LMFDB - Embedded newform 2070.2.d.b.829.2

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			829.2
Root			$1.61803i$ of defining polynomial
Character	$\chi$	$=$	2070.829
Dual form			2070.2.d.b.829.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{4} +2.23607 q^{5} +4.00000i q^{7} +1.00000i q^{8} +O(q^{10})$	$q-1.00000i q^{2} -1.00000 q^{4} +2.23607 q^{5} +4.00000i q^{7} +1.00000i q^{8} -2.23607i q^{10} -6.47214i q^{13} +4.00000 q^{14} +1.00000 q^{16} +6.47214i q^{17} -2.00000 q^{19} -2.23607 q^{20} -1.00000i q^{23} +5.00000 q^{25} -6.47214 q^{26} -4.00000i q^{28} +8.47214 q^{29} -1.00000i q^{32} +6.47214 q^{34} +8.94427i q^{35} -8.47214i q^{37} +2.00000i q^{38} +2.23607i q^{40} +6.94427 q^{41} -1.00000 q^{46} +12.9443i q^{47} -9.00000 q^{49} -5.00000i q^{50} +6.47214i q^{52} +8.94427i q^{53} -4.00000 q^{56} -8.47214i q^{58} -6.00000 q^{59} +8.47214 q^{61} -1.00000 q^{64} -14.4721i q^{65} +12.9443i q^{67} -6.47214i q^{68} +8.94427 q^{70} +16.4721 q^{71} +12.9443i q^{73} -8.47214 q^{74} +2.00000 q^{76} -3.52786 q^{79} +2.23607 q^{80} -6.94427i q^{82} +10.4721i q^{83} +14.4721i q^{85} -7.52786 q^{89} +25.8885 q^{91} +1.00000i q^{92} +12.9443 q^{94} -4.47214 q^{95} +13.4164i q^{97} +9.00000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$13$	− 6.47214i	− 1.79505i	−0.440966	−	0.897524i	$-0.645364\pi$
$13$	− 6.47214i	− 1.79505i	0.440966	−	0.897524i	$-0.354636\pi$
$14$	4.00000	1.06904
$15$	0	0
$16$	1.00000	0.250000
$17$	6.47214i	1.56972i	0.619671	+	0.784862i	$0.287266\pi$
$17$	6.47214i	1.56972i	−0.619671	+	0.784862i	$0.712734\pi$
$18$	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
$26$	−6.47214	−1.26929
$27$	0	0
$28$	− 4.00000i	− 0.755929i
$29$	8.47214	1.57324	0.786618	−	0.617440i	$-0.211830\pi$
$29$	8.47214	1.57324	0.786618	+	0.617440i	$0.211830\pi$
$30$	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
$34$	6.47214	1.10996
$35$	8.94427i	1.51186i
$36$	0	0
$37$	− 8.47214i	− 1.39281i	−0.717649	−	0.696405i	$-0.754782\pi$
$37$	− 8.47214i	− 1.39281i	0.717649	−	0.696405i	$-0.245218\pi$
$38$	2.00000i	0.324443i
$39$	0	0
$40$	2.23607i	0.353553i
$41$	6.94427	1.08451	0.542257	−	0.840213i	$-0.317570\pi$
$41$	6.94427	1.08451	0.542257	+	0.840213i	$0.317570\pi$
$42$	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
$47$	12.9443i	1.88812i	0.329779	+	0.944058i	$0.393026\pi$
$47$	12.9443i	1.88812i	−0.329779	+	0.944058i	$0.606974\pi$
$48$	0	0
$49$	−9.00000	−1.28571
$50$	− 5.00000i	− 0.707107i
$51$	0	0
$52$	6.47214i	0.897524i
$53$	8.94427i	1.22859i	0.789076	+	0.614295i	$0.210560\pi$
$53$	8.94427i	1.22859i	−0.789076	+	0.614295i	$0.789440\pi$
$54$	0	0
$55$	0	0
$56$	−4.00000	−0.534522
$57$	0	0
$58$	− 8.47214i	− 1.11245i
$59$	−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
$61$	8.47214	1.08475	0.542373	−	0.840138i	$-0.317526\pi$
$61$	8.47214	1.08475	0.542373	+	0.840138i	$0.317526\pi$
$62$	0	0
$63$	0	0
$64$	−1.00000	−0.125000
$65$	− 14.4721i	− 1.79505i
$66$	0	0
$67$	12.9443i	1.58139i	0.612207	+	0.790697i	$0.290282\pi$
$67$	12.9443i	1.58139i	−0.612207	+	0.790697i	$0.709718\pi$
$68$	− 6.47214i	− 0.784862i
$69$	0	0
$70$	8.94427	1.06904
$71$	16.4721	1.95488	0.977441	−	0.211207i	$-0.0677393\pi$
$71$	16.4721	1.95488	0.977441	+	0.211207i	$0.0677393\pi$
$72$	0	0
$73$	12.9443i	1.51501i	0.652828	+	0.757506i	$0.273582\pi$
$73$	12.9443i	1.51501i	−0.652828	+	0.757506i	$0.726418\pi$
$74$	−8.47214	−0.984866
$75$	0	0
$76$	2.00000	0.229416
$77$	0	0
$78$	0	0
$79$	−3.52786	−0.396916	−0.198458	−	0.980109i	$-0.563593\pi$
$79$	−3.52786	−0.396916	−0.198458	+	0.980109i	$0.563593\pi$
$80$	2.23607	0.250000
$81$	0	0
$82$	− 6.94427i	− 0.766867i
$83$	10.4721i	1.14947i	0.818341	+	0.574733i	$0.194894\pi$
$83$	10.4721i	1.14947i	−0.818341	+	0.574733i	$0.805106\pi$
$84$	0	0
$85$	14.4721i	1.56972i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−7.52786	−0.797952	−0.398976	−	0.916961i	$-0.630634\pi$
$89$	−7.52786	−0.797952	−0.398976	+	0.916961i	$0.630634\pi$
$90$	0	0
$91$	25.8885	2.71386
$92$	1.00000i	0.104257i
$93$	0	0
$94$	12.9443	1.33510
$95$	−4.47214	−0.458831
$96$	0	0
$97$	13.4164i	1.36223i	0.732177	+	0.681115i	$0.238505\pi$
$97$	13.4164i	1.36223i	−0.732177	+	0.681115i	$0.761495\pi$
$98$	9.00000i	0.909137i
$99$	0	0
$100$	−5.00000	−0.500000
$101$	8.47214	0.843009	0.421505	−	0.906826i	$-0.361502\pi$
$101$	8.47214	0.843009	0.421505	+	0.906826i	$0.361502\pi$
$102$	0	0
$103$	0.944272i	0.0930419i	0.998917	+	0.0465209i	$0.0148134\pi$
$103$	0.944272i	0.0930419i	−0.998917	+	0.0465209i	$0.985187\pi$
$104$	6.47214	0.634645
$105$	0	0
$106$	8.94427	0.868744
$107$	− 6.47214i	− 0.625685i	−0.949805	−	0.312842i	$-0.898719\pi$
$107$	− 6.47214i	− 0.625685i	0.949805	−	0.312842i	$-0.101281\pi$
$108$	0	0
$109$	8.47214	0.811483	0.405742	−	0.913988i	$-0.367013\pi$
$109$	8.47214	0.811483	0.405742	+	0.913988i	$0.367013\pi$
$110$	0	0
$111$	0	0
$112$	4.00000i	0.377964i
$113$	− 7.41641i	− 0.697677i	−0.937183	−	0.348838i	$-0.886576\pi$
$113$	− 7.41641i	− 0.697677i	0.937183	−	0.348838i	$-0.113424\pi$
$114$	0	0
$115$	− 2.23607i	− 0.208514i
$116$	−8.47214	−0.786618
$117$	0	0
$118$	6.00000i	0.552345i
$119$	−25.8885	−2.37320
$120$	0	0
$121$	−11.0000	−1.00000
$122$	− 8.47214i	− 0.767031i
$123$	0	0
$124$	0	0
$125$	11.1803	1.00000
$126$	0	0
$127$	− 15.4164i	− 1.36798i	−0.729489	−	0.683992i	$-0.760242\pi$
$127$	− 15.4164i	− 1.36798i	0.729489	−	0.683992i	$-0.239758\pi$
$128$	1.00000i	0.0883883i
$129$	0	0
$130$	−14.4721	−1.26929
$131$	−10.9443	−0.956205	−0.478103	−	0.878304i	$-0.658675\pi$
$131$	−10.9443	−0.956205	−0.478103	+	0.878304i	$0.658675\pi$
$132$	0	0
$133$	− 8.00000i	− 0.693688i
$134$	12.9443	1.11821
$135$	0	0
$136$	−6.47214	−0.554981
$137$	− 7.41641i	− 0.633626i	−0.948488	−	0.316813i	$-0.897387\pi$
$137$	− 7.41641i	− 0.633626i	0.948488	−	0.316813i	$-0.102613\pi$
$138$	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
$142$	− 16.4721i	− 1.38231i
$143$	0	0
$144$	0	0
$145$	18.9443	1.57324
$146$	12.9443	1.07128
$147$	0	0
$148$	8.47214i	0.696405i
$149$	13.4164	1.09911	0.549557	−	0.835456i	$-0.314796\pi$
$149$	13.4164	1.09911	0.549557	+	0.835456i	$0.314796\pi$
$150$	0	0
$151$	−0.944272	−0.0768438	−0.0384219	−	0.999262i	$-0.512233\pi$
$151$	−0.944272	−0.0768438	−0.0384219	+	0.999262i	$0.512233\pi$
$152$	− 2.00000i	− 0.162221i
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 20.4721i	− 1.63385i	−0.576741	−	0.816927i	$-0.695676\pi$
$157$	− 20.4721i	− 1.63385i	0.576741	−	0.816927i	$-0.304324\pi$
$158$	3.52786i	0.280662i
$159$	0	0
$160$	− 2.23607i	− 0.176777i
$161$	4.00000	0.315244
$162$	0	0
$163$	− 12.9443i	− 1.01387i	−0.861983	−	0.506937i	$-0.830778\pi$
$163$	− 12.9443i	− 1.01387i	0.861983	−	0.506937i	$-0.169222\pi$
$164$	−6.94427	−0.542257
$165$	0	0
$166$	10.4721	0.812795
$167$	− 20.9443i	− 1.62072i	−0.585935	−	0.810358i	$-0.699273\pi$
$167$	− 20.9443i	− 1.62072i	0.585935	−	0.810358i	$-0.300727\pi$
$168$	0	0
$169$	−28.8885	−2.22220
$170$	14.4721	1.10996
$171$	0	0
$172$	0	0
$173$	2.94427i	0.223849i	0.993717	+	0.111924i	$0.0357015\pi$
$173$	2.94427i	0.223849i	−0.993717	+	0.111924i	$0.964299\pi$
$174$	0	0
$175$	20.0000i	1.51186i
$176$	0	0
$177$	0	0
$178$	7.52786i	0.564237i
$179$	−1.05573	−0.0789088	−0.0394544	−	0.999221i	$-0.512562\pi$
$179$	−1.05573	−0.0789088	−0.0394544	+	0.999221i	$0.512562\pi$
$180$	0	0
$181$	−0.472136	−0.0350936	−0.0175468	−	0.999846i	$-0.505586\pi$
$181$	−0.472136	−0.0350936	−0.0175468	+	0.999846i	$0.505586\pi$
$182$	− 25.8885i	− 1.91899i
$183$	0	0
$184$	1.00000	0.0737210
$185$	− 18.9443i	− 1.39281i
$186$	0	0
$187$	0	0
$188$	− 12.9443i	− 0.944058i
$189$	0	0
$190$	4.47214i	0.324443i
$191$	−9.88854	−0.715510	−0.357755	−	0.933816i	$-0.616458\pi$
$191$	−9.88854	−0.715510	−0.357755	+	0.933816i	$0.616458\pi$
$192$	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
$197$	− 1.05573i	− 0.0752175i	−0.999293	−	0.0376088i	$-0.988026\pi$
$197$	− 1.05573i	− 0.0752175i	0.999293	−	0.0376088i	$-0.0119741\pi$
$198$	0	0
$199$	9.41641	0.667511	0.333756	−	0.942660i	$-0.391684\pi$
$199$	9.41641	0.667511	0.333756	+	0.942660i	$0.391684\pi$
$200$	5.00000i	0.353553i
$201$	0	0
$202$	− 8.47214i	− 0.596097i
$203$	33.8885i	2.37851i
$204$	0	0
$205$	15.5279	1.08451
$206$	0.944272	0.0657905
$207$	0	0
$208$	− 6.47214i	− 0.448762i
$209$	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
$214$	−6.47214	−0.442426
$215$	0	0
$216$	0	0
$217$	0	0
$218$	− 8.47214i	− 0.573805i
$219$	0	0
$220$	0	0
$221$	41.8885	2.81773
$222$	0	0
$223$	15.4164i	1.03236i	0.856480	+	0.516180i	$0.172646\pi$
$223$	15.4164i	1.03236i	−0.856480	+	0.516180i	$0.827354\pi$
$224$	4.00000	0.267261
$225$	0	0
$226$	−7.41641	−0.493332
$227$	− 9.52786i	− 0.632387i	−0.948695	−	0.316193i	$-0.897595\pi$
$227$	− 9.52786i	− 0.632387i	0.948695	−	0.316193i	$-0.102405\pi$
$228$	0	0
$229$	1.41641	0.0935989	0.0467994	−	0.998904i	$-0.485098\pi$
$229$	1.41641	0.0935989	0.0467994	+	0.998904i	$0.485098\pi$
$230$	−2.23607	−0.147442
$231$	0	0
$232$	8.47214i	0.556223i
$233$	− 2.94427i	− 0.192886i	−0.995339	−	0.0964428i	$-0.969254\pi$
$233$	− 2.94427i	− 0.192886i	0.995339	−	0.0964428i	$-0.0307465\pi$
$234$	0	0
$235$	28.9443i	1.88812i
$236$	6.00000	0.390567
$237$	0	0
$238$	25.8885i	1.67811i
$239$	−3.52786	−0.228199	−0.114099	−	0.993469i	$-0.536398\pi$
$239$	−3.52786	−0.228199	−0.114099	+	0.993469i	$0.536398\pi$
$240$	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
$244$	−8.47214	−0.542373
$245$	−20.1246	−1.28571
$246$	0	0
$247$	12.9443i	0.823624i
$248$	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
$254$	−15.4164	−0.967311
$255$	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
$259$	33.8885	2.10573
$260$	14.4721i	0.897524i
$261$	0	0
$262$	10.9443i	0.676139i
$263$	3.05573i	0.188424i	0.995552	+	0.0942121i	$0.0300332\pi$
$263$	3.05573i	0.188424i	−0.995552	+	0.0942121i	$0.969967\pi$
$264$	0	0
$265$	20.0000i	1.22859i
$266$	−8.00000	−0.490511
$267$	0	0
$268$	− 12.9443i	− 0.790697i
$269$	−8.47214	−0.516555	−0.258278	−	0.966071i	$-0.583155\pi$
$269$	−8.47214	−0.516555	−0.258278	+	0.966071i	$0.583155\pi$
$270$	0	0
$271$	−16.9443	−1.02929	−0.514646	−	0.857403i	$-0.672076\pi$
$271$	−16.9443	−1.02929	−0.514646	+	0.857403i	$0.672076\pi$
$272$	6.47214i	0.392431i
$273$	0	0
$274$	−7.41641	−0.448042
$275$	0	0
$276$	0	0
$277$	− 0.583592i	− 0.0350647i	−0.999846	−	0.0175323i	$-0.994419\pi$
$277$	− 0.583592i	− 0.0350647i	0.999846	−	0.0175323i	$-0.00558100\pi$
$278$	0	0
$279$	0	0
$280$	−8.94427	−0.534522
$281$	−12.4721	−0.744025	−0.372013	−	0.928228i	$-0.621332\pi$
$281$	−12.4721	−0.744025	−0.372013	+	0.928228i	$0.621332\pi$
$282$	0	0
$283$	− 24.0000i	− 1.42665i	−0.700832	−	0.713326i	$-0.747188\pi$
$283$	− 24.0000i	− 1.42665i	0.700832	−	0.713326i	$-0.252812\pi$
$284$	−16.4721	−0.977441
$285$	0	0
$286$	0	0
$287$	27.7771i	1.63963i
$288$	0	0
$289$	−24.8885	−1.46403
$290$	− 18.9443i	− 1.11245i
$291$	0	0
$292$	− 12.9443i	− 0.757506i
$293$	16.9443i	0.989895i	0.868923	+	0.494947i	$0.164813\pi$
$293$	16.9443i	0.989895i	−0.868923	+	0.494947i	$0.835187\pi$
$294$	0	0
$295$	−13.4164	−0.781133
$296$	8.47214	0.492433
$297$	0	0
$298$	− 13.4164i	− 0.777192i
$299$	−6.47214	−0.374293
$300$	0	0
$301$	0	0
$302$	0.944272i	0.0543367i
$303$	0	0
$304$	−2.00000	−0.114708
$305$	18.9443	1.08475
$306$	0	0
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	15.5279	0.880504	0.440252	−	0.897874i	$-0.354889\pi$
$311$	15.5279	0.880504	0.440252	+	0.897874i	$0.354889\pi$
$312$	0	0
$313$	− 1.41641i	− 0.0800601i	−0.999198	−	0.0400301i	$-0.987255\pi$
$313$	− 1.41641i	− 0.0800601i	0.999198	−	0.0400301i	$-0.0127454\pi$
$314$	−20.4721	−1.15531
$315$	0	0
$316$	3.52786	0.198458
$317$	− 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
$323$	− 12.9443i	− 0.720239i
$324$	0	0
$325$	− 32.3607i	− 1.79505i
$326$	−12.9443	−0.716917
$327$	0	0
$328$	6.94427i	0.383433i
$329$	−51.7771	−2.85456
$330$	0	0
$331$	0	0		−	1.00000i	$-0.5\pi$
$331$	0	0			1.00000i	$0.5\pi$
$332$	− 10.4721i	− 0.574733i
$333$	0	0
$334$	−20.9443	−1.14602
$335$	28.9443i	1.58139i
$336$	0	0
$337$	− 1.41641i	− 0.0771567i	−0.999256	−	0.0385783i	$-0.987717\pi$
$337$	− 1.41641i	− 0.0771567i	0.999256	−	0.0385783i	$-0.0122829\pi$
$338$	28.8885i	1.57133i
$339$	0	0
$340$	− 14.4721i	− 0.784862i
$341$	0	0
$342$	0	0
$343$	− 8.00000i	− 0.431959i
$344$	0	0
$345$	0	0
$346$	2.94427	0.158285
$347$	8.94427i	0.480154i	0.970754	+	0.240077i	$0.0771726\pi$
$347$	8.94427i	0.480154i	−0.970754	+	0.240077i	$0.922827\pi$
$348$	0	0
$349$	−7.88854	−0.422264	−0.211132	−	0.977458i	$-0.567715\pi$
$349$	−7.88854	−0.422264	−0.211132	+	0.977458i	$0.567715\pi$
$350$	20.0000	1.06904
$351$	0	0
$352$	0	0
$353$	− 14.9443i	− 0.795403i	−0.917515	−	0.397702i	$-0.869808\pi$
$353$	− 14.9443i	− 0.795403i	0.917515	−	0.397702i	$-0.130192\pi$
$354$	0	0
$355$	36.8328	1.95488
$356$	7.52786	0.398976
$357$	0	0
$358$	1.05573i	0.0557970i
$359$	−15.0557	−0.794611	−0.397305	−	0.917686i	$-0.630055\pi$
$359$	−15.0557	−0.794611	−0.397305	+	0.917686i	$0.630055\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	0.472136i	0.0248149i
$363$	0	0
$364$	−25.8885	−1.35693
$365$	28.9443i	1.51501i
$366$	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
$370$	−18.9443	−0.984866
$371$	−35.7771	−1.85745
$372$	0	0
$373$	− 5.41641i	− 0.280451i	−0.990120	−	0.140225i	$-0.955217\pi$
$373$	− 5.41641i	− 0.280451i	0.990120	−	0.140225i	$-0.0447827\pi$
$374$	0	0
$375$	0	0
$376$	−12.9443	−0.667550
$377$	− 54.8328i	− 2.82403i
$378$	0	0
$379$	36.8328	1.89197	0.945987	−	0.324204i	$-0.105096\pi$
$379$	36.8328	1.89197	0.945987	+	0.324204i	$0.105096\pi$
$380$	4.47214	0.229416
$381$	0	0
$382$	9.88854i	0.505942i
$383$	21.8885i	1.11845i	0.829015	+	0.559226i	$0.188902\pi$
$383$	21.8885i	1.11845i	−0.829015	+	0.559226i	$0.811098\pi$
$384$	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	$-0.536166\pi$
$389$	−4.47214	−0.226746	−0.113373	+	0.993552i	$0.536166\pi$
$390$	0	0
$391$	6.47214	0.327310
$392$	− 9.00000i	− 0.454569i
$393$	0	0
$394$	−1.05573	−0.0531868
$395$	−7.88854	−0.396916
$396$	0	0
$397$	− 33.3050i	− 1.67153i	−0.549089	−	0.835764i	$-0.685025\pi$
$397$	− 33.3050i	− 1.67153i	0.549089	−	0.835764i	$-0.314975\pi$
$398$	− 9.41641i	− 0.472002i
$399$	0	0
$400$	5.00000	0.250000
$401$	1.41641	0.0707320	0.0353660	−	0.999374i	$-0.488740\pi$
$401$	1.41641	0.0707320	0.0353660	+	0.999374i	$0.488740\pi$
$402$	0	0
$403$	0	0
$404$	−8.47214	−0.421505
$405$	0	0
$406$	33.8885	1.68186
$407$	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	$0.353196\pi$
$410$	− 15.5279i	− 0.766867i
$411$	0	0
$412$	− 0.944272i	− 0.0465209i
$413$	− 24.0000i	− 1.18096i
$414$	0	0
$415$	23.4164i	1.14947i
$416$	−6.47214	−0.317323
$417$	0	0
$418$	0	0
$419$	−2.11146	−0.103151	−0.0515757	−	0.998669i	$-0.516424\pi$
$419$	−2.11146	−0.103151	−0.0515757	+	0.998669i	$0.516424\pi$
$420$	0	0
$421$	−10.5836	−0.515813	−0.257906	−	0.966170i	$-0.583033\pi$
$421$	−10.5836	−0.515813	−0.257906	+	0.966170i	$0.583033\pi$
$422$	12.0000i	0.584151i
$423$	0	0
$424$	−8.94427	−0.434372
$425$	32.3607i	1.56972i
$426$	0	0
$427$	33.8885i	1.63998i
$428$	6.47214i	0.312842i
$429$	0	0
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	0	0
$433$	− 5.41641i	− 0.260296i	−0.991495	−	0.130148i	$-0.958455\pi$
$433$	− 5.41641i	− 0.260296i	0.991495	−	0.130148i	$-0.0415452\pi$
$434$	0	0
$435$	0	0
$436$	−8.47214	−0.405742
$437$	2.00000i	0.0956730i
$438$	0	0
$439$	−26.8328	−1.28066	−0.640330	−	0.768100i	$-0.721202\pi$
$439$	−26.8328	−1.28066	−0.640330	+	0.768100i	$0.721202\pi$
$440$	0	0
$441$	0	0
$442$	− 41.8885i	− 1.99243i
$443$	8.94427i	0.424955i	0.977166	+	0.212478i	$0.0681533\pi$
$443$	8.94427i	0.424955i	−0.977166	+	0.212478i	$0.931847\pi$
$444$	0	0
$445$	−16.8328	−0.797952
$446$	15.4164	0.729988
$447$	0	0
$448$	− 4.00000i	− 0.188982i
$449$	−17.0557	−0.804910	−0.402455	−	0.915440i	$-0.631843\pi$
$449$	−17.0557	−0.804910	−0.402455	+	0.915440i	$0.631843\pi$
$450$	0	0
$451$	0	0
$452$	7.41641i	0.348838i
$453$	0	0
$454$	−9.52786	−0.447165
$455$	57.8885	2.71386
$456$	0	0
$457$	− 1.41641i	− 0.0662568i	−0.999451	−	0.0331284i	$-0.989453\pi$
$457$	− 1.41641i	− 0.0662568i	0.999451	−	0.0331284i	$-0.0105470\pi$
$458$	− 1.41641i	− 0.0661844i
$459$	0	0
$460$	2.23607i	0.104257i
$461$	−30.3607	−1.41404	−0.707019	−	0.707195i	$-0.749960\pi$
$461$	−30.3607	−1.41404	−0.707019	+	0.707195i	$0.749960\pi$
$462$	0	0
$463$	− 11.4164i	− 0.530565i	−0.964171	−	0.265283i	$-0.914535\pi$
$463$	− 11.4164i	− 0.530565i	0.964171	−	0.265283i	$-0.0854653\pi$
$464$	8.47214	0.393309
$465$	0	0
$466$	−2.94427	−0.136391
$467$	− 17.5279i	− 0.811093i	−0.914074	−	0.405546i	$-0.867081\pi$
$467$	− 17.5279i	− 0.811093i	0.914074	−	0.405546i	$-0.132919\pi$
$468$	0	0
$469$	−51.7771	−2.39084
$470$	28.9443	1.33510
$471$	0	0
$472$	− 6.00000i	− 0.276172i
$473$	0	0
$474$	0	0
$475$	−10.0000	−0.458831
$476$	25.8885	1.18660
$477$	0	0
$478$	3.52786i	0.161361i
$479$	0.944272	0.0431449	0.0215724	−	0.999767i	$-0.493133\pi$
$479$	0.944272	0.0431449	0.0215724	+	0.999767i	$0.493133\pi$
$480$	0	0
$481$	−54.8328	−2.50016
$482$	2.00000i	0.0910975i
$483$	0	0
$484$	11.0000	0.500000
$485$	30.0000i	1.36223i
$486$	0	0
$487$	− 4.58359i	− 0.207702i	−0.994593	−	0.103851i	$-0.966883\pi$
$487$	− 4.58359i	− 0.207702i	0.994593	−	0.103851i	$-0.0331166\pi$
$488$	8.47214i	0.383516i
$489$	0	0
$490$	20.1246i	0.909137i
$491$	7.88854	0.356005	0.178002	−	0.984030i	$-0.443036\pi$
$491$	7.88854	0.356005	0.178002	+	0.984030i	$0.443036\pi$
$492$	0	0
$493$	54.8328i	2.46955i
$494$	12.9443	0.582390
$495$	0	0
$496$	0	0
$497$	65.8885i	2.95551i
$498$	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
$503$	− 21.8885i	− 0.975962i	−0.872854	−	0.487981i	$-0.837734\pi$
$503$	− 21.8885i	− 0.975962i	0.872854	−	0.487981i	$-0.162266\pi$
$504$	0	0
$505$	18.9443	0.843009
$506$	0	0
$507$	0	0
$508$	15.4164i	0.683992i
$509$	−25.4164	−1.12656	−0.563281	−	0.826265i	$-0.690461\pi$
$509$	−25.4164	−1.12656	−0.563281	+	0.826265i	$0.690461\pi$
$510$	0	0
$511$	−51.7771	−2.29048
$512$	− 1.00000i	− 0.0441942i
$513$	0	0
$514$	18.0000	0.793946
$515$	2.11146i	0.0930419i
$516$	0	0
$517$	0	0
$518$	− 33.8885i	− 1.48898i
$519$	0	0
$520$	14.4721	0.634645
$521$	28.4721	1.24739	0.623693	−	0.781669i	$-0.285631\pi$
$521$	28.4721	1.24739	0.623693	+	0.781669i	$0.285631\pi$
$522$	0	0
$523$	12.9443i	0.566013i	0.959118	+	0.283007i	$0.0913319\pi$
$523$	12.9443i	0.566013i	−0.959118	+	0.283007i	$0.908668\pi$
$524$	10.9443	0.478103
$525$	0	0
$526$	3.05573	0.133236
$527$	0	0
$528$	0	0
$529$	−1.00000	−0.0434783
$530$	20.0000	0.868744
$531$	0	0
$532$	8.00000i	0.346844i
$533$	− 44.9443i	− 1.94675i
$534$	0	0
$535$	− 14.4721i	− 0.625685i
$536$	−12.9443	−0.559107
$537$	0	0
$538$	8.47214i	0.365260i
$539$	0	0
$540$	0	0
$541$	−12.8328	−0.551726	−0.275863	−	0.961197i	$-0.588964\pi$
$541$	−12.8328	−0.551726	−0.275863	+	0.961197i	$0.588964\pi$
$542$	16.9443i	0.727819i
$543$	0	0
$544$	6.47214	0.277491
$545$	18.9443	0.811483
$546$	0	0
$547$	30.8328i	1.31832i	0.752004	+	0.659158i	$0.229087\pi$
$547$	30.8328i	1.31832i	−0.752004	+	0.659158i	$0.770913\pi$
$548$	7.41641i	0.316813i
$549$	0	0
$550$	0	0
$551$	−16.9443	−0.721850
$552$	0	0
$553$	− 14.1115i	− 0.600080i
$554$	−0.583592	−0.0247945
$555$	0	0
$556$	0	0
$557$	− 5.88854i	− 0.249506i	−0.992188	−	0.124753i	$-0.960186\pi$
$557$	− 5.88854i	− 0.249506i	0.992188	−	0.124753i	$-0.0398138\pi$
$558$	0	0
$559$	0	0
$560$	8.94427i	0.377964i
$561$	0	0
$562$	12.4721i	0.526105i
$563$	25.3050i	1.06648i	0.845965	+	0.533238i	$0.179025\pi$
$563$	25.3050i	1.06648i	−0.845965	+	0.533238i	$0.820975\pi$
$564$	0	0
$565$	− 16.5836i	− 0.697677i
$566$	−24.0000	−1.00880
$567$	0	0
$568$	16.4721i	0.691155i
$569$	−23.3050	−0.976994	−0.488497	−	0.872565i	$-0.662455\pi$
$569$	−23.3050	−0.976994	−0.488497	+	0.872565i	$0.662455\pi$
$570$	0	0
$571$	−20.8328	−0.871826	−0.435913	−	0.899989i	$-0.643575\pi$
$571$	−20.8328	−0.871826	−0.435913	+	0.899989i	$0.643575\pi$
$572$	0	0
$573$	0	0
$574$	27.7771	1.15939
$575$	− 5.00000i	− 0.208514i
$576$	0	0
$577$	− 0.944272i	− 0.0393106i	−0.999807	−	0.0196553i	$-0.993743\pi$
$577$	− 0.944272i	− 0.0393106i	0.999807	−	0.0196553i	$-0.00625687\pi$
$578$	24.8885i	1.03523i
$579$	0	0
$580$	−18.9443	−0.786618
$581$	−41.8885	−1.73783
$582$	0	0
$583$	0	0
$584$	−12.9443	−0.535638
$585$	0	0
$586$	16.9443	0.699961
$587$	− 5.88854i	− 0.243046i	−0.992589	−	0.121523i	$-0.961222\pi$
$587$	− 5.88854i	− 0.243046i	0.992589	−	0.121523i	$-0.0387779\pi$
$588$	0	0
$589$	0	0
$590$	13.4164i	0.552345i
$591$	0	0
$592$	− 8.47214i	− 0.348203i
$593$	− 11.8885i	− 0.488204i	−0.969750	−	0.244102i	$-0.921507\pi$
$593$	− 11.8885i	− 0.488204i	0.969750	−	0.244102i	$-0.0784932\pi$
$594$	0	0
$595$	−57.8885	−2.37320
$596$	−13.4164	−0.549557
$597$	0	0
$598$	6.47214i	0.264665i
$599$	−18.3607	−0.750197	−0.375099	−	0.926985i	$-0.622391\pi$
$599$	−18.3607	−0.750197	−0.375099	+	0.926985i	$0.622391\pi$
$600$	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
$604$	0.944272	0.0384219
$605$	−24.5967	−1.00000
$606$	0	0
$607$	− 19.4164i	− 0.788088i	−0.919092	−	0.394044i	$-0.871076\pi$
$607$	− 19.4164i	− 0.788088i	0.919092	−	0.394044i	$-0.128924\pi$
$608$	2.00000i	0.0811107i
$609$	0	0
$610$	− 18.9443i	− 0.767031i
$611$	83.7771	3.38926
$612$	0	0
$613$	17.4164i	0.703442i	0.936105	+	0.351721i	$0.114403\pi$
$613$	17.4164i	0.703442i	−0.936105	+	0.351721i	$0.885597\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 5.52786i	− 0.222543i	−0.993790	−	0.111272i	$-0.964508\pi$
$617$	− 5.52786i	− 0.222543i	0.993790	−	0.111272i	$-0.0354924\pi$
$618$	0	0
$619$	12.8328	0.515794	0.257897	−	0.966172i	$-0.416970\pi$
$619$	12.8328	0.515794	0.257897	+	0.966172i	$0.416970\pi$
$620$	0	0
$621$	0	0
$622$	− 15.5279i	− 0.622611i
$623$	− 30.1115i	− 1.20639i
$624$	0	0
$625$	25.0000	1.00000
$626$	−1.41641	−0.0566110
$627$	0	0
$628$	20.4721i	0.816927i
$629$	54.8328	2.18633
$630$	0	0
$631$	42.3607	1.68635	0.843176	−	0.537638i	$-0.180683\pi$
$631$	42.3607	1.68635	0.843176	+	0.537638i	$0.180683\pi$
$632$	− 3.52786i	− 0.140331i
$633$	0	0
$634$	−18.0000	−0.714871
$635$	− 34.4721i	− 1.36798i
$636$	0	0
$637$	58.2492i	2.30792i
$638$	0	0
$639$	0	0
$640$	2.23607i	0.0883883i
$641$	20.4721	0.808601	0.404300	−	0.914626i	$-0.367515\pi$
$641$	20.4721	0.808601	0.404300	+	0.914626i	$0.367515\pi$
$642$	0	0
$643$	4.94427i	0.194983i	0.995236	+	0.0974915i	$0.0310819\pi$
$643$	4.94427i	0.194983i	−0.995236	+	0.0974915i	$0.968918\pi$
$644$	−4.00000	−0.157622
$645$	0	0
$646$	−12.9443	−0.509286
$647$	6.11146i	0.240266i	0.992758	+	0.120133i	$0.0383321\pi$
$647$	6.11146i	0.240266i	−0.992758	+	0.120133i	$0.961668\pi$
$648$	0	0
$649$	0	0
$650$	−32.3607	−1.26929
$651$	0	0
$652$	12.9443i	0.506937i
$653$	− 32.8328i	− 1.28485i	−0.766350	−	0.642424i	$-0.777929\pi$
$653$	− 32.8328i	− 1.28485i	0.766350	−	0.642424i	$-0.222071\pi$
$654$	0	0
$655$	−24.4721	−0.956205
$656$	6.94427	0.271128
$657$	0	0
$658$	51.7771i	2.01848i
$659$	−17.8885	−0.696839	−0.348419	−	0.937339i	$-0.613281\pi$
$659$	−17.8885	−0.696839	−0.348419	+	0.937339i	$0.613281\pi$
$660$	0	0
$661$	44.2492	1.72110	0.860548	−	0.509370i	$-0.170121\pi$
$661$	44.2492	1.72110	0.860548	+	0.509370i	$0.170121\pi$
$662$	0	0
$663$	0	0
$664$	−10.4721	−0.406398
$665$	− 17.8885i	− 0.693688i
$666$	0	0
$667$	− 8.47214i	− 0.328042i
$668$	20.9443i	0.810358i
$669$	0	0
$670$	28.9443	1.11821
$671$	0	0
$672$	0	0
$673$	− 14.8328i	− 0.571763i	−0.958265	−	0.285882i	$-0.907714\pi$
$673$	− 14.8328i	− 0.571763i	0.958265	−	0.285882i	$-0.0922864\pi$
$674$	−1.41641	−0.0545580
$675$	0	0
$676$	28.8885	1.11110
$677$	− 9.88854i	− 0.380048i	−0.981779	−	0.190024i	$-0.939143\pi$
$677$	− 9.88854i	− 0.380048i	0.981779	−	0.190024i	$-0.0608565\pi$
$678$	0	0
$679$	−53.6656	−2.05950
$680$	−14.4721	−0.554981
$681$	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
$685$	− 16.5836i	− 0.633626i
$686$	−8.00000	−0.305441
$687$	0	0
$688$	0	0
$689$	57.8885	2.20538
$690$	0	0
$691$	31.7771	1.20886	0.604429	−	0.796659i	$-0.293401\pi$
$691$	31.7771	1.20886	0.604429	+	0.796659i	$0.293401\pi$
$692$	− 2.94427i	− 0.111924i
$693$	0	0
$694$	8.94427	0.339520
$695$	0	0
$696$	0	0
$697$	44.9443i	1.70239i
$698$	7.88854i	0.298586i
$699$	0	0
$700$	− 20.0000i	− 0.755929i
$701$	−10.5836	−0.399737	−0.199868	−	0.979823i	$-0.564051\pi$
$701$	−10.5836	−0.399737	−0.199868	+	0.979823i	$0.564051\pi$
$702$	0	0
$703$	16.9443i	0.639065i
$704$	0	0
$705$	0	0
$706$	−14.9443	−0.562435
$707$	33.8885i	1.27451i
$708$	0	0
$709$	−11.5279	−0.432938	−0.216469	−	0.976289i	$-0.569454\pi$
$709$	−11.5279	−0.432938	−0.216469	+	0.976289i	$0.569454\pi$
$710$	− 36.8328i	− 1.38231i
$711$	0	0
$712$	− 7.52786i	− 0.282119i
$713$	0	0
$714$	0	0
$715$	0	0
$716$	1.05573	0.0394544
$717$	0	0
$718$	15.0557i	0.561875i
$719$	−44.4721	−1.65853	−0.829265	−	0.558855i	$-0.811241\pi$
$719$	−44.4721	−1.65853	−0.829265	+	0.558855i	$0.811241\pi$
$720$	0	0
$721$	−3.77709	−0.140666
$722$	15.0000i	0.558242i
$723$	0	0
$724$	0.472136	0.0175468
$725$	42.3607	1.57324
$726$	0	0
$727$	− 16.9443i	− 0.628428i	−0.949352	−	0.314214i	$-0.898259\pi$
$727$	− 16.9443i	− 0.628428i	0.949352	−	0.314214i	$-0.101741\pi$
$728$	25.8885i	0.959493i
$729$	0	0
$730$	28.9443	1.07128
$731$	0	0
$732$	0	0
$733$	− 8.47214i	− 0.312925i	−0.987684	−	0.156463i	$-0.949991\pi$
$733$	− 8.47214i	− 0.312925i	0.987684	−	0.156463i	$-0.0500091\pi$
$734$	−12.0000	−0.442928
$735$	0	0
$736$	−1.00000	−0.0368605
$737$	0	0
$738$	0	0
$739$	33.8885	1.24661	0.623305	−	0.781979i	$-0.285789\pi$
$739$	33.8885	1.24661	0.623305	+	0.781979i	$0.285789\pi$
$740$	18.9443i	0.696405i
$741$	0	0
$742$	35.7771i	1.31342i
$743$	37.8885i	1.39000i	0.719012	+	0.694998i	$0.244595\pi$
$743$	37.8885i	1.39000i	−0.719012	+	0.694998i	$0.755405\pi$
$744$	0	0
$745$	30.0000	1.09911
$746$	−5.41641	−0.198309
$747$	0	0
$748$	0	0
$749$	25.8885	0.945947
$750$	0	0
$751$	27.5279	1.00451	0.502253	−	0.864721i	$-0.332505\pi$
$751$	27.5279	1.00451	0.502253	+	0.864721i	$0.332505\pi$
$752$	12.9443i	0.472029i
$753$	0	0
$754$	−54.8328	−1.99689
$755$	−2.11146	−0.0768438
$756$	0	0
$757$	18.5836i	0.675432i	0.941248	+	0.337716i	$0.109654\pi$
$757$	18.5836i	0.675432i	−0.941248	+	0.337716i	$0.890346\pi$
$758$	− 36.8328i	− 1.33783i
$759$	0	0
$760$	− 4.47214i	− 0.162221i
$761$	−53.7771	−1.94942	−0.974709	−	0.223478i	$-0.928259\pi$
$761$	−53.7771	−1.94942	−0.974709	+	0.223478i	$0.928259\pi$
$762$	0	0
$763$	33.8885i	1.22685i
$764$	9.88854	0.357755
$765$	0	0
$766$	21.8885	0.790865
$767$	38.8328i	1.40217i
$768$	0	0
$769$	−17.0557	−0.615045	−0.307523	−	0.951541i	$-0.599500\pi$
$769$	−17.0557	−0.615045	−0.307523	+	0.951541i	$0.599500\pi$
$770$	0	0
$771$	0	0
$772$	20.0000i	0.719816i
$773$	− 20.9443i	− 0.753313i	−0.926353	−	0.376657i	$-0.877074\pi$
$773$	− 20.9443i	− 0.753313i	0.926353	−	0.376657i	$-0.122926\pi$
$774$	0	0
$775$	0	0
$776$	−13.4164	−0.481621
$777$	0	0
$778$	4.47214i	0.160334i
$779$	−13.8885	−0.497609
$780$	0	0
$781$	0	0
$782$	− 6.47214i	− 0.231443i
$783$	0	0
$784$	−9.00000	−0.321429
$785$	− 45.7771i	− 1.63385i
$786$	0	0
$787$	− 54.8328i	− 1.95458i	−0.211909	−	0.977289i	$-0.567968\pi$
$787$	− 54.8328i	− 1.95458i	0.211909	−	0.977289i	$-0.432032\pi$
$788$	1.05573i	0.0376088i
$789$	0	0
$790$	7.88854i	0.280662i
$791$	29.6656	1.05479
$792$	0	0
$793$	− 54.8328i	− 1.94717i
$794$	−33.3050	−1.18195
$795$	0	0
$796$	−9.41641	−0.333756
$797$	2.11146i	0.0747916i	0.999301	+	0.0373958i	$0.0119062\pi$
$797$	2.11146i	0.0747916i	−0.999301	+	0.0373958i	$0.988094\pi$
$798$	0	0
$799$	−83.7771	−2.96382
$800$	− 5.00000i	− 0.176777i
$801$	0	0
$802$	− 1.41641i	− 0.0500151i
$803$	0	0
$804$	0	0
$805$	8.94427	0.315244
$806$	0	0
$807$	0	0
$808$	8.47214i	0.298049i
$809$	31.8885	1.12114	0.560571	−	0.828107i	$-0.310582\pi$
$809$	31.8885	1.12114	0.560571	+	0.828107i	$0.310582\pi$
$810$	0	0
$811$	−14.1115	−0.495520	−0.247760	−	0.968821i	$-0.579694\pi$
$811$	−14.1115	−0.495520	−0.247760	+	0.968821i	$0.579694\pi$
$812$	− 33.8885i	− 1.18925i
$813$	0	0
$814$	0	0
$815$	− 28.9443i	− 1.01387i
$816$	0	0
$817$	0	0
$818$	− 18.0000i	− 0.629355i
$819$	0	0
$820$	−15.5279	−0.542257
$821$	7.52786	0.262724	0.131362	−	0.991334i	$-0.458065\pi$
$821$	7.52786	0.262724	0.131362	+	0.991334i	$0.458065\pi$
$822$	0	0
$823$	24.3607i	0.849160i	0.905390	+	0.424580i	$0.139578\pi$
$823$	24.3607i	0.849160i	−0.905390	+	0.424580i	$0.860422\pi$
$824$	−0.944272	−0.0328953
$825$	0	0
$826$	−24.0000	−0.835067
$827$	33.5279i	1.16588i	0.812516	+	0.582939i	$0.198097\pi$
$827$	33.5279i	1.16588i	−0.812516	+	0.582939i	$0.801903\pi$
$828$	0	0
$829$	−13.0557	−0.453444	−0.226722	−	0.973959i	$-0.572801\pi$
$829$	−13.0557	−0.453444	−0.226722	+	0.973959i	$0.572801\pi$
$830$	23.4164	0.812795
$831$	0	0
$832$	6.47214i	0.224381i
$833$	− 58.2492i	− 2.01822i
$834$	0	0
$835$	− 46.8328i	− 1.62072i
$836$	0	0
$837$	0	0
$838$	2.11146i	0.0729390i
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	42.7771	1.47507
$842$	10.5836i	0.364735i
$843$	0	0
$844$	12.0000	0.413057
$845$	−64.5967	−2.22220
$846$	0	0
$847$	− 44.0000i	− 1.51186i
$848$	8.94427i	0.307148i
$849$	0	0
$850$	32.3607	1.10996
$851$	−8.47214	−0.290421
$852$	0	0
$853$	− 18.4721i	− 0.632474i	−0.948680	−	0.316237i	$-0.897581\pi$
$853$	− 18.4721i	− 0.632474i	0.948680	−	0.316237i	$-0.102419\pi$
$854$	33.8885	1.15964
$855$	0	0
$856$	6.47214	0.221213
$857$	− 11.8885i	− 0.406105i	−0.979168	−	0.203052i	$-0.934914\pi$
$857$	− 11.8885i	− 0.406105i	0.979168	−	0.203052i	$-0.0650862\pi$
$858$	0	0
$859$	−31.7771	−1.08422	−0.542110	−	0.840307i	$-0.682374\pi$
$859$	−31.7771	−1.08422	−0.542110	+	0.840307i	$0.682374\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	20.9443i	0.712951i	0.934305	+	0.356476i	$0.116022\pi$
$863$	20.9443i	0.712951i	−0.934305	+	0.356476i	$0.883978\pi$
$864$	0	0
$865$	6.58359i	0.223849i
$866$	−5.41641	−0.184057
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	83.7771	2.83868
$872$	8.47214i	0.286903i
$873$	0	0
$874$	2.00000	0.0676510
$875$	44.7214i	1.51186i
$876$	0	0
$877$	− 24.3607i	− 0.822602i	−0.911500	−	0.411301i	$-0.865075\pi$
$877$	− 24.3607i	− 0.822602i	0.911500	−	0.411301i	$-0.134925\pi$
$878$	26.8328i	0.905564i
$879$	0	0
$880$	0	0
$881$	−4.47214	−0.150670	−0.0753350	−	0.997158i	$-0.524003\pi$
$881$	−4.47214	−0.150670	−0.0753350	+	0.997158i	$0.524003\pi$
$882$	0	0
$883$	− 51.7771i	− 1.74244i	−0.490895	−	0.871219i	$-0.663330\pi$
$883$	− 51.7771i	− 1.74244i	0.490895	−	0.871219i	$-0.336670\pi$
$884$	−41.8885	−1.40886
$885$	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
$889$	61.6656	2.06820
$890$	16.8328i	0.564237i
$891$	0	0
$892$	− 15.4164i	− 0.516180i
$893$	− 25.8885i	− 0.866327i
$894$	0	0
$895$	−2.36068	−0.0789088
$896$	−4.00000	−0.133631
$897$	0	0
$898$	17.0557i	0.569157i
$899$	0	0
$900$	0	0
$901$	−57.8885	−1.92855
$902$	0	0
$903$	0	0
$904$	7.41641	0.246666
$905$	−1.05573	−0.0350936
$906$	0	0
$907$	− 38.8328i	− 1.28942i	−0.764426	−	0.644711i	$-0.776978\pi$
$907$	− 38.8328i	− 1.28942i	0.764426	−	0.644711i	$-0.223022\pi$
$908$	9.52786i	0.316193i
$909$	0	0
$910$	− 57.8885i	− 1.91899i
$911$	−7.05573	−0.233767	−0.116883	−	0.993146i	$-0.537290\pi$
$911$	−7.05573	−0.233767	−0.116883	+	0.993146i	$0.537290\pi$
$912$	0	0
$913$	0	0
$914$	−1.41641	−0.0468506
$915$	0	0
$916$	−1.41641	−0.0467994
$917$	− 43.7771i	− 1.44565i
$918$	0	0
$919$	−17.4164	−0.574514	−0.287257	−	0.957854i	$-0.592743\pi$
$919$	−17.4164	−0.574514	−0.287257	+	0.957854i	$0.592743\pi$
$920$	2.23607	0.0737210
$921$	0	0
$922$	30.3607i	0.999876i
$923$	− 106.610i	− 3.50911i
$924$	0	0
$925$	− 42.3607i	− 1.39281i
$926$	−11.4164	−0.375166
$927$	0	0
$928$	− 8.47214i	− 0.278111i
$929$	6.94427	0.227834	0.113917	−	0.993490i	$-0.463660\pi$
$929$	6.94427	0.227834	0.113917	+	0.993490i	$0.463660\pi$
$930$	0	0
$931$	18.0000	0.589926
$932$	2.94427i	0.0964428i
$933$	0	0
$934$	−17.5279	−0.573529
$935$	0	0
$936$	0	0
$937$	33.4164i	1.09167i	0.837894	+	0.545833i	$0.183787\pi$
$937$	33.4164i	1.09167i	−0.837894	+	0.545833i	$0.816213\pi$
$938$	51.7771i	1.69058i
$939$	0	0
$940$	− 28.9443i	− 0.944058i
$941$	−27.5279	−0.897383	−0.448691	−	0.893687i	$-0.648110\pi$
$941$	−27.5279	−0.897383	−0.448691	+	0.893687i	$0.648110\pi$
$942$	0	0
$943$	− 6.94427i	− 0.226137i
$944$	−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
$949$	83.7771	2.71952
$950$	10.0000i	0.324443i
$951$	0	0
$952$	− 25.8885i	− 0.839053i
$953$	32.3607i	1.04827i	0.851637	+	0.524133i	$0.175610\pi$
$953$	32.3607i	1.04827i	−0.851637	+	0.524133i	$0.824390\pi$
$954$	0	0
$955$	−22.1115	−0.715510
$956$	3.52786	0.114099
$957$	0	0
$958$	− 0.944272i	− 0.0305080i
$959$	29.6656	0.957953
$960$	0	0
$961$	−31.0000	−1.00000
$962$	54.8328i	1.76788i
$963$	0	0
$964$	2.00000	0.0644157
$965$	− 44.7214i	− 1.43963i
$966$	0	0
$967$	28.5836i	0.919186i	0.888130	+	0.459593i	$0.152005\pi$
$967$	28.5836i	0.919186i	−0.888130	+	0.459593i	$0.847995\pi$
$968$	− 11.0000i	− 0.353553i
$969$	0	0
$970$	30.0000	0.963242
$971$	53.8885	1.72937	0.864683	−	0.502318i	$-0.167519\pi$
$971$	53.8885	1.72937	0.864683	+	0.502318i	$0.167519\pi$
$972$	0	0
$973$	0	0
$974$	−4.58359	−0.146868
$975$	0	0
$976$	8.47214	0.271186
$977$	− 10.4721i	− 0.335033i	−0.985869	−	0.167517i	$-0.946425\pi$
$977$	− 10.4721i	− 0.335033i	0.985869	−	0.167517i	$-0.0535748\pi$
$978$	0	0
$979$	0	0
$980$	20.1246	0.642857
$981$	0	0
$982$	− 7.88854i	− 0.251734i
$983$	− 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
$985$	− 2.36068i	− 0.0752175i
$986$	54.8328	1.74623
$987$	0	0
$988$	− 12.9443i	− 0.411812i
$989$	0	0
$990$	0	0
$991$	−53.6656	−1.70474	−0.852372	−	0.522935i	$-0.824837\pi$
$991$	−53.6656	−1.70474	−0.852372	+	0.522935i	$0.824837\pi$
$992$	0	0
$993$	0	0
$994$	65.8885	2.08986
$995$	21.0557	0.667511
$996$	0	0
$997$	19.4164i	0.614924i	0.951560	+	0.307462i	$0.0994797\pi$
$997$	19.4164i	0.614924i	−0.951560	+	0.307462i	$0.900520\pi$
$998$	− 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.829.2		4
3.2	odd	2		690.2.d.a.139.3	yes	4
5.4	even	2	inner	2070.2.d.b.829.4		4
15.2	even	4		3450.2.a.bc.1.1		2
15.8	even	4		3450.2.a.bn.1.2		2
15.14	odd	2		690.2.d.a.139.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
690.2.d.a.139.1	✓	4	15.14	odd	2
690.2.d.a.139.3	yes	4	3.2	odd	2
2070.2.d.b.829.2		4	1.1	even	1	trivial
2070.2.d.b.829.4		4	5.4	even	2	inner
3450.2.a.bc.1.1		2	15.2	even	4
3450.2.a.bn.1.2		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)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} +2.23607 q^{5} +4.00000i q^{7} +1.00000i q^{8} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000 q^{4} +2.23607 q^{5} +4.00000i q^{7} +1.00000i q^{8} -2.23607i q^{10} -6.47214i q^{13} +4.00000 q^{14} +1.00000 q^{16} +6.47214i q^{17} -2.00000 q^{19} -2.23607 q^{20} -1.00000i q^{23} +5.00000 q^{25} -6.47214 q^{26} -4.00000i q^{28} +8.47214 q^{29} -1.00000i q^{32} +6.47214 q^{34} +8.94427i q^{35} -8.47214i q^{37} +2.00000i q^{38} +2.23607i q^{40} +6.94427 q^{41} -1.00000 q^{46} +12.9443i q^{47} -9.00000 q^{49} -5.00000i q^{50} +6.47214i q^{52} +8.94427i q^{53} -4.00000 q^{56} -8.47214i q^{58} -6.00000 q^{59} +8.47214 q^{61} -1.00000 q^{64} -14.4721i q^{65} +12.9443i q^{67} -6.47214i q^{68} +8.94427 q^{70} +16.4721 q^{71} +12.9443i q^{73} -8.47214 q^{74} +2.00000 q^{76} -3.52786 q^{79} +2.23607 q^{80} -6.94427i q^{82} +10.4721i q^{83} +14.4721i q^{85} -7.52786 q^{89} +25.8885 q^{91} +1.00000i q^{92} +12.9443 q^{94} -4.47214 q^{95} +13.4164i q^{97} +9.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4}+O(q^{10}) \) 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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