LMFDB - Embedded newform 1530.2.d.f.919.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1530,2,Mod(919,1530)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1530, 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("1530.919");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1530 = 2 \cdot 3^{2} \cdot 5 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1530.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:	$12.2171115093$
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 510)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			919.3
Root			$0.618034i$ of defining polynomial
Character	$\chi$	$=$	1530.919
Dual form			1530.2.d.f.919.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000i q^{2} -1.00000 q^{4} -2.23607i q^{5} +1.23607i q^{7} -1.00000i q^{8} +2.23607 q^{10} -2.00000 q^{11} +5.23607i q^{13} -1.23607 q^{14} +1.00000 q^{16} +1.00000i q^{17} +6.47214 q^{19} +2.23607i q^{20} -2.00000i q^{22} -4.00000i q^{23} -5.00000 q^{25} -5.23607 q^{26} -1.23607i q^{28} -4.00000 q^{29} +2.47214 q^{31} +1.00000i q^{32} -1.00000 q^{34} +2.76393 q^{35} -2.00000i q^{37} +6.47214i q^{38} -2.23607 q^{40} +7.70820 q^{41} +12.1803i q^{43} +2.00000 q^{44} +4.00000 q^{46} +10.4721i q^{47} +5.47214 q^{49} -5.00000i q^{50} -5.23607i q^{52} +10.9443i q^{53} +4.47214i q^{55} +1.23607 q^{56} -4.00000i q^{58} +11.7082 q^{59} +4.47214 q^{61} +2.47214i q^{62} -1.00000 q^{64} +11.7082 q^{65} +1.70820i q^{67} -1.00000i q^{68} +2.76393i q^{70} -11.2361 q^{71} +8.76393i q^{73} +2.00000 q^{74} -6.47214 q^{76} -2.47214i q^{77} -0.944272 q^{79} -2.23607i q^{80} +7.70820i q^{82} -10.4721i q^{83} +2.23607 q^{85} -12.1803 q^{86} +2.00000i q^{88} +12.4721 q^{89} -6.47214 q^{91} +4.00000i q^{92} -10.4721 q^{94} -14.4721i q^{95} +1.70820i q^{97} +5.47214i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4} - 8 q^{11} + 4 q^{14} + 4 q^{16} + 8 q^{19} - 20 q^{25} - 12 q^{26} - 16 q^{29} - 8 q^{31} - 4 q^{34} + 20 q^{35} + 4 q^{41} + 8 q^{44} + 16 q^{46} + 4 q^{49} - 4 q^{56} + 20 q^{59} - 4 q^{64}+ \cdots - 24 q^{94}+O(q^{100}) $ 4 * q - 4 * q^4 - 8 * q^11 + 4 * q^14 + 4 * q^16 + 8 * q^19 - 20 * q^25 - 12 * q^26 - 16 * q^29 - 8 * q^31 - 4 * q^34 + 20 * q^35 + 4 * q^41 + 8 * q^44 + 16 * q^46 + 4 * q^49 - 4 * q^56 + 20 * q^59 - 4 * q^64 + 20 * q^65 - 36 * q^71 + 8 * q^74 - 8 * q^76 + 32 * q^79 - 4 * q^86 + 32 * q^89 - 8 * q^91 - 24 * q^94

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1530\mathbb{Z}\right)^\times$.

$n$	$307$	$1261$	$1361$
$\chi(n)$	$-1$	$1$	$1$

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
$5$	− 2.23607i	− 1.00000i
$5$	− 2.23607i	− 1.00000i
$6$	0	0
$7$	1.23607i	0.467190i	0.972334	+	0.233595i	$0.0750489\pi$
$7$	1.23607i	0.467190i	−0.972334	+	0.233595i	$0.924951\pi$
$8$	− 1.00000i	− 0.353553i
$9$	0	0
$10$	2.23607	0.707107
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	0	0
$13$	5.23607i	1.45222i	0.687576	+	0.726112i	$0.258675\pi$
$13$	5.23607i	1.45222i	−0.687576	+	0.726112i	$0.741325\pi$
$14$	−1.23607	−0.330353
$15$	0	0
$16$	1.00000	0.250000
$17$	1.00000i	0.242536i
$17$	1.00000i	0.242536i
$18$	0	0
$19$	6.47214	1.48481	0.742405	−	0.669951i	$-0.233685\pi$
$19$	6.47214	1.48481	0.742405	+	0.669951i	$0.233685\pi$
$20$	2.23607i	0.500000i
$21$	0	0
$22$	− 2.00000i	− 0.426401i
$23$	− 4.00000i	− 0.834058i	−0.908893	−	0.417029i	$-0.863071\pi$
$23$	− 4.00000i	− 0.834058i	0.908893	−	0.417029i	$-0.136929\pi$
$24$	0	0
$25$	−5.00000	−1.00000
$26$	−5.23607	−1.02688
$27$	0	0
$28$	− 1.23607i	− 0.233595i
$29$	−4.00000	−0.742781	−0.371391	−	0.928477i	$-0.621119\pi$
$29$	−4.00000	−0.742781	−0.371391	+	0.928477i	$0.621119\pi$
$30$	0	0
$31$	2.47214	0.444009	0.222004	−	0.975046i	$-0.428740\pi$
$31$	2.47214	0.444009	0.222004	+	0.975046i	$0.428740\pi$
$32$	1.00000i	0.176777i
$33$	0	0
$34$	−1.00000	−0.171499
$35$	2.76393	0.467190
$36$	0	0
$37$	− 2.00000i	− 0.328798i	−0.986394	−	0.164399i	$-0.947432\pi$
$37$	− 2.00000i	− 0.328798i	0.986394	−	0.164399i	$-0.0525685\pi$
$38$	6.47214i	1.04992i
$39$	0	0
$40$	−2.23607	−0.353553
$41$	7.70820	1.20382	0.601910	−	0.798564i	$-0.294407\pi$
$41$	7.70820	1.20382	0.601910	+	0.798564i	$0.294407\pi$
$42$	0	0
$43$	12.1803i	1.85748i	0.370726	+	0.928742i	$0.379109\pi$
$43$	12.1803i	1.85748i	−0.370726	+	0.928742i	$0.620891\pi$
$44$	2.00000	0.301511
$45$	0	0
$46$	4.00000	0.589768
$47$	10.4721i	1.52752i	0.645501	+	0.763759i	$0.276648\pi$
$47$	10.4721i	1.52752i	−0.645501	+	0.763759i	$0.723352\pi$
$48$	0	0
$49$	5.47214	0.781734
$50$	− 5.00000i	− 0.707107i
$51$	0	0
$52$	− 5.23607i	− 0.726112i
$53$	10.9443i	1.50331i	0.659556	+	0.751656i	$0.270744\pi$
$53$	10.9443i	1.50331i	−0.659556	+	0.751656i	$0.729256\pi$
$54$	0	0
$55$	4.47214i	0.603023i
$56$	1.23607	0.165177
$57$	0	0
$58$	− 4.00000i	− 0.525226i
$59$	11.7082	1.52428	0.762139	−	0.647413i	$-0.224149\pi$
$59$	11.7082	1.52428	0.762139	+	0.647413i	$0.224149\pi$
$60$	0	0
$61$	4.47214	0.572598	0.286299	−	0.958140i	$-0.407575\pi$
$61$	4.47214	0.572598	0.286299	+	0.958140i	$0.407575\pi$
$62$	2.47214i	0.313962i
$63$	0	0
$64$	−1.00000	−0.125000
$65$	11.7082	1.45222
$66$	0	0
$67$	1.70820i	0.208690i	0.994541	+	0.104345i	$0.0332747\pi$
$67$	1.70820i	0.208690i	−0.994541	+	0.104345i	$0.966725\pi$
$68$	− 1.00000i	− 0.121268i
$69$	0	0
$70$	2.76393i	0.330353i
$71$	−11.2361	−1.33348	−0.666738	−	0.745292i	$-0.732310\pi$
$71$	−11.2361	−1.33348	−0.666738	+	0.745292i	$0.732310\pi$
$72$	0	0
$73$	8.76393i	1.02574i	0.858466	+	0.512870i	$0.171418\pi$
$73$	8.76393i	1.02574i	−0.858466	+	0.512870i	$0.828582\pi$
$74$	2.00000	0.232495
$75$	0	0
$76$	−6.47214	−0.742405
$77$	− 2.47214i	− 0.281726i
$78$	0	0
$79$	−0.944272	−0.106239	−0.0531194	−	0.998588i	$-0.516916\pi$
$79$	−0.944272	−0.106239	−0.0531194	+	0.998588i	$0.516916\pi$
$80$	− 2.23607i	− 0.250000i
$81$	0	0
$82$	7.70820i	0.851229i
$83$	− 10.4721i	− 1.14947i	−0.818341	−	0.574733i	$-0.805106\pi$
$83$	− 10.4721i	− 1.14947i	0.818341	−	0.574733i	$-0.194894\pi$
$84$	0	0
$85$	2.23607	0.242536
$86$	−12.1803	−1.31344
$87$	0	0
$88$	2.00000i	0.213201i
$89$	12.4721	1.32204	0.661022	−	0.750367i	$-0.270123\pi$
$89$	12.4721	1.32204	0.661022	+	0.750367i	$0.270123\pi$
$90$	0	0
$91$	−6.47214	−0.678464
$92$	4.00000i	0.417029i
$93$	0	0
$94$	−10.4721	−1.08012
$95$	− 14.4721i	− 1.48481i
$96$	0	0
$97$	1.70820i	0.173442i	0.996233	+	0.0867209i	$0.0276388\pi$
$97$	1.70820i	0.173442i	−0.996233	+	0.0867209i	$0.972361\pi$
$98$	5.47214i	0.552769i
$99$	0	0
$100$	5.00000	0.500000
$101$	1.70820	0.169973	0.0849863	−	0.996382i	$-0.472915\pi$
$101$	1.70820	0.169973	0.0849863	+	0.996382i	$0.472915\pi$
$102$	0	0
$103$	10.9443i	1.07837i	0.842187	+	0.539186i	$0.181268\pi$
$103$	10.9443i	1.07837i	−0.842187	+	0.539186i	$0.818732\pi$
$104$	5.23607	0.513439
$105$	0	0
$106$	−10.9443	−1.06300
$107$	8.00000i	0.773389i	0.922208	+	0.386695i	$0.126383\pi$
$107$	8.00000i	0.773389i	−0.922208	+	0.386695i	$0.873617\pi$
$108$	0	0
$109$	7.52786	0.721039	0.360519	−	0.932752i	$-0.382599\pi$
$109$	7.52786	0.721039	0.360519	+	0.932752i	$0.382599\pi$
$110$	−4.47214	−0.426401
$111$	0	0
$112$	1.23607i	0.116797i
$113$	10.0000i	0.940721i	0.882474	+	0.470360i	$0.155876\pi$
$113$	10.0000i	0.940721i	−0.882474	+	0.470360i	$0.844124\pi$
$114$	0	0
$115$	−8.94427	−0.834058
$116$	4.00000	0.371391
$117$	0	0
$118$	11.7082i	1.07783i
$119$	−1.23607	−0.113310
$120$	0	0
$121$	−7.00000	−0.636364
$122$	4.47214i	0.404888i
$123$	0	0
$124$	−2.47214	−0.222004
$125$	11.1803i	1.00000i
$126$	0	0
$127$	− 14.0000i	− 1.24230i	−0.783692	−	0.621150i	$-0.786666\pi$
$127$	− 14.0000i	− 1.24230i	0.783692	−	0.621150i	$-0.213334\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	0	0
$130$	11.7082i	1.02688i
$131$	−12.4721	−1.08970	−0.544848	−	0.838535i	$-0.683413\pi$
$131$	−12.4721	−1.08970	−0.544848	+	0.838535i	$0.683413\pi$
$132$	0	0
$133$	8.00000i	0.693688i
$134$	−1.70820	−0.147566
$135$	0	0
$136$	1.00000	0.0857493
$137$	− 2.00000i	− 0.170872i	−0.996344	−	0.0854358i	$-0.972772\pi$
$137$	− 2.00000i	− 0.170872i	0.996344	−	0.0854358i	$-0.0272282\pi$
$138$	0	0
$139$	−1.52786	−0.129592	−0.0647959	−	0.997899i	$-0.520640\pi$
$139$	−1.52786	−0.129592	−0.0647959	+	0.997899i	$0.520640\pi$
$140$	−2.76393	−0.233595
$141$	0	0
$142$	− 11.2361i	− 0.942910i
$143$	− 10.4721i	− 0.875724i
$144$	0	0
$145$	8.94427i	0.742781i
$146$	−8.76393	−0.725308
$147$	0	0
$148$	2.00000i	0.164399i
$149$	12.1803	0.997852	0.498926	−	0.866644i	$-0.333728\pi$
$149$	12.1803	0.997852	0.498926	+	0.866644i	$0.333728\pi$
$150$	0	0
$151$	−4.00000	−0.325515	−0.162758	−	0.986666i	$-0.552039\pi$
$151$	−4.00000	−0.325515	−0.162758	+	0.986666i	$0.552039\pi$
$152$	− 6.47214i	− 0.524960i
$153$	0	0
$154$	2.47214	0.199210
$155$	− 5.52786i	− 0.444009i
$156$	0	0
$157$	− 10.1803i	− 0.812480i	−0.913767	−	0.406240i	$-0.866840\pi$
$157$	− 10.1803i	− 0.812480i	0.913767	−	0.406240i	$-0.133160\pi$
$158$	− 0.944272i	− 0.0751222i
$159$	0	0
$160$	2.23607	0.176777
$161$	4.94427	0.389663
$162$	0	0
$163$	− 22.4721i	− 1.76015i	−0.474831	−	0.880077i	$-0.657491\pi$
$163$	− 22.4721i	− 1.76015i	0.474831	−	0.880077i	$-0.342509\pi$
$164$	−7.70820	−0.601910
$165$	0	0
$166$	10.4721	0.812795
$167$	− 13.8885i	− 1.07473i	−0.843350	−	0.537364i	$-0.819420\pi$
$167$	− 13.8885i	− 1.07473i	0.843350	−	0.537364i	$-0.180580\pi$
$168$	0	0
$169$	−14.4164	−1.10895
$170$	2.23607i	0.171499i
$171$	0	0
$172$	− 12.1803i	− 0.928742i
$173$	15.8885i	1.20798i	0.796991	+	0.603992i	$0.206424\pi$
$173$	15.8885i	1.20798i	−0.796991	+	0.603992i	$0.793576\pi$
$174$	0	0
$175$	− 6.18034i	− 0.467190i
$176$	−2.00000	−0.150756
$177$	0	0
$178$	12.4721i	0.934826i
$179$	−7.70820	−0.576138	−0.288069	−	0.957610i	$-0.593013\pi$
$179$	−7.70820	−0.576138	−0.288069	+	0.957610i	$0.593013\pi$
$180$	0	0
$181$	0.472136	0.0350936	0.0175468	−	0.999846i	$-0.494414\pi$
$181$	0.472136	0.0350936	0.0175468	+	0.999846i	$0.494414\pi$
$182$	− 6.47214i	− 0.479747i
$183$	0	0
$184$	−4.00000	−0.294884
$185$	−4.47214	−0.328798
$186$	0	0
$187$	− 2.00000i	− 0.146254i
$188$	− 10.4721i	− 0.763759i
$189$	0	0
$190$	14.4721	1.04992
$191$	−5.52786	−0.399982	−0.199991	−	0.979798i	$-0.564091\pi$
$191$	−5.52786	−0.399982	−0.199991	+	0.979798i	$0.564091\pi$
$192$	0	0
$193$	− 23.5967i	− 1.69853i	−0.527966	−	0.849266i	$-0.677045\pi$
$193$	− 23.5967i	− 1.69853i	0.527966	−	0.849266i	$-0.322955\pi$
$194$	−1.70820	−0.122642
$195$	0	0
$196$	−5.47214	−0.390867
$197$	15.8885i	1.13201i	0.824401	+	0.566006i	$0.191512\pi$
$197$	15.8885i	1.13201i	−0.824401	+	0.566006i	$0.808488\pi$
$198$	0	0
$199$	8.94427	0.634043	0.317021	−	0.948418i	$-0.397317\pi$
$199$	8.94427	0.634043	0.317021	+	0.948418i	$0.397317\pi$
$200$	5.00000i	0.353553i
$201$	0	0
$202$	1.70820i	0.120189i
$203$	− 4.94427i	− 0.347020i
$204$	0	0
$205$	− 17.2361i	− 1.20382i
$206$	−10.9443	−0.762524
$207$	0	0
$208$	5.23607i	0.363056i
$209$	−12.9443	−0.895374
$210$	0	0
$211$	−24.9443	−1.71723	−0.858617	−	0.512617i	$-0.828676\pi$
$211$	−24.9443	−1.71723	−0.858617	+	0.512617i	$0.828676\pi$
$212$	− 10.9443i	− 0.751656i
$213$	0	0
$214$	−8.00000	−0.546869
$215$	27.2361	1.85748
$216$	0	0
$217$	3.05573i	0.207436i
$218$	7.52786i	0.509851i
$219$	0	0
$220$	− 4.47214i	− 0.301511i
$221$	−5.23607	−0.352216
$222$	0	0
$223$	9.05573i	0.606416i	0.952924	+	0.303208i	$0.0980578\pi$
$223$	9.05573i	0.606416i	−0.952924	+	0.303208i	$0.901942\pi$
$224$	−1.23607	−0.0825883
$225$	0	0
$226$	−10.0000	−0.665190
$227$	9.88854i	0.656326i	0.944621	+	0.328163i	$0.106429\pi$
$227$	9.88854i	0.656326i	−0.944621	+	0.328163i	$0.893571\pi$
$228$	0	0
$229$	21.4164	1.41524	0.707618	−	0.706595i	$-0.249770\pi$
$229$	21.4164	1.41524	0.707618	+	0.706595i	$0.249770\pi$
$230$	− 8.94427i	− 0.589768i
$231$	0	0
$232$	4.00000i	0.262613i
$233$	− 5.41641i	− 0.354841i	−0.984135	−	0.177420i	$-0.943225\pi$
$233$	− 5.41641i	− 0.354841i	0.984135	−	0.177420i	$-0.0567752\pi$
$234$	0	0
$235$	23.4164	1.52752
$236$	−11.7082	−0.762139
$237$	0	0
$238$	− 1.23607i	− 0.0801224i
$239$	−21.8885	−1.41585	−0.707926	−	0.706287i	$-0.750369\pi$
$239$	−21.8885	−1.41585	−0.707926	+	0.706287i	$0.750369\pi$
$240$	0	0
$241$	9.41641	0.606564	0.303282	−	0.952901i	$-0.401918\pi$
$241$	9.41641	0.606564	0.303282	+	0.952901i	$0.401918\pi$
$242$	− 7.00000i	− 0.449977i
$243$	0	0
$244$	−4.47214	−0.286299
$245$	− 12.2361i	− 0.781734i
$246$	0	0
$247$	33.8885i	2.15628i
$248$	− 2.47214i	− 0.156981i
$249$	0	0
$250$	−11.1803	−0.707107
$251$	10.1803	0.642577	0.321289	−	0.946981i	$-0.395884\pi$
$251$	10.1803	0.642577	0.321289	+	0.946981i	$0.395884\pi$
$252$	0	0
$253$	8.00000i	0.502956i
$254$	14.0000	0.878438
$255$	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
$259$	2.47214	0.153611
$260$	−11.7082	−0.726112
$261$	0	0
$262$	− 12.4721i	− 0.770531i
$263$	4.00000i	0.246651i	0.992366	+	0.123325i	$0.0393559\pi$
$263$	4.00000i	0.246651i	−0.992366	+	0.123325i	$0.960644\pi$
$264$	0	0
$265$	24.4721	1.50331
$266$	−8.00000	−0.490511
$267$	0	0
$268$	− 1.70820i	− 0.104345i
$269$	−7.05573	−0.430195	−0.215098	−	0.976593i	$-0.569007\pi$
$269$	−7.05573	−0.430195	−0.215098	+	0.976593i	$0.569007\pi$
$270$	0	0
$271$	4.00000	0.242983	0.121491	−	0.992592i	$-0.461232\pi$
$271$	4.00000	0.242983	0.121491	+	0.992592i	$0.461232\pi$
$272$	1.00000i	0.0606339i
$273$	0	0
$274$	2.00000	0.120824
$275$	10.0000	0.603023
$276$	0	0
$277$	− 26.3607i	− 1.58386i	−0.610612	−	0.791930i	$-0.709077\pi$
$277$	− 26.3607i	− 1.58386i	0.610612	−	0.791930i	$-0.290923\pi$
$278$	− 1.52786i	− 0.0916352i
$279$	0	0
$280$	− 2.76393i	− 0.165177i
$281$	−1.41641	−0.0844958	−0.0422479	−	0.999107i	$-0.513452\pi$
$281$	−1.41641	−0.0844958	−0.0422479	+	0.999107i	$0.513452\pi$
$282$	0	0
$283$	− 25.8885i	− 1.53891i	−0.638699	−	0.769457i	$-0.720527\pi$
$283$	− 25.8885i	− 1.53891i	0.638699	−	0.769457i	$-0.279473\pi$
$284$	11.2361	0.666738
$285$	0	0
$286$	10.4721	0.619230
$287$	9.52786i	0.562412i
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	−8.94427	−0.525226
$291$	0	0
$292$	− 8.76393i	− 0.512870i
$293$	− 29.4164i	− 1.71852i	−0.511535	−	0.859262i	$-0.670923\pi$
$293$	− 29.4164i	− 1.71852i	0.511535	−	0.859262i	$-0.329077\pi$
$294$	0	0
$295$	− 26.1803i	− 1.52428i
$296$	−2.00000	−0.116248
$297$	0	0
$298$	12.1803i	0.705588i
$299$	20.9443	1.21124
$300$	0	0
$301$	−15.0557	−0.867798
$302$	− 4.00000i	− 0.230174i
$303$	0	0
$304$	6.47214	0.371202
$305$	− 10.0000i	− 0.572598i
$306$	0	0
$307$	− 4.18034i	− 0.238585i	−0.992859	−	0.119292i	$-0.961937\pi$
$307$	− 4.18034i	− 0.238585i	0.992859	−	0.119292i	$-0.0380626\pi$
$308$	2.47214i	0.140863i
$309$	0	0
$310$	5.52786	0.313962
$311$	−19.5967	−1.11123	−0.555615	−	0.831440i	$-0.687517\pi$
$311$	−19.5967	−1.11123	−0.555615	+	0.831440i	$0.687517\pi$
$312$	0	0
$313$	− 21.7082i	− 1.22702i	−0.789687	−	0.613510i	$-0.789757\pi$
$313$	− 21.7082i	− 1.22702i	0.789687	−	0.613510i	$-0.210243\pi$
$314$	10.1803	0.574510
$315$	0	0
$316$	0.944272	0.0531194
$317$	11.8885i	0.667727i	0.942621	+	0.333864i	$0.108352\pi$
$317$	11.8885i	0.667727i	−0.942621	+	0.333864i	$0.891648\pi$
$318$	0	0
$319$	8.00000	0.447914
$320$	2.23607i	0.125000i
$321$	0	0
$322$	4.94427i	0.275534i
$323$	6.47214i	0.360119i
$324$	0	0
$325$	− 26.1803i	− 1.45222i
$326$	22.4721	1.24462
$327$	0	0
$328$	− 7.70820i	− 0.425614i
$329$	−12.9443	−0.713641
$330$	0	0
$331$	−3.41641	−0.187783	−0.0938914	−	0.995582i	$-0.529931\pi$
$331$	−3.41641	−0.187783	−0.0938914	+	0.995582i	$0.529931\pi$
$332$	10.4721i	0.574733i
$333$	0	0
$334$	13.8885	0.759947
$335$	3.81966	0.208690
$336$	0	0
$337$	20.1803i	1.09929i	0.835397	+	0.549647i	$0.185238\pi$
$337$	20.1803i	1.09929i	−0.835397	+	0.549647i	$0.814762\pi$
$338$	− 14.4164i	− 0.784149i
$339$	0	0
$340$	−2.23607	−0.121268
$341$	−4.94427	−0.267747
$342$	0	0
$343$	15.4164i	0.832408i
$344$	12.1803	0.656720
$345$	0	0
$346$	−15.8885	−0.854173
$347$	− 22.8328i	− 1.22573i	−0.790188	−	0.612865i	$-0.790017\pi$
$347$	− 22.8328i	− 1.22573i	0.790188	−	0.612865i	$-0.209983\pi$
$348$	0	0
$349$	28.4721	1.52408	0.762039	−	0.647531i	$-0.224198\pi$
$349$	28.4721	1.52408	0.762039	+	0.647531i	$0.224198\pi$
$350$	6.18034	0.330353
$351$	0	0
$352$	− 2.00000i	− 0.106600i
$353$	14.9443i	0.795403i	0.917515	+	0.397702i	$0.130192\pi$
$353$	14.9443i	0.795403i	−0.917515	+	0.397702i	$0.869808\pi$
$354$	0	0
$355$	25.1246i	1.33348i
$356$	−12.4721	−0.661022
$357$	0	0
$358$	− 7.70820i	− 0.407391i
$359$	−9.52786	−0.502861	−0.251431	−	0.967875i	$-0.580901\pi$
$359$	−9.52786	−0.502861	−0.251431	+	0.967875i	$0.580901\pi$
$360$	0	0
$361$	22.8885	1.20466
$362$	0.472136i	0.0248149i
$363$	0	0
$364$	6.47214	0.339232
$365$	19.5967	1.02574
$366$	0	0
$367$	5.81966i	0.303784i	0.988397	+	0.151892i	$0.0485366\pi$
$367$	5.81966i	0.303784i	−0.988397	+	0.151892i	$0.951463\pi$
$368$	− 4.00000i	− 0.208514i
$369$	0	0
$370$	− 4.47214i	− 0.232495i
$371$	−13.5279	−0.702332
$372$	0	0
$373$	− 1.23607i	− 0.0640012i	−0.999488	−	0.0320006i	$-0.989812\pi$
$373$	− 1.23607i	− 0.0640012i	0.999488	−	0.0320006i	$-0.0101878\pi$
$374$	2.00000	0.103418
$375$	0	0
$376$	10.4721	0.540059
$377$	− 20.9443i	− 1.07868i
$378$	0	0
$379$	1.52786	0.0784811	0.0392406	−	0.999230i	$-0.487506\pi$
$379$	1.52786	0.0784811	0.0392406	+	0.999230i	$0.487506\pi$
$380$	14.4721i	0.742405i
$381$	0	0
$382$	− 5.52786i	− 0.282830i
$383$	− 17.8885i	− 0.914062i	−0.889451	−	0.457031i	$-0.848913\pi$
$383$	− 17.8885i	− 0.914062i	0.889451	−	0.457031i	$-0.151087\pi$
$384$	0	0
$385$	−5.52786	−0.281726
$386$	23.5967	1.20104
$387$	0	0
$388$	− 1.70820i	− 0.0867209i
$389$	20.1803	1.02318	0.511592	−	0.859229i	$-0.329056\pi$
$389$	20.1803	1.02318	0.511592	+	0.859229i	$0.329056\pi$
$390$	0	0
$391$	4.00000	0.202289
$392$	− 5.47214i	− 0.276385i
$393$	0	0
$394$	−15.8885	−0.800453
$395$	2.11146i	0.106239i
$396$	0	0
$397$	− 3.88854i	− 0.195160i	−0.995228	−	0.0975802i	$-0.968890\pi$
$397$	− 3.88854i	− 0.195160i	0.995228	−	0.0975802i	$-0.0311103\pi$
$398$	8.94427i	0.448336i
$399$	0	0
$400$	−5.00000	−0.250000
$401$	36.0689	1.80119	0.900597	−	0.434655i	$-0.143130\pi$
$401$	36.0689	1.80119	0.900597	+	0.434655i	$0.143130\pi$
$402$	0	0
$403$	12.9443i	0.644800i
$404$	−1.70820	−0.0849863
$405$	0	0
$406$	4.94427	0.245380
$407$	4.00000i	0.198273i
$408$	0	0
$409$	−26.3607	−1.30345	−0.651726	−	0.758455i	$-0.725955\pi$
$409$	−26.3607	−1.30345	−0.651726	+	0.758455i	$0.725955\pi$
$410$	17.2361	0.851229
$411$	0	0
$412$	− 10.9443i	− 0.539186i
$413$	14.4721i	0.712127i
$414$	0	0
$415$	−23.4164	−1.14947
$416$	−5.23607	−0.256719
$417$	0	0
$418$	− 12.9443i	− 0.633125i
$419$	6.36068	0.310740	0.155370	−	0.987856i	$-0.450343\pi$
$419$	6.36068	0.310740	0.155370	+	0.987856i	$0.450343\pi$
$420$	0	0
$421$	−37.4164	−1.82356	−0.911782	−	0.410674i	$-0.865293\pi$
$421$	−37.4164	−1.82356	−0.911782	+	0.410674i	$0.865293\pi$
$422$	− 24.9443i	− 1.21427i
$423$	0	0
$424$	10.9443	0.531501
$425$	− 5.00000i	− 0.242536i
$426$	0	0
$427$	5.52786i	0.267512i
$428$	− 8.00000i	− 0.386695i
$429$	0	0
$430$	27.2361i	1.31344i
$431$	−31.5967	−1.52196	−0.760981	−	0.648774i	$-0.775282\pi$
$431$	−31.5967	−1.52196	−0.760981	+	0.648774i	$0.775282\pi$
$432$	0	0
$433$	28.3607i	1.36293i	0.731852	+	0.681464i	$0.238656\pi$
$433$	28.3607i	1.36293i	−0.731852	+	0.681464i	$0.761344\pi$
$434$	−3.05573	−0.146680
$435$	0	0
$436$	−7.52786	−0.360519
$437$	− 25.8885i	− 1.23842i
$438$	0	0
$439$	−13.5279	−0.645650	−0.322825	−	0.946459i	$-0.604632\pi$
$439$	−13.5279	−0.645650	−0.322825	+	0.946459i	$0.604632\pi$
$440$	4.47214	0.213201
$441$	0	0
$442$	− 5.23607i	− 0.249054i
$443$	− 4.00000i	− 0.190046i	−0.995475	−	0.0950229i	$-0.969708\pi$
$443$	− 4.00000i	− 0.190046i	0.995475	−	0.0950229i	$-0.0302924\pi$
$444$	0	0
$445$	− 27.8885i	− 1.32204i
$446$	−9.05573	−0.428801
$447$	0	0
$448$	− 1.23607i	− 0.0583987i
$449$	17.5967	0.830442	0.415221	−	0.909721i	$-0.363704\pi$
$449$	17.5967	0.830442	0.415221	+	0.909721i	$0.363704\pi$
$450$	0	0
$451$	−15.4164	−0.725930
$452$	− 10.0000i	− 0.470360i
$453$	0	0
$454$	−9.88854	−0.464092
$455$	14.4721i	0.678464i
$456$	0	0
$457$	16.3607i	0.765320i	0.923889	+	0.382660i	$0.124992\pi$
$457$	16.3607i	0.765320i	−0.923889	+	0.382660i	$0.875008\pi$
$458$	21.4164i	1.00072i
$459$	0	0
$460$	8.94427	0.417029
$461$	30.0689	1.40045	0.700224	−	0.713923i	$-0.253084\pi$
$461$	30.0689	1.40045	0.700224	+	0.713923i	$0.253084\pi$
$462$	0	0
$463$	− 18.9443i	− 0.880415i	−0.897896	−	0.440207i	$-0.854905\pi$
$463$	− 18.9443i	− 0.880415i	0.897896	−	0.440207i	$-0.145095\pi$
$464$	−4.00000	−0.185695
$465$	0	0
$466$	5.41641	0.250910
$467$	0.944272i	0.0436957i	0.999761	+	0.0218478i	$0.00695494\pi$
$467$	0.944272i	0.0436957i	−0.999761	+	0.0218478i	$0.993045\pi$
$468$	0	0
$469$	−2.11146	−0.0974980
$470$	23.4164i	1.08012i
$471$	0	0
$472$	− 11.7082i	− 0.538914i
$473$	− 24.3607i	− 1.12011i
$474$	0	0
$475$	−32.3607	−1.48481
$476$	1.23607	0.0566551
$477$	0	0
$478$	− 21.8885i	− 1.00116i
$479$	−35.5967	−1.62646	−0.813228	−	0.581945i	$-0.802292\pi$
$479$	−35.5967	−1.62646	−0.813228	+	0.581945i	$0.802292\pi$
$480$	0	0
$481$	10.4721	0.477488
$482$	9.41641i	0.428906i
$483$	0	0
$484$	7.00000	0.318182
$485$	3.81966	0.173442
$486$	0	0
$487$	16.2918i	0.738252i	0.929379	+	0.369126i	$0.120343\pi$
$487$	16.2918i	0.738252i	−0.929379	+	0.369126i	$0.879657\pi$
$488$	− 4.47214i	− 0.202444i
$489$	0	0
$490$	12.2361	0.552769
$491$	−12.2918	−0.554721	−0.277360	−	0.960766i	$-0.589460\pi$
$491$	−12.2918	−0.554721	−0.277360	+	0.960766i	$0.589460\pi$
$492$	0	0
$493$	− 4.00000i	− 0.180151i
$494$	−33.8885	−1.52472
$495$	0	0
$496$	2.47214	0.111002
$497$	− 13.8885i	− 0.622986i
$498$	0	0
$499$	16.0000	0.716258	0.358129	−	0.933672i	$-0.383415\pi$
$499$	16.0000	0.716258	0.358129	+	0.933672i	$0.383415\pi$
$500$	− 11.1803i	− 0.500000i
$501$	0	0
$502$	10.1803i	0.454371i
$503$	27.4164i	1.22244i	0.791462	+	0.611219i	$0.209320\pi$
$503$	27.4164i	1.22244i	−0.791462	+	0.611219i	$0.790680\pi$
$504$	0	0
$505$	− 3.81966i	− 0.169973i
$506$	−8.00000	−0.355643
$507$	0	0
$508$	14.0000i	0.621150i
$509$	1.34752	0.0597280	0.0298640	−	0.999554i	$-0.490493\pi$
$509$	1.34752	0.0597280	0.0298640	+	0.999554i	$0.490493\pi$
$510$	0	0
$511$	−10.8328	−0.479216
$512$	1.00000i	0.0441942i
$513$	0	0
$514$	−18.0000	−0.793946
$515$	24.4721	1.07837
$516$	0	0
$517$	− 20.9443i	− 0.921128i
$518$	2.47214i	0.108619i
$519$	0	0
$520$	− 11.7082i	− 0.513439i
$521$	−5.81966	−0.254964	−0.127482	−	0.991841i	$-0.540689\pi$
$521$	−5.81966	−0.254964	−0.127482	+	0.991841i	$0.540689\pi$
$522$	0	0
$523$	− 14.2918i	− 0.624937i	−0.949928	−	0.312468i	$-0.898844\pi$
$523$	− 14.2918i	− 0.624937i	0.949928	−	0.312468i	$-0.101156\pi$
$524$	12.4721	0.544848
$525$	0	0
$526$	−4.00000	−0.174408
$527$	2.47214i	0.107688i
$528$	0	0
$529$	7.00000	0.304348
$530$	24.4721i	1.06300i
$531$	0	0
$532$	− 8.00000i	− 0.346844i
$533$	40.3607i	1.74822i
$534$	0	0
$535$	17.8885	0.773389
$536$	1.70820	0.0737832
$537$	0	0
$538$	− 7.05573i	− 0.304194i
$539$	−10.9443	−0.471403
$540$	0	0
$541$	−31.3050	−1.34590	−0.672952	−	0.739686i	$-0.734974\pi$
$541$	−31.3050	−1.34590	−0.672952	+	0.739686i	$0.734974\pi$
$542$	4.00000i	0.171815i
$543$	0	0
$544$	−1.00000	−0.0428746
$545$	− 16.8328i	− 0.721039i
$546$	0	0
$547$	− 0.944272i	− 0.0403742i	−0.999796	−	0.0201871i	$-0.993574\pi$
$547$	− 0.944272i	− 0.0403742i	0.999796	−	0.0201871i	$-0.00642618\pi$
$548$	2.00000i	0.0854358i
$549$	0	0
$550$	10.0000i	0.426401i
$551$	−25.8885	−1.10289
$552$	0	0
$553$	− 1.16718i	− 0.0496337i
$554$	26.3607	1.11996
$555$	0	0
$556$	1.52786	0.0647959
$557$	− 25.0557i	− 1.06165i	−0.847483	−	0.530823i	$-0.821883\pi$
$557$	− 25.0557i	− 1.06165i	0.847483	−	0.530823i	$-0.178117\pi$
$558$	0	0
$559$	−63.7771	−2.69748
$560$	2.76393	0.116797
$561$	0	0
$562$	− 1.41641i	− 0.0597476i
$563$	− 2.11146i	− 0.0889873i	−0.999010	−	0.0444936i	$-0.985833\pi$
$563$	− 2.11146i	− 0.0889873i	0.999010	−	0.0444936i	$-0.0141674\pi$
$564$	0	0
$565$	22.3607	0.940721
$566$	25.8885	1.08818
$567$	0	0
$568$	11.2361i	0.471455i
$569$	6.00000	0.251533	0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000	0.251533	0.125767	+	0.992060i	$0.459861\pi$
$570$	0	0
$571$	36.3607	1.52165	0.760824	−	0.648959i	$-0.224795\pi$
$571$	36.3607	1.52165	0.760824	+	0.648959i	$0.224795\pi$
$572$	10.4721i	0.437862i
$573$	0	0
$574$	−9.52786	−0.397685
$575$	20.0000i	0.834058i
$576$	0	0
$577$	35.4164i	1.47440i	0.675672	+	0.737202i	$0.263853\pi$
$577$	35.4164i	1.47440i	−0.675672	+	0.737202i	$0.736147\pi$
$578$	− 1.00000i	− 0.0415945i
$579$	0	0
$580$	− 8.94427i	− 0.371391i
$581$	12.9443	0.537019
$582$	0	0
$583$	− 21.8885i	− 0.906531i
$584$	8.76393	0.362654
$585$	0	0
$586$	29.4164	1.21518
$587$	− 5.52786i	− 0.228159i	−0.993472	−	0.114080i	$-0.963608\pi$
$587$	− 5.52786i	− 0.228159i	0.993472	−	0.114080i	$-0.0363919\pi$
$588$	0	0
$589$	16.0000	0.659269
$590$	26.1803	1.07783
$591$	0	0
$592$	− 2.00000i	− 0.0821995i
$593$	− 30.9443i	− 1.27073i	−0.772212	−	0.635364i	$-0.780850\pi$
$593$	− 30.9443i	− 1.27073i	0.772212	−	0.635364i	$-0.219150\pi$
$594$	0	0
$595$	2.76393i	0.113310i
$596$	−12.1803	−0.498926
$597$	0	0
$598$	20.9443i	0.856475i
$599$	25.5279	1.04304	0.521520	−	0.853239i	$-0.325365\pi$
$599$	25.5279	1.04304	0.521520	+	0.853239i	$0.325365\pi$
$600$	0	0
$601$	−28.4721	−1.16140	−0.580701	−	0.814117i	$-0.697222\pi$
$601$	−28.4721	−1.16140	−0.580701	+	0.814117i	$0.697222\pi$
$602$	− 15.0557i	− 0.613626i
$603$	0	0
$604$	4.00000	0.162758
$605$	15.6525i	0.636364i
$606$	0	0
$607$	31.1246i	1.26331i	0.775250	+	0.631655i	$0.217624\pi$
$607$	31.1246i	1.26331i	−0.775250	+	0.631655i	$0.782376\pi$
$608$	6.47214i	0.262480i
$609$	0	0
$610$	10.0000	0.404888
$611$	−54.8328	−2.21830
$612$	0	0
$613$	− 42.1803i	− 1.70365i	−0.523828	−	0.851824i	$-0.675497\pi$
$613$	− 42.1803i	− 1.70365i	0.523828	−	0.851824i	$-0.324503\pi$
$614$	4.18034	0.168705
$615$	0	0
$616$	−2.47214	−0.0996052
$617$	− 20.4721i	− 0.824177i	−0.911144	−	0.412089i	$-0.864799\pi$
$617$	− 20.4721i	− 0.824177i	0.911144	−	0.412089i	$-0.135201\pi$
$618$	0	0
$619$	48.9443	1.96724	0.983618	−	0.180264i	$-0.0576953\pi$
$619$	48.9443	1.96724	0.983618	+	0.180264i	$0.0576953\pi$
$620$	5.52786i	0.222004i
$621$	0	0
$622$	− 19.5967i	− 0.785758i
$623$	15.4164i	0.617645i
$624$	0	0
$625$	25.0000	1.00000
$626$	21.7082	0.867634
$627$	0	0
$628$	10.1803i	0.406240i
$629$	2.00000	0.0797452
$630$	0	0
$631$	−9.88854	−0.393657	−0.196828	−	0.980438i	$-0.563064\pi$
$631$	−9.88854	−0.393657	−0.196828	+	0.980438i	$0.563064\pi$
$632$	0.944272i	0.0375611i
$633$	0	0
$634$	−11.8885	−0.472154
$635$	−31.3050	−1.24230
$636$	0	0
$637$	28.6525i	1.13525i
$638$	8.00000i	0.316723i
$639$	0	0
$640$	−2.23607	−0.0883883
$641$	−11.7082	−0.462446	−0.231223	−	0.972901i	$-0.574273\pi$
$641$	−11.7082	−0.462446	−0.231223	+	0.972901i	$0.574273\pi$
$642$	0	0
$643$	− 27.4164i	− 1.08120i	−0.841281	−	0.540599i	$-0.818198\pi$
$643$	− 27.4164i	− 1.08120i	0.841281	−	0.540599i	$-0.181802\pi$
$644$	−4.94427	−0.194832
$645$	0	0
$646$	−6.47214	−0.254643
$647$	− 20.9443i	− 0.823404i	−0.911318	−	0.411702i	$-0.864934\pi$
$647$	− 20.9443i	− 0.823404i	0.911318	−	0.411702i	$-0.135066\pi$
$648$	0	0
$649$	−23.4164	−0.919174
$650$	26.1803	1.02688
$651$	0	0
$652$	22.4721i	0.880077i
$653$	3.52786i	0.138056i	0.997615	+	0.0690280i	$0.0219898\pi$
$653$	3.52786i	0.138056i	−0.997615	+	0.0690280i	$0.978010\pi$
$654$	0	0
$655$	27.8885i	1.08970i
$656$	7.70820	0.300955
$657$	0	0
$658$	− 12.9443i	− 0.504620i
$659$	28.2918	1.10209	0.551046	−	0.834475i	$-0.314229\pi$
$659$	28.2918	1.10209	0.551046	+	0.834475i	$0.314229\pi$
$660$	0	0
$661$	13.4164	0.521838	0.260919	−	0.965361i	$-0.415974\pi$
$661$	13.4164	0.521838	0.260919	+	0.965361i	$0.415974\pi$
$662$	− 3.41641i	− 0.132782i
$663$	0	0
$664$	−10.4721	−0.406398
$665$	17.8885	0.693688
$666$	0	0
$667$	16.0000i	0.619522i
$668$	13.8885i	0.537364i
$669$	0	0
$670$	3.81966i	0.147566i
$671$	−8.94427	−0.345290
$672$	0	0
$673$	4.18034i	0.161140i	0.996749	+	0.0805701i	$0.0256741\pi$
$673$	4.18034i	0.161140i	−0.996749	+	0.0805701i	$0.974326\pi$
$674$	−20.1803	−0.777318
$675$	0	0
$676$	14.4164	0.554477
$677$	− 9.05573i	− 0.348040i	−0.984742	−	0.174020i	$-0.944324\pi$
$677$	− 9.05573i	− 0.348040i	0.984742	−	0.174020i	$-0.0556757\pi$
$678$	0	0
$679$	−2.11146	−0.0810303
$680$	− 2.23607i	− 0.0857493i
$681$	0	0
$682$	− 4.94427i	− 0.189326i
$683$	32.9443i	1.26058i	0.776361	+	0.630289i	$0.217064\pi$
$683$	32.9443i	1.26058i	−0.776361	+	0.630289i	$0.782936\pi$
$684$	0	0
$685$	−4.47214	−0.170872
$686$	−15.4164	−0.588601
$687$	0	0
$688$	12.1803i	0.464371i
$689$	−57.3050	−2.18314
$690$	0	0
$691$	−4.94427	−0.188089	−0.0940445	−	0.995568i	$-0.529980\pi$
$691$	−4.94427	−0.188089	−0.0940445	+	0.995568i	$0.529980\pi$
$692$	− 15.8885i	− 0.603992i
$693$	0	0
$694$	22.8328	0.866722
$695$	3.41641i	0.129592i
$696$	0	0
$697$	7.70820i	0.291969i
$698$	28.4721i	1.07769i
$699$	0	0
$700$	6.18034i	0.233595i
$701$	−11.5967	−0.438003	−0.219002	−	0.975725i	$-0.570280\pi$
$701$	−11.5967	−0.438003	−0.219002	+	0.975725i	$0.570280\pi$
$702$	0	0
$703$	− 12.9443i	− 0.488202i
$704$	2.00000	0.0753778
$705$	0	0
$706$	−14.9443	−0.562435
$707$	2.11146i	0.0794095i
$708$	0	0
$709$	39.5279	1.48450	0.742250	−	0.670123i	$-0.233759\pi$
$709$	39.5279	1.48450	0.742250	+	0.670123i	$0.233759\pi$
$710$	−25.1246	−0.942910
$711$	0	0
$712$	− 12.4721i	− 0.467413i
$713$	− 9.88854i	− 0.370329i
$714$	0	0
$715$	−23.4164	−0.875724
$716$	7.70820	0.288069
$717$	0	0
$718$	− 9.52786i	− 0.355577i
$719$	−36.5410	−1.36275	−0.681375	−	0.731934i	$-0.738618\pi$
$719$	−36.5410	−1.36275	−0.681375	+	0.731934i	$0.738618\pi$
$720$	0	0
$721$	−13.5279	−0.503804
$722$	22.8885i	0.851823i
$723$	0	0
$724$	−0.472136	−0.0175468
$725$	20.0000	0.742781
$726$	0	0
$727$	23.8885i	0.885977i	0.896527	+	0.442989i	$0.146082\pi$
$727$	23.8885i	0.885977i	−0.896527	+	0.442989i	$0.853918\pi$
$728$	6.47214i	0.239873i
$729$	0	0
$730$	19.5967i	0.725308i
$731$	−12.1803	−0.450506
$732$	0	0
$733$	38.5410i	1.42355i	0.702410	+	0.711773i	$0.252107\pi$
$733$	38.5410i	1.42355i	−0.702410	+	0.711773i	$0.747893\pi$
$734$	−5.81966	−0.214808
$735$	0	0
$736$	4.00000	0.147442
$737$	− 3.41641i	− 0.125845i
$738$	0	0
$739$	−33.5279	−1.23334	−0.616671	−	0.787221i	$-0.711519\pi$
$739$	−33.5279	−1.23334	−0.616671	+	0.787221i	$0.711519\pi$
$740$	4.47214	0.164399
$741$	0	0
$742$	− 13.5279i	− 0.496624i
$743$	− 33.5279i	− 1.23002i	−0.788520	−	0.615009i	$-0.789152\pi$
$743$	− 33.5279i	− 1.23002i	0.788520	−	0.615009i	$-0.210848\pi$
$744$	0	0
$745$	− 27.2361i	− 0.997852i
$746$	1.23607	0.0452557
$747$	0	0
$748$	2.00000i	0.0731272i
$749$	−9.88854	−0.361320
$750$	0	0
$751$	29.8885	1.09065	0.545324	−	0.838225i	$-0.316407\pi$
$751$	29.8885	1.09065	0.545324	+	0.838225i	$0.316407\pi$
$752$	10.4721i	0.381880i
$753$	0	0
$754$	20.9443	0.762745
$755$	8.94427i	0.325515i
$756$	0	0
$757$	37.5967i	1.36648i	0.730195	+	0.683239i	$0.239429\pi$
$757$	37.5967i	1.36648i	−0.730195	+	0.683239i	$0.760571\pi$
$758$	1.52786i	0.0554945i
$759$	0	0
$760$	−14.4721	−0.524960
$761$	−52.8328	−1.91519	−0.957594	−	0.288121i	$-0.906969\pi$
$761$	−52.8328	−1.91519	−0.957594	+	0.288121i	$0.906969\pi$
$762$	0	0
$763$	9.30495i	0.336862i
$764$	5.52786	0.199991
$765$	0	0
$766$	17.8885	0.646339
$767$	61.3050i	2.21359i
$768$	0	0
$769$	34.3607	1.23908	0.619539	−	0.784966i	$-0.287320\pi$
$769$	34.3607	1.23908	0.619539	+	0.784966i	$0.287320\pi$
$770$	− 5.52786i	− 0.199210i
$771$	0	0
$772$	23.5967i	0.849266i
$773$	− 33.4164i	− 1.20190i	−0.799285	−	0.600952i	$-0.794788\pi$
$773$	− 33.4164i	− 1.20190i	0.799285	−	0.600952i	$-0.205212\pi$
$774$	0	0
$775$	−12.3607	−0.444009
$776$	1.70820	0.0613209
$777$	0	0
$778$	20.1803i	0.723500i
$779$	49.8885	1.78744
$780$	0	0
$781$	22.4721	0.804116
$782$	4.00000i	0.143040i
$783$	0	0
$784$	5.47214	0.195433
$785$	−22.7639	−0.812480
$786$	0	0
$787$	− 46.4721i	− 1.65655i	−0.560320	−	0.828276i	$-0.689322\pi$
$787$	− 46.4721i	− 1.65655i	0.560320	−	0.828276i	$-0.310678\pi$
$788$	− 15.8885i	− 0.566006i
$789$	0	0
$790$	−2.11146	−0.0751222
$791$	−12.3607	−0.439495
$792$	0	0
$793$	23.4164i	0.831541i
$794$	3.88854	0.137999
$795$	0	0
$796$	−8.94427	−0.317021
$797$	46.0000i	1.62940i	0.579880	+	0.814702i	$0.303099\pi$
$797$	46.0000i	1.62940i	−0.579880	+	0.814702i	$0.696901\pi$
$798$	0	0
$799$	−10.4721	−0.370478
$800$	− 5.00000i	− 0.176777i
$801$	0	0
$802$	36.0689i	1.27364i
$803$	− 17.5279i	− 0.618545i
$804$	0	0
$805$	− 11.0557i	− 0.389663i
$806$	−12.9443	−0.455943
$807$	0	0
$808$	− 1.70820i	− 0.0600944i
$809$	47.1246	1.65681	0.828407	−	0.560127i	$-0.189248\pi$
$809$	47.1246	1.65681	0.828407	+	0.560127i	$0.189248\pi$
$810$	0	0
$811$	42.2492	1.48357	0.741785	−	0.670637i	$-0.233979\pi$
$811$	42.2492	1.48357	0.741785	+	0.670637i	$0.233979\pi$
$812$	4.94427i	0.173510i
$813$	0	0
$814$	−4.00000	−0.140200
$815$	−50.2492	−1.76015
$816$	0	0
$817$	78.8328i	2.75801i
$818$	− 26.3607i	− 0.921680i
$819$	0	0
$820$	17.2361i	0.601910i
$821$	−15.4164	−0.538036	−0.269018	−	0.963135i	$-0.586699\pi$
$821$	−15.4164	−0.538036	−0.269018	+	0.963135i	$0.586699\pi$
$822$	0	0
$823$	38.1803i	1.33088i	0.746450	+	0.665441i	$0.231757\pi$
$823$	38.1803i	1.33088i	−0.746450	+	0.665441i	$0.768243\pi$
$824$	10.9443	0.381262
$825$	0	0
$826$	−14.4721	−0.503550
$827$	− 23.0557i	− 0.801726i	−0.916138	−	0.400863i	$-0.868710\pi$
$827$	− 23.0557i	− 0.801726i	0.916138	−	0.400863i	$-0.131290\pi$
$828$	0	0
$829$	−34.0000	−1.18087	−0.590434	−	0.807086i	$-0.701044\pi$
$829$	−34.0000	−1.18087	−0.590434	+	0.807086i	$0.701044\pi$
$830$	− 23.4164i	− 0.812795i
$831$	0	0
$832$	− 5.23607i	− 0.181528i
$833$	5.47214i	0.189598i
$834$	0	0
$835$	−31.0557	−1.07473
$836$	12.9443	0.447687
$837$	0	0
$838$	6.36068i	0.219726i
$839$	−22.0689	−0.761902	−0.380951	−	0.924595i	$-0.624403\pi$
$839$	−22.0689	−0.761902	−0.380951	+	0.924595i	$0.624403\pi$
$840$	0	0
$841$	−13.0000	−0.448276
$842$	− 37.4164i	− 1.28945i
$843$	0	0
$844$	24.9443	0.858617
$845$	32.2361i	1.10895i
$846$	0	0
$847$	− 8.65248i	− 0.297303i
$848$	10.9443i	0.375828i
$849$	0	0
$850$	5.00000	0.171499
$851$	−8.00000	−0.274236
$852$	0	0
$853$	− 6.36068i	− 0.217786i	−0.994054	−	0.108893i	$-0.965269\pi$
$853$	− 6.36068i	− 0.217786i	0.994054	−	0.108893i	$-0.0347305\pi$
$854$	−5.52786	−0.189160
$855$	0	0
$856$	8.00000	0.273434
$857$	50.3607i	1.72029i	0.510051	+	0.860144i	$0.329626\pi$
$857$	50.3607i	1.72029i	−0.510051	+	0.860144i	$0.670374\pi$
$858$	0	0
$859$	−7.05573	−0.240738	−0.120369	−	0.992729i	$-0.538408\pi$
$859$	−7.05573	−0.240738	−0.120369	+	0.992729i	$0.538408\pi$
$860$	−27.2361	−0.928742
$861$	0	0
$862$	− 31.5967i	− 1.07619i
$863$	21.3050i	0.725229i	0.931939	+	0.362614i	$0.118116\pi$
$863$	21.3050i	0.725229i	−0.931939	+	0.362614i	$0.881884\pi$
$864$	0	0
$865$	35.5279	1.20798
$866$	−28.3607	−0.963735
$867$	0	0
$868$	− 3.05573i	− 0.103718i
$869$	1.88854	0.0640645
$870$	0	0
$871$	−8.94427	−0.303065
$872$	− 7.52786i	− 0.254926i
$873$	0	0
$874$	25.8885	0.875693
$875$	−13.8197	−0.467190
$876$	0	0
$877$	40.4721i	1.36665i	0.730116	+	0.683323i	$0.239466\pi$
$877$	40.4721i	1.36665i	−0.730116	+	0.683323i	$0.760534\pi$
$878$	− 13.5279i	− 0.456543i
$879$	0	0
$880$	4.47214i	0.150756i
$881$	7.12461	0.240034	0.120017	−	0.992772i	$-0.461705\pi$
$881$	7.12461	0.240034	0.120017	+	0.992772i	$0.461705\pi$
$882$	0	0
$883$	− 27.2361i	− 0.916567i	−0.888806	−	0.458283i	$-0.848465\pi$
$883$	− 27.2361i	− 0.916567i	0.888806	−	0.458283i	$-0.151535\pi$
$884$	5.23607	0.176108
$885$	0	0
$886$	4.00000	0.134383
$887$	8.94427i	0.300319i	0.988662	+	0.150160i	$0.0479788\pi$
$887$	8.94427i	0.300319i	−0.988662	+	0.150160i	$0.952021\pi$
$888$	0	0
$889$	17.3050	0.580389
$890$	27.8885	0.934826
$891$	0	0
$892$	− 9.05573i	− 0.303208i
$893$	67.7771i	2.26807i
$894$	0	0
$895$	17.2361i	0.576138i
$896$	1.23607	0.0412941
$897$	0	0
$898$	17.5967i	0.587211i
$899$	−9.88854	−0.329801
$900$	0	0
$901$	−10.9443	−0.364607
$902$	− 15.4164i	− 0.513310i
$903$	0	0
$904$	10.0000	0.332595
$905$	− 1.05573i	− 0.0350936i
$906$	0	0
$907$	− 4.36068i	− 0.144794i	−0.997376	−	0.0723970i	$-0.976935\pi$
$907$	− 4.36068i	− 0.144794i	0.997376	−	0.0723970i	$-0.0230649\pi$
$908$	− 9.88854i	− 0.328163i
$909$	0	0
$910$	−14.4721	−0.479747
$911$	12.5410	0.415503	0.207751	−	0.978182i	$-0.433386\pi$
$911$	12.5410	0.415503	0.207751	+	0.978182i	$0.433386\pi$
$912$	0	0
$913$	20.9443i	0.693154i
$914$	−16.3607	−0.541163
$915$	0	0
$916$	−21.4164	−0.707618
$917$	− 15.4164i	− 0.509095i
$918$	0	0
$919$	−52.7214	−1.73912	−0.869559	−	0.493830i	$-0.835597\pi$
$919$	−52.7214	−1.73912	−0.869559	+	0.493830i	$0.835597\pi$
$920$	8.94427i	0.294884i
$921$	0	0
$922$	30.0689i	0.990266i
$923$	− 58.8328i	− 1.93651i
$924$	0	0
$925$	10.0000i	0.328798i
$926$	18.9443	0.622547
$927$	0	0
$928$	− 4.00000i	− 0.131306i
$929$	24.0689	0.789674	0.394837	−	0.918751i	$-0.370801\pi$
$929$	24.0689	0.789674	0.394837	+	0.918751i	$0.370801\pi$
$930$	0	0
$931$	35.4164	1.16073
$932$	5.41641i	0.177420i
$933$	0	0
$934$	−0.944272	−0.0308975
$935$	−4.47214	−0.146254
$936$	0	0
$937$	− 17.5279i	− 0.572610i	−0.958138	−	0.286305i	$-0.907573\pi$
$937$	− 17.5279i	− 0.572610i	0.958138	−	0.286305i	$-0.0924271\pi$
$938$	− 2.11146i	− 0.0689415i
$939$	0	0
$940$	−23.4164	−0.763759
$941$	−13.3050	−0.433729	−0.216865	−	0.976202i	$-0.569583\pi$
$941$	−13.3050	−0.433729	−0.216865	+	0.976202i	$0.569583\pi$
$942$	0	0
$943$	− 30.8328i	− 1.00405i
$944$	11.7082	0.381070
$945$	0	0
$946$	24.3607	0.792034
$947$	48.7214i	1.58323i	0.611019	+	0.791616i	$0.290760\pi$
$947$	48.7214i	1.58323i	−0.611019	+	0.791616i	$0.709240\pi$
$948$	0	0
$949$	−45.8885	−1.48961
$950$	− 32.3607i	− 1.04992i
$951$	0	0
$952$	1.23607i	0.0400612i
$953$	− 26.9443i	− 0.872811i	−0.899750	−	0.436405i	$-0.856251\pi$
$953$	− 26.9443i	− 0.872811i	0.899750	−	0.436405i	$-0.143749\pi$
$954$	0	0
$955$	12.3607i	0.399982i
$956$	21.8885	0.707926
$957$	0	0
$958$	− 35.5967i	− 1.15008i
$959$	2.47214	0.0798294
$960$	0	0
$961$	−24.8885	−0.802856
$962$	10.4721i	0.337635i
$963$	0	0
$964$	−9.41641	−0.303282
$965$	−52.7639	−1.69853
$966$	0	0
$967$	0.472136i	0.0151829i	0.999971	+	0.00759143i	$0.00241645\pi$
$967$	0.472136i	0.0151829i	−0.999971	+	0.00759143i	$0.997584\pi$
$968$	7.00000i	0.224989i
$969$	0	0
$970$	3.81966i	0.122642i
$971$	22.7639	0.730529	0.365265	−	0.930904i	$-0.380978\pi$
$971$	22.7639	0.730529	0.365265	+	0.930904i	$0.380978\pi$
$972$	0	0
$973$	− 1.88854i	− 0.0605439i
$974$	−16.2918	−0.522023
$975$	0	0
$976$	4.47214	0.143150
$977$	26.9443i	0.862024i	0.902346	+	0.431012i	$0.141843\pi$
$977$	26.9443i	0.862024i	−0.902346	+	0.431012i	$0.858157\pi$
$978$	0	0
$979$	−24.9443	−0.797222
$980$	12.2361i	0.390867i
$981$	0	0
$982$	− 12.2918i	− 0.392247i
$983$	− 11.4164i	− 0.364127i	−0.983287	−	0.182063i	$-0.941722\pi$
$983$	− 11.4164i	− 0.364127i	0.983287	−	0.182063i	$-0.0582776\pi$
$984$	0	0
$985$	35.5279	1.13201
$986$	4.00000	0.127386
$987$	0	0
$988$	− 33.8885i	− 1.07814i
$989$	48.7214	1.54925
$990$	0	0
$991$	−10.8328	−0.344116	−0.172058	−	0.985087i	$-0.555042\pi$
$991$	−10.8328	−0.344116	−0.172058	+	0.985087i	$0.555042\pi$
$992$	2.47214i	0.0784904i
$993$	0	0
$994$	13.8885	0.440518
$995$	− 20.0000i	− 0.634043i
$996$	0	0
$997$	− 41.0557i	− 1.30025i	−0.759828	−	0.650124i	$-0.774717\pi$
$997$	− 41.0557i	− 1.30025i	0.759828	−	0.650124i	$-0.225283\pi$
$998$	16.0000i	0.506471i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1530.2.d.f.919.3		4
3.2	odd	2		510.2.d.b.409.2	✓	4
5.2	odd	4		7650.2.a.cx.1.1		2
5.3	odd	4		7650.2.a.da.1.2		2
5.4	even	2	inner	1530.2.d.f.919.2		4
12.11	even	2		4080.2.m.m.2449.4		4
15.2	even	4		2550.2.a.bk.1.1		2
15.8	even	4		2550.2.a.bh.1.2		2
15.14	odd	2		510.2.d.b.409.3	yes	4
60.59	even	2		4080.2.m.m.2449.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
510.2.d.b.409.2	✓	4	3.2	odd	2
510.2.d.b.409.3	yes	4	15.14	odd	2
1530.2.d.f.919.2		4	5.4	even	2	inner
1530.2.d.f.919.3		4	1.1	even	1	trivial
2550.2.a.bh.1.2		2	15.8	even	4
2550.2.a.bk.1.1		2	15.2	even	4
4080.2.m.m.2449.1		4	60.59	even	2
4080.2.m.m.2449.4		4	12.11	even	2
7650.2.a.cx.1.1		2	5.2	odd	4
7650.2.a.da.1.2		2	5.3	odd	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:	\( N \)	\(=\)	\( 1530 = 2 \cdot 3^{2} \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1530.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{4} -2.23607i q^{5} +1.23607i q^{7} -1.00000i q^{8} +2.23607 q^{10} -2.00000 q^{11} +5.23607i q^{13} -1.23607 q^{14} +1.00000 q^{16} +1.00000i q^{17} +6.47214 q^{19} +2.23607i q^{20} -2.00000i q^{22} -4.00000i q^{23} -5.00000 q^{25} -5.23607 q^{26} -1.23607i q^{28} -4.00000 q^{29} +2.47214 q^{31} +1.00000i q^{32} -1.00000 q^{34} +2.76393 q^{35} -2.00000i q^{37} +6.47214i q^{38} -2.23607 q^{40} +7.70820 q^{41} +12.1803i q^{43} +2.00000 q^{44} +4.00000 q^{46} +10.4721i q^{47} +5.47214 q^{49} -5.00000i q^{50} -5.23607i q^{52} +10.9443i q^{53} +4.47214i q^{55} +1.23607 q^{56} -4.00000i q^{58} +11.7082 q^{59} +4.47214 q^{61} +2.47214i q^{62} -1.00000 q^{64} +11.7082 q^{65} +1.70820i q^{67} -1.00000i q^{68} +2.76393i q^{70} -11.2361 q^{71} +8.76393i q^{73} +2.00000 q^{74} -6.47214 q^{76} -2.47214i q^{77} -0.944272 q^{79} -2.23607i q^{80} +7.70820i q^{82} -10.4721i q^{83} +2.23607 q^{85} -12.1803 q^{86} +2.00000i q^{88} +12.4721 q^{89} -6.47214 q^{91} +4.00000i q^{92} -10.4721 q^{94} -14.4721i q^{95} +1.70820i q^{97} +5.47214i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4} - 8 q^{11} + 4 q^{14} + 4 q^{16} + 8 q^{19} - 20 q^{25} - 12 q^{26} - 16 q^{29} - 8 q^{31} - 4 q^{34} + 20 q^{35} + 4 q^{41} + 8 q^{44} + 16 q^{46} + 4 q^{49} - 4 q^{56} + 20 q^{59} - 4 q^{64}+ \cdots - 24 q^{94}+O(q^{100}) \) 4 * q - 4 * q^4 - 8 * q^11 + 4 * q^14 + 4 * q^16 + 8 * q^19 - 20 * q^25 - 12 * q^26 - 16 * q^29 - 8 * q^31 - 4 * q^34 + 20 * q^35 + 4 * q^41 + 8 * q^44 + 16 * q^46 + 4 * q^49 - 4 * q^56 + 20 * q^59 - 4 * q^64 + 20 * q^65 - 36 * q^71 + 8 * q^74 - 8 * q^76 + 32 * q^79 - 4 * q^86 + 32 * q^89 - 8 * q^91 - 24 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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