LMFDB - Embedded newform 1575.2.d.d.1324.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1575,2,Mod(1324,1575)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1575, 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("1575.1324");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1575 = 3^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1575.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$12.5764383184$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 105)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1324.1
Root			$1.61803i$ of defining polynomial
Character	$\chi$	$=$	1575.1324
Dual form			1575.2.d.d.1324.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.23607i q^{2} -3.00000 q^{4} -1.00000i q^{7} +2.23607i q^{8} +O(q^{10})$	$q-2.23607i q^{2} -3.00000 q^{4} -1.00000i q^{7} +2.23607i q^{8} -6.47214 q^{11} +4.47214i q^{13} -2.23607 q^{14} -1.00000 q^{16} -2.00000i q^{17} +2.47214 q^{19} +14.4721i q^{22} -4.00000i q^{23} +10.0000 q^{26} +3.00000i q^{28} -2.00000 q^{29} +1.52786 q^{31} +6.70820i q^{32} -4.47214 q^{34} +6.94427i q^{37} -5.52786i q^{38} +2.00000 q^{41} +8.94427i q^{43} +19.4164 q^{44} -8.94427 q^{46} +12.9443i q^{47} -1.00000 q^{49} -13.4164i q^{52} +3.52786i q^{53} +2.23607 q^{56} +4.47214i q^{58} -8.94427 q^{59} -2.00000 q^{61} -3.41641i q^{62} +13.0000 q^{64} +4.00000i q^{67} +6.00000i q^{68} -5.52786 q^{71} -12.4721i q^{73} +15.5279 q^{74} -7.41641 q^{76} +6.47214i q^{77} -12.9443 q^{79} -4.47214i q^{82} +16.9443i q^{83} +20.0000 q^{86} -14.4721i q^{88} -2.00000 q^{89} +4.47214 q^{91} +12.0000i q^{92} +28.9443 q^{94} -8.47214i q^{97} +2.23607i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 12 q^{4}+O(q^{10}) $ 4 * q - 12 * q^4	$ 4 q - 12 q^{4} - 8 q^{11} - 4 q^{16} - 8 q^{19} + 40 q^{26} - 8 q^{29} + 24 q^{31} + 8 q^{41} + 24 q^{44} - 4 q^{49} - 8 q^{61} + 52 q^{64} - 40 q^{71} + 80 q^{74} + 24 q^{76} - 16 q^{79} + 80 q^{86} - 8 q^{89} + 80 q^{94}+O(q^{100}) $ 4 * q - 12 * q^4 - 8 * q^11 - 4 * q^16 - 8 * q^19 + 40 * q^26 - 8 * q^29 + 24 * q^31 + 8 * q^41 + 24 * q^44 - 4 * q^49 - 8 * q^61 + 52 * q^64 - 40 * q^71 + 80 * q^74 + 24 * q^76 - 16 * q^79 + 80 * q^86 - 8 * q^89 + 80 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1575\mathbb{Z}\right)^\times$.

$n$	$127$	$451$	$1226$
$\chi(n)$	$-1$	$1$	$1$

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 $

$2$	− 2.23607i	− 1.58114i	−0.612372	−	0.790569i	$-0.709785\pi$
$2$	− 2.23607i	− 1.58114i	0.612372	−	0.790569i	$-0.290215\pi$
$3$	0	0
$3$	0	0
$4$	−3.00000	−1.50000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 1.00000i	− 0.377964i
$7$	− 1.00000i	− 0.377964i
$8$	2.23607i	0.790569i
$9$	0	0
$10$	0	0
$11$	−6.47214	−1.95142	−0.975711	−	0.219061i	$-0.929701\pi$
$11$	−6.47214	−1.95142	−0.975711	+	0.219061i	$0.929701\pi$
$12$	0	0
$13$	4.47214i	1.24035i	0.784465	+	0.620174i	$0.212938\pi$
$13$	4.47214i	1.24035i	−0.784465	+	0.620174i	$0.787062\pi$
$14$	−2.23607	−0.597614
$15$	0	0
$16$	−1.00000	−0.250000
$17$	− 2.00000i	− 0.485071i	−0.970143	−	0.242536i	$-0.922021\pi$
$17$	− 2.00000i	− 0.485071i	0.970143	−	0.242536i	$-0.0779791\pi$
$18$	0	0
$19$	2.47214	0.567147	0.283573	−	0.958951i	$-0.408480\pi$
$19$	2.47214	0.567147	0.283573	+	0.958951i	$0.408480\pi$
$20$	0	0
$21$	0	0
$22$	14.4721i	3.08547i
$23$	− 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
$25$	0	0
$26$	10.0000	1.96116
$27$	0	0
$28$	3.00000i	0.566947i
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	0	0
$31$	1.52786	0.274412	0.137206	−	0.990543i	$-0.456188\pi$
$31$	1.52786	0.274412	0.137206	+	0.990543i	$0.456188\pi$
$32$	6.70820i	1.18585i
$33$	0	0
$34$	−4.47214	−0.766965
$35$	0	0
$36$	0	0
$37$	6.94427i	1.14163i	0.821078	+	0.570816i	$0.193373\pi$
$37$	6.94427i	1.14163i	−0.821078	+	0.570816i	$0.806627\pi$
$38$	− 5.52786i	− 0.896738i
$39$	0	0
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	0	0
$43$	8.94427i	1.36399i	0.731357	+	0.681994i	$0.238887\pi$
$43$	8.94427i	1.36399i	−0.731357	+	0.681994i	$0.761113\pi$
$44$	19.4164	2.92713
$45$	0	0
$46$	−8.94427	−1.31876
$47$	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
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	− 13.4164i	− 1.86052i
$53$	3.52786i	0.484589i	0.970203	+	0.242295i	$0.0779001\pi$
$53$	3.52786i	0.484589i	−0.970203	+	0.242295i	$0.922100\pi$
$54$	0	0
$55$	0	0
$56$	2.23607	0.298807
$57$	0	0
$58$	4.47214i	0.587220i
$59$	−8.94427	−1.16445	−0.582223	−	0.813029i	$-0.697817\pi$
$59$	−8.94427	−1.16445	−0.582223	+	0.813029i	$0.697817\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	− 3.41641i	− 0.433884i
$63$	0	0
$64$	13.0000	1.62500
$65$	0	0
$66$	0	0
$67$	4.00000i	0.488678i	0.969690	+	0.244339i	$0.0785709\pi$
$67$	4.00000i	0.488678i	−0.969690	+	0.244339i	$0.921429\pi$
$68$	6.00000i	0.727607i
$69$	0	0
$70$	0	0
$71$	−5.52786	−0.656037	−0.328018	−	0.944671i	$-0.606381\pi$
$71$	−5.52786	−0.656037	−0.328018	+	0.944671i	$0.606381\pi$
$72$	0	0
$73$	− 12.4721i	− 1.45975i	−0.683579	−	0.729877i	$-0.739578\pi$
$73$	− 12.4721i	− 1.45975i	0.683579	−	0.729877i	$-0.260422\pi$
$74$	15.5279	1.80508
$75$	0	0
$76$	−7.41641	−0.850720
$77$	6.47214i	0.737568i
$78$	0	0
$79$	−12.9443	−1.45634	−0.728172	−	0.685394i	$-0.759630\pi$
$79$	−12.9443	−1.45634	−0.728172	+	0.685394i	$0.759630\pi$
$80$	0	0
$81$	0	0
$82$	− 4.47214i	− 0.493865i
$83$	16.9443i	1.85988i	0.367717	+	0.929938i	$0.380140\pi$
$83$	16.9443i	1.85988i	−0.367717	+	0.929938i	$0.619860\pi$
$84$	0	0
$85$	0	0
$86$	20.0000	2.15666
$87$	0	0
$88$	− 14.4721i	− 1.54273i
$89$	−2.00000	−0.212000	−0.106000	−	0.994366i	$-0.533804\pi$
$89$	−2.00000	−0.212000	−0.106000	+	0.994366i	$0.533804\pi$
$90$	0	0
$91$	4.47214	0.468807
$92$	12.0000i	1.25109i
$93$	0	0
$94$	28.9443	2.98537
$95$	0	0
$96$	0	0
$97$	− 8.47214i	− 0.860215i	−0.902778	−	0.430108i	$-0.858476\pi$
$97$	− 8.47214i	− 0.860215i	0.902778	−	0.430108i	$-0.141524\pi$
$98$	2.23607i	0.225877i
$99$	0	0
$100$	0	0
$101$	14.0000	1.39305	0.696526	−	0.717532i	$-0.254728\pi$
$101$	14.0000	1.39305	0.696526	+	0.717532i	$0.254728\pi$
$102$	0	0
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	−10.0000	−0.980581
$105$	0	0
$106$	7.88854	0.766203
$107$	− 12.9443i	− 1.25137i	−0.780076	−	0.625685i	$-0.784820\pi$
$107$	− 12.9443i	− 1.25137i	0.780076	−	0.625685i	$-0.215180\pi$
$108$	0	0
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	0	0
$111$	0	0
$112$	1.00000i	0.0944911i
$113$	− 0.472136i	− 0.0444148i	−0.999753	−	0.0222074i	$-0.992931\pi$
$113$	− 0.472136i	− 0.0444148i	0.999753	−	0.0222074i	$-0.00706942\pi$
$114$	0	0
$115$	0	0
$116$	6.00000	0.557086
$117$	0	0
$118$	20.0000i	1.84115i
$119$	−2.00000	−0.183340
$120$	0	0
$121$	30.8885	2.80805
$122$	4.47214i	0.404888i
$123$	0	0
$124$	−4.58359	−0.411619
$125$	0	0
$126$	0	0
$127$	4.94427i	0.438733i	0.975643	+	0.219367i	$0.0703991\pi$
$127$	4.94427i	0.438733i	−0.975643	+	0.219367i	$0.929601\pi$
$128$	− 15.6525i	− 1.38350i
$129$	0	0
$130$	0	0
$131$	−4.00000	−0.349482	−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000	−0.349482	−0.174741	+	0.984614i	$0.555909\pi$
$132$	0	0
$133$	− 2.47214i	− 0.214361i
$134$	8.94427	0.772667
$135$	0	0
$136$	4.47214	0.383482
$137$	3.52786i	0.301406i	0.988579	+	0.150703i	$0.0481537\pi$
$137$	3.52786i	0.301406i	−0.988579	+	0.150703i	$0.951846\pi$
$138$	0	0
$139$	−7.41641	−0.629052	−0.314526	−	0.949249i	$-0.601845\pi$
$139$	−7.41641	−0.629052	−0.314526	+	0.949249i	$0.601845\pi$
$140$	0	0
$141$	0	0
$142$	12.3607i	1.03729i
$143$	− 28.9443i	− 2.42044i
$144$	0	0
$145$	0	0
$146$	−27.8885	−2.30807
$147$	0	0
$148$	− 20.8328i	− 1.71245i
$149$	−14.9443	−1.22428	−0.612141	−	0.790748i	$-0.709692\pi$
$149$	−14.9443	−1.22428	−0.612141	+	0.790748i	$0.709692\pi$
$150$	0	0
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	5.52786i	0.448369i
$153$	0	0
$154$	14.4721	1.16620
$155$	0	0
$156$	0	0
$157$	0.472136i	0.0376806i	0.999823	+	0.0188403i	$0.00599740\pi$
$157$	0.472136i	0.0376806i	−0.999823	+	0.0188403i	$0.994003\pi$
$158$	28.9443i	2.30268i
$159$	0	0
$160$	0	0
$161$	−4.00000	−0.315244
$162$	0	0
$163$	− 16.9443i	− 1.32718i	−0.748097	−	0.663589i	$-0.769032\pi$
$163$	− 16.9443i	− 1.32718i	0.748097	−	0.663589i	$-0.230968\pi$
$164$	−6.00000	−0.468521
$165$	0	0
$166$	37.8885	2.94072
$167$	− 8.00000i	− 0.619059i	−0.950890	−	0.309529i	$-0.899829\pi$
$167$	− 8.00000i	− 0.619059i	0.950890	−	0.309529i	$-0.100171\pi$
$168$	0	0
$169$	−7.00000	−0.538462
$170$	0	0
$171$	0	0
$172$	− 26.8328i	− 2.04598i
$173$	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
$175$	0	0
$176$	6.47214	0.487856
$177$	0	0
$178$	4.47214i	0.335201i
$179$	6.47214	0.483750	0.241875	−	0.970307i	$-0.422238\pi$
$179$	6.47214	0.483750	0.241875	+	0.970307i	$0.422238\pi$
$180$	0	0
$181$	1.05573	0.0784717	0.0392358	−	0.999230i	$-0.487508\pi$
$181$	1.05573	0.0784717	0.0392358	+	0.999230i	$0.487508\pi$
$182$	− 10.0000i	− 0.741249i
$183$	0	0
$184$	8.94427	0.659380
$185$	0	0
$186$	0	0
$187$	12.9443i	0.946579i
$188$	− 38.8328i	− 2.83217i
$189$	0	0
$190$	0	0
$191$	−0.583592	−0.0422272	−0.0211136	−	0.999777i	$-0.506721\pi$
$191$	−0.583592	−0.0422272	−0.0211136	+	0.999777i	$0.506721\pi$
$192$	0	0
$193$	− 14.0000i	− 1.00774i	−0.863779	−	0.503871i	$-0.831909\pi$
$193$	− 14.0000i	− 1.00774i	0.863779	−	0.503871i	$-0.168091\pi$
$194$	−18.9443	−1.36012
$195$	0	0
$196$	3.00000	0.214286
$197$	15.5279i	1.10631i	0.833077	+	0.553157i	$0.186577\pi$
$197$	15.5279i	1.10631i	−0.833077	+	0.553157i	$0.813423\pi$
$198$	0	0
$199$	−27.4164	−1.94350	−0.971749	−	0.236017i	$-0.924158\pi$
$199$	−27.4164	−1.94350	−0.971749	+	0.236017i	$0.924158\pi$
$200$	0	0
$201$	0	0
$202$	− 31.3050i	− 2.20261i
$203$	2.00000i	0.140372i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	− 4.47214i	− 0.310087i
$209$	−16.0000	−1.10674
$210$	0	0
$211$	−16.9443	−1.16649	−0.583246	−	0.812296i	$-0.698218\pi$
$211$	−16.9443	−1.16649	−0.583246	+	0.812296i	$0.698218\pi$
$212$	− 10.5836i	− 0.726884i
$213$	0	0
$214$	−28.9443	−1.97859
$215$	0	0
$216$	0	0
$217$	− 1.52786i	− 0.103718i
$218$	− 4.47214i	− 0.302891i
$219$	0	0
$220$	0	0
$221$	8.94427	0.601657
$222$	0	0
$223$	− 12.9443i	− 0.866813i	−0.901199	−	0.433406i	$-0.857312\pi$
$223$	− 12.9443i	− 0.866813i	0.901199	−	0.433406i	$-0.142688\pi$
$224$	6.70820	0.448211
$225$	0	0
$226$	−1.05573	−0.0702260
$227$	0.944272i	0.0626735i	0.999509	+	0.0313368i	$0.00997644\pi$
$227$	0.944272i	0.0626735i	−0.999509	+	0.0313368i	$0.990024\pi$
$228$	0	0
$229$	−23.8885	−1.57860	−0.789300	−	0.614008i	$-0.789556\pi$
$229$	−23.8885	−1.57860	−0.789300	+	0.614008i	$0.789556\pi$
$230$	0	0
$231$	0	0
$232$	− 4.47214i	− 0.293610i
$233$	9.41641i	0.616889i	0.951242	+	0.308445i	$0.0998085\pi$
$233$	9.41641i	0.616889i	−0.951242	+	0.308445i	$0.900192\pi$
$234$	0	0
$235$	0	0
$236$	26.8328	1.74667
$237$	0	0
$238$	4.47214i	0.289886i
$239$	−10.4721	−0.677386	−0.338693	−	0.940897i	$-0.609985\pi$
$239$	−10.4721	−0.677386	−0.338693	+	0.940897i	$0.609985\pi$
$240$	0	0
$241$	−18.9443	−1.22031	−0.610154	−	0.792283i	$-0.708892\pi$
$241$	−18.9443	−1.22031	−0.610154	+	0.792283i	$0.708892\pi$
$242$	− 69.0689i	− 4.43992i
$243$	0	0
$244$	6.00000	0.384111
$245$	0	0
$246$	0	0
$247$	11.0557i	0.703459i
$248$	3.41641i	0.216942i
$249$	0	0
$250$	0	0
$251$	−16.9443	−1.06951	−0.534756	−	0.845006i	$-0.679597\pi$
$251$	−16.9443	−1.06951	−0.534756	+	0.845006i	$0.679597\pi$
$252$	0	0
$253$	25.8885i	1.62760i
$254$	11.0557	0.693698
$255$	0	0
$256$	−9.00000	−0.562500
$257$	18.9443i	1.18171i	0.806777	+	0.590856i	$0.201210\pi$
$257$	18.9443i	1.18171i	−0.806777	+	0.590856i	$0.798790\pi$
$258$	0	0
$259$	6.94427	0.431496
$260$	0	0
$261$	0	0
$262$	8.94427i	0.552579i
$263$	− 7.05573i	− 0.435075i	−0.976052	−	0.217537i	$-0.930198\pi$
$263$	− 7.05573i	− 0.435075i	0.976052	−	0.217537i	$-0.0698024\pi$
$264$	0	0
$265$	0	0
$266$	−5.52786	−0.338935
$267$	0	0
$268$	− 12.0000i	− 0.733017i
$269$	11.8885	0.724857	0.362429	−	0.932012i	$-0.381948\pi$
$269$	11.8885	0.724857	0.362429	+	0.932012i	$0.381948\pi$
$270$	0	0
$271$	−1.52786	−0.0928111	−0.0464056	−	0.998923i	$-0.514777\pi$
$271$	−1.52786	−0.0928111	−0.0464056	+	0.998923i	$0.514777\pi$
$272$	2.00000i	0.121268i
$273$	0	0
$274$	7.88854	0.476564
$275$	0	0
$276$	0	0
$277$	− 18.9443i	− 1.13825i	−0.822251	−	0.569125i	$-0.807282\pi$
$277$	− 18.9443i	− 1.13825i	0.822251	−	0.569125i	$-0.192718\pi$
$278$	16.5836i	0.994618i
$279$	0	0
$280$	0	0
$281$	10.9443	0.652881	0.326440	−	0.945218i	$-0.394151\pi$
$281$	10.9443	0.652881	0.326440	+	0.945218i	$0.394151\pi$
$282$	0	0
$283$	12.0000i	0.713326i	0.934233	+	0.356663i	$0.116086\pi$
$283$	12.0000i	0.713326i	−0.934233	+	0.356663i	$0.883914\pi$
$284$	16.5836	0.984055
$285$	0	0
$286$	−64.7214	−3.82705
$287$	− 2.00000i	− 0.118056i
$288$	0	0
$289$	13.0000	0.764706
$290$	0	0
$291$	0	0
$292$	37.4164i	2.18963i
$293$	− 5.05573i	− 0.295359i	−0.989035	−	0.147679i	$-0.952820\pi$
$293$	− 5.05573i	− 0.295359i	0.989035	−	0.147679i	$-0.0471804\pi$
$294$	0	0
$295$	0	0
$296$	−15.5279	−0.902539
$297$	0	0
$298$	33.4164i	1.93576i
$299$	17.8885	1.03452
$300$	0	0
$301$	8.94427	0.515539
$302$	35.7771i	2.05874i
$303$	0	0
$304$	−2.47214	−0.141787
$305$	0	0
$306$	0	0
$307$	15.0557i	0.859276i	0.903001	+	0.429638i	$0.141359\pi$
$307$	15.0557i	0.859276i	−0.903001	+	0.429638i	$0.858641\pi$
$308$	− 19.4164i	− 1.10635i
$309$	0	0
$310$	0	0
$311$	−25.8885	−1.46800	−0.734002	−	0.679147i	$-0.762350\pi$
$311$	−25.8885	−1.46800	−0.734002	+	0.679147i	$0.762350\pi$
$312$	0	0
$313$	− 17.4164i	− 0.984434i	−0.870473	−	0.492217i	$-0.836187\pi$
$313$	− 17.4164i	− 0.984434i	0.870473	−	0.492217i	$-0.163813\pi$
$314$	1.05573	0.0595782
$315$	0	0
$316$	38.8328	2.18452
$317$	14.3607i	0.806576i	0.915073	+	0.403288i	$0.132133\pi$
$317$	14.3607i	0.806576i	−0.915073	+	0.403288i	$0.867867\pi$
$318$	0	0
$319$	12.9443	0.724740
$320$	0	0
$321$	0	0
$322$	8.94427i	0.498445i
$323$	− 4.94427i	− 0.275107i
$324$	0	0
$325$	0	0
$326$	−37.8885	−2.09845
$327$	0	0
$328$	4.47214i	0.246932i
$329$	12.9443	0.713641
$330$	0	0
$331$	0.944272	0.0519019	0.0259509	−	0.999663i	$-0.491739\pi$
$331$	0.944272	0.0519019	0.0259509	+	0.999663i	$0.491739\pi$
$332$	− 50.8328i	− 2.78981i
$333$	0	0
$334$	−17.8885	−0.978818
$335$	0	0
$336$	0	0
$337$	23.8885i	1.30129i	0.759381	+	0.650646i	$0.225502\pi$
$337$	23.8885i	1.30129i	−0.759381	+	0.650646i	$0.774498\pi$
$338$	15.6525i	0.851382i
$339$	0	0
$340$	0	0
$341$	−9.88854	−0.535495
$342$	0	0
$343$	1.00000i	0.0539949i
$344$	−20.0000	−1.07833
$345$	0	0
$346$	6.58359	0.353936
$347$	− 8.00000i	− 0.429463i	−0.976673	−	0.214731i	$-0.931112\pi$
$347$	− 8.00000i	− 0.429463i	0.976673	−	0.214731i	$-0.0688876\pi$
$348$	0	0
$349$	11.8885	0.636379	0.318190	−	0.948027i	$-0.396925\pi$
$349$	11.8885	0.636379	0.318190	+	0.948027i	$0.396925\pi$
$350$	0	0
$351$	0	0
$352$	− 43.4164i	− 2.31410i
$353$	− 7.88854i	− 0.419865i	−0.977716	−	0.209932i	$-0.932676\pi$
$353$	− 7.88854i	− 0.419865i	0.977716	−	0.209932i	$-0.0673244\pi$
$354$	0	0
$355$	0	0
$356$	6.00000	0.317999
$357$	0	0
$358$	− 14.4721i	− 0.764876i
$359$	18.4721	0.974922	0.487461	−	0.873145i	$-0.337923\pi$
$359$	18.4721	0.974922	0.487461	+	0.873145i	$0.337923\pi$
$360$	0	0
$361$	−12.8885	−0.678344
$362$	− 2.36068i	− 0.124075i
$363$	0	0
$364$	−13.4164	−0.703211
$365$	0	0
$366$	0	0
$367$	− 3.05573i	− 0.159508i	−0.996815	−	0.0797539i	$-0.974587\pi$
$367$	− 3.05573i	− 0.159508i	0.996815	−	0.0797539i	$-0.0254134\pi$
$368$	4.00000i	0.208514i
$369$	0	0
$370$	0	0
$371$	3.52786	0.183158
$372$	0	0
$373$	6.00000i	0.310668i	0.987862	+	0.155334i	$0.0496454\pi$
$373$	6.00000i	0.310668i	−0.987862	+	0.155334i	$0.950355\pi$
$374$	28.9443	1.49667
$375$	0	0
$376$	−28.9443	−1.49269
$377$	− 8.94427i	− 0.460653i
$378$	0	0
$379$	37.8885	1.94620	0.973102	−	0.230375i	$-0.0739953\pi$
$379$	37.8885	1.94620	0.973102	+	0.230375i	$0.0739953\pi$
$380$	0	0
$381$	0	0
$382$	1.30495i	0.0667671i
$383$	8.00000i	0.408781i	0.978889	+	0.204390i	$0.0655212\pi$
$383$	8.00000i	0.408781i	−0.978889	+	0.204390i	$0.934479\pi$
$384$	0	0
$385$	0	0
$386$	−31.3050	−1.59338
$387$	0	0
$388$	25.4164i	1.29032i
$389$	−6.94427	−0.352089	−0.176044	−	0.984382i	$-0.556330\pi$
$389$	−6.94427	−0.352089	−0.176044	+	0.984382i	$0.556330\pi$
$390$	0	0
$391$	−8.00000	−0.404577
$392$	− 2.23607i	− 0.112938i
$393$	0	0
$394$	34.7214	1.74924
$395$	0	0
$396$	0	0
$397$	13.4164i	0.673350i	0.941621	+	0.336675i	$0.109302\pi$
$397$	13.4164i	0.673350i	−0.941621	+	0.336675i	$0.890698\pi$
$398$	61.3050i	3.07294i
$399$	0	0
$400$	0	0
$401$	−10.0000	−0.499376	−0.249688	−	0.968326i	$-0.580328\pi$
$401$	−10.0000	−0.499376	−0.249688	+	0.968326i	$0.580328\pi$
$402$	0	0
$403$	6.83282i	0.340367i
$404$	−42.0000	−2.08958
$405$	0	0
$406$	4.47214	0.221948
$407$	− 44.9443i	− 2.22780i
$408$	0	0
$409$	−11.8885	−0.587851	−0.293925	−	0.955828i	$-0.594962\pi$
$409$	−11.8885	−0.587851	−0.293925	+	0.955828i	$0.594962\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	8.94427i	0.440119i
$414$	0	0
$415$	0	0
$416$	−30.0000	−1.47087
$417$	0	0
$418$	35.7771i	1.74991i
$419$	−29.8885	−1.46015	−0.730075	−	0.683367i	$-0.760515\pi$
$419$	−29.8885	−1.46015	−0.730075	+	0.683367i	$0.760515\pi$
$420$	0	0
$421$	22.0000	1.07221	0.536107	−	0.844150i	$-0.319894\pi$
$421$	22.0000	1.07221	0.536107	+	0.844150i	$0.319894\pi$
$422$	37.8885i	1.84439i
$423$	0	0
$424$	−7.88854	−0.383102
$425$	0	0
$426$	0	0
$427$	2.00000i	0.0967868i
$428$	38.8328i	1.87705i
$429$	0	0
$430$	0	0
$431$	18.4721	0.889771	0.444886	−	0.895587i	$-0.353244\pi$
$431$	18.4721	0.889771	0.444886	+	0.895587i	$0.353244\pi$
$432$	0	0
$433$	16.4721i	0.791600i	0.918337	+	0.395800i	$0.129533\pi$
$433$	16.4721i	0.791600i	−0.918337	+	0.395800i	$0.870467\pi$
$434$	−3.41641	−0.163993
$435$	0	0
$436$	−6.00000	−0.287348
$437$	− 9.88854i	− 0.473033i
$438$	0	0
$439$	−1.52786	−0.0729210	−0.0364605	−	0.999335i	$-0.511608\pi$
$439$	−1.52786	−0.0729210	−0.0364605	+	0.999335i	$0.511608\pi$
$440$	0	0
$441$	0	0
$442$	− 20.0000i	− 0.951303i
$443$	8.00000i	0.380091i	0.981775	+	0.190046i	$0.0608636\pi$
$443$	8.00000i	0.380091i	−0.981775	+	0.190046i	$0.939136\pi$
$444$	0	0
$445$	0	0
$446$	−28.9443	−1.37055
$447$	0	0
$448$	− 13.0000i	− 0.614192i
$449$	−14.0000	−0.660701	−0.330350	−	0.943858i	$-0.607167\pi$
$449$	−14.0000	−0.660701	−0.330350	+	0.943858i	$0.607167\pi$
$450$	0	0
$451$	−12.9443	−0.609522
$452$	1.41641i	0.0666222i
$453$	0	0
$454$	2.11146	0.0990955
$455$	0	0
$456$	0	0
$457$	− 6.94427i	− 0.324839i	−0.986722	−	0.162420i	$-0.948070\pi$
$457$	− 6.94427i	− 0.324839i	0.986722	−	0.162420i	$-0.0519298\pi$
$458$	53.4164i	2.49598i
$459$	0	0
$460$	0	0
$461$	−3.88854	−0.181108	−0.0905538	−	0.995892i	$-0.528864\pi$
$461$	−3.88854	−0.181108	−0.0905538	+	0.995892i	$0.528864\pi$
$462$	0	0
$463$	20.9443i	0.973363i	0.873580	+	0.486681i	$0.161793\pi$
$463$	20.9443i	0.973363i	−0.873580	+	0.486681i	$0.838207\pi$
$464$	2.00000	0.0928477
$465$	0	0
$466$	21.0557	0.975388
$467$	− 8.94427i	− 0.413892i	−0.978352	−	0.206946i	$-0.933648\pi$
$467$	− 8.94427i	− 0.413892i	0.978352	−	0.206946i	$-0.0663524\pi$
$468$	0	0
$469$	4.00000	0.184703
$470$	0	0
$471$	0	0
$472$	− 20.0000i	− 0.920575i
$473$	− 57.8885i	− 2.66172i
$474$	0	0
$475$	0	0
$476$	6.00000	0.275010
$477$	0	0
$478$	23.4164i	1.07104i
$479$	−17.8885	−0.817348	−0.408674	−	0.912680i	$-0.634009\pi$
$479$	−17.8885	−0.817348	−0.408674	+	0.912680i	$0.634009\pi$
$480$	0	0
$481$	−31.0557	−1.41602
$482$	42.3607i	1.92948i
$483$	0	0
$484$	−92.6656	−4.21207
$485$	0	0
$486$	0	0
$487$	20.9443i	0.949076i	0.880235	+	0.474538i	$0.157385\pi$
$487$	20.9443i	0.949076i	−0.880235	+	0.474538i	$0.842615\pi$
$488$	− 4.47214i	− 0.202444i
$489$	0	0
$490$	0	0
$491$	21.3050	0.961479	0.480740	−	0.876863i	$-0.340368\pi$
$491$	21.3050	0.961479	0.480740	+	0.876863i	$0.340368\pi$
$492$	0	0
$493$	4.00000i	0.180151i
$494$	24.7214	1.11227
$495$	0	0
$496$	−1.52786	−0.0686031
$497$	5.52786i	0.247959i
$498$	0	0
$499$	13.8885	0.621737	0.310868	−	0.950453i	$-0.399380\pi$
$499$	13.8885	0.621737	0.310868	+	0.950453i	$0.399380\pi$
$500$	0	0
$501$	0	0
$502$	37.8885i	1.69105i
$503$	− 32.0000i	− 1.42681i	−0.700752	−	0.713405i	$-0.747152\pi$
$503$	− 32.0000i	− 1.42681i	0.700752	−	0.713405i	$-0.252848\pi$
$504$	0	0
$505$	0	0
$506$	57.8885	2.57346
$507$	0	0
$508$	− 14.8328i	− 0.658100i
$509$	−23.8885	−1.05884	−0.529421	−	0.848360i	$-0.677591\pi$
$509$	−23.8885	−1.05884	−0.529421	+	0.848360i	$0.677591\pi$
$510$	0	0
$511$	−12.4721	−0.551735
$512$	− 11.1803i	− 0.494106i
$513$	0	0
$514$	42.3607	1.86845
$515$	0	0
$516$	0	0
$517$	− 83.7771i	− 3.68451i
$518$	− 15.5279i	− 0.682255i
$519$	0	0
$520$	0	0
$521$	19.8885	0.871333	0.435666	−	0.900108i	$-0.356513\pi$
$521$	19.8885	0.871333	0.435666	+	0.900108i	$0.356513\pi$
$522$	0	0
$523$	− 8.94427i	− 0.391106i	−0.980693	−	0.195553i	$-0.937350\pi$
$523$	− 8.94427i	− 0.391106i	0.980693	−	0.195553i	$-0.0626501\pi$
$524$	12.0000	0.524222
$525$	0	0
$526$	−15.7771	−0.687914
$527$	− 3.05573i	− 0.133110i
$528$	0	0
$529$	7.00000	0.304348
$530$	0	0
$531$	0	0
$532$	7.41641i	0.321542i
$533$	8.94427i	0.387419i
$534$	0	0
$535$	0	0
$536$	−8.94427	−0.386334
$537$	0	0
$538$	− 26.5836i	− 1.14610i
$539$	6.47214	0.278775
$540$	0	0
$541$	−11.8885	−0.511128	−0.255564	−	0.966792i	$-0.582261\pi$
$541$	−11.8885	−0.511128	−0.255564	+	0.966792i	$0.582261\pi$
$542$	3.41641i	0.146747i
$543$	0	0
$544$	13.4164	0.575224
$545$	0	0
$546$	0	0
$547$	− 5.88854i	− 0.251776i	−0.992044	−	0.125888i	$-0.959822\pi$
$547$	− 5.88854i	− 0.251776i	0.992044	−	0.125888i	$-0.0401780\pi$
$548$	− 10.5836i	− 0.452109i
$549$	0	0
$550$	0	0
$551$	−4.94427	−0.210633
$552$	0	0
$553$	12.9443i	0.550446i
$554$	−42.3607	−1.79973
$555$	0	0
$556$	22.2492	0.943577
$557$	20.4721i	0.867432i	0.901050	+	0.433716i	$0.142798\pi$
$557$	20.4721i	0.867432i	−0.901050	+	0.433716i	$0.857202\pi$
$558$	0	0
$559$	−40.0000	−1.69182
$560$	0	0
$561$	0	0
$562$	− 24.4721i	− 1.03229i
$563$	− 13.8885i	− 0.585332i	−0.956215	−	0.292666i	$-0.905458\pi$
$563$	− 13.8885i	− 0.585332i	0.956215	−	0.292666i	$-0.0945425\pi$
$564$	0	0
$565$	0	0
$566$	26.8328	1.12787
$567$	0	0
$568$	− 12.3607i	− 0.518643i
$569$	−39.8885	−1.67221	−0.836107	−	0.548566i	$-0.815174\pi$
$569$	−39.8885	−1.67221	−0.836107	+	0.548566i	$0.815174\pi$
$570$	0	0
$571$	−4.00000	−0.167395	−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000	−0.167395	−0.0836974	+	0.996491i	$0.526673\pi$
$572$	86.8328i	3.63066i
$573$	0	0
$574$	−4.47214	−0.186663
$575$	0	0
$576$	0	0
$577$	− 10.3607i	− 0.431321i	−0.976468	−	0.215660i	$-0.930810\pi$
$577$	− 10.3607i	− 0.431321i	0.976468	−	0.215660i	$-0.0691904\pi$
$578$	− 29.0689i	− 1.20911i
$579$	0	0
$580$	0	0
$581$	16.9443	0.702967
$582$	0	0
$583$	− 22.8328i	− 0.945639i
$584$	27.8885	1.15404
$585$	0	0
$586$	−11.3050	−0.467003
$587$	− 4.00000i	− 0.165098i	−0.996587	−	0.0825488i	$-0.973694\pi$
$587$	− 4.00000i	− 0.165098i	0.996587	−	0.0825488i	$-0.0263060\pi$
$588$	0	0
$589$	3.77709	0.155632
$590$	0	0
$591$	0	0
$592$	− 6.94427i	− 0.285408i
$593$	− 23.8885i	− 0.980985i	−0.871445	−	0.490492i	$-0.836817\pi$
$593$	− 23.8885i	− 0.980985i	0.871445	−	0.490492i	$-0.163183\pi$
$594$	0	0
$595$	0	0
$596$	44.8328	1.83642
$597$	0	0
$598$	− 40.0000i	− 1.63572i
$599$	12.3607	0.505044	0.252522	−	0.967591i	$-0.418740\pi$
$599$	12.3607	0.505044	0.252522	+	0.967591i	$0.418740\pi$
$600$	0	0
$601$	38.9443	1.58857	0.794285	−	0.607545i	$-0.207846\pi$
$601$	38.9443	1.58857	0.794285	+	0.607545i	$0.207846\pi$
$602$	− 20.0000i	− 0.815139i
$603$	0	0
$604$	48.0000	1.95309
$605$	0	0
$606$	0	0
$607$	− 38.8328i	− 1.57618i	−0.615563	−	0.788088i	$-0.711071\pi$
$607$	− 38.8328i	− 1.57618i	0.615563	−	0.788088i	$-0.288929\pi$
$608$	16.5836i	0.672553i
$609$	0	0
$610$	0	0
$611$	−57.8885	−2.34192
$612$	0	0
$613$	− 6.94427i	− 0.280477i	−0.990118	−	0.140238i	$-0.955213\pi$
$613$	− 6.94427i	− 0.280477i	0.990118	−	0.140238i	$-0.0447868\pi$
$614$	33.6656	1.35863
$615$	0	0
$616$	−14.4721	−0.583099
$617$	16.4721i	0.663143i	0.943430	+	0.331572i	$0.107579\pi$
$617$	16.4721i	0.663143i	−0.943430	+	0.331572i	$0.892421\pi$
$618$	0	0
$619$	−39.4164	−1.58428	−0.792140	−	0.610340i	$-0.791033\pi$
$619$	−39.4164	−1.58428	−0.792140	+	0.610340i	$0.791033\pi$
$620$	0	0
$621$	0	0
$622$	57.8885i	2.32112i
$623$	2.00000i	0.0801283i
$624$	0	0
$625$	0	0
$626$	−38.9443	−1.55653
$627$	0	0
$628$	− 1.41641i	− 0.0565208i
$629$	13.8885	0.553773
$630$	0	0
$631$	30.8328	1.22744	0.613718	−	0.789526i	$-0.289673\pi$
$631$	30.8328	1.22744	0.613718	+	0.789526i	$0.289673\pi$
$632$	− 28.9443i	− 1.15134i
$633$	0	0
$634$	32.1115	1.27531
$635$	0	0
$636$	0	0
$637$	− 4.47214i	− 0.177192i
$638$	− 28.9443i	− 1.14591i
$639$	0	0
$640$	0	0
$641$	−16.8328	−0.664856	−0.332428	−	0.943129i	$-0.607868\pi$
$641$	−16.8328	−0.664856	−0.332428	+	0.943129i	$0.607868\pi$
$642$	0	0
$643$	− 15.0557i	− 0.593740i	−0.954918	−	0.296870i	$-0.904057\pi$
$643$	− 15.0557i	− 0.593740i	0.954918	−	0.296870i	$-0.0959428\pi$
$644$	12.0000	0.472866
$645$	0	0
$646$	−11.0557	−0.434982
$647$	− 1.88854i	− 0.0742463i	−0.999311	−	0.0371232i	$-0.988181\pi$
$647$	− 1.88854i	− 0.0742463i	0.999311	−	0.0371232i	$-0.0118194\pi$
$648$	0	0
$649$	57.8885	2.27232
$650$	0	0
$651$	0	0
$652$	50.8328i	1.99077i
$653$	22.5836i	0.883764i	0.897073	+	0.441882i	$0.145689\pi$
$653$	22.5836i	0.883764i	−0.897073	+	0.441882i	$0.854311\pi$
$654$	0	0
$655$	0	0
$656$	−2.00000	−0.0780869
$657$	0	0
$658$	− 28.9443i	− 1.12837i
$659$	21.3050	0.829923	0.414962	−	0.909839i	$-0.363795\pi$
$659$	21.3050	0.829923	0.414962	+	0.909839i	$0.363795\pi$
$660$	0	0
$661$	−35.8885	−1.39590	−0.697951	−	0.716145i	$-0.745905\pi$
$661$	−35.8885	−1.39590	−0.697951	+	0.716145i	$0.745905\pi$
$662$	− 2.11146i	− 0.0820641i
$663$	0	0
$664$	−37.8885	−1.47036
$665$	0	0
$666$	0	0
$667$	8.00000i	0.309761i
$668$	24.0000i	0.928588i
$669$	0	0
$670$	0	0
$671$	12.9443	0.499708
$672$	0	0
$673$	8.83282i	0.340480i	0.985403	+	0.170240i	$0.0544543\pi$
$673$	8.83282i	0.340480i	−0.985403	+	0.170240i	$0.945546\pi$
$674$	53.4164	2.05752
$675$	0	0
$676$	21.0000	0.807692
$677$	21.0557i	0.809237i	0.914485	+	0.404619i	$0.132596\pi$
$677$	21.0557i	0.809237i	−0.914485	+	0.404619i	$0.867404\pi$
$678$	0	0
$679$	−8.47214	−0.325131
$680$	0	0
$681$	0	0
$682$	22.1115i	0.846691i
$683$	1.88854i	0.0722631i	0.999347	+	0.0361316i	$0.0115035\pi$
$683$	1.88854i	0.0722631i	−0.999347	+	0.0361316i	$0.988496\pi$
$684$	0	0
$685$	0	0
$686$	2.23607	0.0853735
$687$	0	0
$688$	− 8.94427i	− 0.340997i
$689$	−15.7771	−0.601059
$690$	0	0
$691$	44.3607	1.68756	0.843780	−	0.536689i	$-0.180325\pi$
$691$	44.3607	1.68756	0.843780	+	0.536689i	$0.180325\pi$
$692$	− 8.83282i	− 0.335773i
$693$	0	0
$694$	−17.8885	−0.679040
$695$	0	0
$696$	0	0
$697$	− 4.00000i	− 0.151511i
$698$	− 26.5836i	− 1.00620i
$699$	0	0
$700$	0	0
$701$	34.0000	1.28416	0.642081	−	0.766637i	$-0.278071\pi$
$701$	34.0000	1.28416	0.642081	+	0.766637i	$0.278071\pi$
$702$	0	0
$703$	17.1672i	0.647473i
$704$	−84.1378	−3.17106
$705$	0	0
$706$	−17.6393	−0.663865
$707$	− 14.0000i	− 0.526524i
$708$	0	0
$709$	−25.7771	−0.968079	−0.484039	−	0.875046i	$-0.660831\pi$
$709$	−25.7771	−0.968079	−0.484039	+	0.875046i	$0.660831\pi$
$710$	0	0
$711$	0	0
$712$	− 4.47214i	− 0.167600i
$713$	− 6.11146i	− 0.228876i
$714$	0	0
$715$	0	0
$716$	−19.4164	−0.725625
$717$	0	0
$718$	− 41.3050i	− 1.54149i
$719$	−6.83282	−0.254821	−0.127411	−	0.991850i	$-0.540667\pi$
$719$	−6.83282	−0.254821	−0.127411	+	0.991850i	$0.540667\pi$
$720$	0	0
$721$	0	0
$722$	28.8197i	1.07256i
$723$	0	0
$724$	−3.16718	−0.117707
$725$	0	0
$726$	0	0
$727$	38.8328i	1.44023i	0.693855	+	0.720115i	$0.255911\pi$
$727$	38.8328i	1.44023i	−0.693855	+	0.720115i	$0.744089\pi$
$728$	10.0000i	0.370625i
$729$	0	0
$730$	0	0
$731$	17.8885	0.661632
$732$	0	0
$733$	10.5836i	0.390914i	0.980712	+	0.195457i	$0.0626190\pi$
$733$	10.5836i	0.390914i	−0.980712	+	0.195457i	$0.937381\pi$
$734$	−6.83282	−0.252204
$735$	0	0
$736$	26.8328	0.989071
$737$	− 25.8885i	− 0.953617i
$738$	0	0
$739$	5.88854	0.216614	0.108307	−	0.994118i	$-0.465457\pi$
$739$	5.88854	0.216614	0.108307	+	0.994118i	$0.465457\pi$
$740$	0	0
$741$	0	0
$742$	− 7.88854i	− 0.289598i
$743$	− 34.8328i	− 1.27789i	−0.769252	−	0.638946i	$-0.779371\pi$
$743$	− 34.8328i	− 1.27789i	0.769252	−	0.638946i	$-0.220629\pi$
$744$	0	0
$745$	0	0
$746$	13.4164	0.491210
$747$	0	0
$748$	− 38.8328i	− 1.41987i
$749$	−12.9443	−0.472973
$750$	0	0
$751$	−20.9443	−0.764267	−0.382134	−	0.924107i	$-0.624811\pi$
$751$	−20.9443	−0.764267	−0.382134	+	0.924107i	$0.624811\pi$
$752$	− 12.9443i	− 0.472029i
$753$	0	0
$754$	−20.0000	−0.728357
$755$	0	0
$756$	0	0
$757$	− 31.8885i	− 1.15901i	−0.814969	−	0.579504i	$-0.803246\pi$
$757$	− 31.8885i	− 1.15901i	0.814969	−	0.579504i	$-0.196754\pi$
$758$	− 84.7214i	− 3.07722i
$759$	0	0
$760$	0	0
$761$	27.8885	1.01096	0.505479	−	0.862839i	$-0.331316\pi$
$761$	27.8885	1.01096	0.505479	+	0.862839i	$0.331316\pi$
$762$	0	0
$763$	− 2.00000i	− 0.0724049i
$764$	1.75078	0.0633409
$765$	0	0
$766$	17.8885	0.646339
$767$	− 40.0000i	− 1.44432i
$768$	0	0
$769$	52.8328	1.90520	0.952600	−	0.304226i	$-0.0983976\pi$
$769$	52.8328	1.90520	0.952600	+	0.304226i	$0.0983976\pi$
$770$	0	0
$771$	0	0
$772$	42.0000i	1.51161i
$773$	42.9443i	1.54460i	0.635259	+	0.772299i	$0.280893\pi$
$773$	42.9443i	1.54460i	−0.635259	+	0.772299i	$0.719107\pi$
$774$	0	0
$775$	0	0
$776$	18.9443	0.680060
$777$	0	0
$778$	15.5279i	0.556701i
$779$	4.94427	0.177147
$780$	0	0
$781$	35.7771	1.28020
$782$	17.8885i	0.639693i
$783$	0	0
$784$	1.00000	0.0357143
$785$	0	0
$786$	0	0
$787$	− 31.0557i	− 1.10702i	−0.832843	−	0.553509i	$-0.813289\pi$
$787$	− 31.0557i	− 1.10702i	0.832843	−	0.553509i	$-0.186711\pi$
$788$	− 46.5836i	− 1.65947i
$789$	0	0
$790$	0	0
$791$	−0.472136	−0.0167872
$792$	0	0
$793$	− 8.94427i	− 0.317620i
$794$	30.0000	1.06466
$795$	0	0
$796$	82.2492	2.91525
$797$	− 18.9443i	− 0.671041i	−0.942033	−	0.335520i	$-0.891088\pi$
$797$	− 18.9443i	− 0.671041i	0.942033	−	0.335520i	$-0.108912\pi$
$798$	0	0
$799$	25.8885	0.915871
$800$	0	0
$801$	0	0
$802$	22.3607i	0.789583i
$803$	80.7214i	2.84859i
$804$	0	0
$805$	0	0
$806$	15.2786	0.538167
$807$	0	0
$808$	31.3050i	1.10130i
$809$	38.9443	1.36921	0.684604	−	0.728915i	$-0.259975\pi$
$809$	38.9443	1.36921	0.684604	+	0.728915i	$0.259975\pi$
$810$	0	0
$811$	55.4164	1.94593	0.972967	−	0.230946i	$-0.0741820\pi$
$811$	55.4164	1.94593	0.972967	+	0.230946i	$0.0741820\pi$
$812$	− 6.00000i	− 0.210559i
$813$	0	0
$814$	−100.498	−3.52247
$815$	0	0
$816$	0	0
$817$	22.1115i	0.773582i
$818$	26.5836i	0.929474i
$819$	0	0
$820$	0	0
$821$	−33.7771	−1.17883	−0.589414	−	0.807831i	$-0.700641\pi$
$821$	−33.7771	−1.17883	−0.589414	+	0.807831i	$0.700641\pi$
$822$	0	0
$823$	44.9443i	1.56666i	0.621607	+	0.783329i	$0.286480\pi$
$823$	44.9443i	1.56666i	−0.621607	+	0.783329i	$0.713520\pi$
$824$	0	0
$825$	0	0
$826$	20.0000	0.695889
$827$	12.9443i	0.450116i	0.974345	+	0.225058i	$0.0722572\pi$
$827$	12.9443i	0.450116i	−0.974345	+	0.225058i	$0.927743\pi$
$828$	0	0
$829$	13.0557	0.453444	0.226722	−	0.973959i	$-0.427199\pi$
$829$	13.0557	0.453444	0.226722	+	0.973959i	$0.427199\pi$
$830$	0	0
$831$	0	0
$832$	58.1378i	2.01556i
$833$	2.00000i	0.0692959i
$834$	0	0
$835$	0	0
$836$	48.0000	1.66011
$837$	0	0
$838$	66.8328i	2.30870i
$839$	54.8328	1.89304	0.946520	−	0.322647i	$-0.104573\pi$
$839$	54.8328	1.89304	0.946520	+	0.322647i	$0.104573\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	− 49.1935i	− 1.69532i
$843$	0	0
$844$	50.8328	1.74974
$845$	0	0
$846$	0	0
$847$	− 30.8885i	− 1.06134i
$848$	− 3.52786i	− 0.121147i
$849$	0	0
$850$	0	0
$851$	27.7771	0.952186
$852$	0	0
$853$	− 31.3050i	− 1.07186i	−0.844262	−	0.535931i	$-0.819961\pi$
$853$	− 31.3050i	− 1.07186i	0.844262	−	0.535931i	$-0.180039\pi$
$854$	4.47214	0.153033
$855$	0	0
$856$	28.9443	0.989295
$857$	36.8328i	1.25819i	0.777330	+	0.629093i	$0.216573\pi$
$857$	36.8328i	1.25819i	−0.777330	+	0.629093i	$0.783427\pi$
$858$	0	0
$859$	−50.4721	−1.72209	−0.861044	−	0.508531i	$-0.830189\pi$
$859$	−50.4721	−1.72209	−0.861044	+	0.508531i	$0.830189\pi$
$860$	0	0
$861$	0	0
$862$	− 41.3050i	− 1.40685i
$863$	− 21.8885i	− 0.745095i	−0.928013	−	0.372547i	$-0.878484\pi$
$863$	− 21.8885i	− 0.745095i	0.928013	−	0.372547i	$-0.121516\pi$
$864$	0	0
$865$	0	0
$866$	36.8328	1.25163
$867$	0	0
$868$	4.58359i	0.155577i
$869$	83.7771	2.84194
$870$	0	0
$871$	−17.8885	−0.606130
$872$	4.47214i	0.151446i
$873$	0	0
$874$	−22.1115	−0.747931
$875$	0	0
$876$	0	0
$877$	56.8328i	1.91911i	0.281525	+	0.959554i	$0.409160\pi$
$877$	56.8328i	1.91911i	−0.281525	+	0.959554i	$0.590840\pi$
$878$	3.41641i	0.115298i
$879$	0	0
$880$	0	0
$881$	27.8885	0.939589	0.469794	−	0.882776i	$-0.344328\pi$
$881$	27.8885	0.939589	0.469794	+	0.882776i	$0.344328\pi$
$882$	0	0
$883$	− 37.8885i	− 1.27505i	−0.770429	−	0.637526i	$-0.779958\pi$
$883$	− 37.8885i	− 1.27505i	0.770429	−	0.637526i	$-0.220042\pi$
$884$	−26.8328	−0.902485
$885$	0	0
$886$	17.8885	0.600977
$887$	− 30.8328i	− 1.03526i	−0.855603	−	0.517632i	$-0.826814\pi$
$887$	− 30.8328i	− 1.03526i	0.855603	−	0.517632i	$-0.173186\pi$
$888$	0	0
$889$	4.94427	0.165826
$890$	0	0
$891$	0	0
$892$	38.8328i	1.30022i
$893$	32.0000i	1.07084i
$894$	0	0
$895$	0	0
$896$	−15.6525	−0.522913
$897$	0	0
$898$	31.3050i	1.04466i
$899$	−3.05573	−0.101914
$900$	0	0
$901$	7.05573	0.235060
$902$	28.9443i	0.963739i
$903$	0	0
$904$	1.05573	0.0351130
$905$	0	0
$906$	0	0
$907$	− 53.8885i	− 1.78934i	−0.446728	−	0.894670i	$-0.647411\pi$
$907$	− 53.8885i	− 1.78934i	0.446728	−	0.894670i	$-0.352589\pi$
$908$	− 2.83282i	− 0.0940103i
$909$	0	0
$910$	0	0
$911$	−46.2492	−1.53231	−0.766153	−	0.642659i	$-0.777831\pi$
$911$	−46.2492	−1.53231	−0.766153	+	0.642659i	$0.777831\pi$
$912$	0	0
$913$	− 109.666i	− 3.62940i
$914$	−15.5279	−0.513616
$915$	0	0
$916$	71.6656	2.36790
$917$	4.00000i	0.132092i
$918$	0	0
$919$	35.0557	1.15638	0.578191	−	0.815902i	$-0.303759\pi$
$919$	35.0557	1.15638	0.578191	+	0.815902i	$0.303759\pi$
$920$	0	0
$921$	0	0
$922$	8.69505i	0.286356i
$923$	− 24.7214i	− 0.813713i
$924$	0	0
$925$	0	0
$926$	46.8328	1.53902
$927$	0	0
$928$	− 13.4164i	− 0.440415i
$929$	−16.1115	−0.528600	−0.264300	−	0.964441i	$-0.585141\pi$
$929$	−16.1115	−0.528600	−0.264300	+	0.964441i	$0.585141\pi$
$930$	0	0
$931$	−2.47214	−0.0810210
$932$	− 28.2492i	− 0.925334i
$933$	0	0
$934$	−20.0000	−0.654420
$935$	0	0
$936$	0	0
$937$	52.4721i	1.71419i	0.515158	+	0.857095i	$0.327733\pi$
$937$	52.4721i	1.71419i	−0.515158	+	0.857095i	$0.672267\pi$
$938$	− 8.94427i	− 0.292041i
$939$	0	0
$940$	0	0
$941$	30.0000	0.977972	0.488986	−	0.872292i	$-0.337367\pi$
$941$	30.0000	0.977972	0.488986	+	0.872292i	$0.337367\pi$
$942$	0	0
$943$	− 8.00000i	− 0.260516i
$944$	8.94427	0.291111
$945$	0	0
$946$	−129.443	−4.20855
$947$	− 17.8885i	− 0.581300i	−0.956830	−	0.290650i	$-0.906129\pi$
$947$	− 17.8885i	− 0.581300i	0.956830	−	0.290650i	$-0.0938715\pi$
$948$	0	0
$949$	55.7771	1.81060
$950$	0	0
$951$	0	0
$952$	− 4.47214i	− 0.144943i
$953$	33.4164i	1.08246i	0.840873	+	0.541232i	$0.182042\pi$
$953$	33.4164i	1.08246i	−0.840873	+	0.541232i	$0.817958\pi$
$954$	0	0
$955$	0	0
$956$	31.4164	1.01608
$957$	0	0
$958$	40.0000i	1.29234i
$959$	3.52786	0.113921
$960$	0	0
$961$	−28.6656	−0.924698
$962$	69.4427i	2.23892i
$963$	0	0
$964$	56.8328	1.83046
$965$	0	0
$966$	0	0
$967$	− 25.8885i	− 0.832519i	−0.909246	−	0.416260i	$-0.863341\pi$
$967$	− 25.8885i	− 0.832519i	0.909246	−	0.416260i	$-0.136659\pi$
$968$	69.0689i	2.21996i
$969$	0	0
$970$	0	0
$971$	−40.9443	−1.31396	−0.656982	−	0.753906i	$-0.728167\pi$
$971$	−40.9443	−1.31396	−0.656982	+	0.753906i	$0.728167\pi$
$972$	0	0
$973$	7.41641i	0.237759i
$974$	46.8328	1.50062
$975$	0	0
$976$	2.00000	0.0640184
$977$	30.5836i	0.978456i	0.872156	+	0.489228i	$0.162721\pi$
$977$	30.5836i	0.978456i	−0.872156	+	0.489228i	$0.837279\pi$
$978$	0	0
$979$	12.9443	0.413701
$980$	0	0
$981$	0	0
$982$	− 47.6393i	− 1.52023i
$983$	22.8328i	0.728254i	0.931349	+	0.364127i	$0.118633\pi$
$983$	22.8328i	0.728254i	−0.931349	+	0.364127i	$0.881367\pi$
$984$	0	0
$985$	0	0
$986$	8.94427	0.284844
$987$	0	0
$988$	− 33.1672i	− 1.05519i
$989$	35.7771	1.13765
$990$	0	0
$991$	4.94427	0.157060	0.0785300	−	0.996912i	$-0.474977\pi$
$991$	4.94427	0.157060	0.0785300	+	0.996912i	$0.474977\pi$
$992$	10.2492i	0.325413i
$993$	0	0
$994$	12.3607	0.392057
$995$	0	0
$996$	0	0
$997$	5.41641i	0.171539i	0.996315	+	0.0857697i	$0.0273349\pi$
$997$	5.41641i	0.171539i	−0.996315	+	0.0857697i	$0.972665\pi$
$998$	− 31.0557i	− 0.983052i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1575.2.d.d.1324.1		4
3.2	odd	2		525.2.d.c.274.3		4
5.2	odd	4		315.2.a.d.1.2		2
5.3	odd	4		1575.2.a.r.1.1		2
5.4	even	2	inner	1575.2.d.d.1324.4		4
15.2	even	4		105.2.a.b.1.1	✓	2
15.8	even	4		525.2.a.g.1.2		2
15.14	odd	2		525.2.d.c.274.2		4
20.7	even	4		5040.2.a.bw.1.2		2
35.27	even	4		2205.2.a.w.1.2		2
60.23	odd	4		8400.2.a.cx.1.1		2
60.47	odd	4		1680.2.a.v.1.1		2
105.2	even	12		735.2.i.k.361.2		4
105.17	odd	12		735.2.i.i.226.2		4
105.32	even	12		735.2.i.k.226.2		4
105.47	odd	12		735.2.i.i.361.2		4
105.62	odd	4		735.2.a.k.1.1		2
105.83	odd	4		3675.2.a.y.1.2		2
120.77	even	4		6720.2.a.cx.1.1		2
120.107	odd	4		6720.2.a.cs.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.2.a.b.1.1	✓	2	15.2	even	4
315.2.a.d.1.2		2	5.2	odd	4
525.2.a.g.1.2		2	15.8	even	4
525.2.d.c.274.2		4	15.14	odd	2
525.2.d.c.274.3		4	3.2	odd	2
735.2.a.k.1.1		2	105.62	odd	4
735.2.i.i.226.2		4	105.17	odd	12
735.2.i.i.361.2		4	105.47	odd	12
735.2.i.k.226.2		4	105.32	even	12
735.2.i.k.361.2		4	105.2	even	12
1575.2.a.r.1.1		2	5.3	odd	4
1575.2.d.d.1324.1		4	1.1	even	1	trivial
1575.2.d.d.1324.4		4	5.4	even	2	inner
1680.2.a.v.1.1		2	60.47	odd	4
2205.2.a.w.1.2		2	35.27	even	4
3675.2.a.y.1.2		2	105.83	odd	4
5040.2.a.bw.1.2		2	20.7	even	4
6720.2.a.cs.1.2		2	120.107	odd	4
6720.2.a.cx.1.1		2	120.77	even	4
8400.2.a.cx.1.1		2	60.23	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:	\( N \)	\(=\)	\( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1575.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.23607i q^{2} -3.00000 q^{4} -1.00000i q^{7} +2.23607i q^{8} +O(q^{10})\)	\(q-2.23607i q^{2} -3.00000 q^{4} -1.00000i q^{7} +2.23607i q^{8} -6.47214 q^{11} +4.47214i q^{13} -2.23607 q^{14} -1.00000 q^{16} -2.00000i q^{17} +2.47214 q^{19} +14.4721i q^{22} -4.00000i q^{23} +10.0000 q^{26} +3.00000i q^{28} -2.00000 q^{29} +1.52786 q^{31} +6.70820i q^{32} -4.47214 q^{34} +6.94427i q^{37} -5.52786i q^{38} +2.00000 q^{41} +8.94427i q^{43} +19.4164 q^{44} -8.94427 q^{46} +12.9443i q^{47} -1.00000 q^{49} -13.4164i q^{52} +3.52786i q^{53} +2.23607 q^{56} +4.47214i q^{58} -8.94427 q^{59} -2.00000 q^{61} -3.41641i q^{62} +13.0000 q^{64} +4.00000i q^{67} +6.00000i q^{68} -5.52786 q^{71} -12.4721i q^{73} +15.5279 q^{74} -7.41641 q^{76} +6.47214i q^{77} -12.9443 q^{79} -4.47214i q^{82} +16.9443i q^{83} +20.0000 q^{86} -14.4721i q^{88} -2.00000 q^{89} +4.47214 q^{91} +12.0000i q^{92} +28.9443 q^{94} -8.47214i q^{97} +2.23607i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 12 q^{4}+O(q^{10}) \) 4 * q - 12 * q^4	\( 4 q - 12 q^{4} - 8 q^{11} - 4 q^{16} - 8 q^{19} + 40 q^{26} - 8 q^{29} + 24 q^{31} + 8 q^{41} + 24 q^{44} - 4 q^{49} - 8 q^{61} + 52 q^{64} - 40 q^{71} + 80 q^{74} + 24 q^{76} - 16 q^{79} + 80 q^{86} - 8 q^{89} + 80 q^{94}+O(q^{100}) \) 4 * q - 12 * q^4 - 8 * q^11 - 4 * q^16 - 8 * q^19 + 40 * q^26 - 8 * q^29 + 24 * q^31 + 8 * q^41 + 24 * q^44 - 4 * q^49 - 8 * q^61 + 52 * q^64 - 40 * q^71 + 80 * q^74 + 24 * q^76 - 16 * q^79 + 80 * q^86 - 8 * q^89 + 80 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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