LMFDB - Embedded newform 2100.4.k.p.1849.2

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:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 395x^{4} + 40849x^{2} + 1040400 $ x^6 + 395x^4 + 40849x^2 + 1040400
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1849.2
Root			$10.5100i$ of defining polynomial
Character	$\chi$	$=$	2100.1849
Dual form			2100.4.k.p.1849.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000i q^{3} +7.00000i q^{7} -9.00000 q^{9} -26.6969 q^{11} -66.0486i q^{13} +49.4026i q^{17} +145.077 q^{19} +21.0000 q^{21} +79.0177i q^{23} +27.0000i q^{27} +108.038 q^{29} -107.051 q^{31} +80.0907i q^{33} -174.007i q^{37} -198.146 q^{39} -470.555 q^{41} +157.914i q^{43} +558.644i q^{47} -49.0000 q^{49} +148.208 q^{51} +262.604i q^{53} -435.232i q^{57} -101.734 q^{59} +227.179 q^{61} -63.0000i q^{63} -405.265i q^{67} +237.053 q^{69} +621.175 q^{71} +76.8430i q^{73} -186.878i q^{77} +168.925 q^{79} +81.0000 q^{81} -840.723i q^{83} -324.113i q^{87} +1028.04 q^{89} +462.340 q^{91} +321.153i q^{93} -1365.07i q^{97} +240.272 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 54 q^{9} - 16 q^{11} + 148 q^{19} + 126 q^{21} + 170 q^{29} - 454 q^{31} + 78 q^{39} - 678 q^{41} - 294 q^{49} - 510 q^{51} + 2634 q^{59} - 850 q^{61} + 354 q^{69} + 652 q^{71} + 3084 q^{79} + 486 q^{81}+ \cdots + 144 q^{99}+O(q^{100}) $ 6 * q - 54 * q^9 - 16 * q^11 + 148 * q^19 + 126 * q^21 + 170 * q^29 - 454 * q^31 + 78 * q^39 - 678 * q^41 - 294 * q^49 - 510 * q^51 + 2634 * q^59 - 850 * q^61 + 354 * q^69 + 652 * q^71 + 3084 * q^79 + 486 * q^81 + 3536 * q^89 - 182 * q^91 + 144 * 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
$11$	−26.6969	−0.731765	−0.365883	−	0.930661i	$-0.619233\pi$
$11$	−26.6969	−0.731765	−0.365883	+	0.930661i	$0.619233\pi$
$12$	0	0
$13$	− 66.0486i	− 1.40912i	−0.709643	−	0.704561i	$-0.751144\pi$
$13$	− 66.0486i	− 1.40912i	0.709643	−	0.704561i	$-0.248856\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	49.4026i	0.704817i	0.935846	+	0.352409i	$0.114637\pi$
$17$	49.4026i	0.704817i	−0.935846	+	0.352409i	$0.885363\pi$
$18$	0	0
$19$	145.077	1.75174	0.875869	−	0.482549i	$-0.160289\pi$
$19$	145.077	1.75174	0.875869	+	0.482549i	$0.160289\pi$
$20$	0	0
$21$	21.0000	0.218218
$22$	0	0
$23$	79.0177i	0.716362i	0.933652	+	0.358181i	$0.116603\pi$
$23$	79.0177i	0.716362i	−0.933652	+	0.358181i	$0.883397\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	27.0000i	0.192450i
$28$	0	0
$29$	108.038	0.691796	0.345898	−	0.938272i	$-0.387574\pi$
$29$	108.038	0.691796	0.345898	+	0.938272i	$0.387574\pi$
$30$	0	0
$31$	−107.051	−0.620223	−0.310111	−	0.950700i	$-0.600366\pi$
$31$	−107.051	−0.620223	−0.310111	+	0.950700i	$0.600366\pi$
$32$	0	0
$33$	80.0907i	0.422485i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 174.007i	− 0.773149i	−0.922258	−	0.386574i	$-0.873658\pi$
$37$	− 174.007i	− 0.773149i	0.922258	−	0.386574i	$-0.126342\pi$
$38$	0	0
$39$	−198.146	−0.813557
$40$	0	0
$41$	−470.555	−1.79240	−0.896199	−	0.443651i	$-0.853683\pi$
$41$	−470.555	−1.79240	−0.896199	+	0.443651i	$0.853683\pi$
$42$	0	0
$43$	157.914i	0.560037i	0.959995	+	0.280019i	$0.0903407\pi$
$43$	157.914i	0.560037i	−0.959995	+	0.280019i	$0.909659\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	558.644i	1.73376i	0.498520	+	0.866878i	$0.333877\pi$
$47$	558.644i	1.73376i	−0.498520	+	0.866878i	$0.666123\pi$
$48$	0	0
$49$	−49.0000	−0.142857
$50$	0	0
$51$	148.208	0.406926
$52$	0	0
$53$	262.604i	0.680593i	0.940318	+	0.340296i	$0.110527\pi$
$53$	262.604i	0.680593i	−0.940318	+	0.340296i	$0.889473\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 435.232i	− 1.01137i
$58$	0	0
$59$	−101.734	−0.224486	−0.112243	−	0.993681i	$-0.535803\pi$
$59$	−101.734	−0.224486	−0.112243	+	0.993681i	$0.535803\pi$
$60$	0	0
$61$	227.179	0.476841	0.238420	−	0.971162i	$-0.423370\pi$
$61$	227.179	0.476841	0.238420	+	0.971162i	$0.423370\pi$
$62$	0	0
$63$	− 63.0000i	− 0.125988i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 405.265i	− 0.738971i	−0.929237	−	0.369485i	$-0.879534\pi$
$67$	− 405.265i	− 0.738971i	0.929237	−	0.369485i	$-0.120466\pi$
$68$	0	0
$69$	237.053	0.413592
$70$	0	0
$71$	621.175	1.03831	0.519154	−	0.854681i	$-0.326247\pi$
$71$	621.175	1.03831	0.519154	+	0.854681i	$0.326247\pi$
$72$	0	0
$73$	76.8430i	0.123203i	0.998101	+	0.0616013i	$0.0196207\pi$
$73$	76.8430i	0.123203i	−0.998101	+	0.0616013i	$0.980379\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 186.878i	− 0.276581i
$78$	0	0
$79$	168.925	0.240577	0.120288	−	0.992739i	$-0.461618\pi$
$79$	168.925	0.240577	0.120288	+	0.992739i	$0.461618\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	− 840.723i	− 1.11182i	−0.831241	−	0.555912i	$-0.812369\pi$
$83$	− 840.723i	− 1.11182i	0.831241	−	0.555912i	$-0.187631\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 324.113i	− 0.399408i
$88$	0	0
$89$	1028.04	1.22441	0.612203	−	0.790700i	$-0.290283\pi$
$89$	1028.04	1.22441	0.612203	+	0.790700i	$0.290283\pi$
$90$	0	0
$91$	462.340	0.532598
$92$	0	0
$93$	321.153i	0.358086i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 1365.07i	− 1.42889i	−0.699692	−	0.714445i	$-0.746679\pi$
$97$	− 1365.07i	− 1.42889i	0.699692	−	0.714445i	$-0.253321\pi$
$98$	0	0
$99$	240.272	0.243922
$100$	0	0
$101$	−157.878	−0.155539	−0.0777693	−	0.996971i	$-0.524780\pi$
$101$	−157.878	−0.155539	−0.0777693	+	0.996971i	$0.524780\pi$
$102$	0	0
$103$	− 860.849i	− 0.823514i	−0.911294	−	0.411757i	$-0.864915\pi$
$103$	− 860.849i	− 0.823514i	0.911294	−	0.411757i	$-0.135085\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1102.76i	0.996339i	0.867080	+	0.498170i	$0.165994\pi$
$107$	1102.76i	0.996339i	−0.867080	+	0.498170i	$0.834006\pi$
$108$	0	0
$109$	1340.97	1.17836	0.589182	−	0.808000i	$-0.299450\pi$
$109$	1340.97	1.17836	0.589182	+	0.808000i	$0.299450\pi$
$110$	0	0
$111$	−522.020	−0.446378
$112$	0	0
$113$	989.907i	0.824094i	0.911163	+	0.412047i	$0.135186\pi$
$113$	989.907i	0.824094i	−0.911163	+	0.412047i	$0.864814\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	594.438i	0.469708i
$118$	0	0
$119$	−345.818	−0.266396
$120$	0	0
$121$	−618.276	−0.464520
$122$	0	0
$123$	1411.67i	1.03484i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	1267.32i	0.885483i	0.896649	+	0.442741i	$0.145994\pi$
$127$	1267.32i	0.885483i	−0.896649	+	0.442741i	$0.854006\pi$
$128$	0	0
$129$	473.741	0.323338
$130$	0	0
$131$	2670.85	1.78132	0.890660	−	0.454670i	$-0.150243\pi$
$131$	2670.85	1.78132	0.890660	+	0.454670i	$0.150243\pi$
$132$	0	0
$133$	1015.54i	0.662095i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 1010.11i	− 0.629921i	−0.949105	−	0.314960i	$-0.898009\pi$
$137$	− 1010.11i	− 0.629921i	0.949105	−	0.314960i	$-0.101991\pi$
$138$	0	0
$139$	2837.56	1.73150	0.865750	−	0.500476i	$-0.166842\pi$
$139$	2837.56	1.73150	0.865750	+	0.500476i	$0.166842\pi$
$140$	0	0
$141$	1675.93	1.00098
$142$	0	0
$143$	1763.29i	1.03115i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	147.000i	0.0824786i
$148$	0	0
$149$	−351.183	−0.193088	−0.0965438	−	0.995329i	$-0.530779\pi$
$149$	−351.183	−0.193088	−0.0965438	+	0.995329i	$0.530779\pi$
$150$	0	0
$151$	−3372.87	−1.81775	−0.908875	−	0.417069i	$-0.863057\pi$
$151$	−3372.87	−1.81775	−0.908875	+	0.417069i	$0.863057\pi$
$152$	0	0
$153$	− 444.624i	− 0.234939i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	515.402i	0.261997i	0.991383	+	0.130999i	$0.0418183\pi$
$157$	515.402i	0.261997i	−0.991383	+	0.130999i	$0.958182\pi$
$158$	0	0
$159$	787.812	0.392940
$160$	0	0
$161$	−553.124	−0.270759
$162$	0	0
$163$	2366.85i	1.13734i	0.822566	+	0.568670i	$0.192542\pi$
$163$	2366.85i	1.13734i	−0.822566	+	0.568670i	$0.807458\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 3107.82i	− 1.44006i	−0.693943	−	0.720030i	$-0.744128\pi$
$167$	− 3107.82i	− 1.44006i	0.693943	−	0.720030i	$-0.255872\pi$
$168$	0	0
$169$	−2165.42	−0.985627
$170$	0	0
$171$	−1305.70	−0.583913
$172$	0	0
$173$	− 2388.14i	− 1.04952i	−0.851251	−	0.524759i	$-0.824155\pi$
$173$	− 2388.14i	− 1.04952i	0.851251	−	0.524759i	$-0.175845\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	305.203i	0.129607i
$178$	0	0
$179$	3544.62	1.48010	0.740049	−	0.672553i	$-0.234802\pi$
$179$	3544.62	1.48010	0.740049	+	0.672553i	$0.234802\pi$
$180$	0	0
$181$	2330.01	0.956841	0.478421	−	0.878131i	$-0.341209\pi$
$181$	2330.01	0.956841	0.478421	+	0.878131i	$0.341209\pi$
$182$	0	0
$183$	− 681.537i	− 0.275304i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 1318.90i	− 0.515761i
$188$	0	0
$189$	−189.000	−0.0727393
$190$	0	0
$191$	2625.69	0.994702	0.497351	−	0.867549i	$-0.334306\pi$
$191$	2625.69	0.994702	0.497351	+	0.867549i	$0.334306\pi$
$192$	0	0
$193$	4660.89i	1.73833i	0.494521	+	0.869166i	$0.335343\pi$
$193$	4660.89i	1.73833i	−0.494521	+	0.869166i	$0.664657\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 1677.43i	− 0.606660i	−0.952886	−	0.303330i	$-0.901902\pi$
$197$	− 1677.43i	− 0.606660i	0.952886	−	0.303330i	$-0.0980985\pi$
$198$	0	0
$199$	−2835.45	−1.01005	−0.505024	−	0.863105i	$-0.668516\pi$
$199$	−2835.45	−1.01005	−0.505024	+	0.863105i	$0.668516\pi$
$200$	0	0
$201$	−1215.80	−0.426645
$202$	0	0
$203$	756.263i	0.261474i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 711.159i	− 0.238787i
$208$	0	0
$209$	−3873.11	−1.28186
$210$	0	0
$211$	2798.05	0.912918	0.456459	−	0.889744i	$-0.349117\pi$
$211$	2798.05	0.912918	0.456459	+	0.889744i	$0.349117\pi$
$212$	0	0
$213$	− 1863.52i	− 0.599467i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 749.356i	− 0.234422i
$218$	0	0
$219$	230.529	0.0711311
$220$	0	0
$221$	3262.98	0.993174
$222$	0	0
$223$	− 2474.79i	− 0.743158i	−0.928401	−	0.371579i	$-0.878816\pi$
$223$	− 2474.79i	− 0.743158i	0.928401	−	0.371579i	$-0.121184\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 3024.51i	− 0.884335i	−0.896933	−	0.442167i	$-0.854210\pi$
$227$	− 3024.51i	− 0.884335i	0.896933	−	0.442167i	$-0.145790\pi$
$228$	0	0
$229$	3588.88	1.03563	0.517816	−	0.855492i	$-0.326745\pi$
$229$	3588.88	1.03563	0.517816	+	0.855492i	$0.326745\pi$
$230$	0	0
$231$	−560.635	−0.159684
$232$	0	0
$233$	707.221i	0.198848i	0.995045	+	0.0994240i	$0.0317000\pi$
$233$	707.221i	0.198848i	−0.995045	+	0.0994240i	$0.968300\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 506.775i	− 0.138897i
$238$	0	0
$239$	5655.70	1.53070	0.765349	−	0.643615i	$-0.222566\pi$
$239$	5655.70	1.53070	0.765349	+	0.643615i	$0.222566\pi$
$240$	0	0
$241$	5282.16	1.41184	0.705921	−	0.708291i	$-0.250533\pi$
$241$	5282.16	1.41184	0.705921	+	0.708291i	$0.250533\pi$
$242$	0	0
$243$	− 243.000i	− 0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 9582.16i	− 2.46841i
$248$	0	0
$249$	−2522.17	−0.641911
$250$	0	0
$251$	1483.01	0.372935	0.186467	−	0.982461i	$-0.440296\pi$
$251$	1483.01	0.372935	0.186467	+	0.982461i	$0.440296\pi$
$252$	0	0
$253$	− 2109.53i	− 0.524209i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 3415.10i	− 0.828904i	−0.910071	−	0.414452i	$-0.863973\pi$
$257$	− 3415.10i	− 0.828904i	0.910071	−	0.414452i	$-0.136027\pi$
$258$	0	0
$259$	1218.05	0.292223
$260$	0	0
$261$	−972.338	−0.230599
$262$	0	0
$263$	− 1342.88i	− 0.314850i	−0.987531	−	0.157425i	$-0.949681\pi$
$263$	− 1342.88i	− 0.314850i	0.987531	−	0.157425i	$-0.0503192\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 3084.13i	− 0.706912i
$268$	0	0
$269$	3008.98	0.682011	0.341006	−	0.940061i	$-0.389232\pi$
$269$	3008.98	0.682011	0.341006	+	0.940061i	$0.389232\pi$
$270$	0	0
$271$	−238.967	−0.0535653	−0.0267827	−	0.999641i	$-0.508526\pi$
$271$	−238.967	−0.0535653	−0.0267827	+	0.999641i	$0.508526\pi$
$272$	0	0
$273$	− 1387.02i	− 0.307496i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 2480.97i	− 0.538147i	−0.963120	−	0.269074i	$-0.913283\pi$
$277$	− 2480.97i	− 0.538147i	0.963120	−	0.269074i	$-0.0867175\pi$
$278$	0	0
$279$	963.458	0.206741
$280$	0	0
$281$	695.444	0.147640	0.0738198	−	0.997272i	$-0.476481\pi$
$281$	695.444	0.147640	0.0738198	+	0.997272i	$0.476481\pi$
$282$	0	0
$283$	− 7410.59i	− 1.55659i	−0.627901	−	0.778293i	$-0.716086\pi$
$283$	− 7410.59i	− 1.55659i	0.627901	−	0.778293i	$-0.283914\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 3293.89i	− 0.677463i
$288$	0	0
$289$	2472.38	0.503233
$290$	0	0
$291$	−4095.22	−0.824970
$292$	0	0
$293$	8002.90i	1.59568i	0.602869	+	0.797840i	$0.294024\pi$
$293$	8002.90i	1.59568i	−0.602869	+	0.797840i	$0.705976\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 720.816i	− 0.140828i
$298$	0	0
$299$	5219.01	1.00944
$300$	0	0
$301$	−1105.40	−0.211674
$302$	0	0
$303$	473.633i	0.0898003i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	3454.92i	0.642288i	0.947030	+	0.321144i	$0.104067\pi$
$307$	3454.92i	0.642288i	−0.947030	+	0.321144i	$0.895933\pi$
$308$	0	0
$309$	−2582.55	−0.475456
$310$	0	0
$311$	4649.17	0.847685	0.423843	−	0.905736i	$-0.360681\pi$
$311$	4649.17	0.847685	0.423843	+	0.905736i	$0.360681\pi$
$312$	0	0
$313$	6506.84i	1.17504i	0.809209	+	0.587522i	$0.199896\pi$
$313$	6506.84i	1.17504i	−0.809209	+	0.587522i	$0.800104\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 1664.13i	− 0.294847i	−0.989073	−	0.147424i	$-0.952902\pi$
$317$	− 1664.13i	− 0.294847i	0.989073	−	0.147424i	$-0.0470981\pi$
$318$	0	0
$319$	−2884.27	−0.506232
$320$	0	0
$321$	3308.29	0.575237
$322$	0	0
$323$	7167.20i	1.23466i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 4022.91i	− 0.680329i
$328$	0	0
$329$	−3910.50	−0.655298
$330$	0	0
$331$	−1145.95	−0.190293	−0.0951467	−	0.995463i	$-0.530332\pi$
$331$	−1145.95	−0.190293	−0.0951467	+	0.995463i	$0.530332\pi$
$332$	0	0
$333$	1566.06i	0.257716i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 8602.86i	− 1.39059i	−0.718726	−	0.695294i	$-0.755274\pi$
$337$	− 8602.86i	− 1.39059i	0.718726	−	0.695294i	$-0.244726\pi$
$338$	0	0
$339$	2969.72	0.475791
$340$	0	0
$341$	2857.93	0.453857
$342$	0	0
$343$	− 343.000i	− 0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	7072.88i	1.09421i	0.837063	+	0.547107i	$0.184271\pi$
$347$	7072.88i	1.09421i	−0.837063	+	0.547107i	$0.815729\pi$
$348$	0	0
$349$	−497.466	−0.0763001	−0.0381500	−	0.999272i	$-0.512146\pi$
$349$	−497.466	−0.0763001	−0.0381500	+	0.999272i	$0.512146\pi$
$350$	0	0
$351$	1783.31	0.271186
$352$	0	0
$353$	10951.0i	1.65117i	0.564279	+	0.825584i	$0.309154\pi$
$353$	10951.0i	1.65117i	−0.564279	+	0.825584i	$0.690846\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	1037.45i	0.153804i
$358$	0	0
$359$	−3379.45	−0.496825	−0.248413	−	0.968654i	$-0.579909\pi$
$359$	−3379.45	−0.496825	−0.248413	+	0.968654i	$0.579909\pi$
$360$	0	0
$361$	14188.4	2.06859
$362$	0	0
$363$	1854.83i	0.268191i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 11428.8i	− 1.62555i	−0.582575	−	0.812777i	$-0.697955\pi$
$367$	− 11428.8i	− 1.62555i	0.582575	−	0.812777i	$-0.302045\pi$
$368$	0	0
$369$	4235.00	0.597466
$370$	0	0
$371$	−1838.23	−0.257240
$372$	0	0
$373$	− 9751.24i	− 1.35362i	−0.736158	−	0.676810i	$-0.763362\pi$
$373$	− 9751.24i	− 1.35362i	0.736158	−	0.676810i	$-0.236638\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 7135.73i	− 0.974825i
$378$	0	0
$379$	5629.75	0.763010	0.381505	−	0.924367i	$-0.375406\pi$
$379$	5629.75	0.763010	0.381505	+	0.924367i	$0.375406\pi$
$380$	0	0
$381$	3801.95	0.511234
$382$	0	0
$383$	− 5254.80i	− 0.701064i	−0.936551	−	0.350532i	$-0.886001\pi$
$383$	− 5254.80i	− 0.701064i	0.936551	−	0.350532i	$-0.113999\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 1421.22i	− 0.186679i
$388$	0	0
$389$	3964.29	0.516703	0.258351	−	0.966051i	$-0.416821\pi$
$389$	3964.29	0.516703	0.258351	+	0.966051i	$0.416821\pi$
$390$	0	0
$391$	−3903.68	−0.504904
$392$	0	0
$393$	− 8012.54i	− 1.02845i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	7404.95i	0.936130i	0.883694	+	0.468065i	$0.155049\pi$
$397$	7404.95i	0.936130i	−0.883694	+	0.468065i	$0.844951\pi$
$398$	0	0
$399$	3046.62	0.382261
$400$	0	0
$401$	5243.57	0.652996	0.326498	−	0.945198i	$-0.394131\pi$
$401$	5243.57	0.652996	0.326498	+	0.945198i	$0.394131\pi$
$402$	0	0
$403$	7070.56i	0.873970i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	4645.44i	0.565763i
$408$	0	0
$409$	12986.1	1.56998	0.784989	−	0.619509i	$-0.212668\pi$
$409$	12986.1	1.56998	0.784989	+	0.619509i	$0.212668\pi$
$410$	0	0
$411$	−3030.32	−0.363685
$412$	0	0
$413$	− 712.139i	− 0.0848477i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 8512.68i	− 0.999682i
$418$	0	0
$419$	−14172.7	−1.65246	−0.826232	−	0.563329i	$-0.809520\pi$
$419$	−14172.7	−1.65246	−0.826232	+	0.563329i	$0.809520\pi$
$420$	0	0
$421$	−10699.3	−1.23860	−0.619299	−	0.785155i	$-0.712583\pi$
$421$	−10699.3	−1.23860	−0.619299	+	0.785155i	$0.712583\pi$
$422$	0	0
$423$	− 5027.79i	− 0.577919i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	1590.25i	0.180229i
$428$	0	0
$429$	5289.88	0.595333
$430$	0	0
$431$	7858.53	0.878265	0.439132	−	0.898422i	$-0.355286\pi$
$431$	7858.53	0.878265	0.439132	+	0.898422i	$0.355286\pi$
$432$	0	0
$433$	1445.41i	0.160420i	0.996778	+	0.0802100i	$0.0255591\pi$
$433$	1445.41i	0.160420i	−0.996778	+	0.0802100i	$0.974441\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	11463.7i	1.25488i
$438$	0	0
$439$	−7774.54	−0.845236	−0.422618	−	0.906308i	$-0.638889\pi$
$439$	−7774.54	−0.845236	−0.422618	+	0.906308i	$0.638889\pi$
$440$	0	0
$441$	441.000	0.0476190
$442$	0	0
$443$	− 14920.4i	− 1.60020i	−0.599868	−	0.800099i	$-0.704780\pi$
$443$	− 14920.4i	− 1.60020i	0.599868	−	0.800099i	$-0.295220\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	1053.55i	0.111479i
$448$	0	0
$449$	−13013.1	−1.36777	−0.683884	−	0.729591i	$-0.739711\pi$
$449$	−13013.1	−1.36777	−0.683884	+	0.729591i	$0.739711\pi$
$450$	0	0
$451$	12562.4	1.31162
$452$	0	0
$453$	10118.6i	1.04948i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 3446.24i	− 0.352753i	−0.984323	−	0.176376i	$-0.943562\pi$
$457$	− 3446.24i	− 0.352753i	0.984323	−	0.176376i	$-0.0564376\pi$
$458$	0	0
$459$	−1333.87	−0.135642
$460$	0	0
$461$	15096.6	1.52520	0.762601	−	0.646869i	$-0.223922\pi$
$461$	15096.6	1.52520	0.762601	+	0.646869i	$0.223922\pi$
$462$	0	0
$463$	915.782i	0.0919223i	0.998943	+	0.0459611i	$0.0146350\pi$
$463$	915.782i	0.0919223i	−0.998943	+	0.0459611i	$0.985365\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 14075.8i	− 1.39475i	−0.716705	−	0.697376i	$-0.754351\pi$
$467$	− 14075.8i	− 1.39475i	0.716705	−	0.697376i	$-0.245649\pi$
$468$	0	0
$469$	2836.86	0.279305
$470$	0	0
$471$	1546.21	0.151264
$472$	0	0
$473$	− 4215.80i	− 0.409816i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 2363.43i	− 0.226864i
$478$	0	0
$479$	11686.4	1.11475	0.557375	−	0.830261i	$-0.311809\pi$
$479$	11686.4	1.11475	0.557375	+	0.830261i	$0.311809\pi$
$480$	0	0
$481$	−11492.9	−1.08946
$482$	0	0
$483$	1659.37i	0.156323i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 7829.75i	− 0.728542i	−0.931293	−	0.364271i	$-0.881318\pi$
$487$	− 7829.75i	− 0.728542i	0.931293	−	0.364271i	$-0.118682\pi$
$488$	0	0
$489$	7100.56	0.656643
$490$	0	0
$491$	−4957.17	−0.455629	−0.227815	−	0.973705i	$-0.573158\pi$
$491$	−4957.17	−0.455629	−0.227815	+	0.973705i	$0.573158\pi$
$492$	0	0
$493$	5337.34i	0.487589i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	4348.22i	0.392443i
$498$	0	0
$499$	10245.1	0.919106	0.459553	−	0.888150i	$-0.348010\pi$
$499$	10245.1	0.919106	0.459553	+	0.888150i	$0.348010\pi$
$500$	0	0
$501$	−9323.45	−0.831419
$502$	0	0
$503$	11782.4i	1.04444i	0.852812	+	0.522218i	$0.174895\pi$
$503$	11782.4i	1.04444i	−0.852812	+	0.522218i	$0.825105\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	6496.26i	0.569052i
$508$	0	0
$509$	−18384.1	−1.60091	−0.800453	−	0.599396i	$-0.795408\pi$
$509$	−18384.1	−1.60091	−0.800453	+	0.599396i	$0.795408\pi$
$510$	0	0
$511$	−537.901	−0.0465662
$512$	0	0
$513$	3917.09i	0.337122i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 14914.0i	− 1.26870i
$518$	0	0
$519$	−7164.41	−0.605940
$520$	0	0
$521$	−3880.67	−0.326325	−0.163163	−	0.986599i	$-0.552170\pi$
$521$	−3880.67	−0.326325	−0.163163	+	0.986599i	$0.552170\pi$
$522$	0	0
$523$	− 155.376i	− 0.0129906i	−0.999979	−	0.00649532i	$-0.997932\pi$
$523$	− 155.376i	− 0.0129906i	0.999979	−	0.00649532i	$-0.00206754\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 5288.59i	− 0.437144i
$528$	0	0
$529$	5923.21	0.486826
$530$	0	0
$531$	915.608	0.0748286
$532$	0	0
$533$	31079.5i	2.52571i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 10633.9i	− 0.854535i
$538$	0	0
$539$	1308.15	0.104538
$540$	0	0
$541$	868.144	0.0689915	0.0344958	−	0.999405i	$-0.489017\pi$
$541$	868.144	0.0689915	0.0344958	+	0.999405i	$0.489017\pi$
$542$	0	0
$543$	− 6990.03i	− 0.552433i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 14593.6i	− 1.14073i	−0.821391	−	0.570365i	$-0.806802\pi$
$547$	− 14593.6i	− 1.14073i	0.821391	−	0.570365i	$-0.193198\pi$
$548$	0	0
$549$	−2044.61	−0.158947
$550$	0	0
$551$	15673.8	1.21184
$552$	0	0
$553$	1182.48i	0.0909294i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	6133.73i	0.466597i	0.972405	+	0.233298i	$0.0749519\pi$
$557$	6133.73i	0.466597i	−0.972405	+	0.233298i	$0.925048\pi$
$558$	0	0
$559$	10430.0	0.789161
$560$	0	0
$561$	−3956.69	−0.297775
$562$	0	0
$563$	− 10042.7i	− 0.751774i	−0.926665	−	0.375887i	$-0.877338\pi$
$563$	− 10042.7i	− 0.751774i	0.926665	−	0.375887i	$-0.122662\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	567.000i	0.0419961i
$568$	0	0
$569$	2535.10	0.186778	0.0933891	−	0.995630i	$-0.470230\pi$
$569$	2535.10	0.186778	0.0933891	+	0.995630i	$0.470230\pi$
$570$	0	0
$571$	−16755.7	−1.22803	−0.614015	−	0.789294i	$-0.710447\pi$
$571$	−16755.7	−1.22803	−0.614015	+	0.789294i	$0.710447\pi$
$572$	0	0
$573$	− 7877.06i	− 0.574291i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	7228.14i	0.521510i	0.965405	+	0.260755i	$0.0839715\pi$
$577$	7228.14i	0.521510i	−0.965405	+	0.260755i	$0.916029\pi$
$578$	0	0
$579$	13982.7	1.00363
$580$	0	0
$581$	5885.06	0.420230
$582$	0	0
$583$	− 7010.71i	− 0.498034i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	7434.57i	0.522755i	0.965237	+	0.261378i	$0.0841768\pi$
$587$	7434.57i	0.522755i	−0.965237	+	0.261378i	$0.915823\pi$
$588$	0	0
$589$	−15530.7	−1.08647
$590$	0	0
$591$	−5032.29	−0.350255
$592$	0	0
$593$	20473.3i	1.41777i	0.705325	+	0.708884i	$0.250801\pi$
$593$	20473.3i	1.41777i	−0.705325	+	0.708884i	$0.749199\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	8506.35i	0.583152i
$598$	0	0
$599$	2136.87	0.145760	0.0728800	−	0.997341i	$-0.476781\pi$
$599$	2136.87	0.145760	0.0728800	+	0.997341i	$0.476781\pi$
$600$	0	0
$601$	−28627.3	−1.94298	−0.971489	−	0.237083i	$-0.923809\pi$
$601$	−28627.3	−1.94298	−0.971489	+	0.237083i	$0.923809\pi$
$602$	0	0
$603$	3647.39i	0.246324i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	11526.8i	0.770768i	0.922756	+	0.385384i	$0.125931\pi$
$607$	11526.8i	0.770768i	−0.922756	+	0.385384i	$0.874069\pi$
$608$	0	0
$609$	2268.79	0.150962
$610$	0	0
$611$	36897.6	2.44307
$612$	0	0
$613$	− 807.212i	− 0.0531859i	−0.999646	−	0.0265930i	$-0.991534\pi$
$613$	− 807.212i	− 0.0531859i	0.999646	−	0.0265930i	$-0.00846580\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	17858.5i	1.16524i	0.812744	+	0.582621i	$0.197973\pi$
$617$	17858.5i	1.16524i	−0.812744	+	0.582621i	$0.802027\pi$
$618$	0	0
$619$	14091.9	0.915028	0.457514	−	0.889202i	$-0.348740\pi$
$619$	14091.9	0.915028	0.457514	+	0.889202i	$0.348740\pi$
$620$	0	0
$621$	−2133.48	−0.137864
$622$	0	0
$623$	7196.29i	0.462782i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	11619.3i	0.740083i
$628$	0	0
$629$	8596.38	0.544929
$630$	0	0
$631$	2161.37	0.136359	0.0681795	−	0.997673i	$-0.478281\pi$
$631$	2161.37	0.136359	0.0681795	+	0.997673i	$0.478281\pi$
$632$	0	0
$633$	− 8394.15i	− 0.527073i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	3236.38i	0.201303i
$638$	0	0
$639$	−5590.57	−0.346103
$640$	0	0
$641$	−4374.38	−0.269544	−0.134772	−	0.990877i	$-0.543030\pi$
$641$	−4374.38	−0.269544	−0.134772	+	0.990877i	$0.543030\pi$
$642$	0	0
$643$	− 9691.44i	− 0.594390i	−0.954817	−	0.297195i	$-0.903949\pi$
$643$	− 9691.44i	− 0.594390i	0.954817	−	0.297195i	$-0.0960512\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	17690.0i	1.07491i	0.843294	+	0.537453i	$0.180613\pi$
$647$	17690.0i	1.07491i	−0.843294	+	0.537453i	$0.819387\pi$
$648$	0	0
$649$	2715.99	0.164271
$650$	0	0
$651$	−2248.07	−0.135344
$652$	0	0
$653$	6777.53i	0.406164i	0.979162	+	0.203082i	$0.0650959\pi$
$653$	6777.53i	0.406164i	−0.979162	+	0.203082i	$0.934904\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 691.587i	− 0.0410676i
$658$	0	0
$659$	29526.8	1.74537	0.872686	−	0.488283i	$-0.162377\pi$
$659$	29526.8	1.74537	0.872686	+	0.488283i	$0.162377\pi$
$660$	0	0
$661$	−9505.64	−0.559344	−0.279672	−	0.960096i	$-0.590226\pi$
$661$	−9505.64	−0.559344	−0.279672	+	0.960096i	$0.590226\pi$
$662$	0	0
$663$	− 9788.93i	− 0.573409i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	8536.87i	0.495576i
$668$	0	0
$669$	−7424.37	−0.429062
$670$	0	0
$671$	−6064.97	−0.348935
$672$	0	0
$673$	100.054i	0.00573077i	0.999996	+	0.00286538i	$0.000912081\pi$
$673$	100.054i	0.00573077i	−0.999996	+	0.00286538i	$0.999088\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	21975.7i	1.24756i	0.781602	+	0.623778i	$0.214403\pi$
$677$	21975.7i	1.24756i	−0.781602	+	0.623778i	$0.785597\pi$
$678$	0	0
$679$	9555.52	0.540070
$680$	0	0
$681$	−9073.54	−0.510571
$682$	0	0
$683$	1891.73i	0.105981i	0.998595	+	0.0529904i	$0.0168753\pi$
$683$	1891.73i	0.105981i	−0.998595	+	0.0529904i	$0.983125\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 10766.6i	− 0.597923i
$688$	0	0
$689$	17344.6	0.959039
$690$	0	0
$691$	−7174.11	−0.394958	−0.197479	−	0.980307i	$-0.563275\pi$
$691$	−7174.11	−0.394958	−0.197479	+	0.980307i	$0.563275\pi$
$692$	0	0
$693$	1681.90i	0.0921937i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 23246.7i	− 1.26331i
$698$	0	0
$699$	2121.66	0.114805
$700$	0	0
$701$	2177.55	0.117325	0.0586626	−	0.998278i	$-0.481316\pi$
$701$	2177.55	0.117325	0.0586626	+	0.998278i	$0.481316\pi$
$702$	0	0
$703$	− 25244.4i	− 1.35435i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 1105.14i	− 0.0587881i
$708$	0	0
$709$	−23644.8	−1.25247	−0.626233	−	0.779636i	$-0.715404\pi$
$709$	−23644.8	−1.25247	−0.626233	+	0.779636i	$0.715404\pi$
$710$	0	0
$711$	−1520.33	−0.0801922
$712$	0	0
$713$	− 8458.91i	− 0.444304i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 16967.1i	− 0.883749i
$718$	0	0
$719$	10678.3	0.553869	0.276934	−	0.960889i	$-0.410682\pi$
$719$	10678.3	0.553869	0.276934	+	0.960889i	$0.410682\pi$
$720$	0	0
$721$	6025.94	0.311259
$722$	0	0
$723$	− 15846.5i	− 0.815127i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	28246.1i	1.44098i	0.693467	+	0.720489i	$0.256083\pi$
$727$	28246.1i	1.44098i	−0.693467	+	0.720489i	$0.743917\pi$
$728$	0	0
$729$	−729.000	−0.0370370
$730$	0	0
$731$	−7801.35	−0.394724
$732$	0	0
$733$	− 3557.32i	− 0.179253i	−0.995975	−	0.0896265i	$-0.971433\pi$
$733$	− 3557.32i	− 0.179253i	0.995975	−	0.0896265i	$-0.0285673\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	10819.3i	0.540753i
$738$	0	0
$739$	−7184.71	−0.357637	−0.178818	−	0.983882i	$-0.557227\pi$
$739$	−7184.71	−0.357637	−0.178818	+	0.983882i	$0.557227\pi$
$740$	0	0
$741$	−28746.5	−1.42514
$742$	0	0
$743$	14989.1i	0.740106i	0.929011	+	0.370053i	$0.120660\pi$
$743$	14989.1i	0.740106i	−0.929011	+	0.370053i	$0.879340\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	7566.51i	0.370608i
$748$	0	0
$749$	−7719.35	−0.376581
$750$	0	0
$751$	16705.0	0.811685	0.405842	−	0.913943i	$-0.366978\pi$
$751$	16705.0	0.811685	0.405842	+	0.913943i	$0.366978\pi$
$752$	0	0
$753$	− 4449.03i	− 0.215314i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	13196.1i	0.633581i	0.948496	+	0.316790i	$0.102605\pi$
$757$	13196.1i	0.633581i	−0.948496	+	0.316790i	$0.897395\pi$
$758$	0	0
$759$	−6328.58	−0.302652
$760$	0	0
$761$	11359.7	0.541117	0.270558	−	0.962704i	$-0.412792\pi$
$761$	11359.7	0.541117	0.270558	+	0.962704i	$0.412792\pi$
$762$	0	0
$763$	9386.80i	0.445380i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	6719.40i	0.316328i
$768$	0	0
$769$	−31970.8	−1.49921	−0.749607	−	0.661883i	$-0.769758\pi$
$769$	−31970.8	−1.49921	−0.749607	+	0.661883i	$0.769758\pi$
$770$	0	0
$771$	−10245.3	−0.478568
$772$	0	0
$773$	− 11528.7i	− 0.536426i	−0.963360	−	0.268213i	$-0.913567\pi$
$773$	− 11528.7i	− 0.536426i	0.963360	−	0.268213i	$-0.0864332\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 3654.14i	− 0.168715i
$778$	0	0
$779$	−68266.9	−3.13981
$780$	0	0
$781$	−16583.4	−0.759797
$782$	0	0
$783$	2917.01i	0.133136i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	26360.4i	1.19396i	0.802256	+	0.596981i	$0.203633\pi$
$787$	26360.4i	1.19396i	−0.802256	+	0.596981i	$0.796367\pi$
$788$	0	0
$789$	−4028.64	−0.181779
$790$	0	0
$791$	−6929.35	−0.311478
$792$	0	0
$793$	− 15004.9i	− 0.671927i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 23128.8i	− 1.02793i	−0.857810	−	0.513967i	$-0.828175\pi$
$797$	− 23128.8i	− 1.02793i	0.857810	−	0.513967i	$-0.171825\pi$
$798$	0	0
$799$	−27598.5	−1.22198
$800$	0	0
$801$	−9252.38	−0.408136
$802$	0	0
$803$	− 2051.47i	− 0.0901554i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 9026.95i	− 0.393759i
$808$	0	0
$809$	−16689.9	−0.725323	−0.362661	−	0.931921i	$-0.618132\pi$
$809$	−16689.9	−0.725323	−0.362661	+	0.931921i	$0.618132\pi$
$810$	0	0
$811$	1784.32	0.0772578	0.0386289	−	0.999254i	$-0.487701\pi$
$811$	1784.32	0.0772578	0.0386289	+	0.999254i	$0.487701\pi$
$812$	0	0
$813$	716.901i	0.0309260i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	22909.7i	0.981039i
$818$	0	0
$819$	−4161.06	−0.177533
$820$	0	0
$821$	−3943.02	−0.167615	−0.0838077	−	0.996482i	$-0.526708\pi$
$821$	−3943.02	−0.167615	−0.0838077	+	0.996482i	$0.526708\pi$
$822$	0	0
$823$	− 20456.9i	− 0.866441i	−0.901288	−	0.433221i	$-0.857377\pi$
$823$	− 20456.9i	− 0.866441i	0.901288	−	0.433221i	$-0.142623\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 960.218i	− 0.0403749i	−0.999796	−	0.0201875i	$-0.993574\pi$
$827$	− 960.218i	− 0.0403749i	0.999796	−	0.0201875i	$-0.00642630\pi$
$828$	0	0
$829$	2875.01	0.120450	0.0602251	−	0.998185i	$-0.480818\pi$
$829$	2875.01	0.120450	0.0602251	+	0.998185i	$0.480818\pi$
$830$	0	0
$831$	−7442.90	−0.310699
$832$	0	0
$833$	− 2420.73i	− 0.100688i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 2890.37i	− 0.119362i
$838$	0	0
$839$	−16620.8	−0.683927	−0.341964	−	0.939713i	$-0.611092\pi$
$839$	−16620.8	−0.683927	−0.341964	+	0.939713i	$0.611092\pi$
$840$	0	0
$841$	−12716.9	−0.521419
$842$	0	0
$843$	− 2086.33i	− 0.0852397i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 4327.93i	− 0.175572i
$848$	0	0
$849$	−22231.8	−0.898695
$850$	0	0
$851$	13749.6	0.553854
$852$	0	0
$853$	32589.4i	1.30813i	0.756437	+	0.654067i	$0.226939\pi$
$853$	32589.4i	1.30813i	−0.756437	+	0.654067i	$0.773061\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	31152.2i	1.24170i	0.783928	+	0.620851i	$0.213213\pi$
$857$	31152.2i	1.24170i	−0.783928	+	0.620851i	$0.786787\pi$
$858$	0	0
$859$	31785.9	1.26254	0.631269	−	0.775564i	$-0.282534\pi$
$859$	31785.9	1.26254	0.631269	+	0.775564i	$0.282534\pi$
$860$	0	0
$861$	−9881.66	−0.391134
$862$	0	0
$863$	− 8290.40i	− 0.327009i	−0.986543	−	0.163504i	$-0.947720\pi$
$863$	− 8290.40i	− 0.327009i	0.986543	−	0.163504i	$-0.0522798\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 7417.14i	− 0.290541i
$868$	0	0
$869$	−4509.77	−0.176046
$870$	0	0
$871$	−26767.2	−1.04130
$872$	0	0
$873$	12285.7i	0.476297i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 8950.92i	− 0.344642i	−0.985041	−	0.172321i	$-0.944873\pi$
$877$	− 8950.92i	− 0.344642i	0.985041	−	0.172321i	$-0.0551266\pi$
$878$	0	0
$879$	24008.7	0.921267
$880$	0	0
$881$	33961.1	1.29873	0.649363	−	0.760478i	$-0.275036\pi$
$881$	33961.1	1.29873	0.649363	+	0.760478i	$0.275036\pi$
$882$	0	0
$883$	− 32361.6i	− 1.23336i	−0.787215	−	0.616679i	$-0.788478\pi$
$883$	− 32361.6i	− 1.23336i	0.787215	−	0.616679i	$-0.211522\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	28712.4i	1.08689i	0.839446	+	0.543443i	$0.182880\pi$
$887$	28712.4i	1.08689i	−0.839446	+	0.543443i	$0.817120\pi$
$888$	0	0
$889$	−8871.23	−0.334681
$890$	0	0
$891$	−2162.45	−0.0813072
$892$	0	0
$893$	81046.5i	3.03709i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 15657.0i	− 0.582801i
$898$	0	0
$899$	−11565.5	−0.429067
$900$	0	0
$901$	−12973.3	−0.479694
$902$	0	0
$903$	3316.19i	0.122210i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	14814.9i	0.542360i	0.962529	+	0.271180i	$0.0874138\pi$
$907$	14814.9i	0.542360i	−0.962529	+	0.271180i	$0.912586\pi$
$908$	0	0
$909$	1420.90	0.0518462
$910$	0	0
$911$	25721.2	0.935436	0.467718	−	0.883878i	$-0.345076\pi$
$911$	25721.2	0.935436	0.467718	+	0.883878i	$0.345076\pi$
$912$	0	0
$913$	22444.7i	0.813594i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	18695.9i	0.673276i
$918$	0	0
$919$	18520.0	0.664765	0.332382	−	0.943145i	$-0.392148\pi$
$919$	18520.0	0.664765	0.332382	+	0.943145i	$0.392148\pi$
$920$	0	0
$921$	10364.8	0.370825
$922$	0	0
$923$	− 41027.7i	− 1.46310i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	7747.64i	0.274505i
$928$	0	0
$929$	−22153.2	−0.782370	−0.391185	−	0.920312i	$-0.627935\pi$
$929$	−22153.2	−0.782370	−0.391185	+	0.920312i	$0.627935\pi$
$930$	0	0
$931$	−7108.79	−0.250248
$932$	0	0
$933$	− 13947.5i	− 0.489411i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 36393.2i	− 1.26885i	−0.772984	−	0.634426i	$-0.781237\pi$
$937$	− 36393.2i	− 1.26885i	0.772984	−	0.634426i	$-0.218763\pi$
$938$	0	0
$939$	19520.5	0.678411
$940$	0	0
$941$	54740.4	1.89637	0.948186	−	0.317716i	$-0.102916\pi$
$941$	54740.4	1.89637	0.948186	+	0.317716i	$0.102916\pi$
$942$	0	0
$943$	− 37182.2i	− 1.28401i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	39249.7i	1.34682i	0.739267	+	0.673412i	$0.235172\pi$
$947$	39249.7i	1.34682i	−0.739267	+	0.673412i	$0.764828\pi$
$948$	0	0
$949$	5075.38	0.173608
$950$	0	0
$951$	−4992.38	−0.170230
$952$	0	0
$953$	− 42849.0i	− 1.45647i	−0.685328	−	0.728234i	$-0.740341\pi$
$953$	− 42849.0i	− 1.45647i	0.685328	−	0.728234i	$-0.259659\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	8652.80i	0.292273i
$958$	0	0
$959$	7070.74	0.238088
$960$	0	0
$961$	−18331.1	−0.615324
$962$	0	0
$963$	− 9924.88i	− 0.332113i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	18397.1i	0.611801i	0.952064	+	0.305900i	$0.0989574\pi$
$967$	18397.1i	0.611801i	−0.952064	+	0.305900i	$0.901043\pi$
$968$	0	0
$969$	21501.6	0.712829
$970$	0	0
$971$	1379.79	0.0456021	0.0228011	−	0.999740i	$-0.492742\pi$
$971$	1379.79	0.0456021	0.0228011	+	0.999740i	$0.492742\pi$
$972$	0	0
$973$	19862.9i	0.654446i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	15827.3i	0.518281i	0.965840	+	0.259141i	$0.0834393\pi$
$977$	15827.3i	0.518281i	−0.965840	+	0.259141i	$0.916561\pi$
$978$	0	0
$979$	−27445.5	−0.895978
$980$	0	0
$981$	−12068.7	−0.392788
$982$	0	0
$983$	5439.58i	0.176496i	0.996099	+	0.0882480i	$0.0281268\pi$
$983$	5439.58i	0.176496i	−0.996099	+	0.0882480i	$0.971873\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	11731.5i	0.378337i
$988$	0	0
$989$	−12478.0	−0.401189
$990$	0	0
$991$	−33148.5	−1.06256	−0.531280	−	0.847196i	$-0.678289\pi$
$991$	−33148.5	−1.06256	−0.531280	+	0.847196i	$0.678289\pi$
$992$	0	0
$993$	3437.85i	0.109866i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	6846.42i	0.217481i	0.994070	+	0.108740i	$0.0346817\pi$
$997$	6846.42i	0.217481i	−0.994070	+	0.108740i	$0.965318\pi$
$998$	0	0
$999$	4698.18	0.148793

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2100.4.k.p.1849.2		6
5.2	odd	4		2100.4.a.v.1.2	✓	3
5.3	odd	4		2100.4.a.z.1.2	yes	3
5.4	even	2	inner	2100.4.k.p.1849.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2100.4.a.v.1.2	✓	3	5.2	odd	4
2100.4.a.z.1.2	yes	3	5.3	odd	4
2100.4.k.p.1849.2		6	1.1	even	1	trivial
2100.4.k.p.1849.5		6	5.4	even	2	inner

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)

\(f(q)\)	\(=\)	\(q-3.00000i q^{3} +7.00000i q^{7} -9.00000 q^{9} -26.6969 q^{11} -66.0486i q^{13} +49.4026i q^{17} +145.077 q^{19} +21.0000 q^{21} +79.0177i q^{23} +27.0000i q^{27} +108.038 q^{29} -107.051 q^{31} +80.0907i q^{33} -174.007i q^{37} -198.146 q^{39} -470.555 q^{41} +157.914i q^{43} +558.644i q^{47} -49.0000 q^{49} +148.208 q^{51} +262.604i q^{53} -435.232i q^{57} -101.734 q^{59} +227.179 q^{61} -63.0000i q^{63} -405.265i q^{67} +237.053 q^{69} +621.175 q^{71} +76.8430i q^{73} -186.878i q^{77} +168.925 q^{79} +81.0000 q^{81} -840.723i q^{83} -324.113i q^{87} +1028.04 q^{89} +462.340 q^{91} +321.153i q^{93} -1365.07i q^{97} +240.272 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 54 q^{9} - 16 q^{11} + 148 q^{19} + 126 q^{21} + 170 q^{29} - 454 q^{31} + 78 q^{39} - 678 q^{41} - 294 q^{49} - 510 q^{51} + 2634 q^{59} - 850 q^{61} + 354 q^{69} + 652 q^{71} + 3084 q^{79} + 486 q^{81}+ \cdots + 144 q^{99}+O(q^{100}) \) 6 * q - 54 * q^9 - 16 * q^11 + 148 * q^19 + 126 * q^21 + 170 * q^29 - 454 * q^31 + 78 * q^39 - 678 * q^41 - 294 * q^49 - 510 * q^51 + 2634 * q^59 - 850 * q^61 + 354 * q^69 + 652 * q^71 + 3084 * q^79 + 486 * q^81 + 3536 * q^89 - 182 * q^91 + 144 * q^99

Introduction

L-functions

Modular forms

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Database

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