$
$422$	0	0
$423$	2070.00i	0.237936i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 2413.00i	− 0.273474i
$428$	0	0
$429$	−66.0000	−0.00742776
$430$	0	0
$431$	2254.00	0.251906	0.125953	−	0.992036i	$-0.459801\pi$
$431$	2254.00	0.251906	0.125953	+	0.992036i	$0.459801\pi$
$432$	0	0
$433$	− 6301.00i	− 0.699323i	−0.936876	−	0.349661i	$-0.886297\pi$
$433$	− 6301.00i	− 0.699323i	0.936876	−	0.349661i	$-0.113703\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	3074.00i	0.336497i
$438$	0	0
$439$	−3779.00	−0.410847	−0.205423	−	0.978673i	$-0.565857\pi$
$439$	−3779.00	−0.410847	−0.205423	+	0.978673i	$0.565857\pi$
$440$	0	0
$441$	162.000	0.0174927
$442$	0	0
$443$	− 7632.00i	− 0.818527i	−0.912416	−	0.409263i	$-0.865786\pi$
$443$	− 7632.00i	− 0.818527i	0.912416	−	0.409263i	$-0.134214\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 7194.00i	− 0.761218i
$448$	0	0
$449$	−2988.00	−0.314059	−0.157029	−	0.987594i	$-0.550192\pi$
$449$	−2988.00	−0.314059	−0.157029	+	0.987594i	$0.550192\pi$
$450$	0	0
$451$	10296.0	1.07499
$452$	0	0
$453$	5613.00i	0.582167i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	1370.00i	0.140232i	0.997539	+	0.0701159i	$0.0223369\pi$
$457$	1370.00i	0.140232i	−0.997539	+	0.0701159i	$0.977663\pi$
$458$	0	0
$459$	−1566.00	−0.159248
$460$	0	0
$461$	−7992.00	−0.807429	−0.403714	−	0.914885i	$-0.632281\pi$
$461$	−7992.00	−0.807429	−0.403714	+	0.914885i	$0.632281\pi$
$462$	0	0
$463$	3096.00i	0.310763i	0.987855	+	0.155382i	$0.0496607\pi$
$463$	3096.00i	0.310763i	−0.987855	+	0.155382i	$0.950339\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 8338.00i	− 0.826203i	−0.910685	−	0.413101i	$-0.864446\pi$
$467$	− 8338.00i	− 0.826203i	0.910685	−	0.413101i	$-0.135554\pi$
$468$	0	0
$469$	−15409.0	−1.51710
$470$	0	0
$471$	9879.00	0.966455
$472$	0	0
$473$	− 9482.00i	− 0.921740i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	4918.00	0.469121	0.234561	−	0.972101i	$-0.424635\pi$
$479$	4918.00	0.469121	0.234561	+	0.972101i	$0.424635\pi$
$480$	0	0
$481$	−270.000	−0.0255945
$482$	0	0
$483$	3306.00i	0.311446i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 19637.0i	− 1.82718i	−0.406635	−	0.913591i	$-0.633298\pi$
$487$	− 19637.0i	− 1.82718i	0.406635	−	0.913591i	$-0.366702\pi$
$488$	0	0
$489$	−2649.00	−0.244973
$490$	0	0
$491$	−3096.00	−0.284563	−0.142282	−	0.989826i	$-0.545444\pi$
$491$	−3096.00	−0.284563	−0.142282	+	0.989826i	$0.545444\pi$
$492$	0	0
$493$	− 1276.00i	− 0.116568i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	684.000i	0.0617336i
$498$	0	0
$499$	6875.00	0.616768	0.308384	−	0.951262i	$-0.400212\pi$
$499$	6875.00	0.616768	0.308384	+	0.951262i	$0.400212\pi$
$500$	0	0
$501$	12312.0	1.09792
$502$	0	0
$503$	− 11268.0i	− 0.998838i	−0.866361	−	0.499419i	$-0.833547\pi$
$503$	− 11268.0i	− 0.998838i	0.866361	−	0.499419i	$-0.166453\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 6588.00i	− 0.577087i
$508$	0	0
$509$	−16078.0	−1.40009	−0.700044	−	0.714100i	$-0.746836\pi$
$509$	−16078.0	−1.40009	−0.700044	+	0.714100i	$0.746836\pi$
$510$	0	0
$511$	9918.00	0.858604
$512$	0	0
$513$	− 1431.00i	− 0.123158i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	5060.00i	0.430442i
$518$	0	0
$519$	5118.00	0.432862
$520$	0	0
$521$	−17366.0	−1.46030	−0.730152	−	0.683285i	$-0.760551\pi$
$521$	−17366.0	−1.46030	−0.730152	+	0.683285i	$0.760551\pi$
$522$	0	0
$523$	4913.00i	0.410766i	0.978682	+	0.205383i	$0.0658440\pi$
$523$	4913.00i	0.410766i	−0.978682	+	0.205383i	$0.934156\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	2030.00i	0.167795i
$528$	0	0
$529$	8803.00	0.723514
$530$	0	0
$531$	−4014.00	−0.328047
$532$	0	0
$533$	− 468.000i	− 0.0380325i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 1986.00i	− 0.159594i
$538$	0	0
$539$	396.000	0.0316455
$540$	0	0
$541$	17605.0	1.39907	0.699536	−	0.714597i	$-0.253390\pi$
$541$	17605.0	1.39907	0.699536	+	0.714597i	$0.253390\pi$
$542$	0	0
$543$	− 12363.0i	− 0.977067i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 14560.0i	− 1.13810i	−0.822303	−	0.569050i	$-0.807311\pi$
$547$	− 14560.0i	− 1.13810i	0.822303	−	0.569050i	$-0.192689\pi$
$548$	0	0
$549$	−1143.00	−0.0888562
$550$	0	0
$551$	1166.00	0.0901511
$552$	0	0
$553$	− 25992.0i	− 1.99872i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	20614.0i	1.56812i	0.620685	+	0.784060i	$0.286855\pi$
$557$	20614.0i	1.56812i	−0.620685	+	0.784060i	$0.713145\pi$
$558$	0	0
$559$	−431.000	−0.0326107
$560$	0	0
$561$	−3828.00	−0.288090
$562$	0	0
$563$	− 3442.00i	− 0.257661i	−0.991667	−	0.128830i	$-0.958878\pi$
$563$	− 3442.00i	− 0.257661i	0.991667	−	0.128830i	$-0.0411223\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 1539.00i	− 0.113989i
$568$	0	0
$569$	−22082.0	−1.62693	−0.813467	−	0.581611i	$-0.802423\pi$
$569$	−22082.0	−1.62693	−0.813467	+	0.581611i	$0.802423\pi$
$570$	0	0
$571$	−451.000	−0.0330539	−0.0165269	−	0.999863i	$-0.505261\pi$
$571$	−451.000	−0.0330539	−0.0165269	+	0.999863i	$0.505261\pi$
$572$	0	0
$573$	− 2874.00i	− 0.209534i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 20987.0i	− 1.51421i	−0.653292	−	0.757106i	$-0.726613\pi$
$577$	− 20987.0i	− 1.51421i	0.653292	−	0.757106i	$-0.273387\pi$
$578$	0	0
$579$	9561.00	0.686255
$580$	0	0
$581$	21622.0	1.54394
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	3888.00i	0.273381i	0.990614	+	0.136691i	$0.0436467\pi$
$587$	3888.00i	0.273381i	−0.990614	+	0.136691i	$0.956353\pi$
$588$	0	0
$589$	−1855.00	−0.129769
$590$	0	0
$591$	6846.00	0.476492
$592$	0	0
$593$	− 11268.0i	− 0.780306i	−0.920750	−	0.390153i	$-0.872422\pi$
$593$	− 11268.0i	− 0.780306i	0.920750	−	0.390153i	$-0.127578\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 3129.00i	− 0.214508i
$598$	0	0
$599$	−3996.00	−0.272575	−0.136287	−	0.990669i	$-0.543517\pi$
$599$	−3996.00	−0.272575	−0.136287	+	0.990669i	$0.543517\pi$
$600$	0	0
$601$	−24965.0	−1.69442	−0.847208	−	0.531262i	$-0.821718\pi$
$601$	−24965.0	−1.69442	−0.847208	+	0.531262i	$0.821718\pi$
$602$	0	0
$603$	7299.00i	0.492932i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	4176.00i	0.279240i	0.990205	+	0.139620i	$0.0445881\pi$
$607$	4176.00i	0.279240i	−0.990205	+	0.139620i	$0.955412\pi$
$608$	0	0
$609$	1254.00	0.0834395
$610$	0	0
$611$	230.000	0.0152288
$612$	0	0
$613$	− 9558.00i	− 0.629762i	−0.949131	−	0.314881i	$-0.898035\pi$
$613$	− 9558.00i	− 0.629762i	0.949131	−	0.314881i	$-0.101965\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	9000.00i	0.587239i	0.955922	+	0.293619i	$0.0948598\pi$
$617$	9000.00i	0.587239i	−0.955922	+	0.293619i	$0.905140\pi$
$618$	0	0
$619$	−15625.0	−1.01457	−0.507287	−	0.861777i	$-0.669352\pi$
$619$	−15625.0	−1.01457	−0.507287	+	0.861777i	$0.669352\pi$
$620$	0	0
$621$	1566.00	0.101194
$622$	0	0
$623$	2736.00i	0.175948i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 3498.00i	− 0.222802i
$628$	0	0
$629$	−15660.0	−0.992695
$630$	0	0
$631$	31175.0	1.96681	0.983405	−	0.181424i	$-0.0580705\pi$
$631$	31175.0	1.96681	0.983405	+	0.181424i	$0.0580705\pi$
$632$	0	0
$633$	12417.0i	0.779671i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 18.0000i	− 0.00111960i
$638$	0	0
$639$	324.000	0.0200583
$640$	0	0
$641$	−6732.00	−0.414817	−0.207409	−	0.978254i	$-0.566503\pi$
$641$	−6732.00	−0.414817	−0.207409	+	0.978254i	$0.566503\pi$
$642$	0	0
$643$	6228.00i	0.381973i	0.981593	+	0.190986i	$0.0611686\pi$
$643$	6228.00i	0.381973i	−0.981593	+	0.190986i	$0.938831\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 396.000i	− 0.0240624i	−0.999928	−	0.0120312i	$-0.996170\pi$
$647$	− 396.000i	− 0.0240624i	0.999928	−	0.0120312i	$-0.00382974\pi$
$648$	0	0
$649$	−9812.00	−0.593459
$650$	0	0
$651$	−1995.00	−0.120108
$652$	0	0
$653$	− 4702.00i	− 0.281782i	−0.990025	−	0.140891i	$-0.955003\pi$
$653$	− 4702.00i	− 0.281782i	0.990025	−	0.140891i	$-0.0449967\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 4698.00i	− 0.278975i
$658$	0	0
$659$	−22140.0	−1.30873	−0.654364	−	0.756180i	$-0.727064\pi$
$659$	−22140.0	−1.30873	−0.654364	+	0.756180i	$0.727064\pi$
$660$	0	0
$661$	−13518.0	−0.795445	−0.397723	−	0.917506i	$-0.630199\pi$
$661$	−13518.0	−0.795445	−0.397723	+	0.917506i	$0.630199\pi$
$662$	0	0
$663$	174.000i	0.0101925i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	1276.00i	0.0740733i
$668$	0	0
$669$	−1239.00	−0.0716032
$670$	0	0
$671$	−2794.00	−0.160747
$672$	0	0
$673$	− 11250.0i	− 0.644362i	−0.946678	−	0.322181i	$-0.895584\pi$
$673$	− 11250.0i	− 0.644362i	0.946678	−	0.322181i	$-0.104416\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	15170.0i	0.861197i	0.902544	+	0.430599i	$0.141698\pi$
$677$	15170.0i	0.861197i	−0.902544	+	0.430599i	$0.858302\pi$
$678$	0	0
$679$	20501.0	1.15870
$680$	0	0
$681$	16956.0	0.954119
$682$	0	0
$683$	− 22680.0i	− 1.27061i	−0.772262	−	0.635305i	$-0.780875\pi$
$683$	− 22680.0i	− 1.27061i	0.772262	−	0.635305i	$-0.219125\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 13173.0i	− 0.731559i
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−128.000	−0.00704682	−0.00352341	−	0.999994i	$-0.501122\pi$
$691$	−128.000	−0.00704682	−0.00352341	+	0.999994i	$0.501122\pi$
$692$	0	0
$693$	− 3762.00i	− 0.206214i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 27144.0i	− 1.47511i
$698$	0	0
$699$	−6156.00	−0.333106
$700$	0	0
$701$	25682.0	1.38373	0.691866	−	0.722026i	$-0.256789\pi$
$701$	25682.0	1.38373	0.691866	+	0.722026i	$0.256789\pi$
$702$	0	0
$703$	− 14310.0i	− 0.767727i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	27360.0i	1.45542i
$708$	0	0
$709$	4951.00	0.262255	0.131127	−	0.991366i	$-0.458140\pi$
$709$	4951.00	0.262255	0.131127	+	0.991366i	$0.458140\pi$
$710$	0	0
$711$	−12312.0	−0.649418
$712$	0	0
$713$	− 2030.00i	− 0.106626i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	12960.0i	0.675035i
$718$	0	0
$719$	−22190.0	−1.15097	−0.575485	−	0.817812i	$-0.695187\pi$
$719$	−22190.0	−1.15097	−0.575485	+	0.817812i	$0.695187\pi$
$720$	0	0
$721$	−2356.00	−0.121695
$722$	0	0
$723$	− 12795.0i	− 0.658162i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	7685.00i	0.392051i	0.980599	+	0.196025i	$0.0628035\pi$
$727$	7685.00i	0.392051i	−0.980599	+	0.196025i	$0.937197\pi$
$728$	0	0
$729$	−729.000	−0.0370370
$730$	0	0
$731$	−24998.0	−1.26482
$732$	0	0
$733$	29574.0i	1.49023i	0.666934	+	0.745116i	$0.267606\pi$
$733$	29574.0i	1.49023i	−0.666934	+	0.745116i	$0.732394\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	17842.0i	0.891748i
$738$	0	0
$739$	−32580.0	−1.62175	−0.810876	−	0.585218i	$-0.801009\pi$
$739$	−32580.0	−1.62175	−0.810876	+	0.585218i	$0.801009\pi$
$740$	0	0
$741$	−159.000	−0.00788261
$742$	0	0
$743$	3060.00i	0.151091i	0.997142	+	0.0755454i	$0.0240698\pi$
$743$	3060.00i	0.151091i	−0.997142	+	0.0755454i	$0.975930\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 10242.0i	− 0.501654i
$748$	0	0
$749$	−8208.00	−0.400419
$750$	0	0
$751$	7992.00	0.388325	0.194163	−	0.980969i	$-0.437801\pi$
$751$	7992.00	0.388325	0.194163	+	0.980969i	$0.437801\pi$
$752$	0	0
$753$	− 7236.00i	− 0.350192i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 22841.0i	− 1.09666i	−0.836263	−	0.548329i	$-0.815264\pi$
$757$	− 22841.0i	− 1.09666i	0.836263	−	0.548329i	$-0.184736\pi$
$758$	0	0
$759$	3828.00	0.183067
$760$	0	0
$761$	−17172.0	−0.817982	−0.408991	−	0.912538i	$-0.634119\pi$
$761$	−17172.0	−0.817982	−0.408991	+	0.912538i	$0.634119\pi$
$762$	0	0
$763$	13319.0i	0.631953i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	446.000i	0.0209963i
$768$	0	0
$769$	30869.0	1.44755	0.723774	−	0.690037i	$-0.242406\pi$
$769$	30869.0	1.44755	0.723774	+	0.690037i	$0.242406\pi$
$770$	0	0
$771$	−20844.0	−0.973642
$772$	0	0
$773$	34884.0i	1.62314i	0.584252	+	0.811572i	$0.301388\pi$
$773$	34884.0i	1.62314i	−0.584252	+	0.811572i	$0.698612\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 15390.0i	− 0.710570i
$778$	0	0
$779$	24804.0	1.14082
$780$	0	0
$781$	792.000	0.0362868
$782$	0	0
$783$	− 594.000i	− 0.0271109i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	5039.00i	0.228235i	0.993467	+	0.114118i	$0.0364040\pi$
$787$	5039.00i	0.228235i	−0.993467	+	0.114118i	$0.963596\pi$
$788$	0	0
$789$	10692.0	0.482440
$790$	0	0
$791$	−19836.0	−0.891640
$792$	0	0
$793$	127.000i	0.00568714i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 540.000i	− 0.0239997i	−0.999928	−	0.0119999i	$-0.996180\pi$
$797$	− 540.000i	− 0.0239997i	0.999928	−	0.0119999i	$-0.00381977\pi$
$798$	0	0
$799$	13340.0	0.590657
$800$	0	0
$801$	1296.00	0.0571684
$802$	0	0
$803$	− 11484.0i	− 0.504684i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 4602.00i	− 0.200741i
$808$	0	0
$809$	41328.0	1.79606	0.898032	−	0.439931i	$-0.144997\pi$
$809$	41328.0	1.79606	0.898032	+	0.439931i	$0.144997\pi$
$810$	0	0
$811$	12853.0	0.556510	0.278255	−	0.960507i	$-0.410244\pi$
$811$	12853.0	0.556510	0.278255	+	0.960507i	$0.410244\pi$
$812$	0	0
$813$	23112.0i	0.997015i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 22843.0i	− 0.978183i
$818$	0	0
$819$	−171.000	−0.00729576
$820$	0	0
$821$	29470.0	1.25275	0.626376	−	0.779521i	$-0.284537\pi$
$821$	29470.0	1.25275	0.626376	+	0.779521i	$0.284537\pi$
$822$	0	0
$823$	24407.0i	1.03375i	0.856062	+	0.516874i	$0.172904\pi$
$823$	24407.0i	1.03375i	−0.856062	+	0.516874i	$0.827096\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	15048.0i	0.632733i	0.948637	+	0.316367i	$0.102463\pi$
$827$	15048.0i	0.632733i	−0.948637	+	0.316367i	$0.897537\pi$
$828$	0	0
$829$	28406.0	1.19009	0.595043	−	0.803694i	$-0.297135\pi$
$829$	28406.0	1.19009	0.595043	+	0.803694i	$0.297135\pi$
$830$	0	0
$831$	−14631.0	−0.610763
$832$	0	0
$833$	− 1044.00i	− 0.0434243i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	945.000i	0.0390251i
$838$	0	0
$839$	−26914.0	−1.10748	−0.553739	−	0.832690i	$-0.686800\pi$
$839$	−26914.0	−1.10748	−0.553739	+	0.832690i	$0.686800\pi$
$840$	0	0
$841$	−23905.0	−0.980155
$842$	0	0
$843$	11274.0i	0.460614i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	16093.0i	0.652848i
$848$	0	0
$849$	−2805.00	−0.113389
$850$	0	0
$851$	15660.0	0.630808
$852$	0	0
$853$	21275.0i	0.853977i	0.904257	+	0.426988i	$0.140425\pi$
$853$	21275.0i	0.853977i	−0.904257	+	0.426988i	$0.859575\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 39132.0i	− 1.55977i	−0.625922	−	0.779885i	$-0.715277\pi$
$857$	− 39132.0i	− 1.55977i	0.625922	−	0.779885i	$-0.284723\pi$
$858$	0	0
$859$	−448.000	−0.0177946	−0.00889730	−	0.999960i	$-0.502832\pi$
$859$	−448.000	−0.0177946	−0.00889730	+	0.999960i	$0.502832\pi$
$860$	0	0
$861$	26676.0	1.05588
$862$	0	0
$863$	26856.0i	1.05932i	0.848212	+	0.529658i	$0.177680\pi$
$863$	26856.0i	1.05932i	−0.848212	+	0.529658i	$0.822320\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 4647.00i	− 0.182030i
$868$	0	0
$869$	−30096.0	−1.17484
$870$	0	0
$871$	811.000	0.0315496
$872$	0	0
$873$	− 9711.00i	− 0.376481i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	3653.00i	0.140653i	0.997524	+	0.0703267i	$0.0224042\pi$
$877$	3653.00i	0.140653i	−0.997524	+	0.0703267i	$0.977596\pi$
$878$	0	0
$879$	18642.0	0.715335
$880$	0	0
$881$	−6552.00	−0.250559	−0.125280	−	0.992121i	$-0.539983\pi$
$881$	−6552.00	−0.250559	−0.125280	+	0.992121i	$0.539983\pi$
$882$	0	0
$883$	− 4481.00i	− 0.170779i	−0.996348	−	0.0853894i	$-0.972787\pi$
$883$	− 4481.00i	− 0.170779i	0.996348	−	0.0853894i	$-0.0272134\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 14666.0i	− 0.555170i	−0.960701	−	0.277585i	$-0.910466\pi$
$887$	− 14666.0i	− 0.555170i	0.960701	−	0.277585i	$-0.0895341\pi$
$888$	0	0
$889$	9576.00	0.361270
$890$	0	0
$891$	−1782.00	−0.0670025
$892$	0	0
$893$	12190.0i	0.456800i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 174.000i	− 0.00647680i
$898$	0	0
$899$	−770.000	−0.0285661
$900$	0	0
$901$	0	0
$902$	0	0
$903$	− 24567.0i	− 0.905358i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	51804.0i	1.89650i	0.317528	+	0.948249i	$0.397147\pi$
$907$	51804.0i	1.89650i	−0.317528	+	0.948249i	$0.602853\pi$
$908$	0	0
$909$	12960.0	0.472889
$910$	0	0
$911$	−31198.0	−1.13462	−0.567308	−	0.823505i	$-0.692015\pi$
$911$	−31198.0	−1.13462	−0.567308	+	0.823505i	$0.692015\pi$
$912$	0	0
$913$	− 25036.0i	− 0.907525i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1368.00i	0.0492643i
$918$	0	0
$919$	−27001.0	−0.969185	−0.484592	−	0.874740i	$-0.661032\pi$
$919$	−27001.0	−0.969185	−0.484592	+	0.874740i	$0.661032\pi$
$920$	0	0
$921$	−11715.0	−0.419134
$922$	0	0
$923$	− 36.0000i	− 0.00128381i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	1116.00i	0.0395407i
$928$	0	0
$929$	22694.0	0.801470	0.400735	−	0.916194i	$-0.368755\pi$
$929$	22694.0	0.801470	0.400735	+	0.916194i	$0.368755\pi$
$930$	0	0
$931$	954.000	0.0335833
$932$	0	0
$933$	28986.0i	1.01711i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 29503.0i	− 1.02862i	−0.857603	−	0.514312i	$-0.828047\pi$
$937$	− 29503.0i	− 1.02862i	0.857603	−	0.514312i	$-0.171953\pi$
$938$	0	0
$939$	15441.0	0.536633
$940$	0	0
$941$	14566.0	0.504610	0.252305	−	0.967648i	$-0.418811\pi$
$941$	14566.0	0.504610	0.252305	+	0.967648i	$0.418811\pi$
$942$	0	0
$943$	27144.0i	0.937360i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	42322.0i	1.45225i	0.687563	+	0.726125i	$0.258681\pi$
$947$	42322.0i	1.45225i	−0.687563	+	0.726125i	$0.741319\pi$
$948$	0	0
$949$	−522.000	−0.0178555
$950$	0	0
$951$	27648.0	0.942742
$952$	0	0
$953$	43416.0i	1.47574i	0.674942	+	0.737871i	$0.264169\pi$
$953$	43416.0i	1.47574i	−0.674942	+	0.737871i	$0.735831\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 1452.00i	− 0.0490454i
$958$	0	0
$959$	35150.0	1.18358
$960$	0	0
$961$	−28566.0	−0.958880
$962$	0	0
$963$	3888.00i	0.130103i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	21528.0i	0.715919i	0.933737	+	0.357960i	$0.116527\pi$
$967$	21528.0i	0.715919i	−0.933737	+	0.357960i	$0.883473\pi$
$968$	0	0
$969$	−9222.00	−0.305731
$970$	0	0
$971$	−27050.0	−0.894002	−0.447001	−	0.894533i	$-0.647508\pi$
$971$	−27050.0	−0.894002	−0.447001	+	0.894533i	$0.647508\pi$
$972$	0	0
$973$	− 34884.0i	− 1.14936i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	24934.0i	0.816489i	0.912873	+	0.408244i	$0.133859\pi$
$977$	24934.0i	0.816489i	−0.912873	+	0.408244i	$0.866141\pi$
$978$	0	0
$979$	3168.00	0.103422
$980$	0	0
$981$	6309.00	0.205332
$982$	0	0
$983$	− 8388.00i	− 0.272162i	−0.990698	−	0.136081i	$-0.956549\pi$
$983$	− 8388.00i	− 0.272162i	0.990698	−	0.136081i	$-0.0434508\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	13110.0i	0.422792i
$988$	0	0
$989$	24998.0	0.803731
$990$	0	0
$991$	−29033.0	−0.930639	−0.465320	−	0.885143i	$-0.654061\pi$
$991$	−29033.0	−0.930639	−0.465320	+	0.885143i	$0.654061\pi$
$992$	0	0
$993$	6588.00i	0.210538i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	25326.0i	0.804496i	0.915531	+	0.402248i	$0.131771\pi$
$997$	25326.0i	0.804496i	−0.915531	+	0.402248i	$0.868229\pi$
$998$	0	0
$999$	−7290.00	−0.230876

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
600.4.a.g.1.1	✓	1	20.3	even	4
600.4.a.j.1.1	yes	1	20.7	even	4
600.4.f.h.49.1		2	20.19	odd	2
600.4.f.h.49.2		2	4.3	odd	2
1200.4.a.n.1.1		1	5.2	odd	4
1200.4.a.x.1.1		1	5.3	odd	4
1200.4.f.f.49.1		2	1.1	even	1	trivial
1200.4.f.f.49.2		2	5.4	even	2	inner
1800.4.a.g.1.1		1	60.47	odd	4
1800.4.a.bf.1.1		1	60.23	odd	4
1800.4.f.g.649.1		2	60.59	even	2
1800.4.f.g.649.2		2	12.11	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)

\(f(q)\)	\(=\)	\(q-3.00000i q^{3} -19.0000i q^{7} -9.00000 q^{9} +O(q^{10})\)	\(q-3.00000i q^{3} -19.0000i q^{7} -9.00000 q^{9} -22.0000 q^{11} +1.00000i q^{13} +58.0000i q^{17} -53.0000 q^{19} -57.0000 q^{21} -58.0000i q^{23} +27.0000i q^{27} -22.0000 q^{29} +35.0000 q^{31} +66.0000i q^{33} +270.000i q^{37} +3.00000 q^{39} -468.000 q^{41} +431.000i q^{43} -230.000i q^{47} -18.0000 q^{49} +174.000 q^{51} +159.000i q^{57} +446.000 q^{59} +127.000 q^{61} +171.000i q^{63} -811.000i q^{67} -174.000 q^{69} -36.0000 q^{71} +522.000i q^{73} +418.000i q^{77} +1368.00 q^{79} +81.0000 q^{81} +1138.00i q^{83} +66.0000i q^{87} -144.000 q^{89} +19.0000 q^{91} -105.000i q^{93} +1079.00i q^{97} +198.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 18 q^{9}+O(q^{10}) \) 2 * q - 18 * q^9	\( 2 q - 18 q^{9} - 44 q^{11} - 106 q^{19} - 114 q^{21} - 44 q^{29} + 70 q^{31} + 6 q^{39} - 936 q^{41} - 36 q^{49} + 348 q^{51} + 892 q^{59} + 254 q^{61} - 348 q^{69} - 72 q^{71} + 2736 q^{79} + 162 q^{81} - 288 q^{89} + 38 q^{91} + 396 q^{99}+O(q^{100}) \) 2 * q - 18 * q^9 - 44 * q^11 - 106 * q^19 - 114 * q^21 - 44 * q^29 + 70 * q^31 + 6 * q^39 - 936 * q^41 - 36 * q^49 + 348 * q^51 + 892 * q^59 + 254 * q^61 - 348 * q^69 - 72 * q^71 + 2736 * q^79 + 162 * q^81 - 288 * q^89 + 38 * q^91 + 396 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more