LMFDB - Embedded newform 2100.4.k.m.1849.3

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:	$4$
Coefficient field:	$\Q(i, \sqrt{109})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 55x^{2} + 729 $ x^4 + 55*x^2 + 729
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 420)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1849.3
Root			$4.72015i$ of defining polynomial
Character	$\chi$	$=$	2100.1849
Dual form			2100.4.k.m.1849.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000i q^{3} +7.00000i q^{7} -9.00000 q^{9} -56.6418 q^{11} -46.6418i q^{13} -66.0000i q^{17} -52.6418 q^{19} -21.0000 q^{21} -12.0000i q^{23} -27.0000i q^{27} -54.0000 q^{29} -65.9255 q^{31} -169.926i q^{33} +140.716i q^{37} +139.926 q^{39} +270.000 q^{41} -146.716i q^{43} +291.851i q^{47} -49.0000 q^{49} +198.000 q^{51} -374.642i q^{53} -157.926i q^{57} -77.2837 q^{59} +758.000 q^{61} -63.0000i q^{63} +356.000i q^{67} +36.0000 q^{69} +820.344 q^{71} +401.209i q^{73} -396.493i q^{77} -707.851 q^{79} +81.0000 q^{81} +900.986i q^{83} -162.000i q^{87} +636.269 q^{89} +326.493 q^{91} -197.777i q^{93} +120.074i q^{97} +509.777 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 36 q^{9} + 24 q^{11} + 40 q^{19} - 84 q^{21} - 216 q^{29} + 488 q^{31} - 192 q^{39} + 1080 q^{41} - 196 q^{49} + 792 q^{51} + 192 q^{59} + 3032 q^{61} + 144 q^{69} + 24 q^{71} - 1328 q^{79} + 324 q^{81}+ \cdots - 216 q^{99}+O(q^{100}) $ 4 * q - 36 * q^9 + 24 * q^11 + 40 * q^19 - 84 * q^21 - 216 * q^29 + 488 * q^31 - 192 * q^39 + 1080 * q^41 - 196 * q^49 + 792 * q^51 + 192 * q^59 + 3032 * q^61 + 144 * q^69 + 24 * q^71 - 1328 * q^79 + 324 * q^81 - 1464 * q^89 - 448 * q^91 - 216 * 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
$11$	−56.6418	−1.55256	−0.776280	−	0.630388i	$-0.782896\pi$
$11$	−56.6418	−1.55256	−0.776280	+	0.630388i	$0.782896\pi$
$12$	0	0
$13$	− 46.6418i	− 0.995086i	−0.867439	−	0.497543i	$-0.834236\pi$
$13$	− 46.6418i	− 0.995086i	0.867439	−	0.497543i	$-0.165764\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 66.0000i	− 0.941609i	−0.882238	−	0.470804i	$-0.843964\pi$
$17$	− 66.0000i	− 0.941609i	0.882238	−	0.470804i	$-0.156036\pi$
$18$	0	0
$19$	−52.6418	−0.635625	−0.317812	−	0.948154i	$-0.602948\pi$
$19$	−52.6418	−0.635625	−0.317812	+	0.948154i	$0.602948\pi$
$20$	0	0
$21$	−21.0000	−0.218218
$22$	0	0
$23$	− 12.0000i	− 0.108790i	−0.998519	−	0.0543951i	$-0.982677\pi$
$23$	− 12.0000i	− 0.108790i	0.998519	−	0.0543951i	$-0.0173230\pi$
$24$	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
$31$	−65.9255	−0.381954	−0.190977	−	0.981595i	$-0.561166\pi$
$31$	−65.9255	−0.381954	−0.190977	+	0.981595i	$0.561166\pi$
$32$	0	0
$33$	− 169.926i	− 0.896371i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	140.716i	0.625233i	0.949879	+	0.312616i	$0.101205\pi$
$37$	140.716i	0.625233i	−0.949879	+	0.312616i	$0.898795\pi$
$38$	0	0
$39$	139.926	0.574513
$40$	0	0
$41$	270.000	1.02846	0.514231	−	0.857652i	$-0.328078\pi$
$41$	270.000	1.02846	0.514231	+	0.857652i	$0.328078\pi$
$42$	0	0
$43$	− 146.716i	− 0.520326i	−0.965565	−	0.260163i	$-0.916224\pi$
$43$	− 146.716i	− 0.520326i	0.965565	−	0.260163i	$-0.0837763\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	291.851i	0.905763i	0.891571	+	0.452881i	$0.149604\pi$
$47$	291.851i	0.905763i	−0.891571	+	0.452881i	$0.850396\pi$
$48$	0	0
$49$	−49.0000	−0.142857
$50$	0	0
$51$	198.000	0.543638
$52$	0	0
$53$	− 374.642i	− 0.970963i	−0.874247	−	0.485481i	$-0.838644\pi$
$53$	− 374.642i	− 0.970963i	0.874247	−	0.485481i	$-0.161356\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 157.926i	− 0.366978i
$58$	0	0
$59$	−77.2837	−0.170534	−0.0852668	−	0.996358i	$-0.527174\pi$
$59$	−77.2837	−0.170534	−0.0852668	+	0.996358i	$0.527174\pi$
$60$	0	0
$61$	758.000	1.59102	0.795508	−	0.605943i	$-0.207204\pi$
$61$	758.000	1.59102	0.795508	+	0.605943i	$0.207204\pi$
$62$	0	0
$63$	− 63.0000i	− 0.125988i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	356.000i	0.649139i	0.945862	+	0.324570i	$0.105219\pi$
$67$	356.000i	0.649139i	−0.945862	+	0.324570i	$0.894781\pi$
$68$	0	0
$69$	36.0000	0.0628100
$70$	0	0
$71$	820.344	1.37122	0.685612	−	0.727967i	$-0.259535\pi$
$71$	820.344	1.37122	0.685612	+	0.727967i	$0.259535\pi$
$72$	0	0
$73$	401.209i	0.643260i	0.946865	+	0.321630i	$0.104231\pi$
$73$	401.209i	0.643260i	−0.946865	+	0.321630i	$0.895769\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 396.493i	− 0.586812i
$78$	0	0
$79$	−707.851	−1.00809	−0.504047	−	0.863676i	$-0.668156\pi$
$79$	−707.851	−1.00809	−0.504047	+	0.863676i	$0.668156\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	900.986i	1.19152i	0.803163	+	0.595759i	$0.203149\pi$
$83$	900.986i	1.19152i	−0.803163	+	0.595759i	$0.796851\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 162.000i	− 0.199635i
$88$	0	0
$89$	636.269	0.757803	0.378901	−	0.925437i	$-0.376302\pi$
$89$	636.269	0.757803	0.378901	+	0.925437i	$0.376302\pi$
$90$	0	0
$91$	326.493	0.376107
$92$	0	0
$93$	− 197.777i	− 0.220521i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	120.074i	0.125688i	0.998023	+	0.0628439i	$0.0200170\pi$
$97$	120.074i	0.125688i	−0.998023	+	0.0628439i	$0.979983\pi$
$98$	0	0
$99$	509.777	0.517520
$100$	0	0
$101$	1668.27	1.64355	0.821777	−	0.569809i	$-0.192983\pi$
$101$	1668.27	1.64355	0.821777	+	0.569809i	$0.192983\pi$
$102$	0	0
$103$	1450.27i	1.38737i	0.720278	+	0.693686i	$0.244014\pi$
$103$	1450.27i	1.38737i	−0.720278	+	0.693686i	$0.755986\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	257.284i	0.232454i	0.993223	+	0.116227i	$0.0370800\pi$
$107$	257.284i	0.232454i	−0.993223	+	0.116227i	$0.962920\pi$
$108$	0	0
$109$	−2042.54	−1.79486	−0.897430	−	0.441157i	$-0.854568\pi$
$109$	−2042.54	−1.79486	−0.897430	+	0.441157i	$0.854568\pi$
$110$	0	0
$111$	−422.149	−0.360978
$112$	0	0
$113$	− 85.2092i	− 0.0709363i	−0.999371	−	0.0354682i	$-0.988708\pi$
$113$	− 85.2092i	− 0.0709363i	0.999371	−	0.0354682i	$-0.0112922\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	419.777i	0.331695i
$118$	0	0
$119$	462.000	0.355895
$120$	0	0
$121$	1877.30	1.41044
$122$	0	0
$123$	810.000i	0.593782i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	526.418i	0.367812i	0.982944	+	0.183906i	$0.0588741\pi$
$127$	526.418i	0.367812i	−0.982944	+	0.183906i	$0.941126\pi$
$128$	0	0
$129$	440.149	0.300411
$130$	0	0
$131$	−1656.54	−1.10483	−0.552414	−	0.833570i	$-0.686293\pi$
$131$	−1656.54	−1.10483	−0.552414	+	0.833570i	$0.686293\pi$
$132$	0	0
$133$	− 368.493i	− 0.240243i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 177.805i	− 0.110883i	−0.998462	−	0.0554413i	$-0.982343\pi$
$137$	− 177.805i	− 0.110883i	0.998462	−	0.0554413i	$-0.0176566\pi$
$138$	0	0
$139$	963.060	0.587667	0.293833	−	0.955857i	$-0.405069\pi$
$139$	963.060	0.587667	0.293833	+	0.955857i	$0.405069\pi$
$140$	0	0
$141$	−875.553	−0.522942
$142$	0	0
$143$	2641.88i	1.54493i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 147.000i	− 0.0824786i
$148$	0	0
$149$	−852.716	−0.468841	−0.234420	−	0.972135i	$-0.575319\pi$
$149$	−852.716	−0.468841	−0.234420	+	0.972135i	$0.575319\pi$
$150$	0	0
$151$	−3166.81	−1.70670	−0.853348	−	0.521341i	$-0.825432\pi$
$151$	−3166.81	−1.70670	−0.853348	+	0.521341i	$0.825432\pi$
$152$	0	0
$153$	594.000i	0.313870i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 1327.18i	− 0.674653i	−0.941388	−	0.337327i	$-0.890477\pi$
$157$	− 1327.18i	− 0.674653i	0.941388	−	0.337327i	$-0.109523\pi$
$158$	0	0
$159$	1123.93	0.560585
$160$	0	0
$161$	84.0000	0.0411188
$162$	0	0
$163$	− 1759.55i	− 0.845514i	−0.906243	−	0.422757i	$-0.861062\pi$
$163$	− 1759.55i	− 0.845514i	0.906243	−	0.422757i	$-0.138938\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 3186.27i	− 1.47641i	−0.674575	−	0.738206i	$-0.735673\pi$
$167$	− 3186.27i	− 1.47641i	0.674575	−	0.738206i	$-0.264327\pi$
$168$	0	0
$169$	21.5388	0.00980375
$170$	0	0
$171$	473.777	0.211875
$172$	0	0
$173$	− 570.986i	− 0.250932i	−0.992098	−	0.125466i	$-0.959957\pi$
$173$	− 570.986i	− 0.250932i	0.992098	−	0.125466i	$-0.0400426\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 231.851i	− 0.0984576i
$178$	0	0
$179$	1865.24	0.778851	0.389426	−	0.921058i	$-0.372673\pi$
$179$	1865.24	0.778851	0.389426	+	0.921058i	$0.372673\pi$
$180$	0	0
$181$	3651.52	1.49953	0.749767	−	0.661702i	$-0.230166\pi$
$181$	3651.52	1.49953	0.749767	+	0.661702i	$0.230166\pi$
$182$	0	0
$183$	2274.00i	0.918573i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	3738.36i	1.46190i
$188$	0	0
$189$	189.000	0.0727393
$190$	0	0
$191$	1437.63	0.544623	0.272312	−	0.962209i	$-0.412212\pi$
$191$	1437.63	0.544623	0.272312	+	0.962209i	$0.412212\pi$
$192$	0	0
$193$	1347.73i	0.502652i	0.967903	+	0.251326i	$0.0808666\pi$
$193$	1347.73i	0.502652i	−0.967903	+	0.251326i	$0.919133\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1275.09i	0.461149i	0.973055	+	0.230574i	$0.0740605\pi$
$197$	1275.09i	0.461149i	−0.973055	+	0.230574i	$0.925939\pi$
$198$	0	0
$199$	−790.372	−0.281548	−0.140774	−	0.990042i	$-0.544959\pi$
$199$	−790.372	−0.281548	−0.140774	+	0.990042i	$0.544959\pi$
$200$	0	0
$201$	−1068.00	−0.374781
$202$	0	0
$203$	− 378.000i	− 0.130692i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	108.000i	0.0362634i
$208$	0	0
$209$	2981.73	0.986845
$210$	0	0
$211$	3343.55	1.09090	0.545449	−	0.838144i	$-0.316359\pi$
$211$	3343.55	1.09090	0.545449	+	0.838144i	$0.316359\pi$
$212$	0	0
$213$	2461.03i	0.791677i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 461.479i	− 0.144365i
$218$	0	0
$219$	−1203.63	−0.371386
$220$	0	0
$221$	−3078.36	−0.936982
$222$	0	0
$223$	3091.85i	0.928456i	0.885716	+	0.464228i	$0.153668\pi$
$223$	3091.85i	0.928456i	−0.885716	+	0.464228i	$0.846332\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 6024.63i	− 1.76154i	−0.473548	−	0.880768i	$-0.657027\pi$
$227$	− 6024.63i	− 1.76154i	0.473548	−	0.880768i	$-0.342973\pi$
$228$	0	0
$229$	−1470.30	−0.424280	−0.212140	−	0.977239i	$-0.568043\pi$
$229$	−1470.30	−0.424280	−0.212140	+	0.977239i	$0.568043\pi$
$230$	0	0
$231$	1189.48	0.338796
$232$	0	0
$233$	2599.03i	0.730765i	0.930857	+	0.365382i	$0.119062\pi$
$233$	2599.03i	0.730765i	−0.930857	+	0.365382i	$0.880938\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 2123.55i	− 0.582024i
$238$	0	0
$239$	395.060	0.106922	0.0534609	−	0.998570i	$-0.482975\pi$
$239$	395.060	0.106922	0.0534609	+	0.998570i	$0.482975\pi$
$240$	0	0
$241$	4140.66	1.10674	0.553368	−	0.832937i	$-0.313342\pi$
$241$	4140.66	1.10674	0.553368	+	0.832937i	$0.313342\pi$
$242$	0	0
$243$	243.000i	0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	2455.31i	0.632501i
$248$	0	0
$249$	−2702.96	−0.687924
$250$	0	0
$251$	−1013.28	−0.254812	−0.127406	−	0.991851i	$-0.540665\pi$
$251$	−1013.28	−0.254812	−0.127406	+	0.991851i	$0.540665\pi$
$252$	0	0
$253$	679.702i	0.168903i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	6981.50i	1.69453i	0.531172	+	0.847264i	$0.321752\pi$
$257$	6981.50i	1.69453i	−0.531172	+	0.847264i	$0.678248\pi$
$258$	0	0
$259$	−985.014	−0.236316
$260$	0	0
$261$	486.000	0.115259
$262$	0	0
$263$	6093.58i	1.42869i	0.699792	+	0.714347i	$0.253276\pi$
$263$	6093.58i	1.42869i	−0.699792	+	0.714347i	$0.746724\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	1908.81i	0.437517i
$268$	0	0
$269$	−706.298	−0.160088	−0.0800441	−	0.996791i	$-0.525506\pi$
$269$	−706.298	−0.160088	−0.0800441	+	0.996791i	$0.525506\pi$
$270$	0	0
$271$	3520.73	0.789186	0.394593	−	0.918856i	$-0.370886\pi$
$271$	3520.73	0.789186	0.394593	+	0.918856i	$0.370886\pi$
$272$	0	0
$273$	979.479i	0.217146i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	1369.55i	0.297070i	0.988907	+	0.148535i	$0.0474558\pi$
$277$	1369.55i	0.297070i	−0.988907	+	0.148535i	$0.952544\pi$
$278$	0	0
$279$	593.330	0.127318
$280$	0	0
$281$	7204.66	1.52952	0.764758	−	0.644318i	$-0.222859\pi$
$281$	7204.66	1.52952	0.764758	+	0.644318i	$0.222859\pi$
$282$	0	0
$283$	189.192i	0.0397395i	0.999803	+	0.0198698i	$0.00632516\pi$
$283$	189.192i	0.0397395i	−0.999803	+	0.0198698i	$0.993675\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1890.00i	0.388722i
$288$	0	0
$289$	557.000	0.113373
$290$	0	0
$291$	−360.223	−0.0725659
$292$	0	0
$293$	1816.48i	0.362183i	0.983466	+	0.181092i	$0.0579630\pi$
$293$	1816.48i	0.362183i	−0.983466	+	0.181092i	$0.942037\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	1529.33i	0.298790i
$298$	0	0
$299$	−559.702	−0.108256
$300$	0	0
$301$	1027.01	0.196665
$302$	0	0
$303$	5004.81i	0.948907i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	1766.18i	0.328342i	0.986432	+	0.164171i	$0.0524949\pi$
$307$	1766.18i	0.328342i	−0.986432	+	0.164171i	$0.947505\pi$
$308$	0	0
$309$	−4350.81	−0.800999
$310$	0	0
$311$	−2542.03	−0.463489	−0.231745	−	0.972777i	$-0.574443\pi$
$311$	−2542.03	−0.463489	−0.231745	+	0.972777i	$0.574443\pi$
$312$	0	0
$313$	8114.97i	1.46545i	0.680525	+	0.732724i	$0.261752\pi$
$313$	8114.97i	1.46545i	−0.680525	+	0.732724i	$0.738248\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	4997.15i	0.885388i	0.896673	+	0.442694i	$0.145977\pi$
$317$	4997.15i	0.885388i	−0.896673	+	0.442694i	$0.854023\pi$
$318$	0	0
$319$	3058.66	0.536840
$320$	0	0
$321$	−771.851	−0.134207
$322$	0	0
$323$	3474.36i	0.598510i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 6127.62i	− 1.03626i
$328$	0	0
$329$	−2042.96	−0.342346
$330$	0	0
$331$	2593.64	0.430693	0.215346	−	0.976538i	$-0.430912\pi$
$331$	2593.64	0.430693	0.215346	+	0.976538i	$0.430912\pi$
$332$	0	0
$333$	− 1266.45i	− 0.208411i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	9363.97i	1.51361i	0.653638	+	0.756807i	$0.273242\pi$
$337$	9363.97i	1.51361i	−0.653638	+	0.756807i	$0.726758\pi$
$338$	0	0
$339$	255.628	0.0409551
$340$	0	0
$341$	3734.14	0.593006
$342$	0	0
$343$	− 343.000i	− 0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	11371.3i	1.75921i	0.475705	+	0.879605i	$0.342193\pi$
$347$	11371.3i	1.75921i	−0.475705	+	0.879605i	$0.657807\pi$
$348$	0	0
$349$	10479.9	1.60738	0.803692	−	0.595046i	$-0.202866\pi$
$349$	10479.9	1.60738	0.803692	+	0.595046i	$0.202866\pi$
$350$	0	0
$351$	−1259.33	−0.191504
$352$	0	0
$353$	− 4551.49i	− 0.686264i	−0.939287	−	0.343132i	$-0.888512\pi$
$353$	− 4551.49i	− 0.686264i	0.939287	−	0.343132i	$-0.111488\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	1386.00i	0.205476i
$358$	0	0
$359$	−6705.27	−0.985768	−0.492884	−	0.870095i	$-0.664057\pi$
$359$	−6705.27	−0.985768	−0.492884	+	0.870095i	$0.664057\pi$
$360$	0	0
$361$	−4087.84	−0.595981
$362$	0	0
$363$	5631.89i	0.814319i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	5880.57i	0.836413i	0.908352	+	0.418206i	$0.137341\pi$
$367$	5880.57i	0.836413i	−0.908352	+	0.418206i	$0.862659\pi$
$368$	0	0
$369$	−2430.00	−0.342820
$370$	0	0
$371$	2622.49	0.366989
$372$	0	0
$373$	− 13540.7i	− 1.87965i	−0.341652	−	0.939827i	$-0.610986\pi$
$373$	− 13540.7i	− 1.87965i	0.341652	−	0.939827i	$-0.389014\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	2518.66i	0.344078i
$378$	0	0
$379$	−10982.5	−1.48847	−0.744236	−	0.667917i	$-0.767186\pi$
$379$	−10982.5	−1.48847	−0.744236	+	0.667917i	$0.767186\pi$
$380$	0	0
$381$	−1579.26	−0.212356
$382$	0	0
$383$	− 11647.9i	− 1.55399i	−0.629505	−	0.776996i	$-0.716742\pi$
$383$	− 11647.9i	− 1.55399i	0.629505	−	0.776996i	$-0.283258\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	1320.45i	0.173442i
$388$	0	0
$389$	5323.17	0.693819	0.346909	−	0.937899i	$-0.387231\pi$
$389$	5323.17	0.693819	0.346909	+	0.937899i	$0.387231\pi$
$390$	0	0
$391$	−792.000	−0.102438
$392$	0	0
$393$	− 4969.62i	− 0.637873i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 3272.35i	− 0.413689i	−0.978374	−	0.206845i	$-0.933681\pi$
$397$	− 3272.35i	− 0.413689i	0.978374	−	0.206845i	$-0.0663194\pi$
$398$	0	0
$399$	1105.48	0.138705
$400$	0	0
$401$	8255.73	1.02811	0.514054	−	0.857758i	$-0.328143\pi$
$401$	8255.73	1.02811	0.514054	+	0.857758i	$0.328143\pi$
$402$	0	0
$403$	3074.89i	0.380077i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 7970.43i	− 0.970712i
$408$	0	0
$409$	−15949.6	−1.92826	−0.964131	−	0.265427i	$-0.914487\pi$
$409$	−15949.6	−1.92826	−0.964131	+	0.265427i	$0.914487\pi$
$410$	0	0
$411$	533.415	0.0640181
$412$	0	0
$413$	− 540.986i	− 0.0644556i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	2889.18i	0.339290i
$418$	0	0
$419$	5743.16	0.669623	0.334811	−	0.942285i	$-0.391327\pi$
$419$	5743.16	0.669623	0.334811	+	0.942285i	$0.391327\pi$
$420$	0	0
$421$	3396.92	0.393244	0.196622	−	0.980479i	$-0.437003\pi$
$421$	3396.92	0.393244	0.196622	+	0.980479i	$0.437003\pi$
$422$	0	0
$423$	− 2626.66i	− 0.301921i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	5306.00i	0.601347i
$428$	0	0
$429$	−7925.64	−0.891966
$430$	0	0
$431$	15383.0	1.71919	0.859596	−	0.510974i	$-0.170715\pi$
$431$	15383.0	1.71919	0.859596	+	0.510974i	$0.170715\pi$
$432$	0	0
$433$	17906.1i	1.98732i	0.112409	+	0.993662i	$0.464143\pi$
$433$	17906.1i	1.98732i	−0.112409	+	0.993662i	$0.535857\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	631.702i	0.0691497i
$438$	0	0
$439$	−13900.3	−1.51122	−0.755609	−	0.655023i	$-0.772659\pi$
$439$	−13900.3	−1.51122	−0.755609	+	0.655023i	$0.772659\pi$
$440$	0	0
$441$	441.000	0.0476190
$442$	0	0
$443$	4219.35i	0.452522i	0.974067	+	0.226261i	$0.0726502\pi$
$443$	4219.35i	0.452522i	−0.974067	+	0.226261i	$0.927350\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 2558.15i	− 0.270685i
$448$	0	0
$449$	6709.89	0.705254	0.352627	−	0.935764i	$-0.385288\pi$
$449$	6709.89	0.705254	0.352627	+	0.935764i	$0.385288\pi$
$450$	0	0
$451$	−15293.3	−1.59675
$452$	0	0
$453$	− 9500.42i	− 0.985362i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 17330.9i	− 1.77398i	−0.461793	−	0.886988i	$-0.652794\pi$
$457$	− 17330.9i	− 1.77398i	0.461793	−	0.886988i	$-0.347206\pi$
$458$	0	0
$459$	−1782.00	−0.181213
$460$	0	0
$461$	−4891.43	−0.494179	−0.247090	−	0.968993i	$-0.579474\pi$
$461$	−4891.43	−0.494179	−0.247090	+	0.968993i	$0.579474\pi$
$462$	0	0
$463$	16496.6i	1.65585i	0.560835	+	0.827927i	$0.310480\pi$
$463$	16496.6i	1.65585i	−0.560835	+	0.827927i	$0.689520\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	6046.48i	0.599138i	0.954075	+	0.299569i	$0.0968429\pi$
$467$	6046.48i	0.599138i	−0.954075	+	0.299569i	$0.903157\pi$
$468$	0	0
$469$	−2492.00	−0.245352
$470$	0	0
$471$	3981.54	0.389511
$472$	0	0
$473$	8310.28i	0.807838i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	3371.78i	0.323654i
$478$	0	0
$479$	16334.5	1.55813	0.779063	−	0.626945i	$-0.215695\pi$
$479$	16334.5	1.55813	0.779063	+	0.626945i	$0.215695\pi$
$480$	0	0
$481$	6563.27	0.622161
$482$	0	0
$483$	252.000i	0.0237400i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 14770.5i	− 1.37437i	−0.726483	−	0.687184i	$-0.758847\pi$
$487$	− 14770.5i	− 1.37437i	0.726483	−	0.687184i	$-0.241153\pi$
$488$	0	0
$489$	5278.66	0.488158
$490$	0	0
$491$	6343.84	0.583082	0.291541	−	0.956558i	$-0.405832\pi$
$491$	6343.84	0.583082	0.291541	+	0.956558i	$0.405832\pi$
$492$	0	0
$493$	3564.00i	0.325587i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	5742.41i	0.518274i
$498$	0	0
$499$	2904.24	0.260544	0.130272	−	0.991478i	$-0.458415\pi$
$499$	2904.24	0.260544	0.130272	+	0.991478i	$0.458415\pi$
$500$	0	0
$501$	9558.81	0.852407
$502$	0	0
$503$	5224.11i	0.463085i	0.972825	+	0.231542i	$0.0743772\pi$
$503$	5224.11i	0.463085i	−0.972825	+	0.231542i	$0.925623\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	64.6165i	0.00566020i
$508$	0	0
$509$	−8188.94	−0.713101	−0.356550	−	0.934276i	$-0.616047\pi$
$509$	−8188.94	−0.713101	−0.356550	+	0.934276i	$0.616047\pi$
$510$	0	0
$511$	−2808.46	−0.243129
$512$	0	0
$513$	1421.33i	0.122326i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 16531.0i	− 1.40625i
$518$	0	0
$519$	1712.96	0.144876
$520$	0	0
$521$	14331.3	1.20512	0.602559	−	0.798074i	$-0.294148\pi$
$521$	14331.3	1.20512	0.602559	+	0.798074i	$0.294148\pi$
$522$	0	0
$523$	14696.2i	1.22872i	0.789026	+	0.614360i	$0.210586\pi$
$523$	14696.2i	1.22872i	−0.789026	+	0.614360i	$0.789414\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	4351.08i	0.359651i
$528$	0	0
$529$	12023.0	0.988165
$530$	0	0
$531$	695.553	0.0568445
$532$	0	0
$533$	− 12593.3i	− 1.02341i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	5595.71i	0.449670i
$538$	0	0
$539$	2775.45	0.221794
$540$	0	0
$541$	988.510	0.0785571	0.0392785	−	0.999228i	$-0.487494\pi$
$541$	988.510	0.0785571	0.0392785	+	0.999228i	$0.487494\pi$
$542$	0	0
$543$	10954.6i	0.865756i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	19765.7i	1.54501i	0.635007	+	0.772507i	$0.280997\pi$
$547$	19765.7i	1.54501i	−0.635007	+	0.772507i	$0.719003\pi$
$548$	0	0
$549$	−6822.00	−0.530339
$550$	0	0
$551$	2842.66	0.219785
$552$	0	0
$553$	− 4954.96i	− 0.381024i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 4856.65i	− 0.369448i	−0.982790	−	0.184724i	$-0.940861\pi$
$557$	− 4856.65i	− 0.369448i	0.982790	−	0.184724i	$-0.0591392\pi$
$558$	0	0
$559$	−6843.12	−0.517769
$560$	0	0
$561$	−11215.1	−0.844031
$562$	0	0
$563$	− 1008.35i	− 0.0754833i	−0.999288	−	0.0377416i	$-0.987984\pi$
$563$	− 1008.35i	− 0.0754833i	0.999288	−	0.0377416i	$-0.0120164\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	567.000i	0.0419961i
$568$	0	0
$569$	18032.3	1.32857	0.664284	−	0.747480i	$-0.268737\pi$
$569$	18032.3	1.32857	0.664284	+	0.747480i	$0.268737\pi$
$570$	0	0
$571$	−12329.6	−0.903640	−0.451820	−	0.892109i	$-0.649225\pi$
$571$	−12329.6	−0.903640	−0.451820	+	0.892109i	$0.649225\pi$
$572$	0	0
$573$	4312.88i	0.314439i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 19227.0i	− 1.38723i	−0.720348	−	0.693613i	$-0.756018\pi$
$577$	− 19227.0i	− 1.38723i	0.720348	−	0.693613i	$-0.243982\pi$
$578$	0	0
$579$	−4043.19	−0.290206
$580$	0	0
$581$	−6306.90	−0.450352
$582$	0	0
$583$	21220.4i	1.50748i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 5388.00i	− 0.378853i	−0.981895	−	0.189426i	$-0.939337\pi$
$587$	− 5388.00i	− 0.378853i	0.981895	−	0.189426i	$-0.0606628\pi$
$588$	0	0
$589$	3470.44	0.242779
$590$	0	0
$591$	−3825.27	−0.266244
$592$	0	0
$593$	− 14921.6i	− 1.03332i	−0.856191	−	0.516660i	$-0.827175\pi$
$593$	− 14921.6i	− 1.03332i	0.856191	−	0.516660i	$-0.172825\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 2371.12i	− 0.162552i
$598$	0	0
$599$	25631.9	1.74840	0.874199	−	0.485568i	$-0.161387\pi$
$599$	25631.9	1.74840	0.874199	+	0.485568i	$0.161387\pi$
$600$	0	0
$601$	−11924.1	−0.809310	−0.404655	−	0.914470i	$-0.632608\pi$
$601$	−11924.1	−0.809310	−0.404655	+	0.914470i	$0.632608\pi$
$602$	0	0
$603$	− 3204.00i	− 0.216380i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 9262.72i	− 0.619377i	−0.950838	−	0.309689i	$-0.899775\pi$
$607$	− 9262.72i	− 0.619377i	0.950838	−	0.309689i	$-0.100225\pi$
$608$	0	0
$609$	1134.00	0.0754548
$610$	0	0
$611$	13612.5	0.901312
$612$	0	0
$613$	17791.9i	1.17228i	0.810209	+	0.586142i	$0.199354\pi$
$613$	17791.9i	1.17228i	−0.810209	+	0.586142i	$0.800646\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	2381.69i	0.155402i	0.996977	+	0.0777012i	$0.0247580\pi$
$617$	2381.69i	0.155402i	−0.996977	+	0.0777012i	$0.975242\pi$
$618$	0	0
$619$	1439.49	0.0934698	0.0467349	−	0.998907i	$-0.485118\pi$
$619$	1439.49	0.0934698	0.0467349	+	0.998907i	$0.485118\pi$
$620$	0	0
$621$	−324.000	−0.0209367
$622$	0	0
$623$	4453.89i	0.286422i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	8945.19i	0.569755i
$628$	0	0
$629$	9287.28	0.588725
$630$	0	0
$631$	12146.4	0.766306	0.383153	−	0.923685i	$-0.374838\pi$
$631$	12146.4	0.766306	0.383153	+	0.923685i	$0.374838\pi$
$632$	0	0
$633$	10030.7i	0.629831i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	2285.45i	0.142155i
$638$	0	0
$639$	−7383.10	−0.457075
$640$	0	0
$641$	10331.8	0.636635	0.318317	−	0.947984i	$-0.396882\pi$
$641$	10331.8	0.636635	0.318317	+	0.947984i	$0.396882\pi$
$642$	0	0
$643$	− 14572.9i	− 0.893776i	−0.894590	−	0.446888i	$-0.852532\pi$
$643$	− 14572.9i	− 0.893776i	0.894590	−	0.446888i	$-0.147468\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	5365.80i	0.326045i	0.986622	+	0.163023i	$0.0521244\pi$
$647$	5365.80i	0.326045i	−0.986622	+	0.163023i	$0.947876\pi$
$648$	0	0
$649$	4377.49	0.264763
$650$	0	0
$651$	1384.44	0.0833492
$652$	0	0
$653$	− 31532.8i	− 1.88970i	−0.327501	−	0.944851i	$-0.606207\pi$
$653$	− 31532.8i	− 1.88970i	0.327501	−	0.944851i	$-0.393793\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 3610.88i	− 0.214420i
$658$	0	0
$659$	4830.39	0.285532	0.142766	−	0.989756i	$-0.454400\pi$
$659$	4830.39	0.285532	0.142766	+	0.989756i	$0.454400\pi$
$660$	0	0
$661$	−24168.9	−1.42218	−0.711090	−	0.703101i	$-0.751798\pi$
$661$	−24168.9	−1.42218	−0.711090	+	0.703101i	$0.751798\pi$
$662$	0	0
$663$	− 9235.08i	− 0.540967i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	648.000i	0.0376172i
$668$	0	0
$669$	−9275.55	−0.536044
$670$	0	0
$671$	−42934.5	−2.47015
$672$	0	0
$673$	− 5383.64i	− 0.308357i	−0.988043	−	0.154178i	$-0.950727\pi$
$673$	− 5383.64i	− 0.308357i	0.988043	−	0.154178i	$-0.0492730\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 27370.4i	− 1.55381i	−0.629617	−	0.776905i	$-0.716788\pi$
$677$	− 27370.4i	− 1.55381i	0.629617	−	0.776905i	$-0.283212\pi$
$678$	0	0
$679$	−840.521	−0.0475055
$680$	0	0
$681$	18073.9	1.01702
$682$	0	0
$683$	− 33313.3i	− 1.86632i	−0.359461	−	0.933160i	$-0.617040\pi$
$683$	− 33313.3i	− 1.86632i	0.359461	−	0.933160i	$-0.382960\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 4410.89i	− 0.244958i
$688$	0	0
$689$	−17474.0	−0.966191
$690$	0	0
$691$	20185.5	1.11128	0.555639	−	0.831423i	$-0.312474\pi$
$691$	20185.5	1.11128	0.555639	+	0.831423i	$0.312474\pi$
$692$	0	0
$693$	3568.44i	0.195604i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 17820.0i	− 0.968408i
$698$	0	0
$699$	−7797.10	−0.421907
$700$	0	0
$701$	12215.2	0.658148	0.329074	−	0.944304i	$-0.393264\pi$
$701$	12215.2	0.658148	0.329074	+	0.944304i	$0.393264\pi$
$702$	0	0
$703$	− 7407.57i	− 0.397413i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	11677.9i	0.621205i
$708$	0	0
$709$	15402.3	0.815859	0.407929	−	0.913014i	$-0.366251\pi$
$709$	15402.3	0.815859	0.407929	+	0.913014i	$0.366251\pi$
$710$	0	0
$711$	6370.66	0.336031
$712$	0	0
$713$	791.106i	0.0415528i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	1185.18i	0.0617314i
$718$	0	0
$719$	20646.6	1.07091	0.535457	−	0.844563i	$-0.320140\pi$
$719$	20646.6	1.07091	0.535457	+	0.844563i	$0.320140\pi$
$720$	0	0
$721$	−10151.9	−0.524377
$722$	0	0
$723$	12422.0i	0.638974i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	17329.1i	0.884045i	0.897004	+	0.442023i	$0.145739\pi$
$727$	17329.1i	0.884045i	−0.897004	+	0.442023i	$0.854261\pi$
$728$	0	0
$729$	−729.000	−0.0370370
$730$	0	0
$731$	−9683.28	−0.489944
$732$	0	0
$733$	29890.5i	1.50618i	0.657917	+	0.753091i	$0.271438\pi$
$733$	29890.5i	1.50618i	−0.657917	+	0.753091i	$0.728562\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 20164.5i	− 1.00783i
$738$	0	0
$739$	14143.0	0.704006	0.352003	−	0.935999i	$-0.385501\pi$
$739$	14143.0	0.704006	0.352003	+	0.935999i	$0.385501\pi$
$740$	0	0
$741$	−7365.94	−0.365175
$742$	0	0
$743$	− 25944.6i	− 1.28104i	−0.767940	−	0.640522i	$-0.778718\pi$
$743$	− 25944.6i	− 1.28104i	0.767940	−	0.640522i	$-0.221282\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 8108.87i	− 0.397173i
$748$	0	0
$749$	−1800.99	−0.0878593
$750$	0	0
$751$	6551.50	0.318332	0.159166	−	0.987252i	$-0.449119\pi$
$751$	6551.50	0.318332	0.159166	+	0.987252i	$0.449119\pi$
$752$	0	0
$753$	− 3039.85i	− 0.147116i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 14503.3i	− 0.696344i	−0.937431	−	0.348172i	$-0.886803\pi$
$757$	− 14503.3i	− 0.696344i	0.937431	−	0.348172i	$-0.113197\pi$
$758$	0	0
$759$	−2039.11	−0.0975163
$760$	0	0
$761$	−34740.1	−1.65483	−0.827417	−	0.561589i	$-0.810190\pi$
$761$	−34740.1	−1.65483	−0.827417	+	0.561589i	$0.810190\pi$
$762$	0	0
$763$	− 14297.8i	− 0.678393i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	3604.65i	0.169696i
$768$	0	0
$769$	−8998.86	−0.421986	−0.210993	−	0.977488i	$-0.567670\pi$
$769$	−8998.86	−0.421986	−0.210993	+	0.977488i	$0.567670\pi$
$770$	0	0
$771$	−20944.5	−0.978336
$772$	0	0
$773$	− 16966.6i	− 0.789450i	−0.918799	−	0.394725i	$-0.870840\pi$
$773$	− 16966.6i	− 0.789450i	0.918799	−	0.394725i	$-0.129160\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 2955.04i	− 0.136437i
$778$	0	0
$779$	−14213.3	−0.653715
$780$	0	0
$781$	−46465.8	−2.12891
$782$	0	0
$783$	1458.00i	0.0665449i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	14915.1i	0.675562i	0.941225	+	0.337781i	$0.109676\pi$
$787$	14915.1i	0.675562i	−0.941225	+	0.337781i	$0.890324\pi$
$788$	0	0
$789$	−18280.7	−0.824856
$790$	0	0
$791$	596.464	0.0268114
$792$	0	0
$793$	− 35354.5i	− 1.58320i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 12855.4i	− 0.571344i	−0.958327	−	0.285672i	$-0.907783\pi$
$797$	− 12855.4i	− 0.571344i	0.958327	−	0.285672i	$-0.0922169\pi$
$798$	0	0
$799$	19262.2	0.852874
$800$	0	0
$801$	−5726.42	−0.252601
$802$	0	0
$803$	− 22725.2i	− 0.998700i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 2118.89i	− 0.0924270i
$808$	0	0
$809$	−29152.0	−1.26691	−0.633454	−	0.773780i	$-0.718363\pi$
$809$	−29152.0	−1.26691	−0.633454	+	0.773780i	$0.718363\pi$
$810$	0	0
$811$	36924.7	1.59877	0.799385	−	0.600820i	$-0.205159\pi$
$811$	36924.7	1.59877	0.799385	+	0.600820i	$0.205159\pi$
$812$	0	0
$813$	10562.2i	0.455637i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	7723.42i	0.330732i
$818$	0	0
$819$	−2938.44	−0.125369
$820$	0	0
$821$	37844.0	1.60873	0.804363	−	0.594138i	$-0.202507\pi$
$821$	37844.0	1.60873	0.804363	+	0.594138i	$0.202507\pi$
$822$	0	0
$823$	− 7667.85i	− 0.324769i	−0.986728	−	0.162384i	$-0.948082\pi$
$823$	− 7667.85i	− 0.324769i	0.986728	−	0.162384i	$-0.0519184\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	23869.5i	1.00366i	0.864967	+	0.501829i	$0.167339\pi$
$827$	23869.5i	1.00366i	−0.864967	+	0.501829i	$0.832661\pi$
$828$	0	0
$829$	−33433.6	−1.40072	−0.700359	−	0.713791i	$-0.746977\pi$
$829$	−33433.6	−1.40072	−0.700359	+	0.713791i	$0.746977\pi$
$830$	0	0
$831$	−4108.66	−0.171514
$832$	0	0
$833$	3234.00i	0.134516i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	1779.99i	0.0735071i
$838$	0	0
$839$	31273.0	1.28685	0.643423	−	0.765511i	$-0.277514\pi$
$839$	31273.0	1.28685	0.643423	+	0.765511i	$0.277514\pi$
$840$	0	0
$841$	−21473.0	−0.880438
$842$	0	0
$843$	21614.0i	0.883066i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	13141.1i	0.533097i
$848$	0	0
$849$	−567.575	−0.0229436
$850$	0	0
$851$	1688.60	0.0680192
$852$	0	0
$853$	9972.48i	0.400294i	0.979766	+	0.200147i	$0.0641420\pi$
$853$	9972.48i	0.400294i	−0.979766	+	0.200147i	$0.935858\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 25152.9i	− 1.00257i	−0.865281	−	0.501287i	$-0.832860\pi$
$857$	− 25152.9i	− 1.00257i	0.865281	−	0.501287i	$-0.167140\pi$
$858$	0	0
$859$	7874.63	0.312781	0.156390	−	0.987695i	$-0.450014\pi$
$859$	7874.63	0.312781	0.156390	+	0.987695i	$0.450014\pi$
$860$	0	0
$861$	−5670.00	−0.224429
$862$	0	0
$863$	1959.42i	0.0772878i	0.999253	+	0.0386439i	$0.0123038\pi$
$863$	1959.42i	0.0772878i	−0.999253	+	0.0386439i	$0.987696\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	1671.00i	0.0654558i
$868$	0	0
$869$	40094.0	1.56513
$870$	0	0
$871$	16604.5	0.645949
$872$	0	0
$873$	− 1080.67i	− 0.0418959i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 31261.9i	− 1.20369i	−0.798612	−	0.601846i	$-0.794432\pi$
$877$	− 31261.9i	− 1.20369i	0.798612	−	0.601846i	$-0.205568\pi$
$878$	0	0
$879$	−5449.43	−0.209106
$880$	0	0
$881$	42459.0	1.62370	0.811850	−	0.583866i	$-0.198461\pi$
$881$	42459.0	1.62370	0.811850	+	0.583866i	$0.198461\pi$
$882$	0	0
$883$	− 34673.0i	− 1.32145i	−0.750629	−	0.660723i	$-0.770250\pi$
$883$	− 34673.0i	− 1.32145i	0.750629	−	0.660723i	$-0.229750\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	38271.3i	1.44873i	0.689415	+	0.724366i	$0.257868\pi$
$887$	38271.3i	1.44873i	−0.689415	+	0.724366i	$0.742132\pi$
$888$	0	0
$889$	−3684.93	−0.139020
$890$	0	0
$891$	−4587.99	−0.172507
$892$	0	0
$893$	− 15363.6i	− 0.575725i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 1679.11i	− 0.0625014i
$898$	0	0
$899$	3559.98	0.132071
$900$	0	0
$901$	−24726.4	−0.914267
$902$	0	0
$903$	3081.04i	0.113544i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	17.6607i	0 0.000646542i	1.00000		0.000323271i	$0.000102900\pi$
$907$	17.6607i	0 0.000646542i	−1.00000		0.000323271i	$0.999897\pi$
$908$	0	0
$909$	−15014.4	−0.547852
$910$	0	0
$911$	−20346.3	−0.739959	−0.369980	−	0.929040i	$-0.620635\pi$
$911$	−20346.3	−0.739959	−0.369980	+	0.929040i	$0.620635\pi$
$912$	0	0
$913$	− 51033.5i	− 1.84990i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 11595.8i	− 0.417586i
$918$	0	0
$919$	−38074.1	−1.36665	−0.683323	−	0.730116i	$-0.739466\pi$
$919$	−38074.1	−1.36665	−0.683323	+	0.730116i	$0.739466\pi$
$920$	0	0
$921$	−5298.53	−0.189568
$922$	0	0
$923$	− 38262.3i	− 1.36449i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 13052.4i	− 0.462457i
$928$	0	0
$929$	−5169.42	−0.182565	−0.0912826	−	0.995825i	$-0.529097\pi$
$929$	−5169.42	−0.182565	−0.0912826	+	0.995825i	$0.529097\pi$
$930$	0	0
$931$	2579.45	0.0908035
$932$	0	0
$933$	− 7626.09i	− 0.267596i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	28753.8i	1.00250i	0.865302	+	0.501252i	$0.167127\pi$
$937$	28753.8i	1.00250i	−0.865302	+	0.501252i	$0.832873\pi$
$938$	0	0
$939$	−24344.9	−0.846077
$940$	0	0
$941$	−12332.7	−0.427242	−0.213621	−	0.976917i	$-0.568526\pi$
$941$	−12332.7	−0.427242	−0.213621	+	0.976917i	$0.568526\pi$
$942$	0	0
$943$	− 3240.00i	− 0.111886i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 18195.0i	− 0.624350i	−0.950025	−	0.312175i	$-0.898942\pi$
$947$	− 18195.0i	− 0.624350i	0.950025	−	0.312175i	$-0.101058\pi$
$948$	0	0
$949$	18713.1	0.640099
$950$	0	0
$951$	−14991.5	−0.511179
$952$	0	0
$953$	− 1331.58i	− 0.0452614i	−0.999744	−	0.0226307i	$-0.992796\pi$
$953$	− 1331.58i	− 0.0452614i	0.999744	−	0.0226307i	$-0.00720419\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	9175.98i	0.309945i
$958$	0	0
$959$	1244.64	0.0419097
$960$	0	0
$961$	−25444.8	−0.854111
$962$	0	0
$963$	− 2315.55i	− 0.0774846i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	19516.2i	0.649018i	0.945883	+	0.324509i	$0.105199\pi$
$967$	19516.2i	0.649018i	−0.945883	+	0.324509i	$0.894801\pi$
$968$	0	0
$969$	−10423.1	−0.345550
$970$	0	0
$971$	−16350.7	−0.540390	−0.270195	−	0.962806i	$-0.587088\pi$
$971$	−16350.7	−0.540390	−0.270195	+	0.962806i	$0.587088\pi$
$972$	0	0
$973$	6741.42i	0.222117i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	22315.8i	0.730754i	0.930860	+	0.365377i	$0.119060\pi$
$977$	22315.8i	0.730754i	−0.930860	+	0.365377i	$0.880940\pi$
$978$	0	0
$979$	−36039.5	−1.17653
$980$	0	0
$981$	18382.8	0.598287
$982$	0	0
$983$	4913.28i	0.159420i	0.996818	+	0.0797098i	$0.0253994\pi$
$983$	4913.28i	0.159420i	−0.996818	+	0.0797098i	$0.974601\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 6128.87i	− 0.197654i
$988$	0	0
$989$	−1760.60	−0.0566064
$990$	0	0
$991$	−30336.3	−0.972418	−0.486209	−	0.873843i	$-0.661621\pi$
$991$	−30336.3	−0.972418	−0.486209	+	0.873843i	$0.661621\pi$
$992$	0	0
$993$	7780.92i	0.248661i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	34353.9i	1.09127i	0.838022	+	0.545637i	$0.183712\pi$
$997$	34353.9i	1.09127i	−0.838022	+	0.545637i	$0.816288\pi$
$998$	0	0
$999$	3799.34	0.120326

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2100.4.k.m.1849.3		4
5.2	odd	4		2100.4.a.r.1.1		2
5.3	odd	4		420.4.a.g.1.1	✓	2
5.4	even	2	inner	2100.4.k.m.1849.1		4
15.8	even	4		1260.4.a.o.1.2		2
20.3	even	4		1680.4.a.bi.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
420.4.a.g.1.1	✓	2	5.3	odd	4
1260.4.a.o.1.2		2	15.8	even	4
1680.4.a.bi.1.2		2	20.3	even	4
2100.4.a.r.1.1		2	5.2	odd	4
2100.4.k.m.1849.1		4	5.4	even	2	inner
2100.4.k.m.1849.3		4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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)

\(f(q)\)	\(=\)	\(q+3.00000i q^{3} +7.00000i q^{7} -9.00000 q^{9} -56.6418 q^{11} -46.6418i q^{13} -66.0000i q^{17} -52.6418 q^{19} -21.0000 q^{21} -12.0000i q^{23} -27.0000i q^{27} -54.0000 q^{29} -65.9255 q^{31} -169.926i q^{33} +140.716i q^{37} +139.926 q^{39} +270.000 q^{41} -146.716i q^{43} +291.851i q^{47} -49.0000 q^{49} +198.000 q^{51} -374.642i q^{53} -157.926i q^{57} -77.2837 q^{59} +758.000 q^{61} -63.0000i q^{63} +356.000i q^{67} +36.0000 q^{69} +820.344 q^{71} +401.209i q^{73} -396.493i q^{77} -707.851 q^{79} +81.0000 q^{81} +900.986i q^{83} -162.000i q^{87} +636.269 q^{89} +326.493 q^{91} -197.777i q^{93} +120.074i q^{97} +509.777 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 36 q^{9} + 24 q^{11} + 40 q^{19} - 84 q^{21} - 216 q^{29} + 488 q^{31} - 192 q^{39} + 1080 q^{41} - 196 q^{49} + 792 q^{51} + 192 q^{59} + 3032 q^{61} + 144 q^{69} + 24 q^{71} - 1328 q^{79} + 324 q^{81}+ \cdots - 216 q^{99}+O(q^{100}) \) 4 * q - 36 * q^9 + 24 * q^11 + 40 * q^19 - 84 * q^21 - 216 * q^29 + 488 * q^31 - 192 * q^39 + 1080 * q^41 - 196 * q^49 + 792 * q^51 + 192 * q^59 + 3032 * q^61 + 144 * q^69 + 24 * q^71 - 1328 * q^79 + 324 * q^81 - 1464 * q^89 - 448 * q^91 - 216 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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