LMFDB - Embedded newform 2028.4.b.d.337.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2028,4,Mod(337,2028)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2028, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2028.337");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2028 = 2^{2} \cdot 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2028.b (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:	$119.655873492$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 156)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	2028.337
Dual form			2028.4.b.d.337.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000 q^{3} -2.00000i q^{5} +32.0000i q^{7} +9.00000 q^{9} +O(q^{10})$	$q+3.00000 q^{3} -2.00000i q^{5} +32.0000i q^{7} +9.00000 q^{9} +68.0000i q^{11} -6.00000i q^{15} +14.0000 q^{17} +4.00000i q^{19} +96.0000i q^{21} -72.0000 q^{23} +121.000 q^{25} +27.0000 q^{27} +102.000 q^{29} -136.000i q^{31} +204.000i q^{33} +64.0000 q^{35} +386.000i q^{37} +250.000i q^{41} +140.000 q^{43} -18.0000i q^{45} +296.000i q^{47} -681.000 q^{49} +42.0000 q^{51} +526.000 q^{53} +136.000 q^{55} +12.0000i q^{57} -332.000i q^{59} -410.000 q^{61} +288.000i q^{63} +596.000i q^{67} -216.000 q^{69} -880.000i q^{71} -506.000i q^{73} +363.000 q^{75} -2176.00 q^{77} -640.000 q^{79} +81.0000 q^{81} +1380.00i q^{83} -28.0000i q^{85} +306.000 q^{87} -1450.00i q^{89} -408.000i q^{93} +8.00000 q^{95} -446.000i q^{97} +612.000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{3} + 18 q^{9}+O(q^{10}) $ 2 * q + 6 * q^3 + 18 * q^9	$ 2 q + 6 q^{3} + 18 q^{9} + 28 q^{17} - 144 q^{23} + 242 q^{25} + 54 q^{27} + 204 q^{29} + 128 q^{35} + 280 q^{43} - 1362 q^{49} + 84 q^{51} + 1052 q^{53} + 272 q^{55} - 820 q^{61} - 432 q^{69} + 726 q^{75} - 4352 q^{77} - 1280 q^{79} + 162 q^{81} + 612 q^{87} + 16 q^{95}+O(q^{100}) $ 2 * q + 6 * q^3 + 18 * q^9 + 28 * q^17 - 144 * q^23 + 242 * q^25 + 54 * q^27 + 204 * q^29 + 128 * q^35 + 280 * q^43 - 1362 * q^49 + 84 * q^51 + 1052 * q^53 + 272 * q^55 - 820 * q^61 - 432 * q^69 + 726 * q^75 - 4352 * q^77 - 1280 * q^79 + 162 * q^81 + 612 * q^87 + 16 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2028\mathbb{Z}\right)^\times$.

$n$	$677$	$1015$	$1861$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	0	0
$5$	− 2.00000i	− 0.178885i	−0.995992	−	0.0894427i	$-0.971491\pi$
$5$	− 2.00000i	− 0.178885i	0.995992	−	0.0894427i	$-0.0285086\pi$
$6$	0	0
$7$	32.0000i	1.72784i	0.503631	+	0.863919i	$0.331997\pi$
$7$	32.0000i	1.72784i	−0.503631	+	0.863919i	$0.668003\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	68.0000i	1.86389i	0.362602	+	0.931944i	$0.381889\pi$
$11$	68.0000i	1.86389i	−0.362602	+	0.931944i	$0.618111\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	− 6.00000i	− 0.103280i
$16$	0	0
$17$	14.0000	0.199735	0.0998676	−	0.995001i	$-0.468158\pi$
$17$	14.0000	0.199735	0.0998676	+	0.995001i	$0.468158\pi$
$18$	0	0
$19$	4.00000i	0.0482980i	0.999708	+	0.0241490i	$0.00768762\pi$
$19$	4.00000i	0.0482980i	−0.999708	+	0.0241490i	$0.992312\pi$
$20$	0	0
$21$	96.0000i	0.997567i
$22$	0	0
$23$	−72.0000	−0.652741	−0.326370	−	0.945242i	$-0.605826\pi$
$23$	−72.0000	−0.652741	−0.326370	+	0.945242i	$0.605826\pi$
$24$	0	0
$25$	121.000	0.968000
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	102.000	0.653135	0.326568	−	0.945174i	$-0.394108\pi$
$29$	102.000	0.653135	0.326568	+	0.945174i	$0.394108\pi$
$30$	0	0
$31$	− 136.000i	− 0.787946i	−0.919122	−	0.393973i	$-0.871100\pi$
$31$	− 136.000i	− 0.787946i	0.919122	−	0.393973i	$-0.128900\pi$
$32$	0	0
$33$	204.000i	1.07612i
$34$	0	0
$35$	64.0000	0.309085
$36$	0	0
$37$	386.000i	1.71508i	0.514416	+	0.857541i	$0.328009\pi$
$37$	386.000i	1.71508i	−0.514416	+	0.857541i	$0.671991\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	250.000i	0.952279i	0.879370	+	0.476140i	$0.157964\pi$
$41$	250.000i	0.952279i	−0.879370	+	0.476140i	$0.842036\pi$
$42$	0	0
$43$	140.000	0.496507	0.248253	−	0.968695i	$-0.420143\pi$
$43$	140.000	0.496507	0.248253	+	0.968695i	$0.420143\pi$
$44$	0	0
$45$	− 18.0000i	− 0.0596285i
$46$	0	0
$47$	296.000i	0.918639i	0.888271	+	0.459320i	$0.151907\pi$
$47$	296.000i	0.918639i	−0.888271	+	0.459320i	$0.848093\pi$
$48$	0	0
$49$	−681.000	−1.98542
$50$	0	0
$51$	42.0000	0.115317
$52$	0	0
$53$	526.000	1.36324	0.681619	−	0.731707i	$-0.261276\pi$
$53$	526.000	1.36324	0.681619	+	0.731707i	$0.261276\pi$
$54$	0	0
$55$	136.000	0.333422
$56$	0	0
$57$	12.0000i	0.0278849i
$58$	0	0
$59$	− 332.000i	− 0.732588i	−0.930499	−	0.366294i	$-0.880626\pi$
$59$	− 332.000i	− 0.732588i	0.930499	−	0.366294i	$-0.119374\pi$
$60$	0	0
$61$	−410.000	−0.860576	−0.430288	−	0.902692i	$-0.641588\pi$
$61$	−410.000	−0.860576	−0.430288	+	0.902692i	$0.641588\pi$
$62$	0	0
$63$	288.000i	0.575946i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	596.000i	1.08676i	0.839487	+	0.543381i	$0.182856\pi$
$67$	596.000i	1.08676i	−0.839487	+	0.543381i	$0.817144\pi$
$68$	0	0
$69$	−216.000	−0.376860
$70$	0	0
$71$	− 880.000i	− 1.47094i	−0.677557	−	0.735470i	$-0.736961\pi$
$71$	− 880.000i	− 1.47094i	0.677557	−	0.735470i	$-0.263039\pi$
$72$	0	0
$73$	− 506.000i	− 0.811272i	−0.914035	−	0.405636i	$-0.867050\pi$
$73$	− 506.000i	− 0.811272i	0.914035	−	0.405636i	$-0.132950\pi$
$74$	0	0
$75$	363.000	0.558875
$76$	0	0
$77$	−2176.00	−3.22050
$78$	0	0
$79$	−640.000	−0.911464	−0.455732	−	0.890117i	$-0.650622\pi$
$79$	−640.000	−0.911464	−0.455732	+	0.890117i	$0.650622\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	1380.00i	1.82500i	0.409081	+	0.912498i	$0.365849\pi$
$83$	1380.00i	1.82500i	−0.409081	+	0.912498i	$0.634151\pi$
$84$	0	0
$85$	− 28.0000i	− 0.0357297i
$86$	0	0
$87$	306.000	0.377088
$88$	0	0
$89$	− 1450.00i	− 1.72696i	−0.504381	−	0.863481i	$-0.668279\pi$
$89$	− 1450.00i	− 1.72696i	0.504381	−	0.863481i	$-0.331721\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	− 408.000i	− 0.454921i
$94$	0	0
$95$	8.00000	0.00863982
$96$	0	0
$97$	− 446.000i	− 0.466850i	−0.972375	−	0.233425i	$-0.925007\pi$
$97$	− 446.000i	− 0.466850i	0.972375	−	0.233425i	$-0.0749933\pi$
$98$	0	0
$99$	612.000i	0.621296i
$100$	0	0
$101$	610.000	0.600963	0.300482	−	0.953788i	$-0.402853\pi$
$101$	610.000	0.600963	0.300482	+	0.953788i	$0.402853\pi$
$102$	0	0
$103$	1352.00	1.29336	0.646682	−	0.762760i	$-0.276156\pi$
$103$	1352.00	1.29336	0.646682	+	0.762760i	$0.276156\pi$
$104$	0	0
$105$	192.000	0.178450
$106$	0	0
$107$	−732.000	−0.661356	−0.330678	−	0.943744i	$-0.607277\pi$
$107$	−732.000	−0.661356	−0.330678	+	0.943744i	$0.607277\pi$
$108$	0	0
$109$	− 1514.00i	− 1.33041i	−0.746660	−	0.665206i	$-0.768344\pi$
$109$	− 1514.00i	− 1.33041i	0.746660	−	0.665206i	$-0.231656\pi$
$110$	0	0
$111$	1158.00i	0.990203i
$112$	0	0
$113$	−1518.00	−1.26373	−0.631865	−	0.775079i	$-0.717710\pi$
$113$	−1518.00	−1.26373	−0.631865	+	0.775079i	$0.717710\pi$
$114$	0	0
$115$	144.000i	0.116766i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	448.000i	0.345110i
$120$	0	0
$121$	−3293.00	−2.47408
$122$	0	0
$123$	750.000i	0.549799i
$124$	0	0
$125$	− 492.000i	− 0.352047i
$126$	0	0
$127$	96.0000	0.0670758	0.0335379	−	0.999437i	$-0.489323\pi$
$127$	96.0000	0.0670758	0.0335379	+	0.999437i	$0.489323\pi$
$128$	0	0
$129$	420.000	0.286658
$130$	0	0
$131$	−2548.00	−1.69939	−0.849694	−	0.527276i	$-0.823213\pi$
$131$	−2548.00	−1.69939	−0.849694	+	0.527276i	$0.823213\pi$
$132$	0	0
$133$	−128.000	−0.0834512
$134$	0	0
$135$	− 54.0000i	− 0.0344265i
$136$	0	0
$137$	230.000i	0.143432i	0.997425	+	0.0717162i	$0.0228476\pi$
$137$	230.000i	0.143432i	−0.997425	+	0.0717162i	$0.977152\pi$
$138$	0	0
$139$	516.000	0.314867	0.157434	−	0.987530i	$-0.449678\pi$
$139$	516.000	0.314867	0.157434	+	0.987530i	$0.449678\pi$
$140$	0	0
$141$	888.000i	0.530377i
$142$	0	0
$143$	0	0
$144$	0	0
$145$	− 204.000i	− 0.116836i
$146$	0	0
$147$	−2043.00	−1.14628
$148$	0	0
$149$	− 1842.00i	− 1.01277i	−0.862308	−	0.506384i	$-0.830982\pi$
$149$	− 1842.00i	− 1.01277i	0.862308	−	0.506384i	$-0.169018\pi$
$150$	0	0
$151$	528.000i	0.284556i	0.989827	+	0.142278i	$0.0454428\pi$
$151$	528.000i	0.284556i	−0.989827	+	0.142278i	$0.954557\pi$
$152$	0	0
$153$	126.000	0.0665784
$154$	0	0
$155$	−272.000	−0.140952
$156$	0	0
$157$	−1306.00	−0.663886	−0.331943	−	0.943299i	$-0.607704\pi$
$157$	−1306.00	−0.663886	−0.331943	+	0.943299i	$0.607704\pi$
$158$	0	0
$159$	1578.00	0.787066
$160$	0	0
$161$	− 2304.00i	− 1.12783i
$162$	0	0
$163$	3772.00i	1.81255i	0.422687	+	0.906276i	$0.361087\pi$
$163$	3772.00i	1.81255i	−0.422687	+	0.906276i	$0.638913\pi$
$164$	0	0
$165$	408.000	0.192502
$166$	0	0
$167$	− 384.000i	− 0.177933i	−0.996035	−	0.0889665i	$-0.971644\pi$
$167$	− 384.000i	− 0.177933i	0.996035	−	0.0889665i	$-0.0283564\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	36.0000i	0.0160993i
$172$	0	0
$173$	−1462.00	−0.642508	−0.321254	−	0.946993i	$-0.604104\pi$
$173$	−1462.00	−0.642508	−0.321254	+	0.946993i	$0.604104\pi$
$174$	0	0
$175$	3872.00i	1.67255i
$176$	0	0
$177$	− 996.000i	− 0.422960i
$178$	0	0
$179$	1332.00	0.556192	0.278096	−	0.960553i	$-0.410297\pi$
$179$	1332.00	0.556192	0.278096	+	0.960553i	$0.410297\pi$
$180$	0	0
$181$	−2030.00	−0.833639	−0.416820	−	0.908989i	$-0.636855\pi$
$181$	−2030.00	−0.833639	−0.416820	+	0.908989i	$0.636855\pi$
$182$	0	0
$183$	−1230.00	−0.496854
$184$	0	0
$185$	772.000	0.306803
$186$	0	0
$187$	952.000i	0.372284i
$188$	0	0
$189$	864.000i	0.332522i
$190$	0	0
$191$	−16.0000	−0.00606136	−0.00303068	−	0.999995i	$-0.500965\pi$
$191$	−16.0000	−0.00606136	−0.00303068	+	0.999995i	$0.500965\pi$
$192$	0	0
$193$	2078.00i	0.775014i	0.921867	+	0.387507i	$0.126664\pi$
$193$	2078.00i	0.775014i	−0.921867	+	0.387507i	$0.873336\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	3486.00i	1.26075i	0.776292	+	0.630374i	$0.217098\pi$
$197$	3486.00i	1.26075i	−0.776292	+	0.630374i	$0.782902\pi$
$198$	0	0
$199$	−568.000	−0.202334	−0.101167	−	0.994869i	$-0.532258\pi$
$199$	−568.000	−0.202334	−0.101167	+	0.994869i	$0.532258\pi$
$200$	0	0
$201$	1788.00i	0.627442i
$202$	0	0
$203$	3264.00i	1.12851i
$204$	0	0
$205$	500.000	0.170349
$206$	0	0
$207$	−648.000	−0.217580
$208$	0	0
$209$	−272.000	−0.0900222
$210$	0	0
$211$	3804.00	1.24113	0.620564	−	0.784156i	$-0.286904\pi$
$211$	3804.00	1.24113	0.620564	+	0.784156i	$0.286904\pi$
$212$	0	0
$213$	− 2640.00i	− 0.849248i
$214$	0	0
$215$	− 280.000i	− 0.0888179i
$216$	0	0
$217$	4352.00	1.36144
$218$	0	0
$219$	− 1518.00i	− 0.468388i
$220$	0	0
$221$	0	0
$222$	0	0
$223$	5912.00i	1.77532i	0.460498	+	0.887661i	$0.347671\pi$
$223$	5912.00i	1.77532i	−0.460498	+	0.887661i	$0.652329\pi$
$224$	0	0
$225$	1089.00	0.322667
$226$	0	0
$227$	− 3308.00i	− 0.967223i	−0.875283	−	0.483612i	$-0.839325\pi$
$227$	− 3308.00i	− 0.967223i	0.875283	−	0.483612i	$-0.160675\pi$
$228$	0	0
$229$	6050.00i	1.74583i	0.487872	+	0.872915i	$0.337773\pi$
$229$	6050.00i	1.74583i	−0.487872	+	0.872915i	$0.662227\pi$
$230$	0	0
$231$	−6528.00	−1.85935
$232$	0	0
$233$	−4794.00	−1.34792	−0.673960	−	0.738768i	$-0.735408\pi$
$233$	−4794.00	−1.34792	−0.673960	+	0.738768i	$0.735408\pi$
$234$	0	0
$235$	592.000	0.164331
$236$	0	0
$237$	−1920.00	−0.526234
$238$	0	0
$239$	− 4440.00i	− 1.20167i	−0.799372	−	0.600836i	$-0.794834\pi$
$239$	− 4440.00i	− 1.20167i	0.799372	−	0.600836i	$-0.205166\pi$
$240$	0	0
$241$	− 1330.00i	− 0.355489i	−0.984077	−	0.177744i	$-0.943120\pi$
$241$	− 1330.00i	− 0.355489i	0.984077	−	0.177744i	$-0.0568800\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	1362.00i	0.355163i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	4140.00i	1.05366i
$250$	0	0
$251$	5116.00	1.28653	0.643265	−	0.765644i	$-0.277579\pi$
$251$	5116.00	1.28653	0.643265	+	0.765644i	$0.277579\pi$
$252$	0	0
$253$	− 4896.00i	− 1.21664i
$254$	0	0
$255$	− 84.0000i	− 0.0206286i
$256$	0	0
$257$	−642.000	−0.155824	−0.0779122	−	0.996960i	$-0.524825\pi$
$257$	−642.000	−0.155824	−0.0779122	+	0.996960i	$0.524825\pi$
$258$	0	0
$259$	−12352.0	−2.96338
$260$	0	0
$261$	918.000	0.217712
$262$	0	0
$263$	4264.00	0.999732	0.499866	−	0.866103i	$-0.333383\pi$
$263$	4264.00	0.999732	0.499866	+	0.866103i	$0.333383\pi$
$264$	0	0
$265$	− 1052.00i	− 0.243864i
$266$	0	0
$267$	− 4350.00i	− 0.997062i
$268$	0	0
$269$	−3850.00	−0.872634	−0.436317	−	0.899793i	$-0.643717\pi$
$269$	−3850.00	−0.872634	−0.436317	+	0.899793i	$0.643717\pi$
$270$	0	0
$271$	− 2936.00i	− 0.658115i	−0.944310	−	0.329058i	$-0.893269\pi$
$271$	− 2936.00i	− 0.658115i	0.944310	−	0.329058i	$-0.106731\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	8228.00i	1.80424i
$276$	0	0
$277$	2066.00	0.448137	0.224068	−	0.974573i	$-0.428066\pi$
$277$	2066.00	0.448137	0.224068	+	0.974573i	$0.428066\pi$
$278$	0	0
$279$	− 1224.00i	− 0.262649i
$280$	0	0
$281$	214.000i	0.0454312i	0.999742	+	0.0227156i	$0.00723122\pi$
$281$	214.000i	0.0454312i	−0.999742	+	0.0227156i	$0.992769\pi$
$282$	0	0
$283$	2620.00	0.550328	0.275164	−	0.961397i	$-0.411268\pi$
$283$	2620.00	0.550328	0.275164	+	0.961397i	$0.411268\pi$
$284$	0	0
$285$	24.0000	0.00498820
$286$	0	0
$287$	−8000.00	−1.64538
$288$	0	0
$289$	−4717.00	−0.960106
$290$	0	0
$291$	− 1338.00i	− 0.269536i
$292$	0	0
$293$	1154.00i	0.230094i	0.993360	+	0.115047i	$0.0367018\pi$
$293$	1154.00i	0.230094i	−0.993360	+	0.115047i	$0.963298\pi$
$294$	0	0
$295$	−664.000	−0.131049
$296$	0	0
$297$	1836.00i	0.358705i
$298$	0	0
$299$	0	0
$300$	0	0
$301$	4480.00i	0.857883i
$302$	0	0
$303$	1830.00	0.346966
$304$	0	0
$305$	820.000i	0.153944i
$306$	0	0
$307$	4076.00i	0.757751i	0.925448	+	0.378876i	$0.123689\pi$
$307$	4076.00i	0.757751i	−0.925448	+	0.378876i	$0.876311\pi$
$308$	0	0
$309$	4056.00	0.746724
$310$	0	0
$311$	6456.00	1.17713	0.588563	−	0.808451i	$-0.299694\pi$
$311$	6456.00	1.17713	0.588563	+	0.808451i	$0.299694\pi$
$312$	0	0
$313$	−5526.00	−0.997917	−0.498958	−	0.866626i	$-0.666284\pi$
$313$	−5526.00	−0.997917	−0.498958	+	0.866626i	$0.666284\pi$
$314$	0	0
$315$	576.000	0.103028
$316$	0	0
$317$	− 10458.0i	− 1.85293i	−0.376377	−	0.926467i	$-0.622830\pi$
$317$	− 10458.0i	− 1.85293i	0.376377	−	0.926467i	$-0.377170\pi$
$318$	0	0
$319$	6936.00i	1.21737i
$320$	0	0
$321$	−2196.00	−0.381834
$322$	0	0
$323$	56.0000i	0.00964682i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 4542.00i	− 0.768114i
$328$	0	0
$329$	−9472.00	−1.58726
$330$	0	0
$331$	2348.00i	0.389903i	0.980813	+	0.194951i	$0.0624549\pi$
$331$	2348.00i	0.389903i	−0.980813	+	0.194951i	$0.937545\pi$
$332$	0	0
$333$	3474.00i	0.571694i
$334$	0	0
$335$	1192.00	0.194406
$336$	0	0
$337$	−5298.00	−0.856381	−0.428191	−	0.903688i	$-0.640849\pi$
$337$	−5298.00	−0.856381	−0.428191	+	0.903688i	$0.640849\pi$
$338$	0	0
$339$	−4554.00	−0.729615
$340$	0	0
$341$	9248.00	1.46864
$342$	0	0
$343$	− 10816.0i	− 1.70265i
$344$	0	0
$345$	432.000i	0.0674148i
$346$	0	0
$347$	9876.00	1.52787	0.763936	−	0.645292i	$-0.223264\pi$
$347$	9876.00	1.52787	0.763936	+	0.645292i	$0.223264\pi$
$348$	0	0
$349$	5370.00i	0.823638i	0.911266	+	0.411819i	$0.135106\pi$
$349$	5370.00i	0.823638i	−0.911266	+	0.411819i	$0.864894\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	7330.00i	1.10520i	0.833446	+	0.552601i	$0.186365\pi$
$353$	7330.00i	1.10520i	−0.833446	+	0.552601i	$0.813635\pi$
$354$	0	0
$355$	−1760.00	−0.263130
$356$	0	0
$357$	1344.00i	0.199249i
$358$	0	0
$359$	− 7488.00i	− 1.10084i	−0.834888	−	0.550420i	$-0.814468\pi$
$359$	− 7488.00i	− 1.10084i	0.834888	−	0.550420i	$-0.185532\pi$
$360$	0	0
$361$	6843.00	0.997667
$362$	0	0
$363$	−9879.00	−1.42841
$364$	0	0
$365$	−1012.00	−0.145125
$366$	0	0
$367$	1504.00	0.213919	0.106959	−	0.994263i	$-0.465889\pi$
$367$	1504.00	0.213919	0.106959	+	0.994263i	$0.465889\pi$
$368$	0	0
$369$	2250.00i	0.317426i
$370$	0	0
$371$	16832.0i	2.35546i
$372$	0	0
$373$	6702.00	0.930339	0.465169	−	0.885222i	$-0.345993\pi$
$373$	6702.00	0.930339	0.465169	+	0.885222i	$0.345993\pi$
$374$	0	0
$375$	− 1476.00i	− 0.203254i
$376$	0	0
$377$	0	0
$378$	0	0
$379$	− 5700.00i	− 0.772531i	−0.922388	−	0.386266i	$-0.873765\pi$
$379$	− 5700.00i	− 0.772531i	0.922388	−	0.386266i	$-0.126235\pi$
$380$	0	0
$381$	288.000	0.0387262
$382$	0	0
$383$	2328.00i	0.310588i	0.987868	+	0.155294i	$0.0496325\pi$
$383$	2328.00i	0.310588i	−0.987868	+	0.155294i	$0.950367\pi$
$384$	0	0
$385$	4352.00i	0.576100i
$386$	0	0
$387$	1260.00	0.165502
$388$	0	0
$389$	11554.0	1.50594	0.752971	−	0.658054i	$-0.228620\pi$
$389$	11554.0	1.50594	0.752971	+	0.658054i	$0.228620\pi$
$390$	0	0
$391$	−1008.00	−0.130375
$392$	0	0
$393$	−7644.00	−0.981142
$394$	0	0
$395$	1280.00i	0.163048i
$396$	0	0
$397$	− 6486.00i	− 0.819957i	−0.912095	−	0.409979i	$-0.865536\pi$
$397$	− 6486.00i	− 0.819957i	0.912095	−	0.409979i	$-0.134464\pi$
$398$	0	0
$399$	−384.000	−0.0481806
$400$	0	0
$401$	− 7698.00i	− 0.958653i	−0.877637	−	0.479326i	$-0.840881\pi$
$401$	− 7698.00i	− 0.958653i	0.877637	−	0.479326i	$-0.159119\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	− 162.000i	− 0.0198762i
$406$	0	0
$407$	−26248.0	−3.19672
$408$	0	0
$409$	3338.00i	0.403554i	0.979432	+	0.201777i	$0.0646716\pi$
$409$	3338.00i	0.403554i	−0.979432	+	0.201777i	$0.935328\pi$
$410$	0	0
$411$	690.000i	0.0828107i
$412$	0	0
$413$	10624.0	1.26579
$414$	0	0
$415$	2760.00	0.326465
$416$	0	0
$417$	1548.00	0.181789
$418$	0	0
$419$	−52.0000	−0.00606293	−0.00303146	−	0.999995i	$-0.500965\pi$
$419$	−52.0000	−0.00606293	−0.00303146	+	0.999995i	$0.500965\pi$
$420$	0	0
$421$	− 5858.00i	− 0.678151i	−0.940759	−	0.339075i	$-0.889886\pi$
$421$	− 5858.00i	− 0.678151i	0.940759	−	0.339075i	$-0.110114\pi$
$422$	0	0
$423$	2664.00i	0.306213i
$424$	0	0
$425$	1694.00	0.193344
$426$	0	0
$427$	− 13120.0i	− 1.48694i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	8840.00i	0.987953i	0.869475	+	0.493977i	$0.164457\pi$
$431$	8840.00i	0.987953i	−0.869475	+	0.493977i	$0.835543\pi$
$432$	0	0
$433$	−11346.0	−1.25925	−0.629624	−	0.776900i	$-0.716791\pi$
$433$	−11346.0	−1.25925	−0.629624	+	0.776900i	$0.716791\pi$
$434$	0	0
$435$	− 612.000i	− 0.0674555i
$436$	0	0
$437$	− 288.000i	− 0.0315261i
$438$	0	0
$439$	−16456.0	−1.78907	−0.894535	−	0.446997i	$-0.852493\pi$
$439$	−16456.0	−1.78907	−0.894535	+	0.446997i	$0.852493\pi$
$440$	0	0
$441$	−6129.00	−0.661808
$442$	0	0
$443$	−3788.00	−0.406260	−0.203130	−	0.979152i	$-0.565111\pi$
$443$	−3788.00	−0.406260	−0.203130	+	0.979152i	$0.565111\pi$
$444$	0	0
$445$	−2900.00	−0.308929
$446$	0	0
$447$	− 5526.00i	− 0.584722i
$448$	0	0
$449$	− 546.000i	− 0.0573883i	−0.999588	−	0.0286941i	$-0.990865\pi$
$449$	− 546.000i	− 0.0573883i	0.999588	−	0.0286941i	$-0.00913488\pi$
$450$	0	0
$451$	−17000.0	−1.77494
$452$	0	0
$453$	1584.00i	0.164289i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	3546.00i	0.362965i	0.983394	+	0.181482i	$0.0580895\pi$
$457$	3546.00i	0.362965i	−0.983394	+	0.181482i	$0.941910\pi$
$458$	0	0
$459$	378.000	0.0384391
$460$	0	0
$461$	12918.0i	1.30510i	0.757746	+	0.652550i	$0.226301\pi$
$461$	12918.0i	1.30510i	−0.757746	+	0.652550i	$0.773699\pi$
$462$	0	0
$463$	− 18328.0i	− 1.83969i	−0.392287	−	0.919843i	$-0.628316\pi$
$463$	− 18328.0i	− 1.83969i	0.392287	−	0.919843i	$-0.371684\pi$
$464$	0	0
$465$	−816.000	−0.0813787
$466$	0	0
$467$	−11980.0	−1.18708	−0.593542	−	0.804803i	$-0.702271\pi$
$467$	−11980.0	−1.18708	−0.593542	+	0.804803i	$0.702271\pi$
$468$	0	0
$469$	−19072.0	−1.87775
$470$	0	0
$471$	−3918.00	−0.383295
$472$	0	0
$473$	9520.00i	0.925434i
$474$	0	0
$475$	484.000i	0.0467525i
$476$	0	0
$477$	4734.00	0.454413
$478$	0	0
$479$	12344.0i	1.17748i	0.808323	+	0.588739i	$0.200375\pi$
$479$	12344.0i	1.17748i	−0.808323	+	0.588739i	$0.799625\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	− 6912.00i	− 0.651153i
$484$	0	0
$485$	−892.000	−0.0835126
$486$	0	0
$487$	− 80.0000i	− 0.00744383i	−0.999993	−	0.00372192i	$-0.998815\pi$
$487$	− 80.0000i	− 0.00744383i	0.999993	−	0.00372192i	$-0.00118473\pi$
$488$	0	0
$489$	11316.0i	1.04648i
$490$	0	0
$491$	15660.0	1.43936	0.719680	−	0.694306i	$-0.244288\pi$
$491$	15660.0	1.43936	0.719680	+	0.694306i	$0.244288\pi$
$492$	0	0
$493$	1428.00	0.130454
$494$	0	0
$495$	1224.00	0.111141
$496$	0	0
$497$	28160.0	2.54155
$498$	0	0
$499$	− 60.0000i	− 0.00538270i	−0.999996	−	0.00269135i	$-0.999143\pi$
$499$	− 60.0000i	− 0.00538270i	0.999996	−	0.00269135i	$-0.000856685\pi$
$500$	0	0
$501$	− 1152.00i	− 0.102730i
$502$	0	0
$503$	12248.0	1.08571	0.542854	−	0.839827i	$-0.317344\pi$
$503$	12248.0	1.08571	0.542854	+	0.839827i	$0.317344\pi$
$504$	0	0
$505$	− 1220.00i	− 0.107504i
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 90.0000i	− 0.00783729i	−0.999992	−	0.00391864i	$-0.998753\pi$
$509$	− 90.0000i	− 0.00783729i	0.999992	−	0.00391864i	$-0.00124735\pi$
$510$	0	0
$511$	16192.0	1.40175
$512$	0	0
$513$	108.000i	0.00929496i
$514$	0	0
$515$	− 2704.00i	− 0.231364i
$516$	0	0
$517$	−20128.0	−1.71224
$518$	0	0
$519$	−4386.00	−0.370952
$520$	0	0
$521$	9818.00	0.825594	0.412797	−	0.910823i	$-0.364552\pi$
$521$	9818.00	0.825594	0.412797	+	0.910823i	$0.364552\pi$
$522$	0	0
$523$	−20252.0	−1.69323	−0.846614	−	0.532208i	$-0.821363\pi$
$523$	−20252.0	−1.69323	−0.846614	+	0.532208i	$0.821363\pi$
$524$	0	0
$525$	11616.0i	0.965645i
$526$	0	0
$527$	− 1904.00i	− 0.157381i
$528$	0	0
$529$	−6983.00	−0.573929
$530$	0	0
$531$	− 2988.00i	− 0.244196i
$532$	0	0
$533$	0	0
$534$	0	0
$535$	1464.00i	0.118307i
$536$	0	0
$537$	3996.00	0.321118
$538$	0	0
$539$	− 46308.0i	− 3.70061i
$540$	0	0
$541$	12634.0i	1.00403i	0.864860	+	0.502013i	$0.167407\pi$
$541$	12634.0i	1.00403i	−0.864860	+	0.502013i	$0.832593\pi$
$542$	0	0
$543$	−6090.00	−0.481302
$544$	0	0
$545$	−3028.00	−0.237991
$546$	0	0
$547$	11756.0	0.918922	0.459461	−	0.888198i	$-0.348043\pi$
$547$	11756.0	0.918922	0.459461	+	0.888198i	$0.348043\pi$
$548$	0	0
$549$	−3690.00	−0.286859
$550$	0	0
$551$	408.000i	0.0315452i
$552$	0	0
$553$	− 20480.0i	− 1.57486i
$554$	0	0
$555$	2316.00	0.177133
$556$	0	0
$557$	− 17622.0i	− 1.34052i	−0.742128	−	0.670259i	$-0.766183\pi$
$557$	− 17622.0i	− 1.34052i	0.742128	−	0.670259i	$-0.233817\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	2856.00i	0.214938i
$562$	0	0
$563$	23092.0	1.72862	0.864309	−	0.502961i	$-0.167756\pi$
$563$	23092.0	1.72862	0.864309	+	0.502961i	$0.167756\pi$
$564$	0	0
$565$	3036.00i	0.226063i
$566$	0	0
$567$	2592.00i	0.191982i
$568$	0	0
$569$	1302.00	0.0959274	0.0479637	−	0.998849i	$-0.484727\pi$
$569$	1302.00	0.0959274	0.0479637	+	0.998849i	$0.484727\pi$
$570$	0	0
$571$	−24868.0	−1.82258	−0.911290	−	0.411765i	$-0.864913\pi$
$571$	−24868.0	−1.82258	−0.911290	+	0.411765i	$0.864913\pi$
$572$	0	0
$573$	−48.0000	−0.00349953
$574$	0	0
$575$	−8712.00	−0.631853
$576$	0	0
$577$	2562.00i	0.184848i	0.995720	+	0.0924241i	$0.0294616\pi$
$577$	2562.00i	0.184848i	−0.995720	+	0.0924241i	$0.970538\pi$
$578$	0	0
$579$	6234.00i	0.447455i
$580$	0	0
$581$	−44160.0	−3.15330
$582$	0	0
$583$	35768.0i	2.54092i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	13484.0i	0.948116i	0.880494	+	0.474058i	$0.157211\pi$
$587$	13484.0i	0.948116i	−0.880494	+	0.474058i	$0.842789\pi$
$588$	0	0
$589$	544.000	0.0380562
$590$	0	0
$591$	10458.0i	0.727893i
$592$	0	0
$593$	16974.0i	1.17544i	0.809063	+	0.587722i	$0.199975\pi$
$593$	16974.0i	1.17544i	−0.809063	+	0.587722i	$0.800025\pi$
$594$	0	0
$595$	896.000	0.0617352
$596$	0	0
$597$	−1704.00	−0.116818
$598$	0	0
$599$	3864.00	0.263571	0.131785	−	0.991278i	$-0.457929\pi$
$599$	3864.00	0.263571	0.131785	+	0.991278i	$0.457929\pi$
$600$	0	0
$601$	17546.0	1.19088	0.595438	−	0.803401i	$-0.296979\pi$
$601$	17546.0	1.19088	0.595438	+	0.803401i	$0.296979\pi$
$602$	0	0
$603$	5364.00i	0.362254i
$604$	0	0
$605$	6586.00i	0.442577i
$606$	0	0
$607$	9296.00	0.621603	0.310801	−	0.950475i	$-0.399403\pi$
$607$	9296.00	0.621603	0.310801	+	0.950475i	$0.399403\pi$
$608$	0	0
$609$	9792.00i	0.651547i
$610$	0	0
$611$	0	0
$612$	0	0
$613$	− 6914.00i	− 0.455553i	−0.973713	−	0.227776i	$-0.926854\pi$
$613$	− 6914.00i	− 0.455553i	0.973713	−	0.227776i	$-0.0731455\pi$
$614$	0	0
$615$	1500.00	0.0983510
$616$	0	0
$617$	− 25446.0i	− 1.66032i	−0.557525	−	0.830160i	$-0.688249\pi$
$617$	− 25446.0i	− 1.66032i	0.557525	−	0.830160i	$-0.311751\pi$
$618$	0	0
$619$	11236.0i	0.729585i	0.931089	+	0.364792i	$0.118860\pi$
$619$	11236.0i	0.729585i	−0.931089	+	0.364792i	$0.881140\pi$
$620$	0	0
$621$	−1944.00	−0.125620
$622$	0	0
$623$	46400.0	2.98391
$624$	0	0
$625$	14141.0	0.905024
$626$	0	0
$627$	−816.000	−0.0519743
$628$	0	0
$629$	5404.00i	0.342562i
$630$	0	0
$631$	29424.0i	1.85634i	0.372156	+	0.928170i	$0.378619\pi$
$631$	29424.0i	1.85634i	−0.372156	+	0.928170i	$0.621381\pi$
$632$	0	0
$633$	11412.0	0.716566
$634$	0	0
$635$	− 192.000i	− 0.0119989i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	− 7920.00i	− 0.490314i
$640$	0	0
$641$	5054.00	0.311421	0.155711	−	0.987803i	$-0.450233\pi$
$641$	5054.00	0.311421	0.155711	+	0.987803i	$0.450233\pi$
$642$	0	0
$643$	− 1132.00i	− 0.0694273i	−0.999397	−	0.0347136i	$-0.988948\pi$
$643$	− 1132.00i	− 0.0694273i	0.999397	−	0.0347136i	$-0.0110519\pi$
$644$	0	0
$645$	− 840.000i	− 0.0512790i
$646$	0	0
$647$	−1464.00	−0.0889579	−0.0444790	−	0.999010i	$-0.514163\pi$
$647$	−1464.00	−0.0889579	−0.0444790	+	0.999010i	$0.514163\pi$
$648$	0	0
$649$	22576.0	1.36546
$650$	0	0
$651$	13056.0	0.786029
$652$	0	0
$653$	5494.00	0.329245	0.164622	−	0.986357i	$-0.447359\pi$
$653$	5494.00	0.329245	0.164622	+	0.986357i	$0.447359\pi$
$654$	0	0
$655$	5096.00i	0.303996i
$656$	0	0
$657$	− 4554.00i	− 0.270424i
$658$	0	0
$659$	11580.0	0.684511	0.342256	−	0.939607i	$-0.388809\pi$
$659$	11580.0	0.684511	0.342256	+	0.939607i	$0.388809\pi$
$660$	0	0
$661$	1298.00i	0.0763787i	0.999271	+	0.0381894i	$0.0121590\pi$
$661$	1298.00i	0.0763787i	−0.999271	+	0.0381894i	$0.987841\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	256.000i	0.0149282i
$666$	0	0
$667$	−7344.00	−0.426328
$668$	0	0
$669$	17736.0i	1.02498i
$670$	0	0
$671$	− 27880.0i	− 1.60402i
$672$	0	0
$673$	−16162.0	−0.925705	−0.462852	−	0.886435i	$-0.653174\pi$
$673$	−16162.0	−0.925705	−0.462852	+	0.886435i	$0.653174\pi$
$674$	0	0
$675$	3267.00	0.186292
$676$	0	0
$677$	−9890.00	−0.561453	−0.280726	−	0.959788i	$-0.590575\pi$
$677$	−9890.00	−0.561453	−0.280726	+	0.959788i	$0.590575\pi$
$678$	0	0
$679$	14272.0	0.806641
$680$	0	0
$681$	− 9924.00i	− 0.558427i
$682$	0	0
$683$	− 27100.0i	− 1.51823i	−0.650955	−	0.759116i	$-0.725631\pi$
$683$	− 27100.0i	− 1.51823i	0.650955	−	0.759116i	$-0.274369\pi$
$684$	0	0
$685$	460.000	0.0256580
$686$	0	0
$687$	18150.0i	1.00796i
$688$	0	0
$689$	0	0
$690$	0	0
$691$	− 11132.0i	− 0.612853i	−0.951894	−	0.306426i	$-0.900867\pi$
$691$	− 11132.0i	− 0.612853i	0.951894	−	0.306426i	$-0.0991333\pi$
$692$	0	0
$693$	−19584.0	−1.07350
$694$	0	0
$695$	− 1032.00i	− 0.0563252i
$696$	0	0
$697$	3500.00i	0.190204i
$698$	0	0
$699$	−14382.0	−0.778222
$700$	0	0
$701$	−1862.00	−0.100323	−0.0501617	−	0.998741i	$-0.515974\pi$
$701$	−1862.00	−0.100323	−0.0501617	+	0.998741i	$0.515974\pi$
$702$	0	0
$703$	−1544.00	−0.0828351
$704$	0	0
$705$	1776.00	0.0948766
$706$	0	0
$707$	19520.0i	1.03837i
$708$	0	0
$709$	7938.00i	0.420477i	0.977650	+	0.210238i	$0.0674240\pi$
$709$	7938.00i	0.420477i	−0.977650	+	0.210238i	$0.932576\pi$
$710$	0	0
$711$	−5760.00	−0.303821
$712$	0	0
$713$	9792.00i	0.514324i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 13320.0i	− 0.693786i
$718$	0	0
$719$	24240.0	1.25730	0.628651	−	0.777688i	$-0.283608\pi$
$719$	24240.0	1.25730	0.628651	+	0.777688i	$0.283608\pi$
$720$	0	0
$721$	43264.0i	2.23472i
$722$	0	0
$723$	− 3990.00i	− 0.205242i
$724$	0	0
$725$	12342.0	0.632235
$726$	0	0
$727$	13720.0	0.699927	0.349963	−	0.936763i	$-0.386194\pi$
$727$	13720.0	0.699927	0.349963	+	0.936763i	$0.386194\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	1960.00	0.0991699
$732$	0	0
$733$	21958.0i	1.10646i	0.833028	+	0.553231i	$0.186605\pi$
$733$	21958.0i	1.10646i	−0.833028	+	0.553231i	$0.813395\pi$
$734$	0	0
$735$	4086.00i	0.205054i
$736$	0	0
$737$	−40528.0	−2.02560
$738$	0	0
$739$	− 13348.0i	− 0.664430i	−0.943204	−	0.332215i	$-0.892204\pi$
$739$	− 13348.0i	− 0.664430i	0.943204	−	0.332215i	$-0.107796\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	20304.0i	1.00253i	0.865293	+	0.501266i	$0.167132\pi$
$743$	20304.0i	1.00253i	−0.865293	+	0.501266i	$0.832868\pi$
$744$	0	0
$745$	−3684.00	−0.181170
$746$	0	0
$747$	12420.0i	0.608332i
$748$	0	0
$749$	− 23424.0i	− 1.14272i
$750$	0	0
$751$	3952.00	0.192025	0.0960123	−	0.995380i	$-0.469391\pi$
$751$	3952.00	0.192025	0.0960123	+	0.995380i	$0.469391\pi$
$752$	0	0
$753$	15348.0	0.742779
$754$	0	0
$755$	1056.00	0.0509030
$756$	0	0
$757$	−22386.0	−1.07481	−0.537406	−	0.843324i	$-0.680596\pi$
$757$	−22386.0	−1.07481	−0.537406	+	0.843324i	$0.680596\pi$
$758$	0	0
$759$	− 14688.0i	− 0.702425i
$760$	0	0
$761$	− 12458.0i	− 0.593433i	−0.954966	−	0.296716i	$-0.904108\pi$
$761$	− 12458.0i	− 0.593433i	0.954966	−	0.296716i	$-0.0958916\pi$
$762$	0	0
$763$	48448.0	2.29874
$764$	0	0
$765$	− 252.000i	− 0.0119099i
$766$	0	0
$767$	0	0
$768$	0	0
$769$	− 28126.0i	− 1.31892i	−0.751740	−	0.659460i	$-0.770785\pi$
$769$	− 28126.0i	− 1.31892i	0.751740	−	0.659460i	$-0.229215\pi$
$770$	0	0
$771$	−1926.00	−0.0899652
$772$	0	0
$773$	− 35778.0i	− 1.66474i	−0.554219	−	0.832371i	$-0.686983\pi$
$773$	− 35778.0i	− 1.66474i	0.554219	−	0.832371i	$-0.313017\pi$
$774$	0	0
$775$	− 16456.0i	− 0.762732i
$776$	0	0
$777$	−37056.0	−1.71091
$778$	0	0
$779$	−1000.00	−0.0459932
$780$	0	0
$781$	59840.0	2.74167
$782$	0	0
$783$	2754.00	0.125696
$784$	0	0
$785$	2612.00i	0.118760i
$786$	0	0
$787$	− 7252.00i	− 0.328470i	−0.986421	−	0.164235i	$-0.947484\pi$
$787$	− 7252.00i	− 0.328470i	0.986421	−	0.164235i	$-0.0525155\pi$
$788$	0	0
$789$	12792.0	0.577196
$790$	0	0
$791$	− 48576.0i	− 2.18352i
$792$	0	0
$793$	0	0
$794$	0	0
$795$	− 3156.00i	− 0.140795i
$796$	0	0
$797$	−4454.00	−0.197953	−0.0989766	−	0.995090i	$-0.531557\pi$
$797$	−4454.00	−0.197953	−0.0989766	+	0.995090i	$0.531557\pi$
$798$	0	0
$799$	4144.00i	0.183485i
$800$	0	0
$801$	− 13050.0i	− 0.575654i
$802$	0	0
$803$	34408.0	1.51212
$804$	0	0
$805$	−4608.00	−0.201752
$806$	0	0
$807$	−11550.0	−0.503816
$808$	0	0
$809$	−34118.0	−1.48273	−0.741363	−	0.671104i	$-0.765820\pi$
$809$	−34118.0	−1.48273	−0.741363	+	0.671104i	$0.765820\pi$
$810$	0	0
$811$	16428.0i	0.711301i	0.934619	+	0.355650i	$0.115741\pi$
$811$	16428.0i	0.711301i	−0.934619	+	0.355650i	$0.884259\pi$
$812$	0	0
$813$	− 8808.00i	− 0.379963i
$814$	0	0
$815$	7544.00	0.324239
$816$	0	0
$817$	560.000i	0.0239803i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 18738.0i	− 0.796542i	−0.917268	−	0.398271i	$-0.869610\pi$
$821$	− 18738.0i	− 0.796542i	0.917268	−	0.398271i	$-0.130390\pi$
$822$	0	0
$823$	−13928.0	−0.589914	−0.294957	−	0.955510i	$-0.595305\pi$
$823$	−13928.0	−0.589914	−0.294957	+	0.955510i	$0.595305\pi$
$824$	0	0
$825$	24684.0i	1.04168i
$826$	0	0
$827$	− 41804.0i	− 1.75776i	−0.477043	−	0.878880i	$-0.658291\pi$
$827$	− 41804.0i	− 1.75776i	0.477043	−	0.878880i	$-0.341709\pi$
$828$	0	0
$829$	43226.0	1.81098	0.905489	−	0.424369i	$-0.139504\pi$
$829$	43226.0	1.81098	0.905489	+	0.424369i	$0.139504\pi$
$830$	0	0
$831$	6198.00	0.258732
$832$	0	0
$833$	−9534.00	−0.396559
$834$	0	0
$835$	−768.000	−0.0318296
$836$	0	0
$837$	− 3672.00i	− 0.151640i
$838$	0	0
$839$	− 10240.0i	− 0.421364i	−0.977555	−	0.210682i	$-0.932432\pi$
$839$	− 10240.0i	− 0.421364i	0.977555	−	0.210682i	$-0.0675684\pi$
$840$	0	0
$841$	−13985.0	−0.573414
$842$	0	0
$843$	642.000i	0.0262297i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 105376.i	− 4.27481i
$848$	0	0
$849$	7860.00	0.317732
$850$	0	0
$851$	− 27792.0i	− 1.11950i
$852$	0	0
$853$	25682.0i	1.03087i	0.856928	+	0.515437i	$0.172370\pi$
$853$	25682.0i	1.03087i	−0.856928	+	0.515437i	$0.827630\pi$
$854$	0	0
$855$	72.0000	0.00287994
$856$	0	0
$857$	21558.0	0.859285	0.429643	−	0.902999i	$-0.358640\pi$
$857$	21558.0	0.859285	0.429643	+	0.902999i	$0.358640\pi$
$858$	0	0
$859$	−14060.0	−0.558465	−0.279232	−	0.960224i	$-0.590080\pi$
$859$	−14060.0	−0.558465	−0.279232	+	0.960224i	$0.590080\pi$
$860$	0	0
$861$	−24000.0	−0.949963
$862$	0	0
$863$	42008.0i	1.65697i	0.560008	+	0.828487i	$0.310798\pi$
$863$	42008.0i	1.65697i	−0.560008	+	0.828487i	$0.689202\pi$
$864$	0	0
$865$	2924.00i	0.114935i
$866$	0	0
$867$	−14151.0	−0.554317
$868$	0	0
$869$	− 43520.0i	− 1.69887i
$870$	0	0
$871$	0	0
$872$	0	0
$873$	− 4014.00i	− 0.155617i
$874$	0	0
$875$	15744.0	0.608279
$876$	0	0
$877$	23734.0i	0.913843i	0.889507	+	0.456921i	$0.151048\pi$
$877$	23734.0i	0.913843i	−0.889507	+	0.456921i	$0.848952\pi$
$878$	0	0
$879$	3462.00i	0.132845i
$880$	0	0
$881$	37550.0	1.43597	0.717986	−	0.696057i	$-0.245064\pi$
$881$	37550.0	1.43597	0.717986	+	0.696057i	$0.245064\pi$
$882$	0	0
$883$	−12556.0	−0.478531	−0.239266	−	0.970954i	$-0.576907\pi$
$883$	−12556.0	−0.478531	−0.239266	+	0.970954i	$0.576907\pi$
$884$	0	0
$885$	−1992.00	−0.0756614
$886$	0	0
$887$	37368.0	1.41454	0.707269	−	0.706945i	$-0.249927\pi$
$887$	37368.0	1.41454	0.707269	+	0.706945i	$0.249927\pi$
$888$	0	0
$889$	3072.00i	0.115896i
$890$	0	0
$891$	5508.00i	0.207099i
$892$	0	0
$893$	−1184.00	−0.0443685
$894$	0	0
$895$	− 2664.00i	− 0.0994946i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 13872.0i	− 0.514635i
$900$	0	0
$901$	7364.00	0.272287
$902$	0	0
$903$	13440.0i	0.495299i
$904$	0	0
$905$	4060.00i	0.149126i
$906$	0	0
$907$	−33364.0	−1.22143	−0.610713	−	0.791852i	$-0.709117\pi$
$907$	−33364.0	−1.22143	−0.610713	+	0.791852i	$0.709117\pi$
$908$	0	0
$909$	5490.00	0.200321
$910$	0	0
$911$	50432.0	1.83412	0.917062	−	0.398745i	$-0.130554\pi$
$911$	50432.0	1.83412	0.917062	+	0.398745i	$0.130554\pi$
$912$	0	0
$913$	−93840.0	−3.40159
$914$	0	0
$915$	2460.00i	0.0888799i
$916$	0	0
$917$	− 81536.0i	− 2.93627i
$918$	0	0
$919$	11864.0	0.425851	0.212926	−	0.977068i	$-0.431701\pi$
$919$	11864.0	0.425851	0.212926	+	0.977068i	$0.431701\pi$
$920$	0	0
$921$	12228.0i	0.437488i
$922$	0	0
$923$	0	0
$924$	0	0
$925$	46706.0i	1.66020i
$926$	0	0
$927$	12168.0	0.431121
$928$	0	0
$929$	− 5950.00i	− 0.210133i	−0.994465	−	0.105066i	$-0.966495\pi$
$929$	− 5950.00i	− 0.210133i	0.994465	−	0.105066i	$-0.0335055\pi$
$930$	0	0
$931$	− 2724.00i	− 0.0958920i
$932$	0	0
$933$	19368.0	0.679614
$934$	0	0
$935$	1904.00	0.0665962
$936$	0	0
$937$	−20806.0	−0.725403	−0.362701	−	0.931905i	$-0.618146\pi$
$937$	−20806.0	−0.725403	−0.362701	+	0.931905i	$0.618146\pi$
$938$	0	0
$939$	−16578.0	−0.576148
$940$	0	0
$941$	− 22346.0i	− 0.774133i	−0.922052	−	0.387066i	$-0.873488\pi$
$941$	− 22346.0i	− 0.774133i	0.922052	−	0.387066i	$-0.126512\pi$
$942$	0	0
$943$	− 18000.0i	− 0.621591i
$944$	0	0
$945$	1728.00	0.0594834
$946$	0	0
$947$	− 31636.0i	− 1.08557i	−0.839873	−	0.542783i	$-0.817370\pi$
$947$	− 31636.0i	− 1.08557i	0.839873	−	0.542783i	$-0.182630\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	− 31374.0i	− 1.06979i
$952$	0	0
$953$	23190.0	0.788245	0.394123	−	0.919058i	$-0.371049\pi$
$953$	23190.0	0.788245	0.394123	+	0.919058i	$0.371049\pi$
$954$	0	0
$955$	32.0000i	0.00108429i
$956$	0	0
$957$	20808.0i	0.702850i
$958$	0	0
$959$	−7360.00	−0.247828
$960$	0	0
$961$	11295.0	0.379141
$962$	0	0
$963$	−6588.00	−0.220452
$964$	0	0
$965$	4156.00	0.138639
$966$	0	0
$967$	304.000i	0.0101096i	0.999987	+	0.00505480i	$0.00160900\pi$
$967$	304.000i	0.0101096i	−0.999987	+	0.00505480i	$0.998391\pi$
$968$	0	0
$969$	168.000i	0.00556960i
$970$	0	0
$971$	−53372.0	−1.76394	−0.881972	−	0.471302i	$-0.843784\pi$
$971$	−53372.0	−1.76394	−0.881972	+	0.471302i	$0.843784\pi$
$972$	0	0
$973$	16512.0i	0.544039i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	13650.0i	0.446983i	0.974706	+	0.223491i	$0.0717455\pi$
$977$	13650.0i	0.446983i	−0.974706	+	0.223491i	$0.928255\pi$
$978$	0	0
$979$	98600.0	3.21887
$980$	0	0
$981$	− 13626.0i	− 0.443471i
$982$	0	0
$983$	− 34992.0i	− 1.13537i	−0.823245	−	0.567686i	$-0.807839\pi$
$983$	− 34992.0i	− 1.13537i	0.823245	−	0.567686i	$-0.192161\pi$
$984$	0	0
$985$	6972.00	0.225529
$986$	0	0
$987$	−28416.0	−0.916405
$988$	0	0
$989$	−10080.0	−0.324090
$990$	0	0
$991$	18096.0	0.580059	0.290029	−	0.957018i	$-0.406335\pi$
$991$	18096.0	0.580059	0.290029	+	0.957018i	$0.406335\pi$
$992$	0	0
$993$	7044.00i	0.225110i
$994$	0	0
$995$	1136.00i	0.0361946i
$996$	0	0
$997$	−18914.0	−0.600815	−0.300407	−	0.953811i	$-0.597123\pi$
$997$	−18914.0	−0.600815	−0.300407	+	0.953811i	$0.597123\pi$
$998$	0	0
$999$	10422.0i	0.330068i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2028.4.b.d.337.1		2
13.5	odd	4		156.4.a.b.1.1	✓	1
13.8	odd	4		2028.4.a.b.1.1		1
13.12	even	2	inner	2028.4.b.d.337.2		2
39.5	even	4		468.4.a.a.1.1		1
52.31	even	4		624.4.a.b.1.1		1
104.5	odd	4		2496.4.a.d.1.1		1
104.83	even	4		2496.4.a.m.1.1		1
156.83	odd	4		1872.4.a.i.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
156.4.a.b.1.1	✓	1	13.5	odd	4
468.4.a.a.1.1		1	39.5	even	4
624.4.a.b.1.1		1	52.31	even	4
1872.4.a.i.1.1		1	156.83	odd	4
2028.4.a.b.1.1		1	13.8	odd	4
2028.4.b.d.337.1		2	1.1	even	1	trivial
2028.4.b.d.337.2		2	13.12	even	2	inner
2496.4.a.d.1.1		1	104.5	odd	4
2496.4.a.m.1.1		1	104.83	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2028 = 2^{2} \cdot 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2028.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} -2.00000i q^{5} +32.0000i q^{7} +9.00000 q^{9} +O(q^{10})\)	\(q+3.00000 q^{3} -2.00000i q^{5} +32.0000i q^{7} +9.00000 q^{9} +68.0000i q^{11} -6.00000i q^{15} +14.0000 q^{17} +4.00000i q^{19} +96.0000i q^{21} -72.0000 q^{23} +121.000 q^{25} +27.0000 q^{27} +102.000 q^{29} -136.000i q^{31} +204.000i q^{33} +64.0000 q^{35} +386.000i q^{37} +250.000i q^{41} +140.000 q^{43} -18.0000i q^{45} +296.000i q^{47} -681.000 q^{49} +42.0000 q^{51} +526.000 q^{53} +136.000 q^{55} +12.0000i q^{57} -332.000i q^{59} -410.000 q^{61} +288.000i q^{63} +596.000i q^{67} -216.000 q^{69} -880.000i q^{71} -506.000i q^{73} +363.000 q^{75} -2176.00 q^{77} -640.000 q^{79} +81.0000 q^{81} +1380.00i q^{83} -28.0000i q^{85} +306.000 q^{87} -1450.00i q^{89} -408.000i q^{93} +8.00000 q^{95} -446.000i q^{97} +612.000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{3} + 18 q^{9}+O(q^{10}) \) 2 * q + 6 * q^3 + 18 * q^9	\( 2 q + 6 q^{3} + 18 q^{9} + 28 q^{17} - 144 q^{23} + 242 q^{25} + 54 q^{27} + 204 q^{29} + 128 q^{35} + 280 q^{43} - 1362 q^{49} + 84 q^{51} + 1052 q^{53} + 272 q^{55} - 820 q^{61} - 432 q^{69} + 726 q^{75} - 4352 q^{77} - 1280 q^{79} + 162 q^{81} + 612 q^{87} + 16 q^{95}+O(q^{100}) \) 2 * q + 6 * q^3 + 18 * q^9 + 28 * q^17 - 144 * q^23 + 242 * q^25 + 54 * q^27 + 204 * q^29 + 128 * q^35 + 280 * q^43 - 1362 * q^49 + 84 * q^51 + 1052 * q^53 + 272 * q^55 - 820 * q^61 - 432 * q^69 + 726 * q^75 - 4352 * q^77 - 1280 * q^79 + 162 * q^81 + 612 * q^87 + 16 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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