LMFDB - Embedded newform 2100.4.k.j.1849.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2100,4,Mod(1849,2100)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2100, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2100.1849");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2100.k (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:	$123.904011012$
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 84)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1849.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	2100.1849
Dual form			2100.4.k.j.1849.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000i q^{3} -7.00000i q^{7} -9.00000 q^{9} +36.0000 q^{11} +62.0000i q^{13} -114.000i q^{17} +76.0000 q^{19} -21.0000 q^{21} -24.0000i q^{23} +27.0000i q^{27} -54.0000 q^{29} -112.000 q^{31} -108.000i q^{33} +178.000i q^{37} +186.000 q^{39} +378.000 q^{41} -172.000i q^{43} +192.000i q^{47} -49.0000 q^{49} -342.000 q^{51} -402.000i q^{53} -228.000i q^{57} -396.000 q^{59} +254.000 q^{61} +63.0000i q^{63} +1012.00i q^{67} -72.0000 q^{69} +840.000 q^{71} +890.000i q^{73} -252.000i q^{77} -80.0000 q^{79} +81.0000 q^{81} -108.000i q^{83} +162.000i q^{87} +1638.00 q^{89} +434.000 q^{91} +336.000i q^{93} -1010.00i q^{97} -324.000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 18 q^{9} + 72 q^{11} + 152 q^{19} - 42 q^{21} - 108 q^{29} - 224 q^{31} + 372 q^{39} + 756 q^{41} - 98 q^{49} - 684 q^{51} - 792 q^{59} + 508 q^{61} - 144 q^{69} + 1680 q^{71} - 160 q^{79} + 162 q^{81}+ \cdots - 648 q^{99}+O(q^{100}) $ 2 * q - 18 * q^9 + 72 * q^11 + 152 * q^19 - 42 * q^21 - 108 * q^29 - 224 * q^31 + 372 * q^39 + 756 * q^41 - 98 * q^49 - 684 * q^51 - 792 * q^59 + 508 * q^61 - 144 * q^69 + 1680 * q^71 - 160 * q^79 + 162 * q^81 + 3276 * q^89 + 868 * q^91 - 648 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2100\mathbb{Z}\right)^\times$.

$n$	$701$	$1051$	$1177$	$1501$
$\chi(n)$	$1$	$1$	$-1$	$1$

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
$7$	− 7.00000i	− 0.377964i
$7$	− 7.00000i	− 0.377964i
$8$	0	0
$9$	−9.00000	−0.333333
$10$	0	0
$11$	36.0000	0.986764	0.493382	−	0.869813i	$-0.335760\pi$
$11$	36.0000	0.986764	0.493382	+	0.869813i	$0.335760\pi$
$12$	0	0
$13$	62.0000i	1.32275i	0.750057	+	0.661373i	$0.230026\pi$
$13$	62.0000i	1.32275i	−0.750057	+	0.661373i	$0.769974\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 114.000i	− 1.62642i	−0.581974	−	0.813208i	$-0.697719\pi$
$17$	− 114.000i	− 1.62642i	0.581974	−	0.813208i	$-0.302281\pi$
$18$	0	0
$19$	76.0000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	76.0000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	0	0
$21$	−21.0000	−0.218218
$22$	0	0
$23$	− 24.0000i	− 0.217580i	−0.994065	−	0.108790i	$-0.965302\pi$
$23$	− 24.0000i	− 0.217580i	0.994065	−	0.108790i	$-0.0346976\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	27.0000i	0.192450i
$28$	0	0
$29$	−54.0000	−0.345778	−0.172889	−	0.984941i	$-0.555310\pi$
$29$	−54.0000	−0.345778	−0.172889	+	0.984941i	$0.555310\pi$
$30$	0	0
$31$	−112.000	−0.648897	−0.324448	−	0.945903i	$-0.605179\pi$
$31$	−112.000	−0.648897	−0.324448	+	0.945903i	$0.605179\pi$
$32$	0	0
$33$	− 108.000i	− 0.569709i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	178.000i	0.790892i	0.918489	+	0.395446i	$0.129410\pi$
$37$	178.000i	0.790892i	−0.918489	+	0.395446i	$0.870590\pi$
$38$	0	0
$39$	186.000	0.763688
$40$	0	0
$41$	378.000	1.43985	0.719923	−	0.694054i	$-0.244177\pi$
$41$	378.000	1.43985	0.719923	+	0.694054i	$0.244177\pi$
$42$	0	0
$43$	− 172.000i	− 0.609994i	−0.952353	−	0.304997i	$-0.901344\pi$
$43$	− 172.000i	− 0.609994i	0.952353	−	0.304997i	$-0.0986555\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	192.000i	0.595874i	0.954586	+	0.297937i	$0.0962985\pi$
$47$	192.000i	0.595874i	−0.954586	+	0.297937i	$0.903701\pi$
$48$	0	0
$49$	−49.0000	−0.142857
$50$	0	0
$51$	−342.000	−0.939011
$52$	0	0
$53$	− 402.000i	− 1.04187i	−0.853597	−	0.520933i	$-0.825584\pi$
$53$	− 402.000i	− 1.04187i	0.853597	−	0.520933i	$-0.174416\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 228.000i	− 0.529813i
$58$	0	0
$59$	−396.000	−0.873810	−0.436905	−	0.899508i	$-0.643925\pi$
$59$	−396.000	−0.873810	−0.436905	+	0.899508i	$0.643925\pi$
$60$	0	0
$61$	254.000	0.533137	0.266569	−	0.963816i	$-0.414110\pi$
$61$	254.000	0.533137	0.266569	+	0.963816i	$0.414110\pi$
$62$	0	0
$63$	63.0000i	0.125988i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	1012.00i	1.84531i	0.385632	+	0.922653i	$0.373984\pi$
$67$	1012.00i	1.84531i	−0.385632	+	0.922653i	$0.626016\pi$
$68$	0	0
$69$	−72.0000	−0.125620
$70$	0	0
$71$	840.000	1.40408	0.702040	−	0.712138i	$-0.252273\pi$
$71$	840.000	1.40408	0.702040	+	0.712138i	$0.252273\pi$
$72$	0	0
$73$	890.000i	1.42694i	0.700686	+	0.713470i	$0.252878\pi$
$73$	890.000i	1.42694i	−0.700686	+	0.713470i	$0.747122\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 252.000i	− 0.372962i
$78$	0	0
$79$	−80.0000	−0.113933	−0.0569665	−	0.998376i	$-0.518143\pi$
$79$	−80.0000	−0.113933	−0.0569665	+	0.998376i	$0.518143\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	− 108.000i	− 0.142826i	−0.997447	−	0.0714129i	$-0.977249\pi$
$83$	− 108.000i	− 0.142826i	0.997447	−	0.0714129i	$-0.0227508\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	162.000i	0.199635i
$88$	0	0
$89$	1638.00	1.95087	0.975436	−	0.220282i	$-0.0706977\pi$
$89$	1638.00	1.95087	0.975436	+	0.220282i	$0.0706977\pi$
$90$	0	0
$91$	434.000	0.499951
$92$	0	0
$93$	336.000i	0.374641i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 1010.00i	− 1.05722i	−0.848866	−	0.528608i	$-0.822714\pi$
$97$	− 1010.00i	− 1.05722i	0.848866	−	0.528608i	$-0.177286\pi$
$98$	0	0
$99$	−324.000	−0.328921
$100$	0	0
$101$	6.00000	0.00591111	0.00295556	−	0.999996i	$-0.499059\pi$
$101$	6.00000	0.00591111	0.00295556	+	0.999996i	$0.499059\pi$
$102$	0	0
$103$	− 472.000i	− 0.451530i	−0.974182	−	0.225765i	$-0.927512\pi$
$103$	− 472.000i	− 0.451530i	0.974182	−	0.225765i	$-0.0724881\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	972.000i	0.878194i	0.898440	+	0.439097i	$0.144702\pi$
$107$	972.000i	0.878194i	−0.898440	+	0.439097i	$0.855298\pi$
$108$	0	0
$109$	1786.00	1.56943	0.784715	−	0.619857i	$-0.212810\pi$
$109$	1786.00	1.56943	0.784715	+	0.619857i	$0.212810\pi$
$110$	0	0
$111$	534.000	0.456622
$112$	0	0
$113$	− 2286.00i	− 1.90309i	−0.307515	−	0.951543i	$-0.599497\pi$
$113$	− 2286.00i	− 1.90309i	0.307515	−	0.951543i	$-0.400503\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 558.000i	− 0.440916i
$118$	0	0
$119$	−798.000	−0.614727
$120$	0	0
$121$	−35.0000	−0.0262960
$122$	0	0
$123$	− 1134.00i	− 0.831295i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 1328.00i	− 0.927881i	−0.885866	−	0.463941i	$-0.846435\pi$
$127$	− 1328.00i	− 0.927881i	0.885866	−	0.463941i	$-0.153565\pi$
$128$	0	0
$129$	−516.000	−0.352180
$130$	0	0
$131$	−1212.00	−0.808343	−0.404171	−	0.914683i	$-0.632440\pi$
$131$	−1212.00	−0.808343	−0.404171	+	0.914683i	$0.632440\pi$
$132$	0	0
$133$	− 532.000i	− 0.346844i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1254.00i	0.782018i	0.920387	+	0.391009i	$0.127874\pi$
$137$	1254.00i	0.782018i	−0.920387	+	0.391009i	$0.872126\pi$
$138$	0	0
$139$	340.000	0.207471	0.103735	−	0.994605i	$-0.466921\pi$
$139$	340.000	0.207471	0.103735	+	0.994605i	$0.466921\pi$
$140$	0	0
$141$	576.000	0.344028
$142$	0	0
$143$	2232.00i	1.30524i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	147.000i	0.0824786i
$148$	0	0
$149$	−1038.00	−0.570713	−0.285357	−	0.958421i	$-0.592112\pi$
$149$	−1038.00	−0.570713	−0.285357	+	0.958421i	$0.592112\pi$
$150$	0	0
$151$	2936.00	1.58231	0.791153	−	0.611618i	$-0.209481\pi$
$151$	2936.00	1.58231	0.791153	+	0.611618i	$0.209481\pi$
$152$	0	0
$153$	1026.00i	0.542138i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	1330.00i	0.676086i	0.941131	+	0.338043i	$0.109765\pi$
$157$	1330.00i	0.676086i	−0.941131	+	0.338043i	$0.890235\pi$
$158$	0	0
$159$	−1206.00	−0.601522
$160$	0	0
$161$	−168.000	−0.0822376
$162$	0	0
$163$	− 3364.00i	− 1.61650i	−0.588842	−	0.808248i	$-0.700416\pi$
$163$	− 3364.00i	− 1.61650i	0.588842	−	0.808248i	$-0.299584\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 3048.00i	− 1.41234i	−0.708041	−	0.706172i	$-0.750421\pi$
$167$	− 3048.00i	− 1.41234i	0.708041	−	0.706172i	$-0.249579\pi$
$168$	0	0
$169$	−1647.00	−0.749659
$170$	0	0
$171$	−684.000	−0.305888
$172$	0	0
$173$	− 2706.00i	− 1.18921i	−0.804018	−	0.594605i	$-0.797308\pi$
$173$	− 2706.00i	− 1.18921i	0.804018	−	0.594605i	$-0.202692\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	1188.00i	0.504495i
$178$	0	0
$179$	−4716.00	−1.96922	−0.984610	−	0.174766i	$-0.944083\pi$
$179$	−4716.00	−1.96922	−0.984610	+	0.174766i	$0.944083\pi$
$180$	0	0
$181$	1910.00	0.784360	0.392180	−	0.919888i	$-0.371721\pi$
$181$	1910.00	0.784360	0.392180	+	0.919888i	$0.371721\pi$
$182$	0	0
$183$	− 762.000i	− 0.307807i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 4104.00i	− 1.60489i
$188$	0	0
$189$	189.000	0.0727393
$190$	0	0
$191$	4080.00	1.54565	0.772823	−	0.634621i	$-0.218844\pi$
$191$	4080.00	1.54565	0.772823	+	0.634621i	$0.218844\pi$
$192$	0	0
$193$	− 2686.00i	− 1.00177i	−0.865512	−	0.500887i	$-0.833007\pi$
$193$	− 2686.00i	− 1.00177i	0.865512	−	0.500887i	$-0.166993\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 510.000i	− 0.184447i	−0.995738	−	0.0922233i	$-0.970603\pi$
$197$	− 510.000i	− 0.184447i	0.995738	−	0.0922233i	$-0.0293974\pi$
$198$	0	0
$199$	−1352.00	−0.481612	−0.240806	−	0.970573i	$-0.577412\pi$
$199$	−1352.00	−0.481612	−0.240806	+	0.970573i	$0.577412\pi$
$200$	0	0
$201$	3036.00	1.06539
$202$	0	0
$203$	378.000i	0.130692i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	216.000i	0.0725268i
$208$	0	0
$209$	2736.00	0.905517
$210$	0	0
$211$	−3364.00	−1.09757	−0.548785	−	0.835963i	$-0.684909\pi$
$211$	−3364.00	−1.09757	−0.548785	+	0.835963i	$0.684909\pi$
$212$	0	0
$213$	− 2520.00i	− 0.810646i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	784.000i	0.245260i
$218$	0	0
$219$	2670.00	0.823844
$220$	0	0
$221$	7068.00	2.15134
$222$	0	0
$223$	− 4768.00i	− 1.43179i	−0.698209	−	0.715894i	$-0.746019\pi$
$223$	− 4768.00i	− 1.43179i	0.698209	−	0.715894i	$-0.253981\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 420.000i	− 0.122803i	−0.998113	−	0.0614017i	$-0.980443\pi$
$227$	− 420.000i	− 0.122803i	0.998113	−	0.0614017i	$-0.0195571\pi$
$228$	0	0
$229$	1882.00	0.543083	0.271542	−	0.962427i	$-0.412467\pi$
$229$	1882.00	0.543083	0.271542	+	0.962427i	$0.412467\pi$
$230$	0	0
$231$	−756.000	−0.215330
$232$	0	0
$233$	5082.00i	1.42890i	0.699688	+	0.714448i	$0.253322\pi$
$233$	5082.00i	1.42890i	−0.699688	+	0.714448i	$0.746678\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	240.000i	0.0657792i
$238$	0	0
$239$	5424.00	1.46799	0.733995	−	0.679155i	$-0.237654\pi$
$239$	5424.00	1.46799	0.733995	+	0.679155i	$0.237654\pi$
$240$	0	0
$241$	−2590.00	−0.692268	−0.346134	−	0.938185i	$-0.612506\pi$
$241$	−2590.00	−0.692268	−0.346134	+	0.938185i	$0.612506\pi$
$242$	0	0
$243$	− 243.000i	− 0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	4712.00i	1.21384i
$248$	0	0
$249$	−324.000	−0.0824605
$250$	0	0
$251$	−4932.00	−1.24026	−0.620130	−	0.784499i	$-0.712920\pi$
$251$	−4932.00	−1.24026	−0.620130	+	0.784499i	$0.712920\pi$
$252$	0	0
$253$	− 864.000i	− 0.214700i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	3438.00i	0.834461i	0.908801	+	0.417231i	$0.136999\pi$
$257$	3438.00i	0.834461i	−0.908801	+	0.417231i	$0.863001\pi$
$258$	0	0
$259$	1246.00	0.298929
$260$	0	0
$261$	486.000	0.115259
$262$	0	0
$263$	6120.00i	1.43489i	0.696617	+	0.717444i	$0.254688\pi$
$263$	6120.00i	1.43489i	−0.696617	+	0.717444i	$0.745312\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 4914.00i	− 1.12634i
$268$	0	0
$269$	18.0000	0.00407985	0.00203992	−	0.999998i	$-0.499351\pi$
$269$	18.0000	0.00407985	0.00203992	+	0.999998i	$0.499351\pi$
$270$	0	0
$271$	6896.00	1.54576	0.772882	−	0.634549i	$-0.218814\pi$
$271$	6896.00	1.54576	0.772882	+	0.634549i	$0.218814\pi$
$272$	0	0
$273$	− 1302.00i	− 0.288647i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 6254.00i	− 1.35656i	−0.734805	−	0.678279i	$-0.762726\pi$
$277$	− 6254.00i	− 1.35656i	0.734805	−	0.678279i	$-0.237274\pi$
$278$	0	0
$279$	1008.00	0.216299
$280$	0	0
$281$	1194.00	0.253481	0.126740	−	0.991936i	$-0.459549\pi$
$281$	1194.00	0.253481	0.126740	+	0.991936i	$0.459549\pi$
$282$	0	0
$283$	− 7156.00i	− 1.50311i	−0.659671	−	0.751555i	$-0.729304\pi$
$283$	− 7156.00i	− 1.50311i	0.659671	−	0.751555i	$-0.270696\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 2646.00i	− 0.544211i
$288$	0	0
$289$	−8083.00	−1.64523
$290$	0	0
$291$	−3030.00	−0.610384
$292$	0	0
$293$	− 3738.00i	− 0.745312i	−0.927970	−	0.372656i	$-0.878447\pi$
$293$	− 3738.00i	− 0.745312i	0.927970	−	0.372656i	$-0.121553\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	972.000i	0.189903i
$298$	0	0
$299$	1488.00	0.287804
$300$	0	0
$301$	−1204.00	−0.230556
$302$	0	0
$303$	− 18.0000i	− 0.00341278i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	844.000i	0.156904i	0.996918	+	0.0784522i	$0.0249978\pi$
$307$	844.000i	0.156904i	−0.996918	+	0.0784522i	$0.975002\pi$
$308$	0	0
$309$	−1416.00	−0.260691
$310$	0	0
$311$	6312.00	1.15087	0.575435	−	0.817847i	$-0.304833\pi$
$311$	6312.00	1.15087	0.575435	+	0.817847i	$0.304833\pi$
$312$	0	0
$313$	8282.00i	1.49561i	0.663918	+	0.747806i	$0.268892\pi$
$313$	8282.00i	1.49561i	−0.663918	+	0.747806i	$0.731108\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 9318.00i	− 1.65095i	−0.564439	−	0.825475i	$-0.690907\pi$
$317$	− 9318.00i	− 1.65095i	0.564439	−	0.825475i	$-0.309093\pi$
$318$	0	0
$319$	−1944.00	−0.341201
$320$	0	0
$321$	2916.00	0.507026
$322$	0	0
$323$	− 8664.00i	− 1.49250i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 5358.00i	− 0.906110i
$328$	0	0
$329$	1344.00	0.225219
$330$	0	0
$331$	1652.00	0.274327	0.137163	−	0.990548i	$-0.456201\pi$
$331$	1652.00	0.274327	0.137163	+	0.990548i	$0.456201\pi$
$332$	0	0
$333$	− 1602.00i	− 0.263631i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	1294.00i	0.209165i	0.994516	+	0.104583i	$0.0333507\pi$
$337$	1294.00i	0.209165i	−0.994516	+	0.104583i	$0.966649\pi$
$338$	0	0
$339$	−6858.00	−1.09875
$340$	0	0
$341$	−4032.00	−0.640308
$342$	0	0
$343$	343.000i	0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 3636.00i	− 0.562509i	−0.959633	−	0.281255i	$-0.909249\pi$
$347$	− 3636.00i	− 0.562509i	0.959633	−	0.281255i	$-0.0907505\pi$
$348$	0	0
$349$	−10478.0	−1.60709	−0.803545	−	0.595244i	$-0.797055\pi$
$349$	−10478.0	−1.60709	−0.803545	+	0.595244i	$0.797055\pi$
$350$	0	0
$351$	−1674.00	−0.254563
$352$	0	0
$353$	− 7566.00i	− 1.14079i	−0.821372	−	0.570393i	$-0.806791\pi$
$353$	− 7566.00i	− 1.14079i	0.821372	−	0.570393i	$-0.193209\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	2394.00i	0.354913i
$358$	0	0
$359$	8040.00	1.18199	0.590996	−	0.806675i	$-0.298735\pi$
$359$	8040.00	1.18199	0.590996	+	0.806675i	$0.298735\pi$
$360$	0	0
$361$	−1083.00	−0.157895
$362$	0	0
$363$	105.000i	0.0151820i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 7568.00i	− 1.07642i	−0.842811	−	0.538210i	$-0.819101\pi$
$367$	− 7568.00i	− 1.07642i	0.842811	−	0.538210i	$-0.180899\pi$
$368$	0	0
$369$	−3402.00	−0.479949
$370$	0	0
$371$	−2814.00	−0.393789
$372$	0	0
$373$	− 13522.0i	− 1.87706i	−0.345200	−	0.938529i	$-0.612189\pi$
$373$	− 13522.0i	− 1.87706i	0.345200	−	0.938529i	$-0.387811\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 3348.00i	− 0.457376i
$378$	0	0
$379$	−2468.00	−0.334492	−0.167246	−	0.985915i	$-0.553487\pi$
$379$	−2468.00	−0.334492	−0.167246	+	0.985915i	$0.553487\pi$
$380$	0	0
$381$	−3984.00	−0.535713
$382$	0	0
$383$	− 12336.0i	− 1.64580i	−0.568189	−	0.822898i	$-0.692356\pi$
$383$	− 12336.0i	− 1.64580i	0.568189	−	0.822898i	$-0.307644\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	1548.00i	0.203331i
$388$	0	0
$389$	3762.00	0.490337	0.245168	−	0.969481i	$-0.421157\pi$
$389$	3762.00	0.490337	0.245168	+	0.969481i	$0.421157\pi$
$390$	0	0
$391$	−2736.00	−0.353876
$392$	0	0
$393$	3636.00i	0.466697i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	8770.00i	1.10870i	0.832284	+	0.554350i	$0.187033\pi$
$397$	8770.00i	1.10870i	−0.832284	+	0.554350i	$0.812967\pi$
$398$	0	0
$399$	−1596.00	−0.200250
$400$	0	0
$401$	6642.00	0.827146	0.413573	−	0.910471i	$-0.364281\pi$
$401$	6642.00	0.827146	0.413573	+	0.910471i	$0.364281\pi$
$402$	0	0
$403$	− 6944.00i	− 0.858326i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	6408.00i	0.780424i
$408$	0	0
$409$	1510.00	0.182554	0.0912771	−	0.995826i	$-0.470905\pi$
$409$	1510.00	0.182554	0.0912771	+	0.995826i	$0.470905\pi$
$410$	0	0
$411$	3762.00	0.451498
$412$	0	0
$413$	2772.00i	0.330269i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 1020.00i	− 0.119783i
$418$	0	0
$419$	1260.00	0.146909	0.0734547	−	0.997299i	$-0.476598\pi$
$419$	1260.00	0.146909	0.0734547	+	0.997299i	$0.476598\pi$
$420$	0	0
$421$	3998.00	0.462828	0.231414	−	0.972855i	$-0.425665\pi$
$421$	3998.00	0.462828	0.231414	+	0.972855i	$0.425665\pi$
$422$	0	0
$423$	− 1728.00i	− 0.198625i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 1778.00i	− 0.201507i
$428$	0	0
$429$	6696.00	0.753580
$430$	0	0
$431$	−2736.00	−0.305774	−0.152887	−	0.988244i	$-0.548857\pi$
$431$	−2736.00	−0.305774	−0.152887	+	0.988244i	$0.548857\pi$
$432$	0	0
$433$	2690.00i	0.298552i	0.988796	+	0.149276i	$0.0476943\pi$
$433$	2690.00i	0.298552i	−0.988796	+	0.149276i	$0.952306\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 1824.00i	− 0.199665i
$438$	0	0
$439$	1240.00	0.134811	0.0674054	−	0.997726i	$-0.478528\pi$
$439$	1240.00	0.134811	0.0674054	+	0.997726i	$0.478528\pi$
$440$	0	0
$441$	441.000	0.0476190
$442$	0	0
$443$	− 3900.00i	− 0.418272i	−0.977887	−	0.209136i	$-0.932935\pi$
$443$	− 3900.00i	− 0.418272i	0.977887	−	0.209136i	$-0.0670652\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	3114.00i	0.329501i
$448$	0	0
$449$	10878.0	1.14335	0.571675	−	0.820480i	$-0.306294\pi$
$449$	10878.0	1.14335	0.571675	+	0.820480i	$0.306294\pi$
$450$	0	0
$451$	13608.0	1.42079
$452$	0	0
$453$	− 8808.00i	− 0.913545i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 2330.00i	− 0.238496i	−0.992864	−	0.119248i	$-0.961952\pi$
$457$	− 2330.00i	− 0.238496i	0.992864	−	0.119248i	$-0.0380484\pi$
$458$	0	0
$459$	3078.00	0.313004
$460$	0	0
$461$	15150.0	1.53060	0.765299	−	0.643675i	$-0.222591\pi$
$461$	15150.0	1.53060	0.765299	+	0.643675i	$0.222591\pi$
$462$	0	0
$463$	− 2992.00i	− 0.300324i	−0.988661	−	0.150162i	$-0.952020\pi$
$463$	− 2992.00i	− 0.300324i	0.988661	−	0.150162i	$-0.0479795\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 8724.00i	− 0.864451i	−0.901766	−	0.432225i	$-0.857728\pi$
$467$	− 8724.00i	− 0.864451i	0.901766	−	0.432225i	$-0.142272\pi$
$468$	0	0
$469$	7084.00	0.697460
$470$	0	0
$471$	3990.00	0.390339
$472$	0	0
$473$	− 6192.00i	− 0.601921i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	3618.00i	0.347289i
$478$	0	0
$479$	−9744.00	−0.929467	−0.464734	−	0.885451i	$-0.653850\pi$
$479$	−9744.00	−0.929467	−0.464734	+	0.885451i	$0.653850\pi$
$480$	0	0
$481$	−11036.0	−1.04615
$482$	0	0
$483$	504.000i	0.0474799i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 4136.00i	− 0.384846i	−0.981312	−	0.192423i	$-0.938365\pi$
$487$	− 4136.00i	− 0.384846i	0.981312	−	0.192423i	$-0.0616346\pi$
$488$	0	0
$489$	−10092.0	−0.933284
$490$	0	0
$491$	16212.0	1.49010	0.745048	−	0.667011i	$-0.232426\pi$
$491$	16212.0	1.49010	0.745048	+	0.667011i	$0.232426\pi$
$492$	0	0
$493$	6156.00i	0.562378i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 5880.00i	− 0.530692i
$498$	0	0
$499$	−2396.00	−0.214949	−0.107475	−	0.994208i	$-0.534276\pi$
$499$	−2396.00	−0.214949	−0.107475	+	0.994208i	$0.534276\pi$
$500$	0	0
$501$	−9144.00	−0.815417
$502$	0	0
$503$	13128.0i	1.16371i	0.813291	+	0.581857i	$0.197674\pi$
$503$	13128.0i	1.16371i	−0.813291	+	0.581857i	$0.802326\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	4941.00i	0.432816i
$508$	0	0
$509$	−12798.0	−1.11446	−0.557231	−	0.830357i	$-0.688136\pi$
$509$	−12798.0	−1.11446	−0.557231	+	0.830357i	$0.688136\pi$
$510$	0	0
$511$	6230.00	0.539333
$512$	0	0
$513$	2052.00i	0.176604i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	6912.00i	0.587987i
$518$	0	0
$519$	−8118.00	−0.686591
$520$	0	0
$521$	7386.00	0.621087	0.310544	−	0.950559i	$-0.399489\pi$
$521$	7386.00	0.621087	0.310544	+	0.950559i	$0.399489\pi$
$522$	0	0
$523$	5180.00i	0.433089i	0.976273	+	0.216545i	$0.0694787\pi$
$523$	5180.00i	0.433089i	−0.976273	+	0.216545i	$0.930521\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	12768.0i	1.05538i
$528$	0	0
$529$	11591.0	0.952659
$530$	0	0
$531$	3564.00	0.291270
$532$	0	0
$533$	23436.0i	1.90455i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	14148.0i	1.13693i
$538$	0	0
$539$	−1764.00	−0.140966
$540$	0	0
$541$	4070.00	0.323444	0.161722	−	0.986836i	$-0.448295\pi$
$541$	4070.00	0.323444	0.161722	+	0.986836i	$0.448295\pi$
$542$	0	0
$543$	− 5730.00i	− 0.452851i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 14780.0i	− 1.15530i	−0.816286	−	0.577648i	$-0.803971\pi$
$547$	− 14780.0i	− 1.15530i	0.816286	−	0.577648i	$-0.196029\pi$
$548$	0	0
$549$	−2286.00	−0.177712
$550$	0	0
$551$	−4104.00	−0.317307
$552$	0	0
$553$	560.000i	0.0430626i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	6858.00i	0.521693i	0.965380	+	0.260846i	$0.0840016\pi$
$557$	6858.00i	0.521693i	−0.965380	+	0.260846i	$0.915998\pi$
$558$	0	0
$559$	10664.0	0.806868
$560$	0	0
$561$	−12312.0	−0.926583
$562$	0	0
$563$	6660.00i	0.498553i	0.968432	+	0.249277i	$0.0801929\pi$
$563$	6660.00i	0.498553i	−0.968432	+	0.249277i	$0.919807\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 567.000i	− 0.0419961i
$568$	0	0
$569$	150.000	0.0110515	0.00552577	−	0.999985i	$-0.498241\pi$
$569$	150.000	0.0110515	0.00552577	+	0.999985i	$0.498241\pi$
$570$	0	0
$571$	−8188.00	−0.600100	−0.300050	−	0.953923i	$-0.597003\pi$
$571$	−8188.00	−0.600100	−0.300050	+	0.953923i	$0.597003\pi$
$572$	0	0
$573$	− 12240.0i	− 0.892379i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	5854.00i	0.422366i	0.977447	+	0.211183i	$0.0677316\pi$
$577$	5854.00i	0.422366i	−0.977447	+	0.211183i	$0.932268\pi$
$578$	0	0
$579$	−8058.00	−0.578375
$580$	0	0
$581$	−756.000	−0.0539831
$582$	0	0
$583$	− 14472.0i	− 1.02808i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 17580.0i	− 1.23612i	−0.786130	−	0.618062i	$-0.787918\pi$
$587$	− 17580.0i	− 1.23612i	0.786130	−	0.618062i	$-0.212082\pi$
$588$	0	0
$589$	−8512.00	−0.595468
$590$	0	0
$591$	−1530.00	−0.106490
$592$	0	0
$593$	17154.0i	1.18791i	0.804498	+	0.593955i	$0.202434\pi$
$593$	17154.0i	1.18791i	−0.804498	+	0.593955i	$0.797566\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	4056.00i	0.278059i
$598$	0	0
$599$	18120.0	1.23600	0.617999	−	0.786179i	$-0.287943\pi$
$599$	18120.0	1.23600	0.617999	+	0.786179i	$0.287943\pi$
$600$	0	0
$601$	17546.0	1.19088	0.595438	−	0.803401i	$-0.296979\pi$
$601$	17546.0	1.19088	0.595438	+	0.803401i	$0.296979\pi$
$602$	0	0
$603$	− 9108.00i	− 0.615102i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	14560.0i	0.973595i	0.873515	+	0.486798i	$0.161835\pi$
$607$	14560.0i	0.973595i	−0.873515	+	0.486798i	$0.838165\pi$
$608$	0	0
$609$	1134.00	0.0754548
$610$	0	0
$611$	−11904.0	−0.788190
$612$	0	0
$613$	− 4498.00i	− 0.296366i	−0.988960	−	0.148183i	$-0.952657\pi$
$613$	− 4498.00i	− 0.296366i	0.988960	−	0.148183i	$-0.0473425\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	5478.00i	0.357433i	0.983901	+	0.178716i	$0.0571944\pi$
$617$	5478.00i	0.357433i	−0.983901	+	0.178716i	$0.942806\pi$
$618$	0	0
$619$	−6044.00	−0.392454	−0.196227	−	0.980559i	$-0.562869\pi$
$619$	−6044.00	−0.392454	−0.196227	+	0.980559i	$0.562869\pi$
$620$	0	0
$621$	648.000	0.0418733
$622$	0	0
$623$	− 11466.0i	− 0.737360i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 8208.00i	− 0.522801i
$628$	0	0
$629$	20292.0	1.28632
$630$	0	0
$631$	−15352.0	−0.968547	−0.484274	−	0.874917i	$-0.660916\pi$
$631$	−15352.0	−0.968547	−0.484274	+	0.874917i	$0.660916\pi$
$632$	0	0
$633$	10092.0i	0.633682i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 3038.00i	− 0.188964i
$638$	0	0
$639$	−7560.00	−0.468027
$640$	0	0
$641$	−22398.0	−1.38014	−0.690068	−	0.723744i	$-0.742420\pi$
$641$	−22398.0	−1.38014	−0.690068	+	0.723744i	$0.742420\pi$
$642$	0	0
$643$	3764.00i	0.230852i	0.993316	+	0.115426i	$0.0368233\pi$
$643$	3764.00i	0.230852i	−0.993316	+	0.115426i	$0.963177\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	17688.0i	1.07479i	0.843332	+	0.537393i	$0.180591\pi$
$647$	17688.0i	1.07479i	−0.843332	+	0.537393i	$0.819409\pi$
$648$	0	0
$649$	−14256.0	−0.862245
$650$	0	0
$651$	2352.00	0.141601
$652$	0	0
$653$	19878.0i	1.19125i	0.803263	+	0.595625i	$0.203096\pi$
$653$	19878.0i	1.19125i	−0.803263	+	0.595625i	$0.796904\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 8010.00i	− 0.475647i
$658$	0	0
$659$	20004.0	1.18247	0.591233	−	0.806501i	$-0.298641\pi$
$659$	20004.0	1.18247	0.591233	+	0.806501i	$0.298641\pi$
$660$	0	0
$661$	−1306.00	−0.0768495	−0.0384247	−	0.999261i	$-0.512234\pi$
$661$	−1306.00	−0.0768495	−0.0384247	+	0.999261i	$0.512234\pi$
$662$	0	0
$663$	− 21204.0i	− 1.24207i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	1296.00i	0.0752344i
$668$	0	0
$669$	−14304.0	−0.826644
$670$	0	0
$671$	9144.00	0.526081
$672$	0	0
$673$	− 13054.0i	− 0.747689i	−0.927491	−	0.373845i	$-0.878039\pi$
$673$	− 13054.0i	− 0.747689i	0.927491	−	0.373845i	$-0.121961\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 5046.00i	− 0.286460i	−0.989689	−	0.143230i	$-0.954251\pi$
$677$	− 5046.00i	− 0.286460i	0.989689	−	0.143230i	$-0.0457489\pi$
$678$	0	0
$679$	−7070.00	−0.399590
$680$	0	0
$681$	−1260.00	−0.0709006
$682$	0	0
$683$	12468.0i	0.698499i	0.937030	+	0.349249i	$0.113563\pi$
$683$	12468.0i	0.698499i	−0.937030	+	0.349249i	$0.886437\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 5646.00i	− 0.313549i
$688$	0	0
$689$	24924.0	1.37813
$690$	0	0
$691$	−23212.0	−1.27790	−0.638948	−	0.769250i	$-0.720630\pi$
$691$	−23212.0	−1.27790	−0.638948	+	0.769250i	$0.720630\pi$
$692$	0	0
$693$	2268.00i	0.124321i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 43092.0i	− 2.34179i
$698$	0	0
$699$	15246.0	0.824974
$700$	0	0
$701$	35958.0	1.93740	0.968698	−	0.248241i	$-0.0798526\pi$
$701$	35958.0	1.93740	0.968698	+	0.248241i	$0.0798526\pi$
$702$	0	0
$703$	13528.0i	0.725773i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 42.0000i	− 0.00223419i
$708$	0	0
$709$	−6446.00	−0.341445	−0.170723	−	0.985319i	$-0.554610\pi$
$709$	−6446.00	−0.341445	−0.170723	+	0.985319i	$0.554610\pi$
$710$	0	0
$711$	720.000	0.0379777
$712$	0	0
$713$	2688.00i	0.141187i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 16272.0i	− 0.847544i
$718$	0	0
$719$	4704.00	0.243991	0.121996	−	0.992531i	$-0.461071\pi$
$719$	4704.00	0.243991	0.121996	+	0.992531i	$0.461071\pi$
$720$	0	0
$721$	−3304.00	−0.170662
$722$	0	0
$723$	7770.00i	0.399681i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	10600.0i	0.540760i	0.962754	+	0.270380i	$0.0871493\pi$
$727$	10600.0i	0.540760i	−0.962754	+	0.270380i	$0.912851\pi$
$728$	0	0
$729$	−729.000	−0.0370370
$730$	0	0
$731$	−19608.0	−0.992104
$732$	0	0
$733$	12542.0i	0.631991i	0.948761	+	0.315995i	$0.102338\pi$
$733$	12542.0i	0.631991i	−0.948761	+	0.315995i	$0.897662\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	36432.0i	1.82088i
$738$	0	0
$739$	−23324.0	−1.16101	−0.580506	−	0.814256i	$-0.697145\pi$
$739$	−23324.0	−1.16101	−0.580506	+	0.814256i	$0.697145\pi$
$740$	0	0
$741$	14136.0	0.700808
$742$	0	0
$743$	− 6312.00i	− 0.311662i	−0.987784	−	0.155831i	$-0.950194\pi$
$743$	− 6312.00i	− 0.311662i	0.987784	−	0.155831i	$-0.0498055\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	972.000i	0.0476086i
$748$	0	0
$749$	6804.00	0.331926
$750$	0	0
$751$	35840.0	1.74144	0.870719	−	0.491781i	$-0.163654\pi$
$751$	35840.0	1.74144	0.870719	+	0.491781i	$0.163654\pi$
$752$	0	0
$753$	14796.0i	0.716064i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	34594.0i	1.66095i	0.557055	+	0.830476i	$0.311931\pi$
$757$	34594.0i	1.66095i	−0.557055	+	0.830476i	$0.688069\pi$
$758$	0	0
$759$	−2592.00	−0.123957
$760$	0	0
$761$	23946.0	1.14066	0.570330	−	0.821416i	$-0.306815\pi$
$761$	23946.0	1.14066	0.570330	+	0.821416i	$0.306815\pi$
$762$	0	0
$763$	− 12502.0i	− 0.593188i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 24552.0i	− 1.15583i
$768$	0	0
$769$	−18770.0	−0.880187	−0.440093	−	0.897952i	$-0.645055\pi$
$769$	−18770.0	−0.880187	−0.440093	+	0.897952i	$0.645055\pi$
$770$	0	0
$771$	10314.0	0.481776
$772$	0	0
$773$	30342.0i	1.41181i	0.708309	+	0.705903i	$0.249459\pi$
$773$	30342.0i	1.41181i	−0.708309	+	0.705903i	$0.750541\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 3738.00i	− 0.172587i
$778$	0	0
$779$	28728.0	1.32129
$780$	0	0
$781$	30240.0	1.38550
$782$	0	0
$783$	− 1458.00i	− 0.0665449i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	26188.0i	1.18615i	0.805147	+	0.593076i	$0.202087\pi$
$787$	26188.0i	1.18615i	−0.805147	+	0.593076i	$0.797913\pi$
$788$	0	0
$789$	18360.0	0.828433
$790$	0	0
$791$	−16002.0	−0.719299
$792$	0	0
$793$	15748.0i	0.705205i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	34818.0i	1.54745i	0.633522	+	0.773724i	$0.281609\pi$
$797$	34818.0i	1.54745i	−0.633522	+	0.773724i	$0.718391\pi$
$798$	0	0
$799$	21888.0	0.969139
$800$	0	0
$801$	−14742.0	−0.650291
$802$	0	0
$803$	32040.0i	1.40805i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 54.0000i	− 0.00235550i
$808$	0	0
$809$	21702.0	0.943142	0.471571	−	0.881828i	$-0.343687\pi$
$809$	21702.0	0.943142	0.471571	+	0.881828i	$0.343687\pi$
$810$	0	0
$811$	−20356.0	−0.881376	−0.440688	−	0.897660i	$-0.645265\pi$
$811$	−20356.0	−0.881376	−0.440688	+	0.897660i	$0.645265\pi$
$812$	0	0
$813$	− 20688.0i	− 0.892448i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 13072.0i	− 0.559769i
$818$	0	0
$819$	−3906.00	−0.166650
$820$	0	0
$821$	−19890.0	−0.845513	−0.422756	−	0.906243i	$-0.638937\pi$
$821$	−19890.0	−0.845513	−0.422756	+	0.906243i	$0.638937\pi$
$822$	0	0
$823$	4232.00i	0.179245i	0.995976	+	0.0896223i	$0.0285660\pi$
$823$	4232.00i	0.179245i	−0.995976	+	0.0896223i	$0.971434\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 9636.00i	− 0.405171i	−0.979265	−	0.202586i	$-0.935066\pi$
$827$	− 9636.00i	− 0.405171i	0.979265	−	0.202586i	$-0.0649344\pi$
$828$	0	0
$829$	−35294.0	−1.47866	−0.739331	−	0.673342i	$-0.764858\pi$
$829$	−35294.0	−1.47866	−0.739331	+	0.673342i	$0.764858\pi$
$830$	0	0
$831$	−18762.0	−0.783209
$832$	0	0
$833$	5586.00i	0.232345i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 3024.00i	− 0.124880i
$838$	0	0
$839$	3768.00	0.155049	0.0775243	−	0.996990i	$-0.475298\pi$
$839$	3768.00	0.155049	0.0775243	+	0.996990i	$0.475298\pi$
$840$	0	0
$841$	−21473.0	−0.880438
$842$	0	0
$843$	− 3582.00i	− 0.146347i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	245.000i	0.00993896i
$848$	0	0
$849$	−21468.0	−0.867821
$850$	0	0
$851$	4272.00	0.172083
$852$	0	0
$853$	− 39466.0i	− 1.58416i	−0.610416	−	0.792081i	$-0.708998\pi$
$853$	− 39466.0i	− 1.58416i	0.610416	−	0.792081i	$-0.291002\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	34038.0i	1.35673i	0.734726	+	0.678364i	$0.237311\pi$
$857$	34038.0i	1.35673i	−0.734726	+	0.678364i	$0.762689\pi$
$858$	0	0
$859$	3364.00	0.133618	0.0668092	−	0.997766i	$-0.478718\pi$
$859$	3364.00	0.133618	0.0668092	+	0.997766i	$0.478718\pi$
$860$	0	0
$861$	−7938.00	−0.314200
$862$	0	0
$863$	13104.0i	0.516878i	0.966028	+	0.258439i	$0.0832080\pi$
$863$	13104.0i	0.516878i	−0.966028	+	0.258439i	$0.916792\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	24249.0i	0.949872i
$868$	0	0
$869$	−2880.00	−0.112425
$870$	0	0
$871$	−62744.0	−2.44087
$872$	0	0
$873$	9090.00i	0.352405i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	40858.0i	1.57318i	0.617477	+	0.786589i	$0.288155\pi$
$877$	40858.0i	1.57318i	−0.617477	+	0.786589i	$0.711845\pi$
$878$	0	0
$879$	−11214.0	−0.430306
$880$	0	0
$881$	−37374.0	−1.42924	−0.714621	−	0.699512i	$-0.753401\pi$
$881$	−37374.0	−1.42924	−0.714621	+	0.699512i	$0.753401\pi$
$882$	0	0
$883$	9788.00i	0.373038i	0.982451	+	0.186519i	$0.0597206\pi$
$883$	9788.00i	0.373038i	−0.982451	+	0.186519i	$0.940279\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 50424.0i	− 1.90876i	−0.298591	−	0.954381i	$-0.596517\pi$
$887$	− 50424.0i	− 1.90876i	0.298591	−	0.954381i	$-0.403483\pi$
$888$	0	0
$889$	−9296.00	−0.350706
$890$	0	0
$891$	2916.00	0.109640
$892$	0	0
$893$	14592.0i	0.546811i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 4464.00i	− 0.166163i
$898$	0	0
$899$	6048.00	0.224374
$900$	0	0
$901$	−45828.0	−1.69451
$902$	0	0
$903$	3612.00i	0.133112i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	12412.0i	0.454392i	0.973849	+	0.227196i	$0.0729558\pi$
$907$	12412.0i	0.454392i	−0.973849	+	0.227196i	$0.927044\pi$
$908$	0	0
$909$	−54.0000	−0.00197037
$910$	0	0
$911$	6576.00	0.239158	0.119579	−	0.992825i	$-0.461846\pi$
$911$	6576.00	0.239158	0.119579	+	0.992825i	$0.461846\pi$
$912$	0	0
$913$	− 3888.00i	− 0.140935i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	8484.00i	0.305525i
$918$	0	0
$919$	−8264.00	−0.296631	−0.148316	−	0.988940i	$-0.547385\pi$
$919$	−8264.00	−0.296631	−0.148316	+	0.988940i	$0.547385\pi$
$920$	0	0
$921$	2532.00	0.0905887
$922$	0	0
$923$	52080.0i	1.85724i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	4248.00i	0.150510i
$928$	0	0
$929$	−39426.0	−1.39238	−0.696192	−	0.717855i	$-0.745124\pi$
$929$	−39426.0	−1.39238	−0.696192	+	0.717855i	$0.745124\pi$
$930$	0	0
$931$	−3724.00	−0.131095
$932$	0	0
$933$	− 18936.0i	− 0.664455i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	4678.00i	0.163099i	0.996669	+	0.0815494i	$0.0259868\pi$
$937$	4678.00i	0.163099i	−0.996669	+	0.0815494i	$0.974013\pi$
$938$	0	0
$939$	24846.0	0.863492
$940$	0	0
$941$	−17346.0	−0.600918	−0.300459	−	0.953795i	$-0.597140\pi$
$941$	−17346.0	−0.600918	−0.300459	+	0.953795i	$0.597140\pi$
$942$	0	0
$943$	− 9072.00i	− 0.313282i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 19452.0i	− 0.667482i	−0.942665	−	0.333741i	$-0.891689\pi$
$947$	− 19452.0i	− 0.667482i	0.942665	−	0.333741i	$-0.108311\pi$
$948$	0	0
$949$	−55180.0	−1.88748
$950$	0	0
$951$	−27954.0	−0.953176
$952$	0	0
$953$	4458.00i	0.151531i	0.997126	+	0.0757654i	$0.0241400\pi$
$953$	4458.00i	0.151531i	−0.997126	+	0.0757654i	$0.975860\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	5832.00i	0.196992i
$958$	0	0
$959$	8778.00	0.295575
$960$	0	0
$961$	−17247.0	−0.578933
$962$	0	0
$963$	− 8748.00i	− 0.292731i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 52520.0i	− 1.74657i	−0.487213	−	0.873283i	$-0.661987\pi$
$967$	− 52520.0i	− 1.74657i	0.487213	−	0.873283i	$-0.338013\pi$
$968$	0	0
$969$	−25992.0	−0.861696
$970$	0	0
$971$	−10404.0	−0.343852	−0.171926	−	0.985110i	$-0.554999\pi$
$971$	−10404.0	−0.343852	−0.171926	+	0.985110i	$0.554999\pi$
$972$	0	0
$973$	− 2380.00i	− 0.0784165i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	7566.00i	0.247756i	0.992297	+	0.123878i	$0.0395332\pi$
$977$	7566.00i	0.247756i	−0.992297	+	0.123878i	$0.960467\pi$
$978$	0	0
$979$	58968.0	1.92505
$980$	0	0
$981$	−16074.0	−0.523143
$982$	0	0
$983$	44376.0i	1.43985i	0.694051	+	0.719926i	$0.255824\pi$
$983$	44376.0i	1.43985i	−0.694051	+	0.719926i	$0.744176\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 4032.00i	− 0.130030i
$988$	0	0
$989$	−4128.00	−0.132723
$990$	0	0
$991$	−27328.0	−0.875986	−0.437993	−	0.898978i	$-0.644311\pi$
$991$	−27328.0	−0.875986	−0.437993	+	0.898978i	$0.644311\pi$
$992$	0	0
$993$	− 4956.00i	− 0.158383i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 2774.00i	− 0.0881178i	−0.999029	−	0.0440589i	$-0.985971\pi$
$997$	− 2774.00i	− 0.0881178i	0.999029	−	0.0440589i	$-0.0140289\pi$
$998$	0	0
$999$	−4806.00	−0.152207

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2100.4.k.j.1849.1		2
5.2	odd	4		84.4.a.a.1.1	✓	1
5.3	odd	4		2100.4.a.l.1.1		1
5.4	even	2	inner	2100.4.k.j.1849.2		2
15.2	even	4		252.4.a.b.1.1		1
20.7	even	4		336.4.a.k.1.1		1
35.2	odd	12		588.4.i.f.361.1		2
35.12	even	12		588.4.i.c.361.1		2
35.17	even	12		588.4.i.c.373.1		2
35.27	even	4		588.4.a.d.1.1		1
35.32	odd	12		588.4.i.f.373.1		2
40.27	even	4		1344.4.a.d.1.1		1
40.37	odd	4		1344.4.a.q.1.1		1
60.47	odd	4		1008.4.a.h.1.1		1
105.2	even	12		1764.4.k.l.361.1		2
105.17	odd	12		1764.4.k.f.1549.1		2
105.32	even	12		1764.4.k.l.1549.1		2
105.47	odd	12		1764.4.k.f.361.1		2
105.62	odd	4		1764.4.a.j.1.1		1
140.27	odd	4		2352.4.a.d.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.4.a.a.1.1	✓	1	5.2	odd	4
252.4.a.b.1.1		1	15.2	even	4
336.4.a.k.1.1		1	20.7	even	4
588.4.a.d.1.1		1	35.27	even	4
588.4.i.c.361.1		2	35.12	even	12
588.4.i.c.373.1		2	35.17	even	12
588.4.i.f.361.1		2	35.2	odd	12
588.4.i.f.373.1		2	35.32	odd	12
1008.4.a.h.1.1		1	60.47	odd	4
1344.4.a.d.1.1		1	40.27	even	4
1344.4.a.q.1.1		1	40.37	odd	4
1764.4.a.j.1.1		1	105.62	odd	4
1764.4.k.f.361.1		2	105.47	odd	12
1764.4.k.f.1549.1		2	105.17	odd	12
1764.4.k.l.361.1		2	105.2	even	12
1764.4.k.l.1549.1		2	105.32	even	12
2100.4.a.l.1.1		1	5.3	odd	4
2100.4.k.j.1849.1		2	1.1	even	1	trivial
2100.4.k.j.1849.2		2	5.4	even	2	inner
2352.4.a.d.1.1		1	140.27	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2100.k (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.00000i q^{3} -7.00000i q^{7} -9.00000 q^{9} +36.0000 q^{11} +62.0000i q^{13} -114.000i q^{17} +76.0000 q^{19} -21.0000 q^{21} -24.0000i q^{23} +27.0000i q^{27} -54.0000 q^{29} -112.000 q^{31} -108.000i q^{33} +178.000i q^{37} +186.000 q^{39} +378.000 q^{41} -172.000i q^{43} +192.000i q^{47} -49.0000 q^{49} -342.000 q^{51} -402.000i q^{53} -228.000i q^{57} -396.000 q^{59} +254.000 q^{61} +63.0000i q^{63} +1012.00i q^{67} -72.0000 q^{69} +840.000 q^{71} +890.000i q^{73} -252.000i q^{77} -80.0000 q^{79} +81.0000 q^{81} -108.000i q^{83} +162.000i q^{87} +1638.00 q^{89} +434.000 q^{91} +336.000i q^{93} -1010.00i q^{97} -324.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 18 q^{9} + 72 q^{11} + 152 q^{19} - 42 q^{21} - 108 q^{29} - 224 q^{31} + 372 q^{39} + 756 q^{41} - 98 q^{49} - 684 q^{51} - 792 q^{59} + 508 q^{61} - 144 q^{69} + 1680 q^{71} - 160 q^{79} + 162 q^{81}+ \cdots - 648 q^{99}+O(q^{100}) \) 2 * q - 18 * q^9 + 72 * q^11 + 152 * q^19 - 42 * q^21 - 108 * q^29 - 224 * q^31 + 372 * q^39 + 756 * q^41 - 98 * q^49 - 684 * q^51 - 792 * q^59 + 508 * q^61 - 144 * q^69 + 1680 * q^71 - 160 * q^79 + 162 * q^81 + 3276 * q^89 + 868 * q^91 - 648 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

Properties

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