LMFDB - Embedded newform 2070.2.d.b.829.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2070,2,Mod(829,2070)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2070, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2070.829");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2070.d (of order $2$, degree $1$, 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:	$16.5290332184$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 690)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			829.1
Root			$-0.618034i$ of defining polynomial
Character	$\chi$	$=$	2070.829
Dual form			2070.2.d.b.829.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{4} -2.23607 q^{5} +4.00000i q^{7} +1.00000i q^{8} +O(q^{10})$	$q-1.00000i q^{2} -1.00000 q^{4} -2.23607 q^{5} +4.00000i q^{7} +1.00000i q^{8} +2.23607i q^{10} +2.47214i q^{13} +4.00000 q^{14} +1.00000 q^{16} -2.47214i q^{17} -2.00000 q^{19} +2.23607 q^{20} -1.00000i q^{23} +5.00000 q^{25} +2.47214 q^{26} -4.00000i q^{28} -0.472136 q^{29} -1.00000i q^{32} -2.47214 q^{34} -8.94427i q^{35} +0.472136i q^{37} +2.00000i q^{38} -2.23607i q^{40} -10.9443 q^{41} -1.00000 q^{46} -4.94427i q^{47} -9.00000 q^{49} -5.00000i q^{50} -2.47214i q^{52} -8.94427i q^{53} -4.00000 q^{56} +0.472136i q^{58} -6.00000 q^{59} -0.472136 q^{61} -1.00000 q^{64} -5.52786i q^{65} -4.94427i q^{67} +2.47214i q^{68} -8.94427 q^{70} +7.52786 q^{71} -4.94427i q^{73} +0.472136 q^{74} +2.00000 q^{76} -12.4721 q^{79} -2.23607 q^{80} +10.9443i q^{82} +1.52786i q^{83} +5.52786i q^{85} -16.4721 q^{89} -9.88854 q^{91} +1.00000i q^{92} -4.94427 q^{94} +4.47214 q^{95} -13.4164i q^{97} +9.00000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4}+O(q^{10}) $ 4 * q - 4 * q^4	$ 4 q - 4 q^{4} + 16 q^{14} + 4 q^{16} - 8 q^{19} + 20 q^{25} - 8 q^{26} + 16 q^{29} + 8 q^{34} - 8 q^{41} - 4 q^{46} - 36 q^{49} - 16 q^{56} - 24 q^{59} + 16 q^{61} - 4 q^{64} + 48 q^{71} - 16 q^{74} + 8 q^{76} - 32 q^{79} - 48 q^{89} + 32 q^{91} + 16 q^{94}+O(q^{100}) $ 4 * q - 4 * q^4 + 16 * q^14 + 4 * q^16 - 8 * q^19 + 20 * q^25 - 8 * q^26 + 16 * q^29 + 8 * q^34 - 8 * q^41 - 4 * q^46 - 36 * q^49 - 16 * q^56 - 24 * q^59 + 16 * q^61 - 4 * q^64 + 48 * q^71 - 16 * q^74 + 8 * q^76 - 32 * q^79 - 48 * q^89 + 32 * q^91 + 16 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2070\mathbb{Z}\right)^\times$.

$n$	$461$	$1657$	$1891$
$\chi(n)$	$1$	$-1$	$1$

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 $

$2$	− 1.00000i	− 0.707107i
$2$	− 1.00000i	− 0.707107i
$3$	0	0
$3$	0	0
$4$	−1.00000	−0.500000
$5$	−2.23607	−1.00000
$5$	−2.23607	−1.00000
$6$	0	0
$7$	4.00000i	1.51186i	0.654654	+	0.755929i	$0.272814\pi$
$7$	4.00000i	1.51186i	−0.654654	+	0.755929i	$0.727186\pi$
$8$	1.00000i	0.353553i
$9$	0	0
$10$	2.23607i	0.707107i
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	2.47214i	0.685647i	0.939400	+	0.342824i	$0.111383\pi$
$13$	2.47214i	0.685647i	−0.939400	+	0.342824i	$0.888617\pi$
$14$	4.00000	1.06904
$15$	0	0
$16$	1.00000	0.250000
$17$	− 2.47214i	− 0.599581i	−0.954005	−	0.299791i	$-0.903083\pi$
$17$	− 2.47214i	− 0.599581i	0.954005	−	0.299791i	$-0.0969168\pi$
$18$	0	0
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	2.23607	0.500000
$21$	0	0
$22$	0	0
$23$	− 1.00000i	− 0.208514i
$23$	− 1.00000i	− 0.208514i
$24$	0	0
$25$	5.00000	1.00000
$26$	2.47214	0.484826
$27$	0	0
$28$	− 4.00000i	− 0.755929i
$29$	−0.472136	−0.0876734	−0.0438367	−	0.999039i	$-0.513958\pi$
$29$	−0.472136	−0.0876734	−0.0438367	+	0.999039i	$0.513958\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	− 1.00000i	− 0.176777i
$33$	0	0
$34$	−2.47214	−0.423968
$35$	− 8.94427i	− 1.51186i
$36$	0	0
$37$	0.472136i	0.0776187i	0.999247	+	0.0388093i	$0.0123565\pi$
$37$	0.472136i	0.0776187i	−0.999247	+	0.0388093i	$0.987644\pi$
$38$	2.00000i	0.324443i
$39$	0	0
$40$	− 2.23607i	− 0.353553i
$41$	−10.9443	−1.70921	−0.854604	−	0.519280i	$-0.826200\pi$
$41$	−10.9443	−1.70921	−0.854604	+	0.519280i	$0.826200\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	0	0
$45$	0	0
$46$	−1.00000	−0.147442
$47$	− 4.94427i	− 0.721196i	−0.932721	−	0.360598i	$-0.882573\pi$
$47$	− 4.94427i	− 0.721196i	0.932721	−	0.360598i	$-0.117427\pi$
$48$	0	0
$49$	−9.00000	−1.28571
$50$	− 5.00000i	− 0.707107i
$51$	0	0
$52$	− 2.47214i	− 0.342824i
$53$	− 8.94427i	− 1.22859i	−0.789076	−	0.614295i	$-0.789440\pi$
$53$	− 8.94427i	− 1.22859i	0.789076	−	0.614295i	$-0.210560\pi$
$54$	0	0
$55$	0	0
$56$	−4.00000	−0.534522
$57$	0	0
$58$	0.472136i	0.0619945i
$59$	−6.00000	−0.781133	−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000	−0.781133	−0.390567	+	0.920575i	$0.627721\pi$
$60$	0	0
$61$	−0.472136	−0.0604508	−0.0302254	−	0.999543i	$-0.509623\pi$
$61$	−0.472136	−0.0604508	−0.0302254	+	0.999543i	$0.509623\pi$
$62$	0	0
$63$	0	0
$64$	−1.00000	−0.125000
$65$	− 5.52786i	− 0.685647i
$66$	0	0
$67$	− 4.94427i	− 0.604039i	−0.953302	−	0.302019i	$-0.902339\pi$
$67$	− 4.94427i	− 0.604039i	0.953302	−	0.302019i	$-0.0976608\pi$
$68$	2.47214i	0.299791i
$69$	0	0
$70$	−8.94427	−1.06904
$71$	7.52786	0.893393	0.446697	−	0.894686i	$-0.352600\pi$
$71$	7.52786	0.893393	0.446697	+	0.894686i	$0.352600\pi$
$72$	0	0
$73$	− 4.94427i	− 0.578683i	−0.957226	−	0.289342i	$-0.906564\pi$
$73$	− 4.94427i	− 0.578683i	0.957226	−	0.289342i	$-0.0934364\pi$
$74$	0.472136	0.0548847
$75$	0	0
$76$	2.00000	0.229416
$77$	0	0
$78$	0	0
$79$	−12.4721	−1.40322	−0.701612	−	0.712559i	$-0.747536\pi$
$79$	−12.4721	−1.40322	−0.701612	+	0.712559i	$0.747536\pi$
$80$	−2.23607	−0.250000
$81$	0	0
$82$	10.9443i	1.20859i
$83$	1.52786i	0.167705i	0.996478	+	0.0838524i	$0.0267224\pi$
$83$	1.52786i	0.167705i	−0.996478	+	0.0838524i	$0.973278\pi$
$84$	0	0
$85$	5.52786i	0.599581i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−16.4721	−1.74604	−0.873021	−	0.487682i	$-0.837843\pi$
$89$	−16.4721	−1.74604	−0.873021	+	0.487682i	$0.837843\pi$
$90$	0	0
$91$	−9.88854	−1.03660
$92$	1.00000i	0.104257i
$93$	0	0
$94$	−4.94427	−0.509963
$95$	4.47214	0.458831
$96$	0	0
$97$	− 13.4164i	− 1.36223i	−0.732177	−	0.681115i	$-0.761495\pi$
$97$	− 13.4164i	− 1.36223i	0.732177	−	0.681115i	$-0.238505\pi$
$98$	9.00000i	0.909137i
$99$	0	0
$100$	−5.00000	−0.500000
$101$	−0.472136	−0.0469793	−0.0234896	−	0.999724i	$-0.507478\pi$
$101$	−0.472136	−0.0469793	−0.0234896	+	0.999724i	$0.507478\pi$
$102$	0	0
$103$	− 16.9443i	− 1.66957i	−0.550577	−	0.834784i	$-0.685592\pi$
$103$	− 16.9443i	− 1.66957i	0.550577	−	0.834784i	$-0.314408\pi$
$104$	−2.47214	−0.242413
$105$	0	0
$106$	−8.94427	−0.868744
$107$	2.47214i	0.238990i	0.992835	+	0.119495i	$0.0381276\pi$
$107$	2.47214i	0.238990i	−0.992835	+	0.119495i	$0.961872\pi$
$108$	0	0
$109$	−0.472136	−0.0452224	−0.0226112	−	0.999744i	$-0.507198\pi$
$109$	−0.472136	−0.0452224	−0.0226112	+	0.999744i	$0.507198\pi$
$110$	0	0
$111$	0	0
$112$	4.00000i	0.377964i
$113$	19.4164i	1.82654i	0.407353	+	0.913271i	$0.366452\pi$
$113$	19.4164i	1.82654i	−0.407353	+	0.913271i	$0.633548\pi$
$114$	0	0
$115$	2.23607i	0.208514i
$116$	0.472136	0.0438367
$117$	0	0
$118$	6.00000i	0.552345i
$119$	9.88854	0.906481
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0.472136i	0.0427452i
$123$	0	0
$124$	0	0
$125$	−11.1803	−1.00000
$126$	0	0
$127$	11.4164i	1.01304i	0.862228	+	0.506521i	$0.169069\pi$
$127$	11.4164i	1.01304i	−0.862228	+	0.506521i	$0.830931\pi$
$128$	1.00000i	0.0883883i
$129$	0	0
$130$	−5.52786	−0.484826
$131$	6.94427	0.606724	0.303362	−	0.952875i	$-0.401891\pi$
$131$	6.94427	0.606724	0.303362	+	0.952875i	$0.401891\pi$
$132$	0	0
$133$	− 8.00000i	− 0.693688i
$134$	−4.94427	−0.427120
$135$	0	0
$136$	2.47214	0.211984
$137$	19.4164i	1.65886i	0.558614	+	0.829428i	$0.311333\pi$
$137$	19.4164i	1.65886i	−0.558614	+	0.829428i	$0.688667\pi$
$138$	0	0
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	8.94427i	0.755929i
$141$	0	0
$142$	− 7.52786i	− 0.631724i
$143$	0	0
$144$	0	0
$145$	1.05573	0.0876734
$146$	−4.94427	−0.409191
$147$	0	0
$148$	− 0.472136i	− 0.0388093i
$149$	−13.4164	−1.09911	−0.549557	−	0.835456i	$-0.685204\pi$
$149$	−13.4164	−1.09911	−0.549557	+	0.835456i	$0.685204\pi$
$150$	0	0
$151$	16.9443	1.37891	0.689453	−	0.724331i	$-0.257851\pi$
$151$	16.9443	1.37891	0.689453	+	0.724331i	$0.257851\pi$
$152$	− 2.00000i	− 0.162221i
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 11.5279i	− 0.920024i	−0.887913	−	0.460012i	$-0.847845\pi$
$157$	− 11.5279i	− 0.920024i	0.887913	−	0.460012i	$-0.152155\pi$
$158$	12.4721i	0.992230i
$159$	0	0
$160$	2.23607i	0.176777i
$161$	4.00000	0.315244
$162$	0	0
$163$	4.94427i	0.387265i	0.981074	+	0.193633i	$0.0620270\pi$
$163$	4.94427i	0.387265i	−0.981074	+	0.193633i	$0.937973\pi$
$164$	10.9443	0.854604
$165$	0	0
$166$	1.52786	0.118585
$167$	− 3.05573i	− 0.236459i	−0.992986	−	0.118230i	$-0.962278\pi$
$167$	− 3.05573i	− 0.236459i	0.992986	−	0.118230i	$-0.0377219\pi$
$168$	0	0
$169$	6.88854	0.529888
$170$	5.52786	0.423968
$171$	0	0
$172$	0	0
$173$	− 14.9443i	− 1.13619i	−0.822962	−	0.568096i	$-0.807680\pi$
$173$	− 14.9443i	− 1.13619i	0.822962	−	0.568096i	$-0.192320\pi$
$174$	0	0
$175$	20.0000i	1.51186i
$176$	0	0
$177$	0	0
$178$	16.4721i	1.23464i
$179$	−18.9443	−1.41596	−0.707981	−	0.706232i	$-0.750394\pi$
$179$	−18.9443	−1.41596	−0.707981	+	0.706232i	$0.750394\pi$
$180$	0	0
$181$	8.47214	0.629729	0.314864	−	0.949137i	$-0.398041\pi$
$181$	8.47214	0.629729	0.314864	+	0.949137i	$0.398041\pi$
$182$	9.88854i	0.732988i
$183$	0	0
$184$	1.00000	0.0737210
$185$	− 1.05573i	− 0.0776187i
$186$	0	0
$187$	0	0
$188$	4.94427i	0.360598i
$189$	0	0
$190$	− 4.47214i	− 0.324443i
$191$	25.8885	1.87323	0.936615	−	0.350361i	$-0.113941\pi$
$191$	25.8885	1.87323	0.936615	+	0.350361i	$0.113941\pi$
$192$	0	0
$193$	− 20.0000i	− 1.43963i	−0.694165	−	0.719816i	$-0.744226\pi$
$193$	− 20.0000i	− 1.43963i	0.694165	−	0.719816i	$-0.255774\pi$
$194$	−13.4164	−0.963242
$195$	0	0
$196$	9.00000	0.642857
$197$	− 18.9443i	− 1.34972i	−0.737944	−	0.674862i	$-0.764203\pi$
$197$	− 18.9443i	− 1.34972i	0.737944	−	0.674862i	$-0.235797\pi$
$198$	0	0
$199$	−17.4164	−1.23462	−0.617308	−	0.786721i	$-0.711777\pi$
$199$	−17.4164	−1.23462	−0.617308	+	0.786721i	$0.711777\pi$
$200$	5.00000i	0.353553i
$201$	0	0
$202$	0.472136i	0.0332194i
$203$	− 1.88854i	− 0.132550i
$204$	0	0
$205$	24.4721	1.70921
$206$	−16.9443	−1.18056
$207$	0	0
$208$	2.47214i	0.171412i
$209$	0	0
$210$	0	0
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	8.94427i	0.614295i
$213$	0	0
$214$	2.47214	0.168992
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0.472136i	0.0319771i
$219$	0	0
$220$	0	0
$221$	6.11146	0.411101
$222$	0	0
$223$	− 11.4164i	− 0.764499i	−0.924059	−	0.382250i	$-0.875149\pi$
$223$	− 11.4164i	− 0.764499i	0.924059	−	0.382250i	$-0.124851\pi$
$224$	4.00000	0.267261
$225$	0	0
$226$	19.4164	1.29156
$227$	− 18.4721i	− 1.22604i	−0.790068	−	0.613019i	$-0.789955\pi$
$227$	− 18.4721i	− 1.22604i	0.790068	−	0.613019i	$-0.210045\pi$
$228$	0	0
$229$	−25.4164	−1.67956	−0.839782	−	0.542924i	$-0.817317\pi$
$229$	−25.4164	−1.67956	−0.839782	+	0.542924i	$0.817317\pi$
$230$	2.23607	0.147442
$231$	0	0
$232$	− 0.472136i	− 0.0309972i
$233$	14.9443i	0.979032i	0.871994	+	0.489516i	$0.162826\pi$
$233$	14.9443i	0.979032i	−0.871994	+	0.489516i	$0.837174\pi$
$234$	0	0
$235$	11.0557i	0.721196i
$236$	6.00000	0.390567
$237$	0	0
$238$	− 9.88854i	− 0.640979i
$239$	−12.4721	−0.806755	−0.403378	−	0.915034i	$-0.632164\pi$
$239$	−12.4721	−0.806755	−0.403378	+	0.915034i	$0.632164\pi$
$240$	0	0
$241$	−2.00000	−0.128831	−0.0644157	−	0.997923i	$-0.520518\pi$
$241$	−2.00000	−0.128831	−0.0644157	+	0.997923i	$0.520518\pi$
$242$	11.0000i	0.707107i
$243$	0	0
$244$	0.472136	0.0302254
$245$	20.1246	1.28571
$246$	0	0
$247$	− 4.94427i	− 0.314596i
$248$	0	0
$249$	0	0
$250$	11.1803i	0.707107i
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	0	0
$253$	0	0
$254$	11.4164	0.716329
$255$	0	0
$256$	1.00000	0.0625000
$257$	18.0000i	1.12281i	0.827541	+	0.561405i	$0.189739\pi$
$257$	18.0000i	1.12281i	−0.827541	+	0.561405i	$0.810261\pi$
$258$	0	0
$259$	−1.88854	−0.117348
$260$	5.52786i	0.342824i
$261$	0	0
$262$	− 6.94427i	− 0.429019i
$263$	20.9443i	1.29148i	0.763558	+	0.645740i	$0.223451\pi$
$263$	20.9443i	1.29148i	−0.763558	+	0.645740i	$0.776549\pi$
$264$	0	0
$265$	20.0000i	1.22859i
$266$	−8.00000	−0.490511
$267$	0	0
$268$	4.94427i	0.302019i
$269$	0.472136	0.0287866	0.0143933	−	0.999896i	$-0.495418\pi$
$269$	0.472136	0.0287866	0.0143933	+	0.999896i	$0.495418\pi$
$270$	0	0
$271$	0.944272	0.0573604	0.0286802	−	0.999589i	$-0.490870\pi$
$271$	0.944272	0.0573604	0.0286802	+	0.999589i	$0.490870\pi$
$272$	− 2.47214i	− 0.149895i
$273$	0	0
$274$	19.4164	1.17299
$275$	0	0
$276$	0	0
$277$	− 27.4164i	− 1.64729i	−0.567104	−	0.823646i	$-0.691936\pi$
$277$	− 27.4164i	− 1.64729i	0.567104	−	0.823646i	$-0.308064\pi$
$278$	0	0
$279$	0	0
$280$	8.94427	0.534522
$281$	−3.52786	−0.210455	−0.105227	−	0.994448i	$-0.533557\pi$
$281$	−3.52786	−0.210455	−0.105227	+	0.994448i	$0.533557\pi$
$282$	0	0
$283$	− 24.0000i	− 1.42665i	−0.700832	−	0.713326i	$-0.747188\pi$
$283$	− 24.0000i	− 1.42665i	0.700832	−	0.713326i	$-0.252812\pi$
$284$	−7.52786	−0.446697
$285$	0	0
$286$	0	0
$287$	− 43.7771i	− 2.58408i
$288$	0	0
$289$	10.8885	0.640503
$290$	− 1.05573i	− 0.0619945i
$291$	0	0
$292$	4.94427i	0.289342i
$293$	− 0.944272i	− 0.0551650i	−0.999620	−	0.0275825i	$-0.991219\pi$
$293$	− 0.944272i	− 0.0551650i	0.999620	−	0.0275825i	$-0.00878089\pi$
$294$	0	0
$295$	13.4164	0.781133
$296$	−0.472136	−0.0274423
$297$	0	0
$298$	13.4164i	0.777192i
$299$	2.47214	0.142967
$300$	0	0
$301$	0	0
$302$	− 16.9443i	− 0.975033i
$303$	0	0
$304$	−2.00000	−0.114708
$305$	1.05573	0.0604508
$306$	0	0
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	24.4721	1.38769	0.693844	−	0.720126i	$-0.255916\pi$
$311$	24.4721	1.38769	0.693844	+	0.720126i	$0.255916\pi$
$312$	0	0
$313$	25.4164i	1.43662i	0.695723	+	0.718310i	$0.255084\pi$
$313$	25.4164i	1.43662i	−0.695723	+	0.718310i	$0.744916\pi$
$314$	−11.5279	−0.650555
$315$	0	0
$316$	12.4721	0.701612
$317$	− 18.0000i	− 1.01098i	−0.862832	−	0.505490i	$-0.831312\pi$
$317$	− 18.0000i	− 1.01098i	0.862832	−	0.505490i	$-0.168688\pi$
$318$	0	0
$319$	0	0
$320$	2.23607	0.125000
$321$	0	0
$322$	− 4.00000i	− 0.222911i
$323$	4.94427i	0.275107i
$324$	0	0
$325$	12.3607i	0.685647i
$326$	4.94427	0.273838
$327$	0	0
$328$	− 10.9443i	− 0.604296i
$329$	19.7771	1.09035
$330$	0	0
$331$	0	0		−	1.00000i	$-0.5\pi$
$331$	0	0			1.00000i	$0.5\pi$
$332$	− 1.52786i	− 0.0838524i
$333$	0	0
$334$	−3.05573	−0.167202
$335$	11.0557i	0.604039i
$336$	0	0
$337$	25.4164i	1.38452i	0.721648	+	0.692260i	$0.243385\pi$
$337$	25.4164i	1.38452i	−0.721648	+	0.692260i	$0.756615\pi$
$338$	− 6.88854i	− 0.374687i
$339$	0	0
$340$	− 5.52786i	− 0.299791i
$341$	0	0
$342$	0	0
$343$	− 8.00000i	− 0.431959i
$344$	0	0
$345$	0	0
$346$	−14.9443	−0.803409
$347$	− 8.94427i	− 0.480154i	−0.970754	−	0.240077i	$-0.922827\pi$
$347$	− 8.94427i	− 0.480154i	0.970754	−	0.240077i	$-0.0771726\pi$
$348$	0	0
$349$	27.8885	1.49284	0.746420	−	0.665475i	$-0.231771\pi$
$349$	27.8885	1.49284	0.746420	+	0.665475i	$0.231771\pi$
$350$	20.0000	1.06904
$351$	0	0
$352$	0	0
$353$	2.94427i	0.156708i	0.996926	+	0.0783539i	$0.0249664\pi$
$353$	2.94427i	0.156708i	−0.996926	+	0.0783539i	$0.975034\pi$
$354$	0	0
$355$	−16.8328	−0.893393
$356$	16.4721	0.873021
$357$	0	0
$358$	18.9443i	1.00124i
$359$	−32.9443	−1.73873	−0.869366	−	0.494169i	$-0.835473\pi$
$359$	−32.9443	−1.73873	−0.869366	+	0.494169i	$0.835473\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	− 8.47214i	− 0.445286i
$363$	0	0
$364$	9.88854	0.518301
$365$	11.0557i	0.578683i
$366$	0	0
$367$	− 12.0000i	− 0.626395i	−0.949688	−	0.313197i	$-0.898600\pi$
$367$	− 12.0000i	− 0.626395i	0.949688	−	0.313197i	$-0.101400\pi$
$368$	− 1.00000i	− 0.0521286i
$369$	0	0
$370$	−1.05573	−0.0548847
$371$	35.7771	1.85745
$372$	0	0
$373$	21.4164i	1.10890i	0.832217	+	0.554450i	$0.187071\pi$
$373$	21.4164i	1.10890i	−0.832217	+	0.554450i	$0.812929\pi$
$374$	0	0
$375$	0	0
$376$	4.94427	0.254981
$377$	− 1.16718i	− 0.0601130i
$378$	0	0
$379$	−16.8328	−0.864644	−0.432322	−	0.901719i	$-0.642306\pi$
$379$	−16.8328	−0.864644	−0.432322	+	0.901719i	$0.642306\pi$
$380$	−4.47214	−0.229416
$381$	0	0
$382$	− 25.8885i	− 1.32457i
$383$	− 13.8885i	− 0.709671i	−0.934929	−	0.354836i	$-0.884537\pi$
$383$	− 13.8885i	− 0.709671i	0.934929	−	0.354836i	$-0.115463\pi$
$384$	0	0
$385$	0	0
$386$	−20.0000	−1.01797
$387$	0	0
$388$	13.4164i	0.681115i
$389$	4.47214	0.226746	0.113373	−	0.993552i	$-0.463834\pi$
$389$	4.47214	0.226746	0.113373	+	0.993552i	$0.463834\pi$
$390$	0	0
$391$	−2.47214	−0.125021
$392$	− 9.00000i	− 0.454569i
$393$	0	0
$394$	−18.9443	−0.954399
$395$	27.8885	1.40322
$396$	0	0
$397$	29.3050i	1.47077i	0.677648	+	0.735387i	$0.262999\pi$
$397$	29.3050i	1.47077i	−0.677648	+	0.735387i	$0.737001\pi$
$398$	17.4164i	0.873006i
$399$	0	0
$400$	5.00000	0.250000
$401$	−25.4164	−1.26923	−0.634617	−	0.772826i	$-0.718842\pi$
$401$	−25.4164	−1.26923	−0.634617	+	0.772826i	$0.718842\pi$
$402$	0	0
$403$	0	0
$404$	0.472136	0.0234896
$405$	0	0
$406$	−1.88854	−0.0937269
$407$	0	0
$408$	0	0
$409$	18.0000	0.890043	0.445021	−	0.895520i	$-0.353196\pi$
$409$	18.0000	0.890043	0.445021	+	0.895520i	$0.353196\pi$
$410$	− 24.4721i	− 1.20859i
$411$	0	0
$412$	16.9443i	0.834784i
$413$	− 24.0000i	− 1.18096i
$414$	0	0
$415$	− 3.41641i	− 0.167705i
$416$	2.47214	0.121206
$417$	0	0
$418$	0	0
$419$	−37.8885	−1.85098	−0.925488	−	0.378776i	$-0.876345\pi$
$419$	−37.8885	−1.85098	−0.925488	+	0.378776i	$0.876345\pi$
$420$	0	0
$421$	−37.4164	−1.82356	−0.911782	−	0.410674i	$-0.865293\pi$
$421$	−37.4164	−1.82356	−0.911782	+	0.410674i	$0.865293\pi$
$422$	12.0000i	0.584151i
$423$	0	0
$424$	8.94427	0.434372
$425$	− 12.3607i	− 0.599581i
$426$	0	0
$427$	− 1.88854i	− 0.0913930i
$428$	− 2.47214i	− 0.119495i
$429$	0	0
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	0	0
$433$	21.4164i	1.02921i	0.857428	+	0.514603i	$0.172061\pi$
$433$	21.4164i	1.02921i	−0.857428	+	0.514603i	$0.827939\pi$
$434$	0	0
$435$	0	0
$436$	0.472136	0.0226112
$437$	2.00000i	0.0956730i
$438$	0	0
$439$	26.8328	1.28066	0.640330	−	0.768100i	$-0.278798\pi$
$439$	26.8328	1.28066	0.640330	+	0.768100i	$0.278798\pi$
$440$	0	0
$441$	0	0
$442$	− 6.11146i	− 0.290692i
$443$	− 8.94427i	− 0.424955i	−0.977166	−	0.212478i	$-0.931847\pi$
$443$	− 8.94427i	− 0.424955i	0.977166	−	0.212478i	$-0.0681533\pi$
$444$	0	0
$445$	36.8328	1.74604
$446$	−11.4164	−0.540583
$447$	0	0
$448$	− 4.00000i	− 0.188982i
$449$	−34.9443	−1.64912	−0.824561	−	0.565773i	$-0.808578\pi$
$449$	−34.9443	−1.64912	−0.824561	+	0.565773i	$0.808578\pi$
$450$	0	0
$451$	0	0
$452$	− 19.4164i	− 0.913271i
$453$	0	0
$454$	−18.4721	−0.866940
$455$	22.1115	1.03660
$456$	0	0
$457$	25.4164i	1.18893i	0.804122	+	0.594465i	$0.202636\pi$
$457$	25.4164i	1.18893i	−0.804122	+	0.594465i	$0.797364\pi$
$458$	25.4164i	1.18763i
$459$	0	0
$460$	− 2.23607i	− 0.104257i
$461$	14.3607	0.668844	0.334422	−	0.942424i	$-0.391459\pi$
$461$	14.3607	0.668844	0.334422	+	0.942424i	$0.391459\pi$
$462$	0	0
$463$	15.4164i	0.716461i	0.933633	+	0.358231i	$0.116620\pi$
$463$	15.4164i	0.716461i	−0.933633	+	0.358231i	$0.883380\pi$
$464$	−0.472136	−0.0219184
$465$	0	0
$466$	14.9443	0.692280
$467$	− 26.4721i	− 1.22498i	−0.790477	−	0.612492i	$-0.790167\pi$
$467$	− 26.4721i	− 1.22498i	0.790477	−	0.612492i	$-0.209833\pi$
$468$	0	0
$469$	19.7771	0.913221
$470$	11.0557	0.509963
$471$	0	0
$472$	− 6.00000i	− 0.276172i
$473$	0	0
$474$	0	0
$475$	−10.0000	−0.458831
$476$	−9.88854	−0.453241
$477$	0	0
$478$	12.4721i	0.570462i
$479$	−16.9443	−0.774204	−0.387102	−	0.922037i	$-0.626524\pi$
$479$	−16.9443	−0.774204	−0.387102	+	0.922037i	$0.626524\pi$
$480$	0	0
$481$	−1.16718	−0.0532190
$482$	2.00000i	0.0910975i
$483$	0	0
$484$	11.0000	0.500000
$485$	30.0000i	1.36223i
$486$	0	0
$487$	− 31.4164i	− 1.42361i	−0.702375	−	0.711807i	$-0.747877\pi$
$487$	− 31.4164i	− 1.42361i	0.702375	−	0.711807i	$-0.252123\pi$
$488$	− 0.472136i	− 0.0213726i
$489$	0	0
$490$	− 20.1246i	− 0.909137i
$491$	−27.8885	−1.25859	−0.629296	−	0.777166i	$-0.716657\pi$
$491$	−27.8885	−1.25859	−0.629296	+	0.777166i	$0.716657\pi$
$492$	0	0
$493$	1.16718i	0.0525673i
$494$	−4.94427	−0.222453
$495$	0	0
$496$	0	0
$497$	30.1115i	1.35068i
$498$	0	0
$499$	24.0000	1.07439	0.537194	−	0.843459i	$-0.319484\pi$
$499$	24.0000	1.07439	0.537194	+	0.843459i	$0.319484\pi$
$500$	11.1803	0.500000
$501$	0	0
$502$	− 12.0000i	− 0.535586i
$503$	13.8885i	0.619260i	0.950857	+	0.309630i	$0.100205\pi$
$503$	13.8885i	0.619260i	−0.950857	+	0.309630i	$0.899795\pi$
$504$	0	0
$505$	1.05573	0.0469793
$506$	0	0
$507$	0	0
$508$	− 11.4164i	− 0.506521i
$509$	1.41641	0.0627812	0.0313906	−	0.999507i	$-0.490006\pi$
$509$	1.41641	0.0627812	0.0313906	+	0.999507i	$0.490006\pi$
$510$	0	0
$511$	19.7771	0.874887
$512$	− 1.00000i	− 0.0441942i
$513$	0	0
$514$	18.0000	0.793946
$515$	37.8885i	1.66957i
$516$	0	0
$517$	0	0
$518$	1.88854i	0.0829779i
$519$	0	0
$520$	5.52786	0.242413
$521$	19.5279	0.855531	0.427766	−	0.903890i	$-0.359301\pi$
$521$	19.5279	0.855531	0.427766	+	0.903890i	$0.359301\pi$
$522$	0	0
$523$	− 4.94427i	− 0.216198i	−0.994140	−	0.108099i	$-0.965524\pi$
$523$	− 4.94427i	− 0.216198i	0.994140	−	0.108099i	$-0.0344763\pi$
$524$	−6.94427	−0.303362
$525$	0	0
$526$	20.9443	0.913214
$527$	0	0
$528$	0	0
$529$	−1.00000	−0.0434783
$530$	20.0000	0.868744
$531$	0	0
$532$	8.00000i	0.346844i
$533$	− 27.0557i	− 1.17191i
$534$	0	0
$535$	− 5.52786i	− 0.238990i
$536$	4.94427	0.213560
$537$	0	0
$538$	− 0.472136i	− 0.0203552i
$539$	0	0
$540$	0	0
$541$	40.8328	1.75554	0.877770	−	0.479082i	$-0.159030\pi$
$541$	40.8328	1.75554	0.877770	+	0.479082i	$0.159030\pi$
$542$	− 0.944272i	− 0.0405600i
$543$	0	0
$544$	−2.47214	−0.105992
$545$	1.05573	0.0452224
$546$	0	0
$547$	− 22.8328i	− 0.976261i	−0.872771	−	0.488130i	$-0.837679\pi$
$547$	− 22.8328i	− 0.976261i	0.872771	−	0.488130i	$-0.162321\pi$
$548$	− 19.4164i	− 0.829428i
$549$	0	0
$550$	0	0
$551$	0.944272	0.0402273
$552$	0	0
$553$	− 49.8885i	− 2.12148i
$554$	−27.4164	−1.16481
$555$	0	0
$556$	0	0
$557$	29.8885i	1.26642i	0.773981	+	0.633209i	$0.218263\pi$
$557$	29.8885i	1.26642i	−0.773981	+	0.633209i	$0.781737\pi$
$558$	0	0
$559$	0	0
$560$	− 8.94427i	− 0.377964i
$561$	0	0
$562$	3.52786i	0.148814i
$563$	− 37.3050i	− 1.57222i	−0.618089	−	0.786108i	$-0.712093\pi$
$563$	− 37.3050i	− 1.57222i	0.618089	−	0.786108i	$-0.287907\pi$
$564$	0	0
$565$	− 43.4164i	− 1.82654i
$566$	−24.0000	−1.00880
$567$	0	0
$568$	7.52786i	0.315862i
$569$	39.3050	1.64775	0.823875	−	0.566772i	$-0.191808\pi$
$569$	39.3050	1.64775	0.823875	+	0.566772i	$0.191808\pi$
$570$	0	0
$571$	32.8328	1.37401	0.687005	−	0.726652i	$-0.258925\pi$
$571$	32.8328	1.37401	0.687005	+	0.726652i	$0.258925\pi$
$572$	0	0
$573$	0	0
$574$	−43.7771	−1.82722
$575$	− 5.00000i	− 0.208514i
$576$	0	0
$577$	16.9443i	0.705399i	0.935737	+	0.352700i	$0.114736\pi$
$577$	16.9443i	0.705399i	−0.935737	+	0.352700i	$0.885264\pi$
$578$	− 10.8885i	− 0.452904i
$579$	0	0
$580$	−1.05573	−0.0438367
$581$	−6.11146	−0.253546
$582$	0	0
$583$	0	0
$584$	4.94427	0.204595
$585$	0	0
$586$	−0.944272	−0.0390075
$587$	29.8885i	1.23363i	0.787107	+	0.616816i	$0.211578\pi$
$587$	29.8885i	1.23363i	−0.787107	+	0.616816i	$0.788422\pi$
$588$	0	0
$589$	0	0
$590$	− 13.4164i	− 0.552345i
$591$	0	0
$592$	0.472136i	0.0194047i
$593$	23.8885i	0.980985i	0.871445	+	0.490492i	$0.163183\pi$
$593$	23.8885i	0.980985i	−0.871445	+	0.490492i	$0.836817\pi$
$594$	0	0
$595$	−22.1115	−0.906481
$596$	13.4164	0.549557
$597$	0	0
$598$	− 2.47214i	− 0.101093i
$599$	26.3607	1.07707	0.538534	−	0.842604i	$-0.318978\pi$
$599$	26.3607	1.07707	0.538534	+	0.842604i	$0.318978\pi$
$600$	0	0
$601$	26.0000	1.06056	0.530281	−	0.847822i	$-0.322086\pi$
$601$	26.0000	1.06056	0.530281	+	0.847822i	$0.322086\pi$
$602$	0	0
$603$	0	0
$604$	−16.9443	−0.689453
$605$	24.5967	1.00000
$606$	0	0
$607$	7.41641i	0.301023i	0.988608	+	0.150511i	$0.0480920\pi$
$607$	7.41641i	0.301023i	−0.988608	+	0.150511i	$0.951908\pi$
$608$	2.00000i	0.0811107i
$609$	0	0
$610$	− 1.05573i	− 0.0427452i
$611$	12.2229	0.494486
$612$	0	0
$613$	− 9.41641i	− 0.380325i	−0.981753	−	0.190163i	$-0.939098\pi$
$613$	− 9.41641i	− 0.380325i	0.981753	−	0.190163i	$-0.0609015\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 14.4721i	− 0.582626i	−0.956628	−	0.291313i	$-0.905908\pi$
$617$	− 14.4721i	− 0.582626i	0.956628	−	0.291313i	$-0.0940922\pi$
$618$	0	0
$619$	−40.8328	−1.64121	−0.820605	−	0.571496i	$-0.806363\pi$
$619$	−40.8328	−1.64121	−0.820605	+	0.571496i	$0.806363\pi$
$620$	0	0
$621$	0	0
$622$	− 24.4721i	− 0.981243i
$623$	− 65.8885i	− 2.63977i
$624$	0	0
$625$	25.0000	1.00000
$626$	25.4164	1.01584
$627$	0	0
$628$	11.5279i	0.460012i
$629$	1.16718	0.0465387
$630$	0	0
$631$	−2.36068	−0.0939772	−0.0469886	−	0.998895i	$-0.514962\pi$
$631$	−2.36068	−0.0939772	−0.0469886	+	0.998895i	$0.514962\pi$
$632$	− 12.4721i	− 0.496115i
$633$	0	0
$634$	−18.0000	−0.714871
$635$	− 25.5279i	− 1.01304i
$636$	0	0
$637$	− 22.2492i	− 0.881546i
$638$	0	0
$639$	0	0
$640$	− 2.23607i	− 0.0883883i
$641$	11.5279	0.455323	0.227662	−	0.973740i	$-0.426892\pi$
$641$	11.5279	0.455323	0.227662	+	0.973740i	$0.426892\pi$
$642$	0	0
$643$	− 12.9443i	− 0.510472i	−0.966879	−	0.255236i	$-0.917847\pi$
$643$	− 12.9443i	− 0.510472i	0.966879	−	0.255236i	$-0.0821532\pi$
$644$	−4.00000	−0.157622
$645$	0	0
$646$	4.94427	0.194530
$647$	41.8885i	1.64681i	0.567455	+	0.823404i	$0.307928\pi$
$647$	41.8885i	1.64681i	−0.567455	+	0.823404i	$0.692072\pi$
$648$	0	0
$649$	0	0
$650$	12.3607	0.484826
$651$	0	0
$652$	− 4.94427i	− 0.193633i
$653$	20.8328i	0.815251i	0.913149	+	0.407626i	$0.133643\pi$
$653$	20.8328i	0.815251i	−0.913149	+	0.407626i	$0.866357\pi$
$654$	0	0
$655$	−15.5279	−0.606724
$656$	−10.9443	−0.427302
$657$	0	0
$658$	− 19.7771i	− 0.770991i
$659$	17.8885	0.696839	0.348419	−	0.937339i	$-0.386719\pi$
$659$	17.8885	0.696839	0.348419	+	0.937339i	$0.386719\pi$
$660$	0	0
$661$	−36.2492	−1.40993	−0.704966	−	0.709241i	$-0.749038\pi$
$661$	−36.2492	−1.40993	−0.704966	+	0.709241i	$0.749038\pi$
$662$	0	0
$663$	0	0
$664$	−1.52786	−0.0592926
$665$	17.8885i	0.693688i
$666$	0	0
$667$	0.472136i	0.0182812i
$668$	3.05573i	0.118230i
$669$	0	0
$670$	11.0557	0.427120
$671$	0	0
$672$	0	0
$673$	38.8328i	1.49690i	0.663194	+	0.748448i	$0.269200\pi$
$673$	38.8328i	1.49690i	−0.663194	+	0.748448i	$0.730800\pi$
$674$	25.4164	0.979003
$675$	0	0
$676$	−6.88854	−0.264944
$677$	25.8885i	0.994978i	0.867470	+	0.497489i	$0.165744\pi$
$677$	25.8885i	0.994978i	−0.867470	+	0.497489i	$0.834256\pi$
$678$	0	0
$679$	53.6656	2.05950
$680$	−5.52786	−0.211984
$681$	0	0
$682$	0	0
$683$	− 12.0000i	− 0.459167i	−0.973289	−	0.229584i	$-0.926264\pi$
$683$	− 12.0000i	− 0.459167i	0.973289	−	0.229584i	$-0.0737364\pi$
$684$	0	0
$685$	− 43.4164i	− 1.65886i
$686$	−8.00000	−0.305441
$687$	0	0
$688$	0	0
$689$	22.1115	0.842379
$690$	0	0
$691$	−39.7771	−1.51319	−0.756596	−	0.653883i	$-0.773139\pi$
$691$	−39.7771	−1.51319	−0.756596	+	0.653883i	$0.773139\pi$
$692$	14.9443i	0.568096i
$693$	0	0
$694$	−8.94427	−0.339520
$695$	0	0
$696$	0	0
$697$	27.0557i	1.02481i
$698$	− 27.8885i	− 1.05560i
$699$	0	0
$700$	− 20.0000i	− 0.755929i
$701$	−37.4164	−1.41320	−0.706599	−	0.707614i	$-0.749772\pi$
$701$	−37.4164	−1.41320	−0.706599	+	0.707614i	$0.749772\pi$
$702$	0	0
$703$	− 0.944272i	− 0.0356139i
$704$	0	0
$705$	0	0
$706$	2.94427	0.110809
$707$	− 1.88854i	− 0.0710260i
$708$	0	0
$709$	−20.4721	−0.768847	−0.384424	−	0.923157i	$-0.625600\pi$
$709$	−20.4721	−0.768847	−0.384424	+	0.923157i	$0.625600\pi$
$710$	16.8328i	0.631724i
$711$	0	0
$712$	− 16.4721i	− 0.617319i
$713$	0	0
$714$	0	0
$715$	0	0
$716$	18.9443	0.707981
$717$	0	0
$718$	32.9443i	1.22947i
$719$	−35.5279	−1.32497	−0.662483	−	0.749077i	$-0.730497\pi$
$719$	−35.5279	−1.32497	−0.662483	+	0.749077i	$0.730497\pi$
$720$	0	0
$721$	67.7771	2.52415
$722$	15.0000i	0.558242i
$723$	0	0
$724$	−8.47214	−0.314864
$725$	−2.36068	−0.0876734
$726$	0	0
$727$	0.944272i	0.0350211i	0.999847	+	0.0175106i	$0.00557407\pi$
$727$	0.944272i	0.0350211i	−0.999847	+	0.0175106i	$0.994426\pi$
$728$	− 9.88854i	− 0.366494i
$729$	0	0
$730$	11.0557	0.409191
$731$	0	0
$732$	0	0
$733$	0.472136i	0.0174387i	0.999962	+	0.00871937i	$0.00277550\pi$
$733$	0.472136i	0.0174387i	−0.999962	+	0.00871937i	$0.997225\pi$
$734$	−12.0000	−0.442928
$735$	0	0
$736$	−1.00000	−0.0368605
$737$	0	0
$738$	0	0
$739$	−1.88854	−0.0694712	−0.0347356	−	0.999397i	$-0.511059\pi$
$739$	−1.88854	−0.0694712	−0.0347356	+	0.999397i	$0.511059\pi$
$740$	1.05573i	0.0388093i
$741$	0	0
$742$	− 35.7771i	− 1.31342i
$743$	2.11146i	0.0774618i	0.999250	+	0.0387309i	$0.0123315\pi$
$743$	2.11146i	0.0774618i	−0.999250	+	0.0387309i	$0.987668\pi$
$744$	0	0
$745$	30.0000	1.09911
$746$	21.4164	0.784110
$747$	0	0
$748$	0	0
$749$	−9.88854	−0.361320
$750$	0	0
$751$	36.4721	1.33089	0.665444	−	0.746448i	$-0.268242\pi$
$751$	36.4721	1.33089	0.665444	+	0.746448i	$0.268242\pi$
$752$	− 4.94427i	− 0.180299i
$753$	0	0
$754$	−1.16718	−0.0425063
$755$	−37.8885	−1.37891
$756$	0	0
$757$	45.4164i	1.65069i	0.564631	+	0.825344i	$0.309019\pi$
$757$	45.4164i	1.65069i	−0.564631	+	0.825344i	$0.690981\pi$
$758$	16.8328i	0.611395i
$759$	0	0
$760$	4.47214i	0.162221i
$761$	17.7771	0.644419	0.322209	−	0.946668i	$-0.395574\pi$
$761$	17.7771	0.644419	0.322209	+	0.946668i	$0.395574\pi$
$762$	0	0
$763$	− 1.88854i	− 0.0683699i
$764$	−25.8885	−0.936615
$765$	0	0
$766$	−13.8885	−0.501813
$767$	− 14.8328i	− 0.535582i
$768$	0	0
$769$	−34.9443	−1.26012	−0.630061	−	0.776545i	$-0.716970\pi$
$769$	−34.9443	−1.26012	−0.630061	+	0.776545i	$0.716970\pi$
$770$	0	0
$771$	0	0
$772$	20.0000i	0.719816i
$773$	− 3.05573i	− 0.109907i	−0.998489	−	0.0549535i	$-0.982499\pi$
$773$	− 3.05573i	− 0.109907i	0.998489	−	0.0549535i	$-0.0175010\pi$
$774$	0	0
$775$	0	0
$776$	13.4164	0.481621
$777$	0	0
$778$	− 4.47214i	− 0.160334i
$779$	21.8885	0.784238
$780$	0	0
$781$	0	0
$782$	2.47214i	0.0884034i
$783$	0	0
$784$	−9.00000	−0.321429
$785$	25.7771i	0.920024i
$786$	0	0
$787$	− 1.16718i	− 0.0416056i	−0.999784	−	0.0208028i	$-0.993378\pi$
$787$	− 1.16718i	− 0.0416056i	0.999784	−	0.0208028i	$-0.00662222\pi$
$788$	18.9443i	0.674862i
$789$	0	0
$790$	− 27.8885i	− 0.992230i
$791$	−77.6656	−2.76147
$792$	0	0
$793$	− 1.16718i	− 0.0414479i
$794$	29.3050	1.03999
$795$	0	0
$796$	17.4164	0.617308
$797$	37.8885i	1.34208i	0.741421	+	0.671041i	$0.234152\pi$
$797$	37.8885i	1.34208i	−0.741421	+	0.671041i	$0.765848\pi$
$798$	0	0
$799$	−12.2229	−0.432416
$800$	− 5.00000i	− 0.176777i
$801$	0	0
$802$	25.4164i	0.897485i
$803$	0	0
$804$	0	0
$805$	−8.94427	−0.315244
$806$	0	0
$807$	0	0
$808$	− 0.472136i	− 0.0166097i
$809$	−3.88854	−0.136714	−0.0683570	−	0.997661i	$-0.521776\pi$
$809$	−3.88854	−0.136714	−0.0683570	+	0.997661i	$0.521776\pi$
$810$	0	0
$811$	−49.8885	−1.75182	−0.875912	−	0.482471i	$-0.839739\pi$
$811$	−49.8885	−1.75182	−0.875912	+	0.482471i	$0.839739\pi$
$812$	1.88854i	0.0662749i
$813$	0	0
$814$	0	0
$815$	− 11.0557i	− 0.387265i
$816$	0	0
$817$	0	0
$818$	− 18.0000i	− 0.629355i
$819$	0	0
$820$	−24.4721	−0.854604
$821$	16.4721	0.574882	0.287441	−	0.957798i	$-0.407196\pi$
$821$	16.4721	0.574882	0.287441	+	0.957798i	$0.407196\pi$
$822$	0	0
$823$	− 20.3607i	− 0.709729i	−0.934918	−	0.354864i	$-0.884527\pi$
$823$	− 20.3607i	− 0.709729i	0.934918	−	0.354864i	$-0.115473\pi$
$824$	16.9443	0.590282
$825$	0	0
$826$	−24.0000	−0.835067
$827$	42.4721i	1.47690i	0.674308	+	0.738450i	$0.264442\pi$
$827$	42.4721i	1.47690i	−0.674308	+	0.738450i	$0.735558\pi$
$828$	0	0
$829$	−30.9443	−1.07474	−0.537369	−	0.843347i	$-0.680582\pi$
$829$	−30.9443	−1.07474	−0.537369	+	0.843347i	$0.680582\pi$
$830$	−3.41641	−0.118585
$831$	0	0
$832$	− 2.47214i	− 0.0857059i
$833$	22.2492i	0.770890i
$834$	0	0
$835$	6.83282i	0.236459i
$836$	0	0
$837$	0	0
$838$	37.8885i	1.30884i
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	−28.7771	−0.992313
$842$	37.4164i	1.28945i
$843$	0	0
$844$	12.0000	0.413057
$845$	−15.4033	−0.529888
$846$	0	0
$847$	− 44.0000i	− 1.51186i
$848$	− 8.94427i	− 0.307148i
$849$	0	0
$850$	−12.3607	−0.423968
$851$	0.472136	0.0161846
$852$	0	0
$853$	− 9.52786i	− 0.326228i	−0.986607	−	0.163114i	$-0.947846\pi$
$853$	− 9.52786i	− 0.326228i	0.986607	−	0.163114i	$-0.0521538\pi$
$854$	−1.88854	−0.0646246
$855$	0	0
$856$	−2.47214	−0.0844959
$857$	23.8885i	0.816017i	0.912978	+	0.408009i	$0.133777\pi$
$857$	23.8885i	0.816017i	−0.912978	+	0.408009i	$0.866223\pi$
$858$	0	0
$859$	39.7771	1.35718	0.678588	−	0.734519i	$-0.262592\pi$
$859$	39.7771	1.35718	0.678588	+	0.734519i	$0.262592\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	3.05573i	0.104018i	0.998647	+	0.0520091i	$0.0165625\pi$
$863$	3.05573i	0.104018i	−0.998647	+	0.0520091i	$0.983438\pi$
$864$	0	0
$865$	33.4164i	1.13619i
$866$	21.4164	0.727759
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	12.2229	0.414158
$872$	− 0.472136i	− 0.0159885i
$873$	0	0
$874$	2.00000	0.0676510
$875$	− 44.7214i	− 1.51186i
$876$	0	0
$877$	20.3607i	0.687531i	0.939055	+	0.343766i	$0.111703\pi$
$877$	20.3607i	0.687531i	−0.939055	+	0.343766i	$0.888297\pi$
$878$	− 26.8328i	− 0.905564i
$879$	0	0
$880$	0	0
$881$	4.47214	0.150670	0.0753350	−	0.997158i	$-0.475997\pi$
$881$	4.47214	0.150670	0.0753350	+	0.997158i	$0.475997\pi$
$882$	0	0
$883$	19.7771i	0.665552i	0.943006	+	0.332776i	$0.107985\pi$
$883$	19.7771i	0.665552i	−0.943006	+	0.332776i	$0.892015\pi$
$884$	−6.11146	−0.205551
$885$	0	0
$886$	−8.94427	−0.300489
$887$	− 24.0000i	− 0.805841i	−0.915235	−	0.402921i	$-0.867995\pi$
$887$	− 24.0000i	− 0.805841i	0.915235	−	0.402921i	$-0.132005\pi$
$888$	0	0
$889$	−45.6656	−1.53158
$890$	− 36.8328i	− 1.23464i
$891$	0	0
$892$	11.4164i	0.382250i
$893$	9.88854i	0.330908i
$894$	0	0
$895$	42.3607	1.41596
$896$	−4.00000	−0.133631
$897$	0	0
$898$	34.9443i	1.16611i
$899$	0	0
$900$	0	0
$901$	−22.1115	−0.736639
$902$	0	0
$903$	0	0
$904$	−19.4164	−0.645780
$905$	−18.9443	−0.629729
$906$	0	0
$907$	14.8328i	0.492516i	0.969204	+	0.246258i	$0.0792010\pi$
$907$	14.8328i	0.492516i	−0.969204	+	0.246258i	$0.920799\pi$
$908$	18.4721i	0.613019i
$909$	0	0
$910$	− 22.1115i	− 0.732988i
$911$	−24.9443	−0.826441	−0.413220	−	0.910631i	$-0.635596\pi$
$911$	−24.9443	−0.826441	−0.413220	+	0.910631i	$0.635596\pi$
$912$	0	0
$913$	0	0
$914$	25.4164	0.840700
$915$	0	0
$916$	25.4164	0.839782
$917$	27.7771i	0.917280i
$918$	0	0
$919$	9.41641	0.310619	0.155309	−	0.987866i	$-0.450363\pi$
$919$	9.41641	0.310619	0.155309	+	0.987866i	$0.450363\pi$
$920$	−2.23607	−0.0737210
$921$	0	0
$922$	− 14.3607i	− 0.472944i
$923$	18.6099i	0.612552i
$924$	0	0
$925$	2.36068i	0.0776187i
$926$	15.4164	0.506615
$927$	0	0
$928$	0.472136i	0.0154986i
$929$	−10.9443	−0.359070	−0.179535	−	0.983752i	$-0.557459\pi$
$929$	−10.9443	−0.359070	−0.179535	+	0.983752i	$0.557459\pi$
$930$	0	0
$931$	18.0000	0.589926
$932$	− 14.9443i	− 0.489516i
$933$	0	0
$934$	−26.4721	−0.866195
$935$	0	0
$936$	0	0
$937$	6.58359i	0.215077i	0.994201	+	0.107538i	$0.0342968\pi$
$937$	6.58359i	0.215077i	−0.994201	+	0.107538i	$0.965703\pi$
$938$	− 19.7771i	− 0.645745i
$939$	0	0
$940$	− 11.0557i	− 0.360598i
$941$	−36.4721	−1.18896	−0.594479	−	0.804111i	$-0.702642\pi$
$941$	−36.4721	−1.18896	−0.594479	+	0.804111i	$0.702642\pi$
$942$	0	0
$943$	10.9443i	0.356395i
$944$	−6.00000	−0.195283
$945$	0	0
$946$	0	0
$947$	12.0000i	0.389948i	0.980808	+	0.194974i	$0.0624622\pi$
$947$	12.0000i	0.389948i	−0.980808	+	0.194974i	$0.937538\pi$
$948$	0	0
$949$	12.2229	0.396773
$950$	10.0000i	0.324443i
$951$	0	0
$952$	9.88854i	0.320490i
$953$	− 12.3607i	− 0.400402i	−0.979755	−	0.200201i	$-0.935841\pi$
$953$	− 12.3607i	− 0.400402i	0.979755	−	0.200201i	$-0.0641595\pi$
$954$	0	0
$955$	−57.8885	−1.87323
$956$	12.4721	0.403378
$957$	0	0
$958$	16.9443i	0.547445i
$959$	−77.6656	−2.50795
$960$	0	0
$961$	−31.0000	−1.00000
$962$	1.16718i	0.0376315i
$963$	0	0
$964$	2.00000	0.0644157
$965$	44.7214i	1.43963i
$966$	0	0
$967$	55.4164i	1.78207i	0.453933	+	0.891036i	$0.350021\pi$
$967$	55.4164i	1.78207i	−0.453933	+	0.891036i	$0.649979\pi$
$968$	− 11.0000i	− 0.353553i
$969$	0	0
$970$	30.0000	0.963242
$971$	18.1115	0.581224	0.290612	−	0.956841i	$-0.406141\pi$
$971$	18.1115	0.581224	0.290612	+	0.956841i	$0.406141\pi$
$972$	0	0
$973$	0	0
$974$	−31.4164	−1.00665
$975$	0	0
$976$	−0.472136	−0.0151127
$977$	− 1.52786i	− 0.0488807i	−0.999701	−	0.0244404i	$-0.992220\pi$
$977$	− 1.52786i	− 0.0488807i	0.999701	−	0.0244404i	$-0.00778038\pi$
$978$	0	0
$979$	0	0
$980$	−20.1246	−0.642857
$981$	0	0
$982$	27.8885i	0.889959i
$983$	− 24.0000i	− 0.765481i	−0.923856	−	0.382741i	$-0.874980\pi$
$983$	− 24.0000i	− 0.765481i	0.923856	−	0.382741i	$-0.125020\pi$
$984$	0	0
$985$	42.3607i	1.34972i
$986$	1.16718	0.0371707
$987$	0	0
$988$	4.94427i	0.157298i
$989$	0	0
$990$	0	0
$991$	53.6656	1.70474	0.852372	−	0.522935i	$-0.175163\pi$
$991$	53.6656	1.70474	0.852372	+	0.522935i	$0.175163\pi$
$992$	0	0
$993$	0	0
$994$	30.1115	0.955077
$995$	38.9443	1.23462
$996$	0	0
$997$	− 7.41641i	− 0.234880i	−0.993080	−	0.117440i	$-0.962531\pi$
$997$	− 7.41641i	− 0.234880i	0.993080	−	0.117440i	$-0.0374688\pi$
$998$	− 24.0000i	− 0.759707i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2070.2.d.b.829.1		4
3.2	odd	2		690.2.d.a.139.4	yes	4
5.4	even	2	inner	2070.2.d.b.829.3		4
15.2	even	4		3450.2.a.bc.1.2		2
15.8	even	4		3450.2.a.bn.1.1		2
15.14	odd	2		690.2.d.a.139.2	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
690.2.d.a.139.2	✓	4	15.14	odd	2
690.2.d.a.139.4	yes	4	3.2	odd	2
2070.2.d.b.829.1		4	1.1	even	1	trivial
2070.2.d.b.829.3		4	5.4	even	2	inner
3450.2.a.bc.1.2		2	15.2	even	4
3450.2.a.bn.1.1		2	15.8	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:	\( N \)	\(=\)	\( 2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2070.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} -2.23607 q^{5} +4.00000i q^{7} +1.00000i q^{8} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000 q^{4} -2.23607 q^{5} +4.00000i q^{7} +1.00000i q^{8} +2.23607i q^{10} +2.47214i q^{13} +4.00000 q^{14} +1.00000 q^{16} -2.47214i q^{17} -2.00000 q^{19} +2.23607 q^{20} -1.00000i q^{23} +5.00000 q^{25} +2.47214 q^{26} -4.00000i q^{28} -0.472136 q^{29} -1.00000i q^{32} -2.47214 q^{34} -8.94427i q^{35} +0.472136i q^{37} +2.00000i q^{38} -2.23607i q^{40} -10.9443 q^{41} -1.00000 q^{46} -4.94427i q^{47} -9.00000 q^{49} -5.00000i q^{50} -2.47214i q^{52} -8.94427i q^{53} -4.00000 q^{56} +0.472136i q^{58} -6.00000 q^{59} -0.472136 q^{61} -1.00000 q^{64} -5.52786i q^{65} -4.94427i q^{67} +2.47214i q^{68} -8.94427 q^{70} +7.52786 q^{71} -4.94427i q^{73} +0.472136 q^{74} +2.00000 q^{76} -12.4721 q^{79} -2.23607 q^{80} +10.9443i q^{82} +1.52786i q^{83} +5.52786i q^{85} -16.4721 q^{89} -9.88854 q^{91} +1.00000i q^{92} -4.94427 q^{94} +4.47214 q^{95} -13.4164i q^{97} +9.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4}+O(q^{10}) \) 4 * q - 4 * q^4	\( 4 q - 4 q^{4} + 16 q^{14} + 4 q^{16} - 8 q^{19} + 20 q^{25} - 8 q^{26} + 16 q^{29} + 8 q^{34} - 8 q^{41} - 4 q^{46} - 36 q^{49} - 16 q^{56} - 24 q^{59} + 16 q^{61} - 4 q^{64} + 48 q^{71} - 16 q^{74} + 8 q^{76} - 32 q^{79} - 48 q^{89} + 32 q^{91} + 16 q^{94}+O(q^{100}) \) 4 * q - 4 * q^4 + 16 * q^14 + 4 * q^16 - 8 * q^19 + 20 * q^25 - 8 * q^26 + 16 * q^29 + 8 * q^34 - 8 * q^41 - 4 * q^46 - 36 * q^49 - 16 * q^56 - 24 * q^59 + 16 * q^61 - 4 * q^64 + 48 * q^71 - 16 * q^74 + 8 * q^76 - 32 * q^79 - 48 * q^89 + 32 * q^91 + 16 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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