LMFDB - Embedded newform 1400.2.g.f.449.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1400,2,Mod(449,1400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1400, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1400.449");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1400 = 2^{3} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1400.g (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:	$11.1790562830$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	1400.449
Dual form			1400.2.g.f.449.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +1.00000i q^{7} +2.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} +1.00000i q^{7} +2.00000 q^{9} -1.00000 q^{11} -6.00000i q^{13} -7.00000i q^{17} -1.00000 q^{19} -1.00000 q^{21} -8.00000i q^{23} +5.00000i q^{27} +6.00000 q^{29} +4.00000 q^{31} -1.00000i q^{33} +8.00000i q^{37} +6.00000 q^{39} -5.00000 q^{41} +6.00000i q^{47} -1.00000 q^{49} +7.00000 q^{51} -4.00000i q^{53} -1.00000i q^{57} +4.00000 q^{59} +6.00000 q^{61} +2.00000i q^{63} -5.00000i q^{67} +8.00000 q^{69} +14.0000 q^{71} -15.0000i q^{73} -1.00000i q^{77} -14.0000 q^{79} +1.00000 q^{81} -1.00000i q^{83} +6.00000i q^{87} +3.00000 q^{89} +6.00000 q^{91} +4.00000i q^{93} -6.00000i q^{97} -2.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{9}+O(q^{10}) $ 2 * q + 4 * q^9	$ 2 q + 4 q^{9} - 2 q^{11} - 2 q^{19} - 2 q^{21} + 12 q^{29} + 8 q^{31} + 12 q^{39} - 10 q^{41} - 2 q^{49} + 14 q^{51} + 8 q^{59} + 12 q^{61} + 16 q^{69} + 28 q^{71} - 28 q^{79} + 2 q^{81} + 6 q^{89} + 12 q^{91} - 4 q^{99}+O(q^{100}) $ 2 * q + 4 * q^9 - 2 * q^11 - 2 * q^19 - 2 * q^21 + 12 * q^29 + 8 * q^31 + 12 * q^39 - 10 * q^41 - 2 * q^49 + 14 * q^51 + 8 * q^59 + 12 * q^61 + 16 * q^69 + 28 * q^71 - 28 * q^79 + 2 * q^81 + 6 * q^89 + 12 * q^91 - 4 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1400\mathbb{Z}\right)^\times$.

$n$	$351$	$701$	$801$	$1177$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$	1.00000i	0.577350i	0.957427	+	0.288675i	$0.0932147\pi$
$3$	1.00000i	0.577350i	−0.957427	+	0.288675i	$0.906785\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.00000i	0.377964i
$7$	1.00000i	0.377964i
$8$	0	0
$9$	2.00000	0.666667
$10$	0	0
$11$	−1.00000	−0.301511	−0.150756	−	0.988571i	$-0.548171\pi$
$11$	−1.00000	−0.301511	−0.150756	+	0.988571i	$0.548171\pi$
$12$	0	0
$13$	− 6.00000i	− 1.66410i	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	− 6.00000i	− 1.66410i	0.554700	−	0.832050i	$-0.312833\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 7.00000i	− 1.69775i	−0.528594	−	0.848875i	$-0.677281\pi$
$17$	− 7.00000i	− 1.69775i	0.528594	−	0.848875i	$-0.322719\pi$
$18$	0	0
$19$	−1.00000	−0.229416	−0.114708	−	0.993399i	$-0.536593\pi$
$19$	−1.00000	−0.229416	−0.114708	+	0.993399i	$0.536593\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	− 8.00000i	− 1.66812i	−0.551677	−	0.834058i	$-0.686012\pi$
$23$	− 8.00000i	− 1.66812i	0.551677	−	0.834058i	$-0.313988\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	5.00000i	0.962250i
$28$	0	0
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	0	0
$33$	− 1.00000i	− 0.174078i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	8.00000i	1.31519i	0.753371	+	0.657596i	$0.228427\pi$
$37$	8.00000i	1.31519i	−0.753371	+	0.657596i	$0.771573\pi$
$38$	0	0
$39$	6.00000	0.960769
$40$	0	0
$41$	−5.00000	−0.780869	−0.390434	−	0.920631i	$-0.627675\pi$
$41$	−5.00000	−0.780869	−0.390434	+	0.920631i	$0.627675\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	6.00000i	0.875190i	0.899172	+	0.437595i	$0.144170\pi$
$47$	6.00000i	0.875190i	−0.899172	+	0.437595i	$0.855830\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	7.00000	0.980196
$52$	0	0
$53$	− 4.00000i	− 0.549442i	−0.961524	−	0.274721i	$-0.911414\pi$
$53$	− 4.00000i	− 0.549442i	0.961524	−	0.274721i	$-0.0885855\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 1.00000i	− 0.132453i
$58$	0	0
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	0	0
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	0	0
$63$	2.00000i	0.251976i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 5.00000i	− 0.610847i	−0.952217	−	0.305424i	$-0.901202\pi$
$67$	− 5.00000i	− 0.610847i	0.952217	−	0.305424i	$-0.0987981\pi$
$68$	0	0
$69$	8.00000	0.963087
$70$	0	0
$71$	14.0000	1.66149	0.830747	−	0.556650i	$-0.187914\pi$
$71$	14.0000	1.66149	0.830747	+	0.556650i	$0.187914\pi$
$72$	0	0
$73$	− 15.0000i	− 1.75562i	−0.479012	−	0.877809i	$-0.659005\pi$
$73$	− 15.0000i	− 1.75562i	0.479012	−	0.877809i	$-0.340995\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 1.00000i	− 0.113961i
$78$	0	0
$79$	−14.0000	−1.57512	−0.787562	−	0.616236i	$-0.788657\pi$
$79$	−14.0000	−1.57512	−0.787562	+	0.616236i	$0.788657\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 1.00000i	− 0.109764i	−0.998493	−	0.0548821i	$-0.982522\pi$
$83$	− 1.00000i	− 0.109764i	0.998493	−	0.0548821i	$-0.0174783\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	6.00000i	0.643268i
$88$	0	0
$89$	3.00000	0.317999	0.159000	−	0.987279i	$-0.449173\pi$
$89$	3.00000	0.317999	0.159000	+	0.987279i	$0.449173\pi$
$90$	0	0
$91$	6.00000	0.628971
$92$	0	0
$93$	4.00000i	0.414781i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 6.00000i	− 0.609208i	−0.952479	−	0.304604i	$-0.901476\pi$
$97$	− 6.00000i	− 0.609208i	0.952479	−	0.304604i	$-0.0985241\pi$
$98$	0	0
$99$	−2.00000	−0.201008
$100$	0	0
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	− 4.00000i	− 0.394132i	−0.980390	−	0.197066i	$-0.936859\pi$
$103$	− 4.00000i	− 0.394132i	0.980390	−	0.197066i	$-0.0631413\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	9.00000i	0.870063i	0.900415	+	0.435031i	$0.143263\pi$
$107$	9.00000i	0.870063i	−0.900415	+	0.435031i	$0.856737\pi$
$108$	0	0
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	0	0
$111$	−8.00000	−0.759326
$112$	0	0
$113$	− 1.00000i	− 0.0940721i	−0.998893	−	0.0470360i	$-0.985022\pi$
$113$	− 1.00000i	− 0.0940721i	0.998893	−	0.0470360i	$-0.0149776\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 12.0000i	− 1.10940i
$118$	0	0
$119$	7.00000	0.641689
$120$	0	0
$121$	−10.0000	−0.909091
$122$	0	0
$123$	− 5.00000i	− 0.450835i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	22.0000i	1.95218i	0.217357	+	0.976092i	$0.430256\pi$
$127$	22.0000i	1.95218i	−0.217357	+	0.976092i	$0.569744\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	16.0000	1.39793	0.698963	−	0.715158i	$-0.253645\pi$
$131$	16.0000	1.39793	0.698963	+	0.715158i	$0.253645\pi$
$132$	0	0
$133$	− 1.00000i	− 0.0867110i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	3.00000i	0.256307i	0.991754	+	0.128154i	$0.0409051\pi$
$137$	3.00000i	0.256307i	−0.991754	+	0.128154i	$0.959095\pi$
$138$	0	0
$139$	−1.00000	−0.0848189	−0.0424094	−	0.999100i	$-0.513503\pi$
$139$	−1.00000	−0.0848189	−0.0424094	+	0.999100i	$0.513503\pi$
$140$	0	0
$141$	−6.00000	−0.505291
$142$	0	0
$143$	6.00000i	0.501745i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 1.00000i	− 0.0824786i
$148$	0	0
$149$	−4.00000	−0.327693	−0.163846	−	0.986486i	$-0.552390\pi$
$149$	−4.00000	−0.327693	−0.163846	+	0.986486i	$0.552390\pi$
$150$	0	0
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	0	0
$153$	− 14.0000i	− 1.13183i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	4.00000i	0.319235i	0.987179	+	0.159617i	$0.0510260\pi$
$157$	4.00000i	0.319235i	−0.987179	+	0.159617i	$0.948974\pi$
$158$	0	0
$159$	4.00000	0.317221
$160$	0	0
$161$	8.00000	0.630488
$162$	0	0
$163$	25.0000i	1.95815i	0.203497	+	0.979076i	$0.434769\pi$
$163$	25.0000i	1.95815i	−0.203497	+	0.979076i	$0.565231\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 20.0000i	− 1.54765i	−0.633402	−	0.773823i	$-0.718342\pi$
$167$	− 20.0000i	− 1.54765i	0.633402	−	0.773823i	$-0.281658\pi$
$168$	0	0
$169$	−23.0000	−1.76923
$170$	0	0
$171$	−2.00000	−0.152944
$172$	0	0
$173$	10.0000i	0.760286i	0.924928	+	0.380143i	$0.124125\pi$
$173$	10.0000i	0.760286i	−0.924928	+	0.380143i	$0.875875\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	4.00000i	0.300658i
$178$	0	0
$179$	−15.0000	−1.12115	−0.560576	−	0.828103i	$-0.689420\pi$
$179$	−15.0000	−1.12115	−0.560576	+	0.828103i	$0.689420\pi$
$180$	0	0
$181$	−22.0000	−1.63525	−0.817624	−	0.575753i	$-0.804709\pi$
$181$	−22.0000	−1.63525	−0.817624	+	0.575753i	$0.804709\pi$
$182$	0	0
$183$	6.00000i	0.443533i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	7.00000i	0.511891i
$188$	0	0
$189$	−5.00000	−0.363696
$190$	0	0
$191$	18.0000	1.30243	0.651217	−	0.758891i	$-0.274259\pi$
$191$	18.0000	1.30243	0.651217	+	0.758891i	$0.274259\pi$
$192$	0	0
$193$	− 11.0000i	− 0.791797i	−0.918294	−	0.395899i	$-0.870433\pi$
$193$	− 11.0000i	− 0.791797i	0.918294	−	0.395899i	$-0.129567\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	6.00000i	0.427482i	0.976890	+	0.213741i	$0.0685649\pi$
$197$	6.00000i	0.427482i	−0.976890	+	0.213741i	$0.931435\pi$
$198$	0	0
$199$	10.0000	0.708881	0.354441	−	0.935079i	$-0.384671\pi$
$199$	10.0000	0.708881	0.354441	+	0.935079i	$0.384671\pi$
$200$	0	0
$201$	5.00000	0.352673
$202$	0	0
$203$	6.00000i	0.421117i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 16.0000i	− 1.11208i
$208$	0	0
$209$	1.00000	0.0691714
$210$	0	0
$211$	−19.0000	−1.30801	−0.654007	−	0.756489i	$-0.726913\pi$
$211$	−19.0000	−1.30801	−0.654007	+	0.756489i	$0.726913\pi$
$212$	0	0
$213$	14.0000i	0.959264i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	4.00000i	0.271538i
$218$	0	0
$219$	15.0000	1.01361
$220$	0	0
$221$	−42.0000	−2.82523
$222$	0	0
$223$	− 10.0000i	− 0.669650i	−0.942280	−	0.334825i	$-0.891323\pi$
$223$	− 10.0000i	− 0.669650i	0.942280	−	0.334825i	$-0.108677\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 28.0000i	− 1.85843i	−0.369546	−	0.929213i	$-0.620487\pi$
$227$	− 28.0000i	− 1.85843i	0.369546	−	0.929213i	$-0.379513\pi$
$228$	0	0
$229$	22.0000	1.45380	0.726900	−	0.686743i	$-0.240960\pi$
$229$	22.0000	1.45380	0.726900	+	0.686743i	$0.240960\pi$
$230$	0	0
$231$	1.00000	0.0657952
$232$	0	0
$233$	− 10.0000i	− 0.655122i	−0.944830	−	0.327561i	$-0.893773\pi$
$233$	− 10.0000i	− 0.655122i	0.944830	−	0.327561i	$-0.106227\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 14.0000i	− 0.909398i
$238$	0	0
$239$	−12.0000	−0.776215	−0.388108	−	0.921614i	$-0.626871\pi$
$239$	−12.0000	−0.776215	−0.388108	+	0.921614i	$0.626871\pi$
$240$	0	0
$241$	−13.0000	−0.837404	−0.418702	−	0.908124i	$-0.637515\pi$
$241$	−13.0000	−0.837404	−0.418702	+	0.908124i	$0.637515\pi$
$242$	0	0
$243$	16.0000i	1.02640i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	6.00000i	0.381771i
$248$	0	0
$249$	1.00000	0.0633724
$250$	0	0
$251$	−25.0000	−1.57799	−0.788993	−	0.614402i	$-0.789397\pi$
$251$	−25.0000	−1.57799	−0.788993	+	0.614402i	$0.789397\pi$
$252$	0	0
$253$	8.00000i	0.502956i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 18.0000i	− 1.12281i	−0.827541	−	0.561405i	$-0.810261\pi$
$257$	− 18.0000i	− 1.12281i	0.827541	−	0.561405i	$-0.189739\pi$
$258$	0	0
$259$	−8.00000	−0.497096
$260$	0	0
$261$	12.0000	0.742781
$262$	0	0
$263$	14.0000i	0.863277i	0.902047	+	0.431638i	$0.142064\pi$
$263$	14.0000i	0.863277i	−0.902047	+	0.431638i	$0.857936\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	3.00000i	0.183597i
$268$	0	0
$269$	6.00000	0.365826	0.182913	−	0.983129i	$-0.441447\pi$
$269$	6.00000	0.365826	0.182913	+	0.983129i	$0.441447\pi$
$270$	0	0
$271$	−14.0000	−0.850439	−0.425220	−	0.905090i	$-0.639803\pi$
$271$	−14.0000	−0.850439	−0.425220	+	0.905090i	$0.639803\pi$
$272$	0	0
$273$	6.00000i	0.363137i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 2.00000i	− 0.120168i	−0.998193	−	0.0600842i	$-0.980863\pi$
$277$	− 2.00000i	− 0.120168i	0.998193	−	0.0600842i	$-0.0191369\pi$
$278$	0	0
$279$	8.00000	0.478947
$280$	0	0
$281$	30.0000	1.78965	0.894825	−	0.446417i	$-0.147300\pi$
$281$	30.0000	1.78965	0.894825	+	0.446417i	$0.147300\pi$
$282$	0	0
$283$	− 1.00000i	− 0.0594438i	−0.999558	−	0.0297219i	$-0.990538\pi$
$283$	− 1.00000i	− 0.0594438i	0.999558	−	0.0297219i	$-0.00946217\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 5.00000i	− 0.295141i
$288$	0	0
$289$	−32.0000	−1.88235
$290$	0	0
$291$	6.00000	0.351726
$292$	0	0
$293$	− 6.00000i	− 0.350524i	−0.984522	−	0.175262i	$-0.943923\pi$
$293$	− 6.00000i	− 0.350524i	0.984522	−	0.175262i	$-0.0560772\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 5.00000i	− 0.290129i
$298$	0	0
$299$	−48.0000	−2.77591
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	15.0000i	0.856095i	0.903756	+	0.428048i	$0.140798\pi$
$307$	15.0000i	0.856095i	−0.903756	+	0.428048i	$0.859202\pi$
$308$	0	0
$309$	4.00000	0.227552
$310$	0	0
$311$	−10.0000	−0.567048	−0.283524	−	0.958965i	$-0.591504\pi$
$311$	−10.0000	−0.567048	−0.283524	+	0.958965i	$0.591504\pi$
$312$	0	0
$313$	22.0000i	1.24351i	0.783210	+	0.621757i	$0.213581\pi$
$313$	22.0000i	1.24351i	−0.783210	+	0.621757i	$0.786419\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	28.0000i	1.57264i	0.617822	+	0.786318i	$0.288015\pi$
$317$	28.0000i	1.57264i	−0.617822	+	0.786318i	$0.711985\pi$
$318$	0	0
$319$	−6.00000	−0.335936
$320$	0	0
$321$	−9.00000	−0.502331
$322$	0	0
$323$	7.00000i	0.389490i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	10.0000i	0.553001i
$328$	0	0
$329$	−6.00000	−0.330791
$330$	0	0
$331$	3.00000	0.164895	0.0824475	−	0.996595i	$-0.473726\pi$
$331$	3.00000	0.164895	0.0824475	+	0.996595i	$0.473726\pi$
$332$	0	0
$333$	16.0000i	0.876795i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 19.0000i	− 1.03500i	−0.855684	−	0.517498i	$-0.826864\pi$
$337$	− 19.0000i	− 1.03500i	0.855684	−	0.517498i	$-0.173136\pi$
$338$	0	0
$339$	1.00000	0.0543125
$340$	0	0
$341$	−4.00000	−0.216612
$342$	0	0
$343$	− 1.00000i	− 0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 23.0000i	− 1.23470i	−0.786687	−	0.617352i	$-0.788205\pi$
$347$	− 23.0000i	− 1.23470i	0.786687	−	0.617352i	$-0.211795\pi$
$348$	0	0
$349$	−8.00000	−0.428230	−0.214115	−	0.976808i	$-0.568687\pi$
$349$	−8.00000	−0.428230	−0.214115	+	0.976808i	$0.568687\pi$
$350$	0	0
$351$	30.0000	1.60128
$352$	0	0
$353$	14.0000i	0.745145i	0.928003	+	0.372572i	$0.121524\pi$
$353$	14.0000i	0.745145i	−0.928003	+	0.372572i	$0.878476\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	7.00000i	0.370479i
$358$	0	0
$359$	6.00000	0.316668	0.158334	−	0.987386i	$-0.449388\pi$
$359$	6.00000	0.316668	0.158334	+	0.987386i	$0.449388\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	0	0
$363$	− 10.0000i	− 0.524864i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	8.00000i	0.417597i	0.977959	+	0.208798i	$0.0669552\pi$
$367$	8.00000i	0.417597i	−0.977959	+	0.208798i	$0.933045\pi$
$368$	0	0
$369$	−10.0000	−0.520579
$370$	0	0
$371$	4.00000	0.207670
$372$	0	0
$373$	20.0000i	1.03556i	0.855514	+	0.517780i	$0.173242\pi$
$373$	20.0000i	1.03556i	−0.855514	+	0.517780i	$0.826758\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 36.0000i	− 1.85409i
$378$	0	0
$379$	3.00000	0.154100	0.0770498	−	0.997027i	$-0.475450\pi$
$379$	3.00000	0.154100	0.0770498	+	0.997027i	$0.475450\pi$
$380$	0	0
$381$	−22.0000	−1.12709
$382$	0	0
$383$	− 6.00000i	− 0.306586i	−0.988181	−	0.153293i	$-0.951012\pi$
$383$	− 6.00000i	− 0.306586i	0.988181	−	0.153293i	$-0.0489878\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	16.0000	0.811232	0.405616	−	0.914044i	$-0.367057\pi$
$389$	16.0000	0.811232	0.405616	+	0.914044i	$0.367057\pi$
$390$	0	0
$391$	−56.0000	−2.83204
$392$	0	0
$393$	16.0000i	0.807093i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	30.0000i	1.50566i	0.658217	+	0.752828i	$0.271311\pi$
$397$	30.0000i	1.50566i	−0.658217	+	0.752828i	$0.728689\pi$
$398$	0	0
$399$	1.00000	0.0500626
$400$	0	0
$401$	13.0000	0.649189	0.324595	−	0.945853i	$-0.394772\pi$
$401$	13.0000	0.649189	0.324595	+	0.945853i	$0.394772\pi$
$402$	0	0
$403$	− 24.0000i	− 1.19553i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 8.00000i	− 0.396545i
$408$	0	0
$409$	−27.0000	−1.33506	−0.667532	−	0.744581i	$-0.732649\pi$
$409$	−27.0000	−1.33506	−0.667532	+	0.744581i	$0.732649\pi$
$410$	0	0
$411$	−3.00000	−0.147979
$412$	0	0
$413$	4.00000i	0.196827i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 1.00000i	− 0.0489702i
$418$	0	0
$419$	35.0000	1.70986	0.854931	−	0.518742i	$-0.173599\pi$
$419$	35.0000	1.70986	0.854931	+	0.518742i	$0.173599\pi$
$420$	0	0
$421$	36.0000	1.75453	0.877266	−	0.480004i	$-0.159365\pi$
$421$	36.0000	1.75453	0.877266	+	0.480004i	$0.159365\pi$
$422$	0	0
$423$	12.0000i	0.583460i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	6.00000i	0.290360i
$428$	0	0
$429$	−6.00000	−0.289683
$430$	0	0
$431$	−4.00000	−0.192673	−0.0963366	−	0.995349i	$-0.530713\pi$
$431$	−4.00000	−0.192673	−0.0963366	+	0.995349i	$0.530713\pi$
$432$	0	0
$433$	− 9.00000i	− 0.432512i	−0.976337	−	0.216256i	$-0.930615\pi$
$433$	− 9.00000i	− 0.432512i	0.976337	−	0.216256i	$-0.0693846\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	8.00000i	0.382692i
$438$	0	0
$439$	−20.0000	−0.954548	−0.477274	−	0.878755i	$-0.658375\pi$
$439$	−20.0000	−0.954548	−0.477274	+	0.878755i	$0.658375\pi$
$440$	0	0
$441$	−2.00000	−0.0952381
$442$	0	0
$443$	25.0000i	1.18779i	0.804544	+	0.593893i	$0.202410\pi$
$443$	25.0000i	1.18779i	−0.804544	+	0.593893i	$0.797590\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 4.00000i	− 0.189194i
$448$	0	0
$449$	23.0000	1.08544	0.542719	−	0.839915i	$-0.317395\pi$
$449$	23.0000	1.08544	0.542719	+	0.839915i	$0.317395\pi$
$450$	0	0
$451$	5.00000	0.235441
$452$	0	0
$453$	8.00000i	0.375873i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	7.00000i	0.327446i	0.986506	+	0.163723i	$0.0523504\pi$
$457$	7.00000i	0.327446i	−0.986506	+	0.163723i	$0.947650\pi$
$458$	0	0
$459$	35.0000	1.63366
$460$	0	0
$461$	2.00000	0.0931493	0.0465746	−	0.998915i	$-0.485169\pi$
$461$	2.00000	0.0931493	0.0465746	+	0.998915i	$0.485169\pi$
$462$	0	0
$463$	24.0000i	1.11537i	0.830051	+	0.557687i	$0.188311\pi$
$463$	24.0000i	1.11537i	−0.830051	+	0.557687i	$0.811689\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	28.0000i	1.29569i	0.761774	+	0.647843i	$0.224329\pi$
$467$	28.0000i	1.29569i	−0.761774	+	0.647843i	$0.775671\pi$
$468$	0	0
$469$	5.00000	0.230879
$470$	0	0
$471$	−4.00000	−0.184310
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 8.00000i	− 0.366295i
$478$	0	0
$479$	−2.00000	−0.0913823	−0.0456912	−	0.998956i	$-0.514549\pi$
$479$	−2.00000	−0.0913823	−0.0456912	+	0.998956i	$0.514549\pi$
$480$	0	0
$481$	48.0000	2.18861
$482$	0	0
$483$	8.00000i	0.364013i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 22.0000i	− 0.996915i	−0.866914	−	0.498458i	$-0.833900\pi$
$487$	− 22.0000i	− 0.996915i	0.866914	−	0.498458i	$-0.166100\pi$
$488$	0	0
$489$	−25.0000	−1.13054
$490$	0	0
$491$	−20.0000	−0.902587	−0.451294	−	0.892375i	$-0.649037\pi$
$491$	−20.0000	−0.902587	−0.451294	+	0.892375i	$0.649037\pi$
$492$	0	0
$493$	− 42.0000i	− 1.89158i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	14.0000i	0.627986i
$498$	0	0
$499$	−28.0000	−1.25345	−0.626726	−	0.779240i	$-0.715605\pi$
$499$	−28.0000	−1.25345	−0.626726	+	0.779240i	$0.715605\pi$
$500$	0	0
$501$	20.0000	0.893534
$502$	0	0
$503$	− 14.0000i	− 0.624229i	−0.950044	−	0.312115i	$-0.898963\pi$
$503$	− 14.0000i	− 0.624229i	0.950044	−	0.312115i	$-0.101037\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 23.0000i	− 1.02147i
$508$	0	0
$509$	30.0000	1.32973	0.664863	−	0.746965i	$-0.268490\pi$
$509$	30.0000	1.32973	0.664863	+	0.746965i	$0.268490\pi$
$510$	0	0
$511$	15.0000	0.663561
$512$	0	0
$513$	− 5.00000i	− 0.220755i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 6.00000i	− 0.263880i
$518$	0	0
$519$	−10.0000	−0.438951
$520$	0	0
$521$	27.0000	1.18289	0.591446	−	0.806345i	$-0.298557\pi$
$521$	27.0000	1.18289	0.591446	+	0.806345i	$0.298557\pi$
$522$	0	0
$523$	− 23.0000i	− 1.00572i	−0.864368	−	0.502860i	$-0.832281\pi$
$523$	− 23.0000i	− 1.00572i	0.864368	−	0.502860i	$-0.167719\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 28.0000i	− 1.21970i
$528$	0	0
$529$	−41.0000	−1.78261
$530$	0	0
$531$	8.00000	0.347170
$532$	0	0
$533$	30.0000i	1.29944i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 15.0000i	− 0.647298i
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	14.0000	0.601907	0.300954	−	0.953639i	$-0.402695\pi$
$541$	14.0000	0.601907	0.300954	+	0.953639i	$0.402695\pi$
$542$	0	0
$543$	− 22.0000i	− 0.944110i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	41.0000i	1.75303i	0.481371	+	0.876517i	$0.340139\pi$
$547$	41.0000i	1.75303i	−0.481371	+	0.876517i	$0.659861\pi$
$548$	0	0
$549$	12.0000	0.512148
$550$	0	0
$551$	−6.00000	−0.255609
$552$	0	0
$553$	− 14.0000i	− 0.595341i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	36.0000i	1.52537i	0.646771	+	0.762684i	$0.276119\pi$
$557$	36.0000i	1.52537i	−0.646771	+	0.762684i	$0.723881\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−7.00000	−0.295540
$562$	0	0
$563$	− 36.0000i	− 1.51722i	−0.651546	−	0.758610i	$-0.725879\pi$
$563$	− 36.0000i	− 1.51722i	0.651546	−	0.758610i	$-0.274121\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	1.00000i	0.0419961i
$568$	0	0
$569$	11.0000	0.461144	0.230572	−	0.973055i	$-0.425940\pi$
$569$	11.0000	0.461144	0.230572	+	0.973055i	$0.425940\pi$
$570$	0	0
$571$	−44.0000	−1.84134	−0.920671	−	0.390339i	$-0.872358\pi$
$571$	−44.0000	−1.84134	−0.920671	+	0.390339i	$0.872358\pi$
$572$	0	0
$573$	18.0000i	0.751961i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	19.0000i	0.790980i	0.918470	+	0.395490i	$0.129425\pi$
$577$	19.0000i	0.790980i	−0.918470	+	0.395490i	$0.870575\pi$
$578$	0	0
$579$	11.0000	0.457144
$580$	0	0
$581$	1.00000	0.0414870
$582$	0	0
$583$	4.00000i	0.165663i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 15.0000i	− 0.619116i	−0.950881	−	0.309558i	$-0.899819\pi$
$587$	− 15.0000i	− 0.619116i	0.950881	−	0.309558i	$-0.100181\pi$
$588$	0	0
$589$	−4.00000	−0.164817
$590$	0	0
$591$	−6.00000	−0.246807
$592$	0	0
$593$	9.00000i	0.369586i	0.982777	+	0.184793i	$0.0591614\pi$
$593$	9.00000i	0.369586i	−0.982777	+	0.184793i	$0.940839\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	10.0000i	0.409273i
$598$	0	0
$599$	8.00000	0.326871	0.163436	−	0.986554i	$-0.447742\pi$
$599$	8.00000	0.326871	0.163436	+	0.986554i	$0.447742\pi$
$600$	0	0
$601$	−35.0000	−1.42768	−0.713840	−	0.700309i	$-0.753046\pi$
$601$	−35.0000	−1.42768	−0.713840	+	0.700309i	$0.753046\pi$
$602$	0	0
$603$	− 10.0000i	− 0.407231i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	36.0000i	1.46119i	0.682808	+	0.730597i	$0.260758\pi$
$607$	36.0000i	1.46119i	−0.682808	+	0.730597i	$0.739242\pi$
$608$	0	0
$609$	−6.00000	−0.243132
$610$	0	0
$611$	36.0000	1.45640
$612$	0	0
$613$	− 14.0000i	− 0.565455i	−0.959200	−	0.282727i	$-0.908761\pi$
$613$	− 14.0000i	− 0.565455i	0.959200	−	0.282727i	$-0.0912392\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	18.0000i	0.724653i	0.932051	+	0.362326i	$0.118017\pi$
$617$	18.0000i	0.724653i	−0.932051	+	0.362326i	$0.881983\pi$
$618$	0	0
$619$	28.0000	1.12542	0.562708	−	0.826656i	$-0.309760\pi$
$619$	28.0000	1.12542	0.562708	+	0.826656i	$0.309760\pi$
$620$	0	0
$621$	40.0000	1.60514
$622$	0	0
$623$	3.00000i	0.120192i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	1.00000i	0.0399362i
$628$	0	0
$629$	56.0000	2.23287
$630$	0	0
$631$	−8.00000	−0.318475	−0.159237	−	0.987240i	$-0.550904\pi$
$631$	−8.00000	−0.318475	−0.159237	+	0.987240i	$0.550904\pi$
$632$	0	0
$633$	− 19.0000i	− 0.755182i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	6.00000i	0.237729i
$638$	0	0
$639$	28.0000	1.10766
$640$	0	0
$641$	2.00000	0.0789953	0.0394976	−	0.999220i	$-0.487424\pi$
$641$	2.00000	0.0789953	0.0394976	+	0.999220i	$0.487424\pi$
$642$	0	0
$643$	− 40.0000i	− 1.57745i	−0.614749	−	0.788723i	$-0.710743\pi$
$643$	− 40.0000i	− 1.57745i	0.614749	−	0.788723i	$-0.289257\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 8.00000i	− 0.314512i	−0.987558	−	0.157256i	$-0.949735\pi$
$647$	− 8.00000i	− 0.314512i	0.987558	−	0.157256i	$-0.0502649\pi$
$648$	0	0
$649$	−4.00000	−0.157014
$650$	0	0
$651$	−4.00000	−0.156772
$652$	0	0
$653$	36.0000i	1.40879i	0.709809	+	0.704394i	$0.248781\pi$
$653$	36.0000i	1.40879i	−0.709809	+	0.704394i	$0.751219\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 30.0000i	− 1.17041i
$658$	0	0
$659$	−29.0000	−1.12968	−0.564840	−	0.825201i	$-0.691062\pi$
$659$	−29.0000	−1.12968	−0.564840	+	0.825201i	$0.691062\pi$
$660$	0	0
$661$	−18.0000	−0.700119	−0.350059	−	0.936727i	$-0.613839\pi$
$661$	−18.0000	−0.700119	−0.350059	+	0.936727i	$0.613839\pi$
$662$	0	0
$663$	− 42.0000i	− 1.63114i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 48.0000i	− 1.85857i
$668$	0	0
$669$	10.0000	0.386622
$670$	0	0
$671$	−6.00000	−0.231627
$672$	0	0
$673$	− 30.0000i	− 1.15642i	−0.815890	−	0.578208i	$-0.803752\pi$
$673$	− 30.0000i	− 1.15642i	0.815890	−	0.578208i	$-0.196248\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 48.0000i	− 1.84479i	−0.386248	−	0.922395i	$-0.626229\pi$
$677$	− 48.0000i	− 1.84479i	0.386248	−	0.922395i	$-0.373771\pi$
$678$	0	0
$679$	6.00000	0.230259
$680$	0	0
$681$	28.0000	1.07296
$682$	0	0
$683$	− 9.00000i	− 0.344375i	−0.985064	−	0.172188i	$-0.944916\pi$
$683$	− 9.00000i	− 0.344375i	0.985064	−	0.172188i	$-0.0550836\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	22.0000i	0.839352i
$688$	0	0
$689$	−24.0000	−0.914327
$690$	0	0
$691$	−43.0000	−1.63580	−0.817899	−	0.575362i	$-0.804861\pi$
$691$	−43.0000	−1.63580	−0.817899	+	0.575362i	$0.804861\pi$
$692$	0	0
$693$	− 2.00000i	− 0.0759737i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	35.0000i	1.32572i
$698$	0	0
$699$	10.0000	0.378235
$700$	0	0
$701$	24.0000	0.906467	0.453234	−	0.891392i	$-0.350270\pi$
$701$	24.0000	0.906467	0.453234	+	0.891392i	$0.350270\pi$
$702$	0	0
$703$	− 8.00000i	− 0.301726i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	12.0000	0.450669	0.225335	−	0.974281i	$-0.427652\pi$
$709$	12.0000	0.450669	0.225335	+	0.974281i	$0.427652\pi$
$710$	0	0
$711$	−28.0000	−1.05008
$712$	0	0
$713$	− 32.0000i	− 1.19841i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 12.0000i	− 0.448148i
$718$	0	0
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	4.00000	0.148968
$722$	0	0
$723$	− 13.0000i	− 0.483475i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 14.0000i	− 0.519231i	−0.965712	−	0.259616i	$-0.916404\pi$
$727$	− 14.0000i	− 0.519231i	0.965712	−	0.259616i	$-0.0835959\pi$
$728$	0	0
$729$	−13.0000	−0.481481
$730$	0	0
$731$	0	0
$732$	0	0
$733$	16.0000i	0.590973i	0.955347	+	0.295487i	$0.0954818\pi$
$733$	16.0000i	0.590973i	−0.955347	+	0.295487i	$0.904518\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	5.00000i	0.184177i
$738$	0	0
$739$	20.0000	0.735712	0.367856	−	0.929883i	$-0.380092\pi$
$739$	20.0000	0.735712	0.367856	+	0.929883i	$0.380092\pi$
$740$	0	0
$741$	−6.00000	−0.220416
$742$	0	0
$743$	− 48.0000i	− 1.76095i	−0.474093	−	0.880475i	$-0.657224\pi$
$743$	− 48.0000i	− 1.76095i	0.474093	−	0.880475i	$-0.342776\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 2.00000i	− 0.0731762i
$748$	0	0
$749$	−9.00000	−0.328853
$750$	0	0
$751$	42.0000	1.53260	0.766301	−	0.642482i	$-0.222095\pi$
$751$	42.0000	1.53260	0.766301	+	0.642482i	$0.222095\pi$
$752$	0	0
$753$	− 25.0000i	− 0.911051i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	42.0000i	1.52652i	0.646094	+	0.763258i	$0.276401\pi$
$757$	42.0000i	1.52652i	−0.646094	+	0.763258i	$0.723599\pi$
$758$	0	0
$759$	−8.00000	−0.290382
$760$	0	0
$761$	27.0000	0.978749	0.489375	−	0.872074i	$-0.337225\pi$
$761$	27.0000	0.978749	0.489375	+	0.872074i	$0.337225\pi$
$762$	0	0
$763$	10.0000i	0.362024i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 24.0000i	− 0.866590i
$768$	0	0
$769$	53.0000	1.91123	0.955614	−	0.294620i	$-0.0951931\pi$
$769$	53.0000	1.91123	0.955614	+	0.294620i	$0.0951931\pi$
$770$	0	0
$771$	18.0000	0.648254
$772$	0	0
$773$	12.0000i	0.431610i	0.976436	+	0.215805i	$0.0692376\pi$
$773$	12.0000i	0.431610i	−0.976436	+	0.215805i	$0.930762\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 8.00000i	− 0.286998i
$778$	0	0
$779$	5.00000	0.179144
$780$	0	0
$781$	−14.0000	−0.500959
$782$	0	0
$783$	30.0000i	1.07211i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	36.0000i	1.28326i	0.767014	+	0.641631i	$0.221742\pi$
$787$	36.0000i	1.28326i	−0.767014	+	0.641631i	$0.778258\pi$
$788$	0	0
$789$	−14.0000	−0.498413
$790$	0	0
$791$	1.00000	0.0355559
$792$	0	0
$793$	− 36.0000i	− 1.27840i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	22.0000i	0.779280i	0.920967	+	0.389640i	$0.127401\pi$
$797$	22.0000i	0.779280i	−0.920967	+	0.389640i	$0.872599\pi$
$798$	0	0
$799$	42.0000	1.48585
$800$	0	0
$801$	6.00000	0.212000
$802$	0	0
$803$	15.0000i	0.529339i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	6.00000i	0.211210i
$808$	0	0
$809$	26.0000	0.914111	0.457056	−	0.889438i	$-0.348904\pi$
$809$	26.0000	0.914111	0.457056	+	0.889438i	$0.348904\pi$
$810$	0	0
$811$	28.0000	0.983213	0.491606	−	0.870817i	$-0.336410\pi$
$811$	28.0000	0.983213	0.491606	+	0.870817i	$0.336410\pi$
$812$	0	0
$813$	− 14.0000i	− 0.491001i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	12.0000	0.419314
$820$	0	0
$821$	0	0		−	1.00000i	$-0.5\pi$
$821$	0	0			1.00000i	$0.5\pi$
$822$	0	0
$823$	10.0000i	0.348578i	0.984695	+	0.174289i	$0.0557627\pi$
$823$	10.0000i	0.348578i	−0.984695	+	0.174289i	$0.944237\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 13.0000i	− 0.452054i	−0.974121	−	0.226027i	$-0.927426\pi$
$827$	− 13.0000i	− 0.452054i	0.974121	−	0.226027i	$-0.0725738\pi$
$828$	0	0
$829$	−20.0000	−0.694629	−0.347314	−	0.937749i	$-0.612906\pi$
$829$	−20.0000	−0.694629	−0.347314	+	0.937749i	$0.612906\pi$
$830$	0	0
$831$	2.00000	0.0693792
$832$	0	0
$833$	7.00000i	0.242536i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	20.0000i	0.691301i
$838$	0	0
$839$	−34.0000	−1.17381	−0.586905	−	0.809656i	$-0.699654\pi$
$839$	−34.0000	−1.17381	−0.586905	+	0.809656i	$0.699654\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	0	0
$843$	30.0000i	1.03325i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 10.0000i	− 0.343604i
$848$	0	0
$849$	1.00000	0.0343199
$850$	0	0
$851$	64.0000	2.19389
$852$	0	0
$853$	14.0000i	0.479351i	0.970853	+	0.239675i	$0.0770410\pi$
$853$	14.0000i	0.479351i	−0.970853	+	0.239675i	$0.922959\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	45.0000i	1.53717i	0.639747	+	0.768585i	$0.279039\pi$
$857$	45.0000i	1.53717i	−0.639747	+	0.768585i	$0.720961\pi$
$858$	0	0
$859$	23.0000	0.784750	0.392375	−	0.919805i	$-0.371654\pi$
$859$	23.0000	0.784750	0.392375	+	0.919805i	$0.371654\pi$
$860$	0	0
$861$	5.00000	0.170400
$862$	0	0
$863$	− 4.00000i	− 0.136162i	−0.997680	−	0.0680808i	$-0.978312\pi$
$863$	− 4.00000i	− 0.136162i	0.997680	−	0.0680808i	$-0.0216876\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 32.0000i	− 1.08678i
$868$	0	0
$869$	14.0000	0.474917
$870$	0	0
$871$	−30.0000	−1.01651
$872$	0	0
$873$	− 12.0000i	− 0.406138i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	48.0000i	1.62084i	0.585846	+	0.810422i	$0.300762\pi$
$877$	48.0000i	1.62084i	−0.585846	+	0.810422i	$0.699238\pi$
$878$	0	0
$879$	6.00000	0.202375
$880$	0	0
$881$	26.0000	0.875962	0.437981	−	0.898984i	$-0.355694\pi$
$881$	26.0000	0.875962	0.437981	+	0.898984i	$0.355694\pi$
$882$	0	0
$883$	11.0000i	0.370179i	0.982722	+	0.185090i	$0.0592576\pi$
$883$	11.0000i	0.370179i	−0.982722	+	0.185090i	$0.940742\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	18.0000i	0.604381i	0.953248	+	0.302190i	$0.0977178\pi$
$887$	18.0000i	0.604381i	−0.953248	+	0.302190i	$0.902282\pi$
$888$	0	0
$889$	−22.0000	−0.737856
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	0	0
$893$	− 6.00000i	− 0.200782i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 48.0000i	− 1.60267i
$898$	0	0
$899$	24.0000	0.800445
$900$	0	0
$901$	−28.0000	−0.932815
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 4.00000i	− 0.132818i	−0.997792	−	0.0664089i	$-0.978846\pi$
$907$	− 4.00000i	− 0.132818i	0.997792	−	0.0664089i	$-0.0211542\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−12.0000	−0.397578	−0.198789	−	0.980042i	$-0.563701\pi$
$911$	−12.0000	−0.397578	−0.198789	+	0.980042i	$0.563701\pi$
$912$	0	0
$913$	1.00000i	0.0330952i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	16.0000i	0.528367i
$918$	0	0
$919$	42.0000	1.38545	0.692726	−	0.721201i	$-0.256409\pi$
$919$	42.0000	1.38545	0.692726	+	0.721201i	$0.256409\pi$
$920$	0	0
$921$	−15.0000	−0.494267
$922$	0	0
$923$	− 84.0000i	− 2.76489i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 8.00000i	− 0.262754i
$928$	0	0
$929$	−14.0000	−0.459325	−0.229663	−	0.973270i	$-0.573762\pi$
$929$	−14.0000	−0.459325	−0.229663	+	0.973270i	$0.573762\pi$
$930$	0	0
$931$	1.00000	0.0327737
$932$	0	0
$933$	− 10.0000i	− 0.327385i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 1.00000i	− 0.0326686i	−0.999867	−	0.0163343i	$-0.994800\pi$
$937$	− 1.00000i	− 0.0326686i	0.999867	−	0.0163343i	$-0.00519960\pi$
$938$	0	0
$939$	−22.0000	−0.717943
$940$	0	0
$941$	−48.0000	−1.56476	−0.782378	−	0.622804i	$-0.785993\pi$
$941$	−48.0000	−1.56476	−0.782378	+	0.622804i	$0.785993\pi$
$942$	0	0
$943$	40.0000i	1.30258i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 20.0000i	− 0.649913i	−0.945729	−	0.324956i	$-0.894650\pi$
$947$	− 20.0000i	− 0.649913i	0.945729	−	0.324956i	$-0.105350\pi$
$948$	0	0
$949$	−90.0000	−2.92152
$950$	0	0
$951$	−28.0000	−0.907962
$952$	0	0
$953$	25.0000i	0.809829i	0.914354	+	0.404915i	$0.132699\pi$
$953$	25.0000i	0.809829i	−0.914354	+	0.404915i	$0.867301\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 6.00000i	− 0.193952i
$958$	0	0
$959$	−3.00000	−0.0968751
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	18.0000i	0.580042i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	2.00000i	0.0643157i	0.999483	+	0.0321578i	$0.0102379\pi$
$967$	2.00000i	0.0643157i	−0.999483	+	0.0321578i	$0.989762\pi$
$968$	0	0
$969$	−7.00000	−0.224872
$970$	0	0
$971$	−23.0000	−0.738105	−0.369053	−	0.929409i	$-0.620318\pi$
$971$	−23.0000	−0.738105	−0.369053	+	0.929409i	$0.620318\pi$
$972$	0	0
$973$	− 1.00000i	− 0.0320585i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 45.0000i	− 1.43968i	−0.694141	−	0.719839i	$-0.744216\pi$
$977$	− 45.0000i	− 1.43968i	0.694141	−	0.719839i	$-0.255784\pi$
$978$	0	0
$979$	−3.00000	−0.0958804
$980$	0	0
$981$	20.0000	0.638551
$982$	0	0
$983$	− 36.0000i	− 1.14822i	−0.818778	−	0.574111i	$-0.805348\pi$
$983$	− 36.0000i	− 1.14822i	0.818778	−	0.574111i	$-0.194652\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 6.00000i	− 0.190982i
$988$	0	0
$989$	0	0
$990$	0	0
$991$	20.0000	0.635321	0.317660	−	0.948205i	$-0.397103\pi$
$991$	20.0000	0.635321	0.317660	+	0.948205i	$0.397103\pi$
$992$	0	0
$993$	3.00000i	0.0952021i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	2.00000i	0.0633406i	0.999498	+	0.0316703i	$0.0100827\pi$
$997$	2.00000i	0.0633406i	−0.999498	+	0.0316703i	$0.989917\pi$
$998$	0	0
$999$	−40.0000	−1.26554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1400.2.g.f.449.2		2
4.3	odd	2		2800.2.g.k.449.1		2
5.2	odd	4		1400.2.a.i.1.1	yes	1
5.3	odd	4		1400.2.a.e.1.1	✓	1
5.4	even	2	inner	1400.2.g.f.449.1		2
20.3	even	4		2800.2.a.v.1.1		1
20.7	even	4		2800.2.a.j.1.1		1
20.19	odd	2		2800.2.g.k.449.2		2
35.13	even	4		9800.2.a.ba.1.1		1
35.27	even	4		9800.2.a.q.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1400.2.a.e.1.1	✓	1	5.3	odd	4
1400.2.a.i.1.1	yes	1	5.2	odd	4
1400.2.g.f.449.1		2	5.4	even	2	inner
1400.2.g.f.449.2		2	1.1	even	1	trivial
2800.2.a.j.1.1		1	20.7	even	4
2800.2.a.v.1.1		1	20.3	even	4
2800.2.g.k.449.1		2	4.3	odd	2
2800.2.g.k.449.2		2	20.19	odd	2
9800.2.a.q.1.1		1	35.27	even	4
9800.2.a.ba.1.1		1	35.13	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:	\( N \)	\(=\)	\( 1400 = 2^{3} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1400.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +1.00000i q^{7} +2.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} +1.00000i q^{7} +2.00000 q^{9} -1.00000 q^{11} -6.00000i q^{13} -7.00000i q^{17} -1.00000 q^{19} -1.00000 q^{21} -8.00000i q^{23} +5.00000i q^{27} +6.00000 q^{29} +4.00000 q^{31} -1.00000i q^{33} +8.00000i q^{37} +6.00000 q^{39} -5.00000 q^{41} +6.00000i q^{47} -1.00000 q^{49} +7.00000 q^{51} -4.00000i q^{53} -1.00000i q^{57} +4.00000 q^{59} +6.00000 q^{61} +2.00000i q^{63} -5.00000i q^{67} +8.00000 q^{69} +14.0000 q^{71} -15.0000i q^{73} -1.00000i q^{77} -14.0000 q^{79} +1.00000 q^{81} -1.00000i q^{83} +6.00000i q^{87} +3.00000 q^{89} +6.00000 q^{91} +4.00000i q^{93} -6.00000i q^{97} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{9}+O(q^{10}) \) 2 * q + 4 * q^9	\( 2 q + 4 q^{9} - 2 q^{11} - 2 q^{19} - 2 q^{21} + 12 q^{29} + 8 q^{31} + 12 q^{39} - 10 q^{41} - 2 q^{49} + 14 q^{51} + 8 q^{59} + 12 q^{61} + 16 q^{69} + 28 q^{71} - 28 q^{79} + 2 q^{81} + 6 q^{89} + 12 q^{91} - 4 q^{99}+O(q^{100}) \) 2 * q + 4 * q^9 - 2 * q^11 - 2 * q^19 - 2 * q^21 + 12 * q^29 + 8 * q^31 + 12 * q^39 - 10 * q^41 - 2 * q^49 + 14 * q^51 + 8 * q^59 + 12 * q^61 + 16 * q^69 + 28 * q^71 - 28 * q^79 + 2 * q^81 + 6 * q^89 + 12 * q^91 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more