LMFDB - Embedded newform 2100.4.k.l.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:	$4$
Coefficient field:	$\Q(i, \sqrt{421})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 211x^{2} + 11025 $ x^4 + 211*x^2 + 11025
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 420)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1849.1
Root			$9.75914i$ of defining polynomial
Character	$\chi$	$=$	2100.1849
Dual form			2100.4.k.l.1849.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000i q^{3} -7.00000i q^{7} -9.00000 q^{9} -47.0366 q^{11} -61.0366i q^{13} +132.073i q^{17} +105.110 q^{19} -21.0000 q^{21} -170.073i q^{23} +27.0000i q^{27} -260.073 q^{29} +24.9634 q^{31} +141.110i q^{33} +158.000i q^{37} -183.110 q^{39} -226.293 q^{41} -224.146i q^{43} -232.293i q^{47} -49.0000 q^{49} +396.219 q^{51} -11.1097i q^{53} -315.329i q^{57} -734.219 q^{59} +336.512 q^{61} +63.0000i q^{63} +270.219i q^{67} -510.219 q^{69} +116.963 q^{71} +981.622i q^{73} +329.256i q^{77} +974.658 q^{79} +81.0000 q^{81} +146.658i q^{83} +780.219i q^{87} -766.293 q^{89} -427.256 q^{91} -74.8903i q^{93} +1817.33i q^{97} +423.329 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 36 q^{9} - 24 q^{11} - 72 q^{19} - 84 q^{21} - 712 q^{29} + 264 q^{31} - 240 q^{39} + 408 q^{41} - 196 q^{49} + 600 q^{51} - 1952 q^{59} - 952 q^{61} - 1056 q^{69} + 632 q^{71} + 944 q^{79} + 324 q^{81}+ \cdots + 216 q^{99}+O(q^{100}) $ 4 * q - 36 * q^9 - 24 * q^11 - 72 * q^19 - 84 * q^21 - 712 * q^29 + 264 * q^31 - 240 * q^39 + 408 * q^41 - 196 * q^49 + 600 * q^51 - 1952 * q^59 - 952 * q^61 - 1056 * q^69 + 632 * q^71 + 944 * q^79 + 324 * q^81 - 1752 * q^89 - 560 * 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
$11$	−47.0366	−1.28928	−0.644639	−	0.764487i	$-0.722992\pi$
$11$	−47.0366	−1.28928	−0.644639	+	0.764487i	$0.722992\pi$
$12$	0	0
$13$	− 61.0366i	− 1.30219i	−0.758995	−	0.651096i	$-0.774309\pi$
$13$	− 61.0366i	− 1.30219i	0.758995	−	0.651096i	$-0.225691\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	132.073i	1.88426i	0.335246	+	0.942131i	$0.391181\pi$
$17$	132.073i	1.88426i	−0.335246	+	0.942131i	$0.608819\pi$
$18$	0	0
$19$	105.110	1.26915	0.634574	−	0.772862i	$-0.281175\pi$
$19$	105.110	1.26915	0.634574	+	0.772862i	$0.281175\pi$
$20$	0	0
$21$	−21.0000	−0.218218
$22$	0	0
$23$	− 170.073i	− 1.54186i	−0.636922	−	0.770928i	$-0.719793\pi$
$23$	− 170.073i	− 1.54186i	0.636922	−	0.770928i	$-0.280207\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	27.0000i	0.192450i
$28$	0	0
$29$	−260.073	−1.66532	−0.832662	−	0.553782i	$-0.813184\pi$
$29$	−260.073	−1.66532	−0.832662	+	0.553782i	$0.813184\pi$
$30$	0	0
$31$	24.9634	0.144631	0.0723156	−	0.997382i	$-0.476961\pi$
$31$	24.9634	0.144631	0.0723156	+	0.997382i	$0.476961\pi$
$32$	0	0
$33$	141.110i	0.744365i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	158.000i	0.702028i	0.936370	+	0.351014i	$0.114163\pi$
$37$	158.000i	0.702028i	−0.936370	+	0.351014i	$0.885837\pi$
$38$	0	0
$39$	−183.110	−0.751821
$40$	0	0
$41$	−226.293	−0.861975	−0.430987	−	0.902358i	$-0.641835\pi$
$41$	−226.293	−0.861975	−0.430987	+	0.902358i	$0.641835\pi$
$42$	0	0
$43$	− 224.146i	− 0.794930i	−0.917617	−	0.397465i	$-0.869890\pi$
$43$	− 224.146i	− 0.794930i	0.917617	−	0.397465i	$-0.130110\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 232.293i	− 0.720922i	−0.932774	−	0.360461i	$-0.882619\pi$
$47$	− 232.293i	− 0.720922i	0.932774	−	0.360461i	$-0.117381\pi$
$48$	0	0
$49$	−49.0000	−0.142857
$50$	0	0
$51$	396.219	1.08788
$52$	0	0
$53$	− 11.1097i	− 0.0287931i	−0.999896	−	0.0143966i	$-0.995417\pi$
$53$	− 11.1097i	− 0.0287931i	0.999896	−	0.0143966i	$-0.00458273\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 315.329i	− 0.732743i
$58$	0	0
$59$	−734.219	−1.62012	−0.810061	−	0.586345i	$-0.800566\pi$
$59$	−734.219	−1.62012	−0.810061	+	0.586345i	$0.800566\pi$
$60$	0	0
$61$	336.512	0.706327	0.353163	−	0.935562i	$-0.385106\pi$
$61$	336.512	0.706327	0.353163	+	0.935562i	$0.385106\pi$
$62$	0	0
$63$	63.0000i	0.125988i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	270.219i	0.492725i	0.969178	+	0.246362i	$0.0792353\pi$
$67$	270.219i	0.492725i	−0.969178	+	0.246362i	$0.920765\pi$
$68$	0	0
$69$	−510.219	−0.890191
$70$	0	0
$71$	116.963	0.195507	0.0977536	−	0.995211i	$-0.468834\pi$
$71$	116.963	0.195507	0.0977536	+	0.995211i	$0.468834\pi$
$72$	0	0
$73$	981.622i	1.57384i	0.617057	+	0.786919i	$0.288325\pi$
$73$	981.622i	1.57384i	−0.617057	+	0.786919i	$0.711675\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	329.256i	0.487301i
$78$	0	0
$79$	974.658	1.38807	0.694036	−	0.719941i	$-0.255831\pi$
$79$	974.658	1.38807	0.694036	+	0.719941i	$0.255831\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	146.658i	0.193950i	0.995287	+	0.0969749i	$0.0309167\pi$
$83$	146.658i	0.193950i	−0.995287	+	0.0969749i	$0.969083\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	780.219i	0.961475i
$88$	0	0
$89$	−766.293	−0.912661	−0.456331	−	0.889810i	$-0.650837\pi$
$89$	−766.293	−0.912661	−0.456331	+	0.889810i	$0.650837\pi$
$90$	0	0
$91$	−427.256	−0.492182
$92$	0	0
$93$	− 74.8903i	− 0.0835028i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	1817.33i	1.90229i	0.308750	+	0.951143i	$0.400090\pi$
$97$	1817.33i	1.90229i	−0.308750	+	0.951143i	$0.599910\pi$
$98$	0	0
$99$	423.329	0.429759
$100$	0	0
$101$	702.439	0.692032	0.346016	−	0.938229i	$-0.387534\pi$
$101$	702.439	0.692032	0.346016	+	0.938229i	$0.387534\pi$
$102$	0	0
$103$	399.415i	0.382092i	0.981581	+	0.191046i	$0.0611880\pi$
$103$	399.415i	0.382092i	−0.981581	+	0.191046i	$0.938812\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1360.59i	1.22928i	0.788809	+	0.614639i	$0.210698\pi$
$107$	1360.59i	1.22928i	−0.788809	+	0.614639i	$0.789302\pi$
$108$	0	0
$109$	1458.88	1.28197	0.640987	−	0.767552i	$-0.278525\pi$
$109$	1458.88	1.28197	0.640987	+	0.767552i	$0.278525\pi$
$110$	0	0
$111$	474.000	0.405316
$112$	0	0
$113$	− 1912.28i	− 1.59197i	−0.605319	−	0.795983i	$-0.706954\pi$
$113$	− 1912.28i	− 1.59197i	0.605319	−	0.795983i	$-0.293046\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	549.329i	0.434064i
$118$	0	0
$119$	924.512	0.712184
$120$	0	0
$121$	881.439	0.662238
$122$	0	0
$123$	678.878i	0.497661i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 866.219i	− 0.605233i	−0.953112	−	0.302616i	$-0.902140\pi$
$127$	− 866.219i	− 0.605233i	0.953112	−	0.302616i	$-0.0978601\pi$
$128$	0	0
$129$	−672.439	−0.458953
$130$	0	0
$131$	43.7074	0.0291507	0.0145753	−	0.999894i	$-0.495360\pi$
$131$	43.7074	0.0291507	0.0145753	+	0.999894i	$0.495360\pi$
$132$	0	0
$133$	− 735.768i	− 0.479693i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 2297.30i	− 1.43264i	−0.697771	−	0.716321i	$-0.745825\pi$
$137$	− 2297.30i	− 1.43264i	0.697771	−	0.716321i	$-0.254175\pi$
$138$	0	0
$139$	−1668.79	−1.01831	−0.509155	−	0.860675i	$-0.670042\pi$
$139$	−1668.79	−1.01831	−0.509155	+	0.860675i	$0.670042\pi$
$140$	0	0
$141$	−696.878	−0.416225
$142$	0	0
$143$	2870.95i	1.67889i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	147.000i	0.0824786i
$148$	0	0
$149$	114.878	0.0631620	0.0315810	−	0.999501i	$-0.489946\pi$
$149$	114.878	0.0631620	0.0315810	+	0.999501i	$0.489946\pi$
$150$	0	0
$151$	−335.415	−0.180766	−0.0903830	−	0.995907i	$-0.528809\pi$
$151$	−335.415	−0.180766	−0.0903830	+	0.995907i	$0.528809\pi$
$152$	0	0
$153$	− 1188.66i	− 0.628087i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 528.744i	− 0.268779i	−0.990929	−	0.134390i	$-0.957093\pi$
$157$	− 528.744i	− 0.268779i	0.990929	−	0.134390i	$-0.0429074\pi$
$158$	0	0
$159$	−33.3291	−0.0166237
$160$	0	0
$161$	−1190.51	−0.582767
$162$	0	0
$163$	3128.15i	1.50316i	0.659641	+	0.751581i	$0.270708\pi$
$163$	3128.15i	1.50316i	−0.659641	+	0.751581i	$0.729292\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1446.66i	0.670334i	0.942159	+	0.335167i	$0.108793\pi$
$167$	1446.66i	0.670334i	−0.942159	+	0.335167i	$0.891207\pi$
$168$	0	0
$169$	−1528.46	−0.695704
$170$	0	0
$171$	−945.987	−0.423049
$172$	0	0
$173$	2207.93i	0.970321i	0.874425	+	0.485161i	$0.161239\pi$
$173$	2207.93i	0.970321i	−0.874425	+	0.485161i	$0.838761\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	2202.66i	0.935378i
$178$	0	0
$179$	−770.134	−0.321578	−0.160789	−	0.986989i	$-0.551404\pi$
$179$	−770.134	−0.321578	−0.160789	+	0.986989i	$0.551404\pi$
$180$	0	0
$181$	−3117.22	−1.28012	−0.640058	−	0.768327i	$-0.721090\pi$
$181$	−3117.22	−1.28012	−0.640058	+	0.768327i	$0.721090\pi$
$182$	0	0
$183$	− 1009.54i	− 0.407798i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 6212.27i	− 2.42934i
$188$	0	0
$189$	189.000	0.0727393
$190$	0	0
$191$	183.768	0.0696177	0.0348089	−	0.999394i	$-0.488918\pi$
$191$	183.768	0.0696177	0.0348089	+	0.999394i	$0.488918\pi$
$192$	0	0
$193$	673.269i	0.251103i	0.992087	+	0.125552i	$0.0400701\pi$
$193$	673.269i	0.251103i	−0.992087	+	0.125552i	$0.959930\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	517.329i	0.187097i	0.995615	+	0.0935487i	$0.0298211\pi$
$197$	517.329i	0.187097i	−0.995615	+	0.0935487i	$0.970179\pi$
$198$	0	0
$199$	3843.18	1.36903	0.684513	−	0.729001i	$-0.260015\pi$
$199$	3843.18	1.36903	0.684513	+	0.729001i	$0.260015\pi$
$200$	0	0
$201$	810.658	0.284475
$202$	0	0
$203$	1820.51i	0.629433i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	1530.66i	0.513952i
$208$	0	0
$209$	−4944.00	−1.63629
$210$	0	0
$211$	−4402.95	−1.43655	−0.718274	−	0.695760i	$-0.755068\pi$
$211$	−4402.95	−1.43655	−0.718274	+	0.695760i	$0.755068\pi$
$212$	0	0
$213$	− 350.890i	− 0.112876i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 174.744i	− 0.0546654i
$218$	0	0
$219$	2944.87	0.908655
$220$	0	0
$221$	8061.29	2.45367
$222$	0	0
$223$	4218.80i	1.26687i	0.773796	+	0.633435i	$0.218356\pi$
$223$	4218.80i	1.26687i	−0.773796	+	0.633435i	$0.781644\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	4886.17i	1.42866i	0.699808	+	0.714331i	$0.253269\pi$
$227$	4886.17i	1.42866i	−0.699808	+	0.714331i	$0.746731\pi$
$228$	0	0
$229$	3917.39	1.13043	0.565215	−	0.824944i	$-0.308793\pi$
$229$	3917.39	1.13043	0.565215	+	0.824944i	$0.308793\pi$
$230$	0	0
$231$	987.768	0.281344
$232$	0	0
$233$	6058.65i	1.70350i	0.523950	+	0.851749i	$0.324458\pi$
$233$	6058.65i	1.70350i	−0.523950	+	0.851749i	$0.675542\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 2923.97i	− 0.801403i
$238$	0	0
$239$	6221.38	1.68380	0.841898	−	0.539636i	$-0.181438\pi$
$239$	6221.38	1.68380	0.841898	+	0.539636i	$0.181438\pi$
$240$	0	0
$241$	6257.54	1.67254	0.836272	−	0.548314i	$-0.184730\pi$
$241$	6257.54	1.67254	0.836272	+	0.548314i	$0.184730\pi$
$242$	0	0
$243$	− 243.000i	− 0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 6415.54i	− 1.65268i
$248$	0	0
$249$	439.975	0.111977
$250$	0	0
$251$	5030.51	1.26503	0.632516	−	0.774547i	$-0.282022\pi$
$251$	5030.51	1.26503	0.632516	+	0.774547i	$0.282022\pi$
$252$	0	0
$253$	7999.66i	1.98788i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 1968.49i	− 0.477785i	−0.971046	−	0.238893i	$-0.923216\pi$
$257$	− 1968.49i	− 0.477785i	0.971046	−	0.238893i	$-0.0767844\pi$
$258$	0	0
$259$	1106.00	0.265342
$260$	0	0
$261$	2340.66	0.555108
$262$	0	0
$263$	1872.15i	0.438941i	0.975619	+	0.219471i	$0.0704330\pi$
$263$	1872.15i	0.438941i	−0.975619	+	0.219471i	$0.929567\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	2298.88i	0.526925i
$268$	0	0
$269$	−955.513	−0.216575	−0.108287	−	0.994120i	$-0.534537\pi$
$269$	−955.513	−0.216575	−0.108287	+	0.994120i	$0.534537\pi$
$270$	0	0
$271$	7659.13	1.71682	0.858412	−	0.512961i	$-0.171452\pi$
$271$	7659.13	1.71682	0.858412	+	0.512961i	$0.171452\pi$
$272$	0	0
$273$	1281.77i	0.284162i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	6616.05i	1.43509i	0.696512	+	0.717545i	$0.254734\pi$
$277$	6616.05i	1.43509i	−0.696512	+	0.717545i	$0.745266\pi$
$278$	0	0
$279$	−224.671	−0.0482104
$280$	0	0
$281$	5008.66	1.06331	0.531657	−	0.846959i	$-0.321569\pi$
$281$	5008.66	1.06331	0.531657	+	0.846959i	$0.321569\pi$
$282$	0	0
$283$	− 5191.80i	− 1.09053i	−0.838263	−	0.545266i	$-0.816429\pi$
$283$	− 5191.80i	− 1.09053i	0.838263	−	0.545266i	$-0.183571\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1584.05i	0.325796i
$288$	0	0
$289$	−12530.3	−2.55044
$290$	0	0
$291$	5451.99	1.09829
$292$	0	0
$293$	− 5656.32i	− 1.12780i	−0.825843	−	0.563900i	$-0.809300\pi$
$293$	− 5656.32i	− 1.12780i	0.825843	−	0.563900i	$-0.190700\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 1269.99i	− 0.248122i
$298$	0	0
$299$	−10380.7	−2.00779
$300$	0	0
$301$	−1569.02	−0.300455
$302$	0	0
$303$	− 2107.32i	− 0.399545i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	6721.49i	1.24956i	0.780800	+	0.624781i	$0.214812\pi$
$307$	6721.49i	1.24956i	−0.780800	+	0.624781i	$0.785188\pi$
$308$	0	0
$309$	1198.24	0.220601
$310$	0	0
$311$	−3954.58	−0.721041	−0.360521	−	0.932751i	$-0.617401\pi$
$311$	−3954.58	−0.721041	−0.360521	+	0.932751i	$0.617401\pi$
$312$	0	0
$313$	7170.47i	1.29489i	0.762114	+	0.647443i	$0.224162\pi$
$313$	7170.47i	1.29489i	−0.762114	+	0.647443i	$0.775838\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 2169.62i	− 0.384410i	−0.981355	−	0.192205i	$-0.938436\pi$
$317$	− 2169.62i	− 0.384410i	0.981355	−	0.192205i	$-0.0615639\pi$
$318$	0	0
$319$	12232.9	2.14706
$320$	0	0
$321$	4081.76	0.709724
$322$	0	0
$323$	13882.2i	2.39141i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 4376.63i	− 0.740148i
$328$	0	0
$329$	−1626.05	−0.272483
$330$	0	0
$331$	−6194.90	−1.02871	−0.514354	−	0.857578i	$-0.671968\pi$
$331$	−6194.90	−1.02871	−0.514354	+	0.857578i	$0.671968\pi$
$332$	0	0
$333$	− 1422.00i	− 0.234009i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	8902.68i	1.43905i	0.694466	+	0.719525i	$0.255640\pi$
$337$	8902.68i	1.43905i	−0.694466	+	0.719525i	$0.744360\pi$
$338$	0	0
$339$	−5736.84	−0.919122
$340$	0	0
$341$	−1174.19	−0.186470
$342$	0	0
$343$	343.000i	0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 127.193i	− 0.0196774i	−0.999952	−	0.00983871i	$-0.996868\pi$
$347$	− 127.193i	− 0.0196774i	0.999952	−	0.00983871i	$-0.00313181\pi$
$348$	0	0
$349$	−4821.88	−0.739568	−0.369784	−	0.929118i	$-0.620568\pi$
$349$	−4821.88	−0.739568	−0.369784	+	0.929118i	$0.620568\pi$
$350$	0	0
$351$	1647.99	0.250607
$352$	0	0
$353$	1739.34i	0.262254i	0.991366	+	0.131127i	$0.0418596\pi$
$353$	1739.34i	0.262254i	−0.991366	+	0.131127i	$0.958140\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 2773.54i	− 0.411180i
$358$	0	0
$359$	3647.65	0.536255	0.268127	−	0.963383i	$-0.413595\pi$
$359$	3647.65	0.536255	0.268127	+	0.963383i	$0.413595\pi$
$360$	0	0
$361$	4189.05	0.610738
$362$	0	0
$363$	− 2644.32i	− 0.382343i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	31.5864i	0.00449264i	0.999997	+	0.00224632i	$0.000715026\pi$
$367$	31.5864i	0.00449264i	−0.999997	+	0.00224632i	$0.999285\pi$
$368$	0	0
$369$	2036.63	0.287325
$370$	0	0
$371$	−77.7680	−0.0108828
$372$	0	0
$373$	6879.14i	0.954929i	0.878651	+	0.477465i	$0.158444\pi$
$373$	6879.14i	0.954929i	−0.878651	+	0.477465i	$0.841556\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	15874.0i	2.16857i
$378$	0	0
$379$	−685.997	−0.0929744	−0.0464872	−	0.998919i	$-0.514803\pi$
$379$	−685.997	−0.0929744	−0.0464872	+	0.998919i	$0.514803\pi$
$380$	0	0
$381$	−2598.66	−0.349431
$382$	0	0
$383$	− 6316.51i	− 0.842712i	−0.906895	−	0.421356i	$-0.861554\pi$
$383$	− 6316.51i	− 0.842712i	0.906895	−	0.421356i	$-0.138446\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	2017.32i	0.264977i
$388$	0	0
$389$	−14214.5	−1.85271	−0.926357	−	0.376647i	$-0.877077\pi$
$389$	−14214.5	−1.85271	−0.926357	+	0.376647i	$0.877077\pi$
$390$	0	0
$391$	22462.1	2.90526
$392$	0	0
$393$	− 131.122i	− 0.0168301i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	8461.81i	1.06974i	0.844935	+	0.534869i	$0.179639\pi$
$397$	8461.81i	1.06974i	−0.844935	+	0.534869i	$0.820361\pi$
$398$	0	0
$399$	−2207.30	−0.276951
$400$	0	0
$401$	−6394.44	−0.796317	−0.398158	−	0.917317i	$-0.630351\pi$
$401$	−6394.44	−0.796317	−0.398158	+	0.917317i	$0.630351\pi$
$402$	0	0
$403$	− 1523.68i	− 0.188338i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 7431.78i	− 0.905109i
$408$	0	0
$409$	−1171.27	−0.141603	−0.0708015	−	0.997490i	$-0.522556\pi$
$409$	−1171.27	−0.141603	−0.0708015	+	0.997490i	$0.522556\pi$
$410$	0	0
$411$	−6891.91	−0.827136
$412$	0	0
$413$	5139.54i	0.612349i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	5006.38i	0.587922i
$418$	0	0
$419$	8353.80	0.974009	0.487005	−	0.873399i	$-0.338089\pi$
$419$	8353.80	0.974009	0.487005	+	0.873399i	$0.338089\pi$
$420$	0	0
$421$	5803.66	0.671860	0.335930	−	0.941887i	$-0.390949\pi$
$421$	5803.66	0.671860	0.335930	+	0.941887i	$0.390949\pi$
$422$	0	0
$423$	2090.63i	0.240307i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 2355.58i	− 0.266966i
$428$	0	0
$429$	8612.85	0.969306
$430$	0	0
$431$	9433.94	1.05433	0.527166	−	0.849763i	$-0.323255\pi$
$431$	9433.94	1.05433	0.527166	+	0.849763i	$0.323255\pi$
$432$	0	0
$433$	− 5553.67i	− 0.616380i	−0.951325	−	0.308190i	$-0.900277\pi$
$433$	− 5553.67i	− 0.616380i	0.951325	−	0.308190i	$-0.0997232\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 17876.3i	− 1.95684i
$438$	0	0
$439$	7388.06	0.803218	0.401609	−	0.915811i	$-0.368451\pi$
$439$	7388.06	0.803218	0.401609	+	0.915811i	$0.368451\pi$
$440$	0	0
$441$	441.000	0.0476190
$442$	0	0
$443$	− 7823.43i	− 0.839058i	−0.907742	−	0.419529i	$-0.862195\pi$
$443$	− 7823.43i	− 0.839058i	0.907742	−	0.419529i	$-0.137805\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 344.633i	− 0.0364666i
$448$	0	0
$449$	5863.12	0.616253	0.308127	−	0.951345i	$-0.400298\pi$
$449$	5863.12	0.616253	0.308127	+	0.951345i	$0.400298\pi$
$450$	0	0
$451$	10644.0	1.11132
$452$	0	0
$453$	1006.24i	0.104365i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 17072.7i	− 1.74754i	−0.486336	−	0.873772i	$-0.661667\pi$
$457$	− 17072.7i	− 1.74754i	0.486336	−	0.873772i	$-0.338333\pi$
$458$	0	0
$459$	−3565.97	−0.362626
$460$	0	0
$461$	−10383.1	−1.04900	−0.524499	−	0.851411i	$-0.675747\pi$
$461$	−10383.1	−1.04900	−0.524499	+	0.851411i	$0.675747\pi$
$462$	0	0
$463$	− 858.169i	− 0.0861393i	−0.999072	−	0.0430696i	$-0.986286\pi$
$463$	− 858.169i	− 0.0861393i	0.999072	−	0.0430696i	$-0.0137137\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	5420.71i	0.537131i	0.963261	+	0.268566i	$0.0865496\pi$
$467$	5420.71i	0.537131i	−0.963261	+	0.268566i	$0.913450\pi$
$468$	0	0
$469$	1891.54	0.186232
$470$	0	0
$471$	−1586.23	−0.155180
$472$	0	0
$473$	10543.1i	1.02489i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	99.9874i	0.00959771i
$478$	0	0
$479$	−13103.4	−1.24991	−0.624956	−	0.780660i	$-0.714883\pi$
$479$	−13103.4	−1.24991	−0.624956	+	0.780660i	$0.714883\pi$
$480$	0	0
$481$	9643.78	0.914176
$482$	0	0
$483$	3571.54i	0.336461i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	5500.85i	0.511843i	0.966698	+	0.255921i	$0.0823788\pi$
$487$	5500.85i	0.511843i	−0.966698	+	0.255921i	$0.917621\pi$
$488$	0	0
$489$	9384.44	0.867851
$490$	0	0
$491$	5940.67	0.546025	0.273013	−	0.962010i	$-0.411980\pi$
$491$	5940.67	0.546025	0.273013	+	0.962010i	$0.411980\pi$
$492$	0	0
$493$	− 34348.7i	− 3.13790i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 818.744i	− 0.0738947i
$498$	0	0
$499$	14869.1	1.33393	0.666967	−	0.745087i	$-0.267592\pi$
$499$	14869.1	1.33393	0.666967	+	0.745087i	$0.267592\pi$
$500$	0	0
$501$	4339.97	0.387018
$502$	0	0
$503$	13859.9i	1.22860i	0.789074	+	0.614298i	$0.210561\pi$
$503$	13859.9i	1.22860i	−0.789074	+	0.614298i	$0.789439\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	4585.39i	0.401665i
$508$	0	0
$509$	2427.02	0.211348	0.105674	−	0.994401i	$-0.466300\pi$
$509$	2427.02	0.211348	0.105674	+	0.994401i	$0.466300\pi$
$510$	0	0
$511$	6871.35	0.594855
$512$	0	0
$513$	2837.96i	0.244248i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	10926.2i	0.929469i
$518$	0	0
$519$	6623.78	0.560215
$520$	0	0
$521$	15695.0	1.31979	0.659894	−	0.751359i	$-0.270601\pi$
$521$	15695.0	1.31979	0.659894	+	0.751359i	$0.270601\pi$
$522$	0	0
$523$	18552.1i	1.55110i	0.631284	+	0.775552i	$0.282528\pi$
$523$	18552.1i	1.55110i	−0.631284	+	0.775552i	$0.717472\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	3297.00i	0.272523i
$528$	0	0
$529$	−16757.9	−1.37732
$530$	0	0
$531$	6607.97	0.540041
$532$	0	0
$533$	13812.1i	1.12246i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	2310.40i	0.185663i
$538$	0	0
$539$	2304.79	0.184183
$540$	0	0
$541$	−15752.8	−1.25188	−0.625940	−	0.779871i	$-0.715285\pi$
$541$	−15752.8	−1.25188	−0.625940	+	0.779871i	$0.715285\pi$
$542$	0	0
$543$	9351.65i	0.739075i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	5697.49i	0.445351i	0.974893	+	0.222676i	$0.0714790\pi$
$547$	5697.49i	0.445351i	−0.974893	+	0.222676i	$0.928521\pi$
$548$	0	0
$549$	−3028.61	−0.235442
$550$	0	0
$551$	−27336.2	−2.11354
$552$	0	0
$553$	− 6822.61i	− 0.524642i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 2495.82i	− 0.189858i	−0.995484	−	0.0949292i	$-0.969738\pi$
$557$	− 2495.82i	− 0.189858i	0.995484	−	0.0949292i	$-0.0302625\pi$
$558$	0	0
$559$	−13681.1	−1.03515
$560$	0	0
$561$	−18636.8	−1.40258
$562$	0	0
$563$	9315.26i	0.697321i	0.937249	+	0.348660i	$0.113363\pi$
$563$	9315.26i	0.697321i	−0.937249	+	0.348660i	$0.886637\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 567.000i	− 0.0419961i
$568$	0	0
$569$	−13941.5	−1.02717	−0.513584	−	0.858039i	$-0.671683\pi$
$569$	−13941.5	−1.02717	−0.513584	+	0.858039i	$0.671683\pi$
$570$	0	0
$571$	−23166.8	−1.69790	−0.848950	−	0.528472i	$-0.822765\pi$
$571$	−23166.8	−1.69790	−0.848950	+	0.528472i	$0.822765\pi$
$572$	0	0
$573$	− 551.304i	− 0.0401938i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 24466.1i	− 1.76523i	−0.470099	−	0.882614i	$-0.655782\pi$
$577$	− 24466.1i	− 1.76523i	0.470099	−	0.882614i	$-0.344218\pi$
$578$	0	0
$579$	2019.81	0.144975
$580$	0	0
$581$	1026.61	0.0733061
$582$	0	0
$583$	522.563i	0.0371223i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	17270.0i	1.21433i	0.794576	+	0.607165i	$0.207693\pi$
$587$	17270.0i	1.21433i	−0.794576	+	0.607165i	$0.792307\pi$
$588$	0	0
$589$	2623.90	0.183558
$590$	0	0
$591$	1551.99	0.108021
$592$	0	0
$593$	27715.9i	1.91932i	0.281169	+	0.959658i	$0.409278\pi$
$593$	27715.9i	1.91932i	−0.281169	+	0.959658i	$0.590722\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 11529.5i	− 0.790407i
$598$	0	0
$599$	−10382.5	−0.708209	−0.354105	−	0.935206i	$-0.615214\pi$
$599$	−10382.5	−0.708209	−0.354105	+	0.935206i	$0.615214\pi$
$600$	0	0
$601$	11995.0	0.814120	0.407060	−	0.913401i	$-0.366554\pi$
$601$	11995.0	0.814120	0.407060	+	0.913401i	$0.366554\pi$
$602$	0	0
$603$	− 2431.97i	− 0.164242i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	9933.39i	0.664224i	0.943240	+	0.332112i	$0.107761\pi$
$607$	9933.39i	0.664224i	−0.943240	+	0.332112i	$0.892239\pi$
$608$	0	0
$609$	5461.54	0.363403
$610$	0	0
$611$	−14178.3	−0.938779
$612$	0	0
$613$	− 27758.4i	− 1.82896i	−0.404635	−	0.914478i	$-0.632601\pi$
$613$	− 27758.4i	− 1.82896i	0.404635	−	0.914478i	$-0.367399\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 7929.23i	− 0.517373i	−0.965961	−	0.258686i	$-0.916710\pi$
$617$	− 7929.23i	− 0.517373i	0.965961	−	0.258686i	$-0.0832896\pi$
$618$	0	0
$619$	−15219.3	−0.988230	−0.494115	−	0.869397i	$-0.664508\pi$
$619$	−15219.3	−0.988230	−0.494115	+	0.869397i	$0.664508\pi$
$620$	0	0
$621$	4591.97	0.296730
$622$	0	0
$623$	5364.05i	0.344954i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	14832.0i	0.944710i
$628$	0	0
$629$	−20867.6	−1.32280
$630$	0	0
$631$	−17582.4	−1.10926	−0.554630	−	0.832097i	$-0.687140\pi$
$631$	−17582.4	−1.10926	−0.554630	+	0.832097i	$0.687140\pi$
$632$	0	0
$633$	13208.9i	0.829391i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	2990.79i	0.186027i
$638$	0	0
$639$	−1052.67	−0.0651690
$640$	0	0
$641$	−25878.8	−1.59462	−0.797310	−	0.603570i	$-0.793744\pi$
$641$	−25878.8	−1.59462	−0.797310	+	0.603570i	$0.793744\pi$
$642$	0	0
$643$	− 12752.7i	− 0.782140i	−0.920361	−	0.391070i	$-0.872105\pi$
$643$	− 12752.7i	− 0.782140i	0.920361	−	0.391070i	$-0.127895\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 10878.7i	− 0.661030i	−0.943801	−	0.330515i	$-0.892778\pi$
$647$	− 10878.7i	− 0.661030i	0.943801	−	0.330515i	$-0.107222\pi$
$648$	0	0
$649$	34535.2	2.08879
$650$	0	0
$651$	−524.232	−0.0315611
$652$	0	0
$653$	22872.2i	1.37069i	0.728219	+	0.685344i	$0.240348\pi$
$653$	22872.2i	1.37069i	−0.728219	+	0.685344i	$0.759652\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 8834.60i	− 0.524612i
$658$	0	0
$659$	4776.04	0.282319	0.141159	−	0.989987i	$-0.454917\pi$
$659$	4776.04	0.282319	0.141159	+	0.989987i	$0.454917\pi$
$660$	0	0
$661$	−7581.58	−0.446126	−0.223063	−	0.974804i	$-0.571606\pi$
$661$	−7581.58	−0.446126	−0.223063	+	0.974804i	$0.571606\pi$
$662$	0	0
$663$	− 24183.9i	− 1.41663i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	44231.5i	2.56769i
$668$	0	0
$669$	12656.4	0.731428
$670$	0	0
$671$	−15828.4	−0.910652
$672$	0	0
$673$	− 1065.64i	− 0.0610361i	−0.999534	−	0.0305180i	$-0.990284\pi$
$673$	− 1065.64i	− 0.0610361i	0.999534	−	0.0305180i	$-0.00971570\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 2755.35i	− 0.156420i	−0.996937	−	0.0782102i	$-0.975079\pi$
$677$	− 2755.35i	− 0.156420i	0.996937	−	0.0782102i	$-0.0249205\pi$
$678$	0	0
$679$	12721.3	0.718997
$680$	0	0
$681$	14658.5	0.824839
$682$	0	0
$683$	− 5779.63i	− 0.323794i	−0.986808	−	0.161897i	$-0.948239\pi$
$683$	− 5779.63i	− 0.323794i	0.986808	−	0.161897i	$-0.0517613\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 11752.2i	− 0.652654i
$688$	0	0
$689$	−678.098	−0.0374942
$690$	0	0
$691$	13138.1	0.723297	0.361648	−	0.932315i	$-0.382214\pi$
$691$	13138.1	0.723297	0.361648	+	0.932315i	$0.382214\pi$
$692$	0	0
$693$	− 2963.30i	− 0.162434i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 29887.2i	− 1.62419i
$698$	0	0
$699$	18175.9	0.983515
$700$	0	0
$701$	−19665.9	−1.05959	−0.529793	−	0.848127i	$-0.677730\pi$
$701$	−19665.9	−1.05959	−0.529793	+	0.848127i	$0.677730\pi$
$702$	0	0
$703$	16607.3i	0.890978i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 4917.07i	− 0.261564i
$708$	0	0
$709$	−13114.0	−0.694652	−0.347326	−	0.937744i	$-0.612910\pi$
$709$	−13114.0	−0.694652	−0.347326	+	0.937744i	$0.612910\pi$
$710$	0	0
$711$	−8771.92	−0.462690
$712$	0	0
$713$	− 4245.61i	− 0.223000i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 18664.1i	− 0.972141i
$718$	0	0
$719$	−1591.29	−0.0825386	−0.0412693	−	0.999148i	$-0.513140\pi$
$719$	−1591.29	−0.0825386	−0.0412693	+	0.999148i	$0.513140\pi$
$720$	0	0
$721$	2795.90	0.144417
$722$	0	0
$723$	− 18772.6i	− 0.965644i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 8618.60i	− 0.439678i	−0.975536	−	0.219839i	$-0.929447\pi$
$727$	− 8618.60i	− 0.439678i	0.975536	−	0.219839i	$-0.0705533\pi$
$728$	0	0
$729$	−729.000	−0.0370370
$730$	0	0
$731$	29603.7	1.49786
$732$	0	0
$733$	− 20709.0i	− 1.04353i	−0.853091	−	0.521763i	$-0.825275\pi$
$733$	− 20709.0i	− 1.04353i	0.853091	−	0.521763i	$-0.174725\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 12710.2i	− 0.635259i
$738$	0	0
$739$	31806.0	1.58323	0.791613	−	0.611022i	$-0.209242\pi$
$739$	31806.0	1.58323	0.791613	+	0.611022i	$0.209242\pi$
$740$	0	0
$741$	−19246.6	−0.954172
$742$	0	0
$743$	− 16734.7i	− 0.826293i	−0.910665	−	0.413146i	$-0.864430\pi$
$743$	− 16734.7i	− 0.826293i	0.910665	−	0.413146i	$-0.135570\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 1319.92i	− 0.0646499i
$748$	0	0
$749$	9524.10	0.464623
$750$	0	0
$751$	−21859.9	−1.06215	−0.531077	−	0.847323i	$-0.678213\pi$
$751$	−21859.9	−1.06215	−0.531077	+	0.847323i	$0.678213\pi$
$752$	0	0
$753$	− 15091.5i	− 0.730367i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	25203.5i	1.21009i	0.796191	+	0.605045i	$0.206845\pi$
$757$	25203.5i	1.21009i	−0.796191	+	0.605045i	$0.793155\pi$
$758$	0	0
$759$	23999.0	1.14770
$760$	0	0
$761$	−1716.73	−0.0817760	−0.0408880	−	0.999164i	$-0.513019\pi$
$761$	−1716.73	−0.0817760	−0.0408880	+	0.999164i	$0.513019\pi$
$762$	0	0
$763$	− 10212.1i	− 0.484541i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	44814.2i	2.10971i
$768$	0	0
$769$	−17702.8	−0.830140	−0.415070	−	0.909790i	$-0.636243\pi$
$769$	−17702.8	−0.830140	−0.415070	+	0.909790i	$0.636243\pi$
$770$	0	0
$771$	−5905.46	−0.275849
$772$	0	0
$773$	12756.7i	0.593564i	0.954945	+	0.296782i	$0.0959135\pi$
$773$	12756.7i	0.593564i	−0.954945	+	0.296782i	$0.904087\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 3318.00i	− 0.153195i
$778$	0	0
$779$	−23785.5	−1.09397
$780$	0	0
$781$	−5501.56	−0.252063
$782$	0	0
$783$	− 7021.97i	− 0.320492i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	24168.7i	1.09469i	0.836907	+	0.547345i	$0.184361\pi$
$787$	24168.7i	1.09469i	−0.836907	+	0.547345i	$0.815639\pi$
$788$	0	0
$789$	5616.44	0.253423
$790$	0	0
$791$	−13386.0	−0.601707
$792$	0	0
$793$	− 20539.5i	− 0.919773i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 35375.0i	− 1.57221i	−0.618096	−	0.786103i	$-0.712096\pi$
$797$	− 35375.0i	− 1.57221i	0.618096	−	0.786103i	$-0.287904\pi$
$798$	0	0
$799$	30679.6	1.35841
$800$	0	0
$801$	6896.63	0.304220
$802$	0	0
$803$	− 46172.1i	− 2.02911i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	2866.54i	0.125040i
$808$	0	0
$809$	−14463.9	−0.628582	−0.314291	−	0.949327i	$-0.601767\pi$
$809$	−14463.9	−0.628582	−0.314291	+	0.949327i	$0.601767\pi$
$810$	0	0
$811$	−1250.23	−0.0541325	−0.0270663	−	0.999634i	$-0.508617\pi$
$811$	−1250.23	−0.0541325	−0.0270663	+	0.999634i	$0.508617\pi$
$812$	0	0
$813$	− 22977.4i	− 0.991208i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 23559.9i	− 1.00888i
$818$	0	0
$819$	3845.30	0.164061
$820$	0	0
$821$	−12337.5	−0.524462	−0.262231	−	0.965005i	$-0.584458\pi$
$821$	−12337.5	−0.524462	−0.262231	+	0.965005i	$0.584458\pi$
$822$	0	0
$823$	18837.4i	0.797852i	0.916983	+	0.398926i	$0.130617\pi$
$823$	18837.4i	0.797852i	−0.916983	+	0.398926i	$0.869383\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 24463.7i	− 1.02864i	−0.857598	−	0.514320i	$-0.828044\pi$
$827$	− 24463.7i	− 1.02864i	0.857598	−	0.514320i	$-0.171956\pi$
$828$	0	0
$829$	−15847.1	−0.663922	−0.331961	−	0.943293i	$-0.607710\pi$
$829$	−15847.1	−0.663922	−0.331961	+	0.943293i	$0.607710\pi$
$830$	0	0
$831$	19848.1	0.828549
$832$	0	0
$833$	− 6471.58i	− 0.269180i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	674.013i	0.0278343i
$838$	0	0
$839$	−15175.4	−0.624451	−0.312225	−	0.950008i	$-0.601074\pi$
$839$	−15175.4	−0.624451	−0.312225	+	0.950008i	$0.601074\pi$
$840$	0	0
$841$	43249.0	1.77330
$842$	0	0
$843$	− 15026.0i	− 0.613905i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 6170.07i	− 0.250302i
$848$	0	0
$849$	−15575.4	−0.629619
$850$	0	0
$851$	26871.6	1.08243
$852$	0	0
$853$	− 16286.7i	− 0.653745i	−0.945068	−	0.326872i	$-0.894005\pi$
$853$	− 16286.7i	− 0.653745i	0.945068	−	0.326872i	$-0.105995\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 19223.2i	− 0.766221i	−0.923703	−	0.383110i	$-0.874853\pi$
$857$	− 19223.2i	− 0.766221i	0.923703	−	0.383110i	$-0.125147\pi$
$858$	0	0
$859$	−37835.3	−1.50282	−0.751411	−	0.659835i	$-0.770626\pi$
$859$	−37835.3	−1.50282	−0.751411	+	0.659835i	$0.770626\pi$
$860$	0	0
$861$	4752.14	0.188098
$862$	0	0
$863$	4724.16i	0.186341i	0.995650	+	0.0931705i	$0.0297002\pi$
$863$	4724.16i	0.186341i	−0.995650	+	0.0931705i	$0.970300\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	37590.9i	1.47250i
$868$	0	0
$869$	−45844.6	−1.78961
$870$	0	0
$871$	16493.3	0.641622
$872$	0	0
$873$	− 16356.0i	− 0.634096i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 10312.1i	− 0.397054i	−0.980095	−	0.198527i	$-0.936384\pi$
$877$	− 10312.1i	− 0.397054i	0.980095	−	0.198527i	$-0.0636157\pi$
$878$	0	0
$879$	−16968.9	−0.651136
$880$	0	0
$881$	−30026.2	−1.14825	−0.574126	−	0.818767i	$-0.694658\pi$
$881$	−30026.2	−1.14825	−0.574126	+	0.818767i	$0.694658\pi$
$882$	0	0
$883$	− 5078.42i	− 0.193548i	−0.995306	−	0.0967738i	$-0.969148\pi$
$883$	− 5078.42i	− 0.193548i	0.995306	−	0.0967738i	$-0.0308523\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 37482.4i	− 1.41887i	−0.704772	−	0.709434i	$-0.748951\pi$
$887$	− 37482.4i	− 1.41887i	0.704772	−	0.709434i	$-0.251049\pi$
$888$	0	0
$889$	−6063.54	−0.228756
$890$	0	0
$891$	−3809.96	−0.143253
$892$	0	0
$893$	− 24416.2i	− 0.914958i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	31142.0i	1.15920i
$898$	0	0
$899$	−6492.32	−0.240858
$900$	0	0
$901$	1467.29	0.0542538
$902$	0	0
$903$	4707.07i	0.173468i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 27832.8i	− 1.01893i	−0.860490	−	0.509467i	$-0.829843\pi$
$907$	− 27832.8i	− 1.01893i	0.860490	−	0.509467i	$-0.170157\pi$
$908$	0	0
$909$	−6321.95	−0.230677
$910$	0	0
$911$	−10366.9	−0.377026	−0.188513	−	0.982071i	$-0.560367\pi$
$911$	−10366.9	−0.377026	−0.188513	+	0.982071i	$0.560367\pi$
$912$	0	0
$913$	− 6898.30i	− 0.250055i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 305.952i	− 0.0110179i
$918$	0	0
$919$	29978.3	1.07605	0.538026	−	0.842928i	$-0.319170\pi$
$919$	29978.3	1.07605	0.538026	+	0.842928i	$0.319170\pi$
$920$	0	0
$921$	20164.5	0.721435
$922$	0	0
$923$	− 7139.05i	− 0.254588i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 3594.73i	− 0.127364i
$928$	0	0
$929$	−19198.7	−0.678028	−0.339014	−	0.940781i	$-0.610093\pi$
$929$	−19198.7	−0.678028	−0.339014	+	0.940781i	$0.610093\pi$
$930$	0	0
$931$	−5150.38	−0.181307
$932$	0	0
$933$	11863.7i	0.416293i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	20296.3i	0.707631i	0.935315	+	0.353815i	$0.115116\pi$
$937$	20296.3i	0.707631i	−0.935315	+	0.353815i	$0.884884\pi$
$938$	0	0
$939$	21511.4	0.747602
$940$	0	0
$941$	−19254.6	−0.667037	−0.333518	−	0.942744i	$-0.608236\pi$
$941$	−19254.6	−0.667037	−0.333518	+	0.942744i	$0.608236\pi$
$942$	0	0
$943$	38486.3i	1.32904i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	51679.2i	1.77333i	0.462408	+	0.886667i	$0.346986\pi$
$947$	51679.2i	1.77333i	−0.462408	+	0.886667i	$0.653014\pi$
$948$	0	0
$949$	59914.8	2.04944
$950$	0	0
$951$	−6508.87	−0.221939
$952$	0	0
$953$	5456.65i	0.185476i	0.995691	+	0.0927378i	$0.0295618\pi$
$953$	5456.65i	0.185476i	−0.995691	+	0.0927378i	$0.970438\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 36698.8i	− 1.23961i
$958$	0	0
$959$	−16081.1	−0.541488
$960$	0	0
$961$	−29167.8	−0.979082
$962$	0	0
$963$	− 12245.3i	− 0.409759i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	15567.7i	0.517708i	0.965916	+	0.258854i	$0.0833448\pi$
$967$	15567.7i	0.517708i	−0.965916	+	0.258854i	$0.916655\pi$
$968$	0	0
$969$	41646.5	1.38068
$970$	0	0
$971$	13166.3	0.435145	0.217572	−	0.976044i	$-0.430186\pi$
$971$	13166.3	0.435145	0.217572	+	0.976044i	$0.430186\pi$
$972$	0	0
$973$	11681.5i	0.384885i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	17844.2i	0.584325i	0.956369	+	0.292163i	$0.0943748\pi$
$977$	17844.2i	0.584325i	−0.956369	+	0.292163i	$0.905625\pi$
$978$	0	0
$979$	36043.8	1.17667
$980$	0	0
$981$	−13129.9	−0.427325
$982$	0	0
$983$	6809.32i	0.220940i	0.993879	+	0.110470i	$0.0352355\pi$
$983$	6809.32i	0.220940i	−0.993879	+	0.110470i	$0.964764\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	4878.14i	0.157318i
$988$	0	0
$989$	−38121.3	−1.22567
$990$	0	0
$991$	−26981.0	−0.864864	−0.432432	−	0.901667i	$-0.642344\pi$
$991$	−26981.0	−0.864864	−0.432432	+	0.901667i	$0.642344\pi$
$992$	0	0
$993$	18584.7i	0.593925i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	3272.54i	0.103954i	0.998648	+	0.0519771i	$0.0165523\pi$
$997$	3272.54i	0.103954i	−0.998648	+	0.0519771i	$0.983448\pi$
$998$	0	0
$999$	−4266.00	−0.135105

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
420.4.a.i.1.1	✓	2	5.3	odd	4
1260.4.a.n.1.2		2	15.8	even	4
1680.4.a.bc.1.2		2	20.3	even	4
2100.4.a.o.1.1		2	5.2	odd	4
2100.4.k.l.1849.1		4	1.1	even	1	trivial
2100.4.k.l.1849.3		4	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)

\(f(q)\)	\(=\)	\(q-3.00000i q^{3} -7.00000i q^{7} -9.00000 q^{9} -47.0366 q^{11} -61.0366i q^{13} +132.073i q^{17} +105.110 q^{19} -21.0000 q^{21} -170.073i q^{23} +27.0000i q^{27} -260.073 q^{29} +24.9634 q^{31} +141.110i q^{33} +158.000i q^{37} -183.110 q^{39} -226.293 q^{41} -224.146i q^{43} -232.293i q^{47} -49.0000 q^{49} +396.219 q^{51} -11.1097i q^{53} -315.329i q^{57} -734.219 q^{59} +336.512 q^{61} +63.0000i q^{63} +270.219i q^{67} -510.219 q^{69} +116.963 q^{71} +981.622i q^{73} +329.256i q^{77} +974.658 q^{79} +81.0000 q^{81} +146.658i q^{83} +780.219i q^{87} -766.293 q^{89} -427.256 q^{91} -74.8903i q^{93} +1817.33i q^{97} +423.329 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 36 q^{9} - 24 q^{11} - 72 q^{19} - 84 q^{21} - 712 q^{29} + 264 q^{31} - 240 q^{39} + 408 q^{41} - 196 q^{49} + 600 q^{51} - 1952 q^{59} - 952 q^{61} - 1056 q^{69} + 632 q^{71} + 944 q^{79} + 324 q^{81}+ \cdots + 216 q^{99}+O(q^{100}) \) 4 * q - 36 * q^9 - 24 * q^11 - 72 * q^19 - 84 * q^21 - 712 * q^29 + 264 * q^31 - 240 * q^39 + 408 * q^41 - 196 * q^49 + 600 * q^51 - 1952 * q^59 - 952 * q^61 - 1056 * q^69 + 632 * q^71 + 944 * q^79 + 324 * q^81 - 1752 * q^89 - 560 * q^91 + 216 * q^99

Introduction

L-functions

Modular forms

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Database

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