LMFDB - Embedded newform 225.7.c.e.26.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [225,7,Mod(26,225)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(225, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 7, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("225.26"); S:= CuspForms(chi, 7); N := Newforms(S);

Level:	$ N $	$=$	$ 225 = 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	225.c (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [12,0,0,-516,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$51.7621688145$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 654x^{10} + 151557x^{8} + 15450132x^{6} + 718595460x^{4} + 14140615200x^{2} + 82024960000 $ x^12 + 654x^10 + 151557x^8 + 15450132x^6 + 718595460x^4 + 14140615200*x^2 + 82024960000
Coefficient ring:	$\Z[a_1, \ldots, a_{29}]$
Coefficient ring index:	$ 2^{9}\cdot 3^{20}\cdot 5^{4} $
Twist minimal:	no (minimal twist has level 45)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			26.4
Root			$-6.96373i$ of defining polynomial
Character	$\chi$	$=$	225.26
Dual form			225.7.c.e.26.10

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-8.37795i q^{2} -6.18997 q^{4} +270.176 q^{7} -484.329i q^{8} +1187.10i q^{11} -526.105 q^{13} -2263.52i q^{14} -4453.84 q^{16} -6589.09i q^{17} +9088.18 q^{19} +9945.48 q^{22} +8922.54i q^{23} +4407.68i q^{26} -1672.38 q^{28} -7731.97i q^{29} +3114.29 q^{31} +6316.98i q^{32} -55203.1 q^{34} +75996.2 q^{37} -76140.3i q^{38} -56742.3i q^{41} +126636. q^{43} -7348.13i q^{44} +74752.5 q^{46} -182063. i q^{47} -44654.2 q^{49} +3256.57 q^{52} +31815.2i q^{53} -130854. i q^{56} -64778.1 q^{58} -319642. i q^{59} -359529. q^{61} -26091.3i q^{62} -232123. q^{64} +487383. q^{67} +40786.3i q^{68} +370686. i q^{71} -323083. q^{73} -636692. i q^{74} -56255.6 q^{76} +320726. i q^{77} +66000.9 q^{79} -475384. q^{82} +7403.28i q^{83} -1.06095e6i q^{86} +574948. q^{88} -991440. i q^{89} -142141. q^{91} -55230.3i q^{92} -1.52532e6 q^{94} -1.52751e6 q^{97} +374110. i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 516 q^{4} + 36372 q^{16} - 4320 q^{19} + 60192 q^{31} - 106296 q^{34} - 1078968 q^{46} - 711516 q^{49} - 449784 q^{61} - 3964572 q^{64} - 584400 q^{76} - 4324608 q^{79} + 631152 q^{91} - 5793408 q^{94}+O(q^{100}) $ 12 * q - 516 * q^4 + 36372 * q^16 - 4320 * q^19 + 60192 * q^31 - 106296 * q^34 - 1078968 * q^46 - 711516 * q^49 - 449784 * q^61 - 3964572 * q^64 - 584400 * q^76 - 4324608 * q^79 + 631152 * q^91 - 5793408 * q^94

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/225\mathbb{Z}\right)^\times$.

$n$	$101$	$127$
$\chi(n)$	$-1$	$1$

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 $

$2$	− 8.37795i	− 1.04724i	−0.851951	−	0.523622i	$-0.824581\pi$
$2$	− 8.37795i	− 1.04724i	0.851951	−	0.523622i	$-0.175419\pi$
$3$	0	0
$3$	0	0
$4$	−6.18997	−0.0967183
$5$	0	0
$5$	0	0
$6$	0	0
$7$	270.176	0.787684	0.393842	−	0.919178i	$-0.371146\pi$
$7$	270.176	0.787684	0.393842	+	0.919178i	$0.371146\pi$
$8$	− 484.329i	− 0.945956i
$9$	0	0
$10$	0	0
$11$	1187.10i	0.891888i	0.895061	+	0.445944i	$0.147132\pi$
$11$	1187.10i	0.891888i	−0.895061	+	0.445944i	$0.852868\pi$
$12$	0	0
$13$	−526.105	−0.239465	−0.119733	−	0.992806i	$-0.538204\pi$
$13$	−526.105	−0.239465	−0.119733	+	0.992806i	$0.538204\pi$
$14$	− 2263.52i	− 0.824896i
$15$	0	0
$16$	−4453.84	−1.08736
$17$	− 6589.09i	− 1.34115i	−0.741840	−	0.670577i	$-0.766046\pi$
$17$	− 6589.09i	− 1.34115i	0.741840	−	0.670577i	$-0.233954\pi$
$18$	0	0
$19$	9088.18	1.32500	0.662500	−	0.749062i	$-0.269495\pi$
$19$	9088.18	1.32500	0.662500	+	0.749062i	$0.269495\pi$
$20$	0	0
$21$	0	0
$22$	9945.48	0.934023
$23$	8922.54i	0.733339i	0.930351	+	0.366670i	$0.119502\pi$
$23$	8922.54i	0.733339i	−0.930351	+	0.366670i	$0.880498\pi$
$24$	0	0
$25$	0	0
$26$	4407.68i	0.250778i
$27$	0	0
$28$	−1672.38	−0.0761834
$29$	− 7731.97i	− 0.317027i	−0.987357	−	0.158514i	$-0.949330\pi$
$29$	− 7731.97i	− 0.317027i	0.987357	−	0.158514i	$-0.0506702\pi$
$30$	0	0
$31$	3114.29	0.104538	0.0522689	−	0.998633i	$-0.483355\pi$
$31$	3114.29	0.104538	0.0522689	+	0.998633i	$0.483355\pi$
$32$	6316.98i	0.192779i
$33$	0	0
$34$	−55203.1	−1.40451
$35$	0	0
$36$	0	0
$37$	75996.2	1.50033	0.750165	−	0.661251i	$-0.229974\pi$
$37$	75996.2	1.50033	0.750165	+	0.661251i	$0.229974\pi$
$38$	− 76140.3i	− 1.38760i
$39$	0	0
$40$	0	0
$41$	− 56742.3i	− 0.823295i	−0.911343	−	0.411648i	$-0.864953\pi$
$41$	− 56742.3i	− 0.823295i	0.911343	−	0.411648i	$-0.135047\pi$
$42$	0	0
$43$	126636.	1.59277	0.796385	−	0.604790i	$-0.206743\pi$
$43$	126636.	1.59277	0.796385	+	0.604790i	$0.206743\pi$
$44$	− 7348.13i	− 0.0862619i
$45$	0	0
$46$	74752.5	0.767984
$47$	− 182063.i	− 1.75359i	−0.480861	−	0.876797i	$-0.659676\pi$
$47$	− 182063.i	− 1.75359i	0.480861	−	0.876797i	$-0.340324\pi$
$48$	0	0
$49$	−44654.2	−0.379554
$50$	0	0
$51$	0	0
$52$	3256.57	0.0231607
$53$	31815.2i	0.213701i	0.994275	+	0.106851i	$0.0340767\pi$
$53$	31815.2i	0.213701i	−0.994275	+	0.106851i	$0.965923\pi$
$54$	0	0
$55$	0	0
$56$	− 130854.i	− 0.745114i
$57$	0	0
$58$	−64778.1	−0.332005
$59$	− 319642.i	− 1.55635i	−0.628048	−	0.778175i	$-0.716146\pi$
$59$	− 319642.i	− 1.55635i	0.628048	−	0.778175i	$-0.283854\pi$
$60$	0	0
$61$	−359529.	−1.58396	−0.791981	−	0.610546i	$-0.790950\pi$
$61$	−359529.	−1.58396	−0.791981	+	0.610546i	$0.790950\pi$
$62$	− 26091.3i	− 0.109476i
$63$	0	0
$64$	−232123.	−0.885478
$65$	0	0
$66$	0	0
$67$	487383.	1.62049	0.810244	−	0.586093i	$-0.199335\pi$
$67$	487383.	1.62049	0.810244	+	0.586093i	$0.199335\pi$
$68$	40786.3i	0.129714i
$69$	0	0
$70$	0	0
$71$	370686.i	1.03569i	0.855473	+	0.517847i	$0.173266\pi$
$71$	370686.i	1.03569i	−0.855473	+	0.517847i	$0.826734\pi$
$72$	0	0
$73$	−323083.	−0.830511	−0.415255	−	0.909705i	$-0.636308\pi$
$73$	−323083.	−0.830511	−0.415255	+	0.909705i	$0.636308\pi$
$74$	− 636692.i	− 1.57121i
$75$	0	0
$76$	−56255.6	−0.128152
$77$	320726.i	0.702525i
$78$	0	0
$79$	66000.9	0.133866	0.0669328	−	0.997757i	$-0.478679\pi$
$79$	66000.9	0.133866	0.0669328	+	0.997757i	$0.478679\pi$
$80$	0	0
$81$	0	0
$82$	−475384.	−0.862190
$83$	7403.28i	0.0129476i	0.999979	+	0.00647381i	$0.00206069\pi$
$83$	7403.28i	0.0129476i	−0.999979	+	0.00647381i	$0.997939\pi$
$84$	0	0
$85$	0	0
$86$	− 1.06095e6i	− 1.66802i
$87$	0	0
$88$	574948.	0.843686
$89$	− 991440.i	− 1.40636i	−0.711012	−	0.703180i	$-0.751763\pi$
$89$	− 991440.i	− 1.40636i	0.711012	−	0.703180i	$-0.248237\pi$
$90$	0	0
$91$	−142141.	−0.188623
$92$	− 55230.3i	− 0.0709273i
$93$	0	0
$94$	−1.52532e6	−1.83644
$95$	0	0
$96$	0	0
$97$	−1.52751e6	−1.67367	−0.836833	−	0.547458i	$-0.815596\pi$
$97$	−1.52751e6	−1.67367	−0.836833	+	0.547458i	$0.815596\pi$
$98$	374110.i	0.397486i
$99$	0	0
$100$	0	0
$101$	− 452341.i	− 0.439037i	−0.975608	−	0.219519i	$-0.929551\pi$
$101$	− 452341.i	− 0.439037i	0.975608	−	0.219519i	$-0.0704487\pi$
$102$	0	0
$103$	−104961.	−0.0960542	−0.0480271	−	0.998846i	$-0.515293\pi$
$103$	−104961.	−0.0960542	−0.0480271	+	0.998846i	$0.515293\pi$
$104$	254808.i	0.226523i
$105$	0	0
$106$	266546.	0.223797
$107$	657099.i	0.536388i	0.963365	+	0.268194i	$0.0864269\pi$
$107$	657099.i	0.536388i	−0.963365	+	0.268194i	$0.913573\pi$
$108$	0	0
$109$	−1.65657e6	−1.27918	−0.639589	−	0.768717i	$-0.720895\pi$
$109$	−1.65657e6	−1.27918	−0.639589	+	0.768717i	$0.720895\pi$
$110$	0	0
$111$	0	0
$112$	−1.20332e6	−0.856499
$113$	− 1.02621e6i	− 0.711216i	−0.934635	−	0.355608i	$-0.884274\pi$
$113$	− 1.02621e6i	− 0.711216i	0.934635	−	0.355608i	$-0.115726\pi$
$114$	0	0
$115$	0	0
$116$	47860.7i	0.0306623i
$117$	0	0
$118$	−2.67794e6	−1.62988
$119$	− 1.78021e6i	− 1.05641i
$120$	0	0
$121$	362349.	0.204536
$122$	3.01212e6i	1.65879i
$123$	0	0
$124$	−19277.3	−0.0101107
$125$	0	0
$126$	0	0
$127$	375925.	0.183523	0.0917615	−	0.995781i	$-0.470750\pi$
$127$	375925.	0.183523	0.0917615	+	0.995781i	$0.470750\pi$
$128$	2.34900e6i	1.12009i
$129$	0	0
$130$	0	0
$131$	− 1.42739e6i	− 0.634934i	−0.948269	−	0.317467i	$-0.897168\pi$
$131$	− 1.42739e6i	− 0.634934i	0.948269	−	0.317467i	$-0.102832\pi$
$132$	0	0
$133$	2.45540e6	1.04368
$134$	− 4.08327e6i	− 1.69705i
$135$	0	0
$136$	−3.19129e6	−1.26867
$137$	− 2.76280e6i	− 1.07445i	−0.843438	−	0.537226i	$-0.819472\pi$
$137$	− 2.76280e6i	− 1.07445i	0.843438	−	0.537226i	$-0.180528\pi$
$138$	0	0
$139$	2.33540e6	0.869596	0.434798	−	0.900528i	$-0.356820\pi$
$139$	2.33540e6	0.869596	0.434798	+	0.900528i	$0.356820\pi$
$140$	0	0
$141$	0	0
$142$	3.10559e6	1.08462
$143$	− 624540.i	− 0.213576i
$144$	0	0
$145$	0	0
$146$	2.70677e6i	0.869747i
$147$	0	0
$148$	−470414.	−0.145109
$149$	5.57895e6i	1.68653i	0.537499	+	0.843265i	$0.319369\pi$
$149$	5.57895e6i	1.68653i	−0.537499	+	0.843265i	$0.680631\pi$
$150$	0	0
$151$	−187475.	−0.0544520	−0.0272260	−	0.999629i	$-0.508667\pi$
$151$	−187475.	−0.0544520	−0.0272260	+	0.999629i	$0.508667\pi$
$152$	− 4.40167e6i	− 1.25339i
$153$	0	0
$154$	2.68703e6	0.735715
$155$	0	0
$156$	0	0
$157$	2.11041e6	0.545340	0.272670	−	0.962108i	$-0.412093\pi$
$157$	2.11041e6	0.545340	0.272670	+	0.962108i	$0.412093\pi$
$158$	− 552952.i	− 0.140190i
$159$	0	0
$160$	0	0
$161$	2.41065e6i	0.577639i
$162$	0	0
$163$	4.41451e6	1.01934	0.509671	−	0.860370i	$-0.329767\pi$
$163$	4.41451e6	1.01934	0.509671	+	0.860370i	$0.329767\pi$
$164$	351233.i	0.0796277i
$165$	0	0
$166$	62024.3	0.0135593
$167$	− 279043.i	− 0.0599131i	−0.999551	−	0.0299565i	$-0.990463\pi$
$167$	− 279043.i	− 0.0599131i	0.999551	−	0.0299565i	$-0.00953689\pi$
$168$	0	0
$169$	−4.55002e6	−0.942656
$170$	0	0
$171$	0	0
$172$	−783876.	−0.154050
$173$	− 4.97675e6i	− 0.961187i	−0.876943	−	0.480594i	$-0.840421\pi$
$173$	− 4.97675e6i	− 0.961187i	0.876943	−	0.480594i	$-0.159579\pi$
$174$	0	0
$175$	0	0
$176$	− 5.28717e6i	− 0.969806i
$177$	0	0
$178$	−8.30623e6	−1.47280
$179$	3.25828e6i	0.568106i	0.958809	+	0.284053i	$0.0916791\pi$
$179$	3.25828e6i	0.568106i	−0.958809	+	0.284053i	$0.908321\pi$
$180$	0	0
$181$	1.15053e7	1.94027	0.970137	−	0.242559i	$-0.0779869\pi$
$181$	1.15053e7	1.94027	0.970137	+	0.242559i	$0.0779869\pi$
$182$	1.19085e6i	0.197534i
$183$	0	0
$184$	4.32145e6	0.693706
$185$	0	0
$186$	0	0
$187$	7.82193e6	1.19616
$188$	1.12697e6i	0.169605i
$189$	0	0
$190$	0	0
$191$	6.07903e6i	0.872437i	0.899841	+	0.436218i	$0.143683\pi$
$191$	6.07903e6i	0.872437i	−0.899841	+	0.436218i	$0.856317\pi$
$192$	0	0
$193$	1.12931e7	1.57088	0.785438	−	0.618941i	$-0.212438\pi$
$193$	1.12931e7	1.57088	0.785438	+	0.618941i	$0.212438\pi$
$194$	1.27974e7i	1.75274i
$195$	0	0
$196$	276408.	0.0367099
$197$	− 3.46877e6i	− 0.453708i	−0.973929	−	0.226854i	$-0.927156\pi$
$197$	− 3.46877e6i	− 0.453708i	0.973929	−	0.226854i	$-0.0728440\pi$
$198$	0	0
$199$	5.80303e6	0.736369	0.368185	−	0.929753i	$-0.379979\pi$
$199$	5.80303e6	0.736369	0.368185	+	0.929753i	$0.379979\pi$
$200$	0	0
$201$	0	0
$202$	−3.78969e6	−0.459779
$203$	− 2.08899e6i	− 0.249717i
$204$	0	0
$205$	0	0
$206$	879358.i	0.100592i
$207$	0	0
$208$	2.34319e6	0.260386
$209$	1.07886e7i	1.18175i
$210$	0	0
$211$	−1.31986e6	−0.140502	−0.0702509	−	0.997529i	$-0.522380\pi$
$211$	−1.31986e6	−0.140502	−0.0702509	+	0.997529i	$0.522380\pi$
$212$	− 196935.i	− 0.0206688i
$213$	0	0
$214$	5.50514e6	0.561729
$215$	0	0
$216$	0	0
$217$	841404.	0.0823427
$218$	1.38787e7i	1.33961i
$219$	0	0
$220$	0	0
$221$	3.46655e6i	0.321160i
$222$	0	0
$223$	−1.33313e6	−0.120215	−0.0601076	−	0.998192i	$-0.519144\pi$
$223$	−1.33313e6	−0.120215	−0.0601076	+	0.998192i	$0.519144\pi$
$224$	1.70669e6i	0.151849i
$225$	0	0
$226$	−8.59754e6	−0.744816
$227$	7.97225e6i	0.681559i	0.940143	+	0.340779i	$0.110691\pi$
$227$	7.97225e6i	0.681559i	−0.940143	+	0.340779i	$0.889309\pi$
$228$	0	0
$229$	1.11916e7	0.931938	0.465969	−	0.884801i	$-0.345706\pi$
$229$	1.11916e7	0.931938	0.465969	+	0.884801i	$0.345706\pi$
$230$	0	0
$231$	0	0
$232$	−3.74482e6	−0.299894
$233$	− 1.33843e7i	− 1.05810i	−0.848589	−	0.529052i	$-0.822548\pi$
$233$	− 1.33843e7i	− 1.05810i	0.848589	−	0.529052i	$-0.177452\pi$
$234$	0	0
$235$	0	0
$236$	1.97857e6i	0.150528i
$237$	0	0
$238$	−1.49145e7	−1.10631
$239$	− 3.17061e6i	− 0.232247i	−0.993235	−	0.116123i	$-0.962953\pi$
$239$	− 3.17061e6i	− 0.232247i	0.993235	−	0.116123i	$-0.0370468\pi$
$240$	0	0
$241$	3.36319e6	0.240270	0.120135	−	0.992758i	$-0.461667\pi$
$241$	3.36319e6	0.240270	0.120135	+	0.992758i	$0.461667\pi$
$242$	− 3.03574e6i	− 0.214199i
$243$	0	0
$244$	2.22548e6	0.153198
$245$	0	0
$246$	0	0
$247$	−4.78134e6	−0.317291
$248$	− 1.50834e6i	− 0.0988881i
$249$	0	0
$250$	0	0
$251$	2.64563e7i	1.67305i	0.547932	+	0.836523i	$0.315415\pi$
$251$	2.64563e7i	1.67305i	−0.547932	+	0.836523i	$0.684585\pi$
$252$	0	0
$253$	−1.05920e7	−0.654056
$254$	− 3.14948e6i	− 0.192193i
$255$	0	0
$256$	4.82392e6	0.287528
$257$	4.28701e6i	0.252554i	0.991995	+	0.126277i	$0.0403029\pi$
$257$	4.28701e6i	0.252554i	−0.991995	+	0.126277i	$0.959697\pi$
$258$	0	0
$259$	2.05323e7	1.18179
$260$	0	0
$261$	0	0
$262$	−1.19586e7	−0.664930
$263$	2.94305e7i	1.61782i	0.587933	+	0.808910i	$0.299942\pi$
$263$	2.94305e7i	1.61782i	−0.587933	+	0.808910i	$0.700058\pi$
$264$	0	0
$265$	0	0
$266$	− 2.05712e7i	− 1.09299i
$267$	0	0
$268$	−3.01689e6	−0.156731
$269$	1.40558e7i	0.722100i	0.932547	+	0.361050i	$0.117582\pi$
$269$	1.40558e7i	0.722100i	−0.932547	+	0.361050i	$0.882418\pi$
$270$	0	0
$271$	−2.69073e6	−0.135196	−0.0675978	−	0.997713i	$-0.521533\pi$
$271$	−2.69073e6	−0.135196	−0.0675978	+	0.997713i	$0.521533\pi$
$272$	2.93468e7i	1.45832i
$273$	0	0
$274$	−2.31465e7	−1.12521
$275$	0	0
$276$	0	0
$277$	−4.16627e7	−1.96024	−0.980118	−	0.198414i	$-0.936421\pi$
$277$	−4.16627e7	−1.96024	−0.980118	+	0.198414i	$0.936421\pi$
$278$	− 1.95659e7i	− 0.910679i
$279$	0	0
$280$	0	0
$281$	− 1.07292e7i	− 0.483556i	−0.970332	−	0.241778i	$-0.922269\pi$
$281$	− 1.07292e7i	− 0.483556i	0.970332	−	0.241778i	$-0.0777305\pi$
$282$	0	0
$283$	−5.50312e6	−0.242800	−0.121400	−	0.992604i	$-0.538738\pi$
$283$	−5.50312e6	−0.242800	−0.121400	+	0.992604i	$0.538738\pi$
$284$	− 2.29454e6i	− 0.100171i
$285$	0	0
$286$	−5.23236e6	−0.223666
$287$	− 1.53304e7i	− 0.648496i
$288$	0	0
$289$	−1.92786e7	−0.798696
$290$	0	0
$291$	0	0
$292$	1.99987e6	0.0803256
$293$	1.23547e7i	0.491169i	0.969375	+	0.245584i	$0.0789798\pi$
$293$	1.23547e7i	0.491169i	−0.969375	+	0.245584i	$0.921020\pi$
$294$	0	0
$295$	0	0
$296$	− 3.68072e7i	− 1.41925i
$297$	0	0
$298$	4.67402e7	1.76621
$299$	− 4.69419e6i	− 0.175609i
$300$	0	0
$301$	3.42140e7	1.25460
$302$	1.57066e6i	0.0570245i
$303$	0	0
$304$	−4.04773e7	−1.44076
$305$	0	0
$306$	0	0
$307$	4.10198e7	1.41768	0.708840	−	0.705370i	$-0.249219\pi$
$307$	4.10198e7	1.41768	0.708840	+	0.705370i	$0.249219\pi$
$308$	− 1.98529e6i	− 0.0679471i
$309$	0	0
$310$	0	0
$311$	4.55195e7i	1.51327i	0.653837	+	0.756635i	$0.273158\pi$
$311$	4.55195e7i	1.51327i	−0.653837	+	0.756635i	$0.726842\pi$
$312$	0	0
$313$	1.47679e7	0.481598	0.240799	−	0.970575i	$-0.422591\pi$
$313$	1.47679e7	0.481598	0.240799	+	0.970575i	$0.422591\pi$
$314$	− 1.76809e7i	− 0.571103i
$315$	0	0
$316$	−408544.	−0.0129472
$317$	7.38900e6i	0.231957i	0.993252	+	0.115979i	$0.0370004\pi$
$317$	7.38900e6i	0.231957i	−0.993252	+	0.115979i	$0.963000\pi$
$318$	0	0
$319$	9.17865e6	0.282753
$320$	0	0
$321$	0	0
$322$	2.01963e7	0.604929
$323$	− 5.98829e7i	− 1.77703i
$324$	0	0
$325$	0	0
$326$	− 3.69845e7i	− 1.06750i
$327$	0	0
$328$	−2.74820e7	−0.778801
$329$	− 4.91891e7i	− 1.38128i
$330$	0	0
$331$	−4.89533e7	−1.34989	−0.674944	−	0.737869i	$-0.735832\pi$
$331$	−4.89533e7	−1.34989	−0.674944	+	0.737869i	$0.735832\pi$
$332$	− 45826.1i	− 0.00125227i
$333$	0	0
$334$	−2.33781e6	−0.0627436
$335$	0	0
$336$	0	0
$337$	2.52313e7	0.659250	0.329625	−	0.944112i	$-0.393078\pi$
$337$	2.52313e7	0.659250	0.329625	+	0.944112i	$0.393078\pi$
$338$	3.81198e7i	0.987191i
$339$	0	0
$340$	0	0
$341$	3.69698e6i	0.0932360i
$342$	0	0
$343$	−4.38503e7	−1.08665
$344$	− 6.13337e7i	− 1.50669i
$345$	0	0
$346$	−4.16950e7	−1.00660
$347$	− 4.18476e7i	− 1.00157i	−0.865571	−	0.500787i	$-0.833044\pi$
$347$	− 4.18476e7i	− 1.00157i	0.865571	−	0.500787i	$-0.166956\pi$
$348$	0	0
$349$	−1.11598e7	−0.262532	−0.131266	−	0.991347i	$-0.541904\pi$
$349$	−1.11598e7	−0.262532	−0.131266	+	0.991347i	$0.541904\pi$
$350$	0	0
$351$	0	0
$352$	−7.49890e6	−0.171937
$353$	2.13705e7i	0.485837i	0.970047	+	0.242918i	$0.0781047\pi$
$353$	2.13705e7i	0.485837i	−0.970047	+	0.242918i	$0.921895\pi$
$354$	0	0
$355$	0	0
$356$	6.13699e6i	0.136021i
$357$	0	0
$358$	2.72977e7	0.594945
$359$	− 2.27851e7i	− 0.492455i	−0.969212	−	0.246228i	$-0.920809\pi$
$359$	− 2.27851e7i	− 0.492455i	0.969212	−	0.246228i	$-0.0791911\pi$
$360$	0	0
$361$	3.55491e7	0.755627
$362$	− 9.63909e7i	− 2.03194i
$363$	0	0
$364$	879847.	0.0182433
$365$	0	0
$366$	0	0
$367$	3.26916e7	0.661360	0.330680	−	0.943743i	$-0.392722\pi$
$367$	3.26916e7	0.661360	0.330680	+	0.943743i	$0.392722\pi$
$368$	− 3.97396e7i	− 0.797407i
$369$	0	0
$370$	0	0
$371$	8.59569e6i	0.168329i
$372$	0	0
$373$	2.20835e7	0.425542	0.212771	−	0.977102i	$-0.431751\pi$
$373$	2.20835e7	0.425542	0.212771	+	0.977102i	$0.431751\pi$
$374$	− 6.55317e7i	− 1.25267i
$375$	0	0
$376$	−8.81786e7	−1.65882
$377$	4.06783e6i	0.0759169i
$378$	0	0
$379$	−2.12751e7	−0.390800	−0.195400	−	0.980724i	$-0.562601\pi$
$379$	−2.12751e7	−0.390800	−0.195400	+	0.980724i	$0.562601\pi$
$380$	0	0
$381$	0	0
$382$	5.09298e7	0.913654
$383$	7.76311e7i	1.38178i	0.722959	+	0.690891i	$0.242781\pi$
$383$	7.76311e7i	1.38178i	−0.722959	+	0.690891i	$0.757219\pi$
$384$	0	0
$385$	0	0
$386$	− 9.46131e7i	− 1.64509i
$387$	0	0
$388$	9.45525e6	0.161874
$389$	7.43035e7i	1.26229i	0.775663	+	0.631147i	$0.217415\pi$
$389$	7.43035e7i	1.26229i	−0.775663	+	0.631147i	$0.782585\pi$
$390$	0	0
$391$	5.87914e7	0.983521
$392$	2.16273e7i	0.359042i
$393$	0	0
$394$	−2.90611e7	−0.475143
$395$	0	0
$396$	0	0
$397$	−2.25395e7	−0.360224	−0.180112	−	0.983646i	$-0.557646\pi$
$397$	−2.25395e7	−0.360224	−0.180112	+	0.983646i	$0.557646\pi$
$398$	− 4.86175e7i	− 0.771157i
$399$	0	0
$400$	0	0
$401$	4.85810e7i	0.753414i	0.926332	+	0.376707i	$0.122944\pi$
$401$	4.85810e7i	0.753414i	−0.926332	+	0.376707i	$0.877056\pi$
$402$	0	0
$403$	−1.63844e6	−0.0250332
$404$	2.79998e6i	0.0424630i
$405$	0	0
$406$	−1.75014e7	−0.261515
$407$	9.02153e7i	1.33813i
$408$	0	0
$409$	−6.71469e7	−0.981423	−0.490711	−	0.871322i	$-0.663263\pi$
$409$	−6.71469e7	−0.981423	−0.490711	+	0.871322i	$0.663263\pi$
$410$	0	0
$411$	0	0
$412$	649706.	0.00929020
$413$	− 8.63593e7i	− 1.22591i
$414$	0	0
$415$	0	0
$416$	− 3.32339e6i	− 0.0461638i
$417$	0	0
$418$	9.03863e7	1.23758
$419$	− 4.06857e7i	− 0.553095i	−0.961000	−	0.276548i	$-0.910810\pi$
$419$	− 4.06857e7i	− 0.553095i	0.961000	−	0.276548i	$-0.0891904\pi$
$420$	0	0
$421$	9.91589e7	1.32888	0.664440	−	0.747342i	$-0.268670\pi$
$421$	9.91589e7	1.32888	0.664440	+	0.747342i	$0.268670\pi$
$422$	1.10578e7i	0.147140i
$423$	0	0
$424$	1.54090e7	0.202152
$425$	0	0
$426$	0	0
$427$	−9.71360e7	−1.24766
$428$	− 4.06742e6i	− 0.0518786i
$429$	0	0
$430$	0	0
$431$	− 2.37118e7i	− 0.296164i	−0.988975	−	0.148082i	$-0.952690\pi$
$431$	− 2.37118e7i	− 0.296164i	0.988975	−	0.148082i	$-0.0473099\pi$
$432$	0	0
$433$	−6.09661e7	−0.750974	−0.375487	−	0.926828i	$-0.622524\pi$
$433$	−6.09661e7	−0.750974	−0.375487	+	0.926828i	$0.622524\pi$
$434$	− 7.04923e6i	− 0.0862328i
$435$	0	0
$436$	1.02541e7	0.123720
$437$	8.10896e7i	0.971675i
$438$	0	0
$439$	−1.53212e7	−0.181092	−0.0905458	−	0.995892i	$-0.528861\pi$
$439$	−1.53212e7	−0.181092	−0.0905458	+	0.995892i	$0.528861\pi$
$440$	0	0
$441$	0	0
$442$	2.90426e7	0.336332
$443$	1.12121e8i	1.28966i	0.764327	+	0.644828i	$0.223071\pi$
$443$	1.12121e8i	1.28966i	−0.764327	+	0.644828i	$0.776929\pi$
$444$	0	0
$445$	0	0
$446$	1.11689e7i	0.125895i
$447$	0	0
$448$	−6.27138e7	−0.697476
$449$	1.27219e8i	1.40544i	0.711468	+	0.702719i	$0.248031\pi$
$449$	1.27219e8i	1.40544i	−0.711468	+	0.702719i	$0.751969\pi$
$450$	0	0
$451$	6.73589e7	0.734287
$452$	6.35222e6i	0.0687876i
$453$	0	0
$454$	6.67911e7	0.713758
$455$	0	0
$456$	0	0
$457$	−7.74410e7	−0.811377	−0.405688	−	0.914011i	$-0.632968\pi$
$457$	−7.74410e7	−0.811377	−0.405688	+	0.914011i	$0.632968\pi$
$458$	− 9.37629e7i	− 0.975965i
$459$	0	0
$460$	0	0
$461$	2.35645e7i	0.240523i	0.992742	+	0.120261i	$0.0383733\pi$
$461$	2.35645e7i	0.240523i	−0.992742	+	0.120261i	$0.961627\pi$
$462$	0	0
$463$	−4.67479e7	−0.470998	−0.235499	−	0.971875i	$-0.575672\pi$
$463$	−4.67479e7	−0.470998	−0.235499	+	0.971875i	$0.575672\pi$
$464$	3.44370e7i	0.344724i
$465$	0	0
$466$	−1.12133e8	−1.10809
$467$	− 1.04192e8i	− 1.02302i	−0.859277	−	0.511511i	$-0.829086\pi$
$467$	− 1.04192e8i	− 1.02302i	0.859277	−	0.511511i	$-0.170914\pi$
$468$	0	0
$469$	1.31679e8	1.27643
$470$	0	0
$471$	0	0
$472$	−1.54812e8	−1.47224
$473$	1.50330e8i	1.42057i
$474$	0	0
$475$	0	0
$476$	1.10195e7i	0.102174i
$477$	0	0
$478$	−2.65632e7	−0.243219
$479$	9.95625e7i	0.905919i	0.891531	+	0.452959i	$0.149632\pi$
$479$	9.95625e7i	0.905919i	−0.891531	+	0.452959i	$0.850368\pi$
$480$	0	0
$481$	−3.99820e7	−0.359277
$482$	− 2.81766e7i	− 0.251622i
$483$	0	0
$484$	−2.24293e6	−0.0197824
$485$	0	0
$486$	0	0
$487$	−1.90543e8	−1.64971	−0.824853	−	0.565347i	$-0.808742\pi$
$487$	−1.90543e8	−1.64971	−0.824853	+	0.565347i	$0.808742\pi$
$488$	1.74130e8i	1.49836i
$489$	0	0
$490$	0	0
$491$	1.08462e8i	0.916287i	0.888878	+	0.458143i	$0.151485\pi$
$491$	1.08462e8i	0.916287i	−0.888878	+	0.458143i	$0.848515\pi$
$492$	0	0
$493$	−5.09467e7	−0.425182
$494$	4.00578e7i	0.332281i
$495$	0	0
$496$	−1.38705e7	−0.113671
$497$	1.00150e8i	0.815799i
$498$	0	0
$499$	1.85918e8	1.49631	0.748153	−	0.663527i	$-0.230941\pi$
$499$	1.85918e8	1.49631	0.748153	+	0.663527i	$0.230941\pi$
$500$	0	0
$501$	0	0
$502$	2.21649e8	1.75209
$503$	− 1.39688e8i	− 1.09763i	−0.835944	−	0.548815i	$-0.815079\pi$
$503$	− 1.39688e8i	− 1.09763i	0.835944	−	0.548815i	$-0.184921\pi$
$504$	0	0
$505$	0	0
$506$	8.87389e7i	0.684956i
$507$	0	0
$508$	−2.32697e6	−0.0177500
$509$	3.55184e7i	0.269340i	0.990891	+	0.134670i	$0.0429974\pi$
$509$	3.55184e7i	0.269340i	−0.990891	+	0.134670i	$0.957003\pi$
$510$	0	0
$511$	−8.72890e7	−0.654180
$512$	1.09921e8i	0.818977i
$513$	0	0
$514$	3.59163e7	0.264486
$515$	0	0
$516$	0	0
$517$	2.16128e8	1.56401
$518$	− 1.72019e8i	− 1.23762i
$519$	0	0
$520$	0	0
$521$	1.93385e8i	1.36745i	0.729742	+	0.683723i	$0.239640\pi$
$521$	1.93385e8i	1.36745i	−0.729742	+	0.683723i	$0.760360\pi$
$522$	0	0
$523$	9.39198e6	0.0656527	0.0328263	−	0.999461i	$-0.489549\pi$
$523$	9.39198e6	0.0656527	0.0328263	+	0.999461i	$0.489549\pi$
$524$	8.83550e6i	0.0614098i
$525$	0	0
$526$	2.46567e8	1.69425
$527$	− 2.05203e7i	− 0.140201i
$528$	0	0
$529$	6.84242e7	0.462214
$530$	0	0
$531$	0	0
$532$	−1.51989e7	−0.100943
$533$	2.98524e7i	0.197150i
$534$	0	0
$535$	0	0
$536$	− 2.36054e8i	− 1.53291i
$537$	0	0
$538$	1.17758e8	0.756214
$539$	− 5.30091e7i	− 0.338520i
$540$	0	0
$541$	2.07632e7	0.131130	0.0655651	−	0.997848i	$-0.479115\pi$
$541$	2.07632e7	0.131130	0.0655651	+	0.997848i	$0.479115\pi$
$542$	2.25428e7i	0.141583i
$543$	0	0
$544$	4.16231e7	0.258546
$545$	0	0
$546$	0	0
$547$	3.73970e7	0.228494	0.114247	−	0.993452i	$-0.463554\pi$
$547$	3.73970e7	0.228494	0.114247	+	0.993452i	$0.463554\pi$
$548$	1.71016e7i	0.103919i
$549$	0	0
$550$	0	0
$551$	− 7.02696e7i	− 0.420061i
$552$	0	0
$553$	1.78318e7	0.105444
$554$	3.49048e8i	2.05284i
$555$	0	0
$556$	−1.44561e7	−0.0841059
$557$	− 2.13408e8i	− 1.23494i	−0.786596	−	0.617468i	$-0.788159\pi$
$557$	− 2.13408e8i	− 1.23494i	0.786596	−	0.617468i	$-0.211841\pi$
$558$	0	0
$559$	−6.66240e7	−0.381413
$560$	0	0
$561$	0	0
$562$	−8.98882e7	−0.506400
$563$	− 1.01902e8i	− 0.571029i	−0.958374	−	0.285515i	$-0.907835\pi$
$563$	− 1.01902e8i	− 0.571029i	0.958374	−	0.285515i	$-0.0921645\pi$
$564$	0	0
$565$	0	0
$566$	4.61048e7i	0.254271i
$567$	0	0
$568$	1.79534e8	0.979720
$569$	1.12478e8i	0.610566i	0.952262	+	0.305283i	$0.0987510\pi$
$569$	1.12478e8i	0.610566i	−0.952262	+	0.305283i	$0.901249\pi$
$570$	0	0
$571$	1.75933e8	0.945016	0.472508	−	0.881326i	$-0.343349\pi$
$571$	1.75933e8	0.945016	0.472508	+	0.881326i	$0.343349\pi$
$572$	3.86589e6i	0.0206567i
$573$	0	0
$574$	−1.28437e8	−0.679133
$575$	0	0
$576$	0	0
$577$	1.61356e8	0.839957	0.419978	−	0.907534i	$-0.362038\pi$
$577$	1.61356e8	0.839957	0.419978	+	0.907534i	$0.362038\pi$
$578$	1.61515e8i	0.836428i
$579$	0	0
$580$	0	0
$581$	2.00019e6i	0.0101986i
$582$	0	0
$583$	−3.77679e7	−0.190598
$584$	1.56478e8i	0.785626i
$585$	0	0
$586$	1.03507e8	0.514373
$587$	4.39079e7i	0.217084i	0.994092	+	0.108542i	$0.0346182\pi$
$587$	4.39079e7i	0.217084i	−0.994092	+	0.108542i	$0.965382\pi$
$588$	0	0
$589$	2.83032e7	0.138513
$590$	0	0
$591$	0	0
$592$	−3.38475e8	−1.63140
$593$	− 3.94038e7i	− 0.188962i	−0.995527	−	0.0944808i	$-0.969881\pi$
$593$	− 3.94038e7i	− 0.188962i	0.995527	−	0.0944808i	$-0.0301191\pi$
$594$	0	0
$595$	0	0
$596$	− 3.45336e7i	− 0.163118i
$597$	0	0
$598$	−3.93277e7	−0.183905
$599$	1.11994e8i	0.521092i	0.965461	+	0.260546i	$0.0839026\pi$
$599$	1.11994e8i	0.521092i	−0.965461	+	0.260546i	$0.916097\pi$
$600$	0	0
$601$	−3.96028e8	−1.82432	−0.912162	−	0.409830i	$-0.865588\pi$
$601$	−3.96028e8	−1.82432	−0.912162	+	0.409830i	$0.865588\pi$
$602$	− 2.86643e8i	− 1.31387i
$603$	0	0
$604$	1.16047e6	0.00526650
$605$	0	0
$606$	0	0
$607$	3.06455e8	1.37025	0.685125	−	0.728425i	$-0.259747\pi$
$607$	3.06455e8	1.37025	0.685125	+	0.728425i	$0.259747\pi$
$608$	5.74098e7i	0.255432i
$609$	0	0
$610$	0	0
$611$	9.57844e7i	0.419924i
$612$	0	0
$613$	−1.80074e7	−0.0781751	−0.0390875	−	0.999236i	$-0.512445\pi$
$613$	−1.80074e7	−0.0781751	−0.0390875	+	0.999236i	$0.512445\pi$
$614$	− 3.43661e8i	− 1.48465i
$615$	0	0
$616$	1.55337e8	0.664558
$617$	− 1.66217e8i	− 0.707651i	−0.935311	−	0.353825i	$-0.884881\pi$
$617$	− 1.66217e8i	− 0.707651i	0.935311	−	0.353825i	$-0.115119\pi$
$618$	0	0
$619$	−1.89841e8	−0.800422	−0.400211	−	0.916423i	$-0.631063\pi$
$619$	−1.89841e8	−0.800422	−0.400211	+	0.916423i	$0.631063\pi$
$620$	0	0
$621$	0	0
$622$	3.81360e8	1.58476
$623$	− 2.67863e8i	− 1.10777i
$624$	0	0
$625$	0	0
$626$	− 1.23724e8i	− 0.504350i
$627$	0	0
$628$	−1.30634e7	−0.0527443
$629$	− 5.00746e8i	− 2.01217i
$630$	0	0
$631$	2.56171e8	1.01963	0.509814	−	0.860285i	$-0.329714\pi$
$631$	2.56171e8	1.01963	0.509814	+	0.860285i	$0.329714\pi$
$632$	− 3.19662e7i	− 0.126631i
$633$	0	0
$634$	6.19046e7	0.242916
$635$	0	0
$636$	0	0
$637$	2.34928e7	0.0908900
$638$	− 7.68982e7i	− 0.296111i
$639$	0	0
$640$	0	0
$641$	4.41284e7i	0.167550i	0.996485	+	0.0837749i	$0.0266977\pi$
$641$	4.41284e7i	0.167550i	−0.996485	+	0.0837749i	$0.973302\pi$
$642$	0	0
$643$	1.96340e8	0.738545	0.369272	−	0.929321i	$-0.379607\pi$
$643$	1.96340e8	0.738545	0.369272	+	0.929321i	$0.379607\pi$
$644$	− 1.49219e7i	− 0.0558683i
$645$	0	0
$646$	−5.01695e8	−1.86098
$647$	9.99293e7i	0.368961i	0.982836	+	0.184480i	$0.0590602\pi$
$647$	9.99293e7i	0.368961i	−0.982836	+	0.184480i	$0.940940\pi$
$648$	0	0
$649$	3.79447e8	1.38809
$650$	0	0
$651$	0	0
$652$	−2.73257e7	−0.0985890
$653$	− 2.49531e7i	− 0.0896160i	−0.998996	−	0.0448080i	$-0.985732\pi$
$653$	− 2.49531e7i	− 0.0896160i	0.998996	−	0.0448080i	$-0.0142676\pi$
$654$	0	0
$655$	0	0
$656$	2.52721e8i	0.895221i
$657$	0	0
$658$	−4.12103e8	−1.44653
$659$	− 1.49890e8i	− 0.523739i	−0.965103	−	0.261870i	$-0.915661\pi$
$659$	− 1.49890e8i	− 0.523739i	0.965103	−	0.261870i	$-0.0843390\pi$
$660$	0	0
$661$	−4.68646e6	−0.0162271	−0.00811355	−	0.999967i	$-0.502583\pi$
$661$	−4.68646e6	−0.0162271	−0.00811355	+	0.999967i	$0.502583\pi$
$662$	4.10128e8i	1.41366i
$663$	0	0
$664$	3.58563e6	0.0122479
$665$	0	0
$666$	0	0
$667$	6.89888e7	0.232488
$668$	1.72727e6i	0.00579469i
$669$	0	0
$670$	0	0
$671$	− 4.26798e8i	− 1.41272i
$672$	0	0
$673$	3.58399e8	1.17577	0.587884	−	0.808945i	$-0.299961\pi$
$673$	3.58399e8	1.17577	0.587884	+	0.808945i	$0.299961\pi$
$674$	− 2.11387e8i	− 0.690395i
$675$	0	0
$676$	2.81645e7	0.0911721
$677$	− 2.26026e7i	− 0.0728437i	−0.999337	−	0.0364218i	$-0.988404\pi$
$677$	− 2.26026e7i	− 0.0728437i	0.999337	−	0.0364218i	$-0.0115960\pi$
$678$	0	0
$679$	−4.12696e8	−1.31832
$680$	0	0
$681$	0	0
$682$	3.09731e7	0.0976407
$683$	5.32455e8i	1.67117i	0.549361	+	0.835585i	$0.314871\pi$
$683$	5.32455e8i	1.67117i	−0.549361	+	0.835585i	$0.685129\pi$
$684$	0	0
$685$	0	0
$686$	3.67376e8i	1.13799i
$687$	0	0
$688$	−5.64018e8	−1.73192
$689$	− 1.67381e7i	− 0.0511740i
$690$	0	0
$691$	−5.20725e8	−1.57825	−0.789123	−	0.614236i	$-0.789464\pi$
$691$	−5.20725e8	−1.57825	−0.789123	+	0.614236i	$0.789464\pi$
$692$	3.08060e7i	0.0929644i
$693$	0	0
$694$	−3.50597e8	−1.04889
$695$	0	0
$696$	0	0
$697$	−3.73880e8	−1.10417
$698$	9.34965e7i	0.274934i
$699$	0	0
$700$	0	0
$701$	− 2.97763e8i	− 0.864404i	−0.901777	−	0.432202i	$-0.857737\pi$
$701$	− 2.97763e8i	− 0.864404i	0.901777	−	0.432202i	$-0.142263\pi$
$702$	0	0
$703$	6.90667e8	1.98794
$704$	− 2.75553e8i	− 0.789746i
$705$	0	0
$706$	1.79041e8	0.508789
$707$	− 1.22211e8i	− 0.345823i
$708$	0	0
$709$	−1.89039e8	−0.530411	−0.265205	−	0.964192i	$-0.585440\pi$
$709$	−1.89039e8	−0.530411	−0.265205	+	0.964192i	$0.585440\pi$
$710$	0	0
$711$	0	0
$712$	−4.80183e8	−1.33035
$713$	2.77873e7i	0.0766617i
$714$	0	0
$715$	0	0
$716$	− 2.01687e7i	− 0.0549462i
$717$	0	0
$718$	−1.90892e8	−0.515721
$719$	− 1.30902e8i	− 0.352176i	−0.984374	−	0.176088i	$-0.943656\pi$
$719$	− 1.30902e8i	− 0.352176i	0.984374	−	0.176088i	$-0.0563443\pi$
$720$	0	0
$721$	−2.83579e7	−0.0756603
$722$	− 2.97829e8i	− 0.791325i
$723$	0	0
$724$	−7.12176e7	−0.187660
$725$	0	0
$726$	0	0
$727$	−4.44143e8	−1.15590	−0.577949	−	0.816073i	$-0.696147\pi$
$727$	−4.44143e8	−1.15590	−0.577949	+	0.816073i	$0.696147\pi$
$728$	6.88429e7i	0.178429i
$729$	0	0
$730$	0	0
$731$	− 8.34419e8i	− 2.13615i
$732$	0	0
$733$	−4.71631e8	−1.19754	−0.598770	−	0.800921i	$-0.704344\pi$
$733$	−4.71631e8	−1.19754	−0.598770	+	0.800921i	$0.704344\pi$
$734$	− 2.73888e8i	− 0.692605i
$735$	0	0
$736$	−5.63635e7	−0.141372
$737$	5.78573e8i	1.44529i
$738$	0	0
$739$	−6.40610e8	−1.58731	−0.793653	−	0.608371i	$-0.791823\pi$
$739$	−6.40610e8	−1.58731	−0.793653	+	0.608371i	$0.791823\pi$
$740$	0	0
$741$	0	0
$742$	7.20142e7	0.176281
$743$	− 5.44941e8i	− 1.32857i	−0.747481	−	0.664283i	$-0.768737\pi$
$743$	− 5.44941e8i	− 1.32857i	0.747481	−	0.664283i	$-0.231263\pi$
$744$	0	0
$745$	0	0
$746$	− 1.85015e8i	− 0.445646i
$747$	0	0
$748$	−4.84175e7	−0.115691
$749$	1.77532e8i	0.422504i
$750$	0	0
$751$	5.56224e8	1.31320	0.656598	−	0.754241i	$-0.271995\pi$
$751$	5.56224e8	1.31320	0.656598	+	0.754241i	$0.271995\pi$
$752$	8.10881e8i	1.90679i
$753$	0	0
$754$	3.40801e7	0.0795035
$755$	0	0
$756$	0	0
$757$	−4.02524e8	−0.927906	−0.463953	−	0.885860i	$-0.653569\pi$
$757$	−4.02524e8	−0.927906	−0.463953	+	0.885860i	$0.653569\pi$
$758$	1.78242e8i	0.409263i
$759$	0	0
$760$	0	0
$761$	1.44544e8i	0.327979i	0.986462	+	0.163989i	$0.0524363\pi$
$761$	1.44544e8i	0.327979i	−0.986462	+	0.163989i	$0.947564\pi$
$762$	0	0
$763$	−4.47565e8	−1.00759
$764$	− 3.76290e7i	− 0.0843806i
$765$	0	0
$766$	6.50389e8	1.44706
$767$	1.68165e8i	0.372691i
$768$	0	0
$769$	2.64281e8	0.581148	0.290574	−	0.956853i	$-0.406154\pi$
$769$	2.64281e8	0.581148	0.290574	+	0.956853i	$0.406154\pi$
$770$	0	0
$771$	0	0
$772$	−6.99040e7	−0.151932
$773$	− 1.66034e8i	− 0.359466i	−0.983715	−	0.179733i	$-0.942477\pi$
$773$	− 1.66034e8i	− 0.359466i	0.983715	−	0.179733i	$-0.0575234\pi$
$774$	0	0
$775$	0	0
$776$	7.39818e8i	1.58321i
$777$	0	0
$778$	6.22511e8	1.32193
$779$	− 5.15684e8i	− 1.09087i
$780$	0	0
$781$	−4.40043e8	−0.923723
$782$	− 4.92551e8i	− 1.02999i
$783$	0	0
$784$	1.98883e8	0.412714
$785$	0	0
$786$	0	0
$787$	−2.25184e8	−0.461970	−0.230985	−	0.972957i	$-0.574195\pi$
$787$	−2.25184e8	−0.461970	−0.230985	+	0.972957i	$0.574195\pi$
$788$	2.14716e7i	0.0438819i
$789$	0	0
$790$	0	0
$791$	− 2.77257e8i	− 0.560213i
$792$	0	0
$793$	1.89150e8	0.379303
$794$	1.88834e8i	0.377242i
$795$	0	0
$796$	−3.59206e7	−0.0712204
$797$	− 5.51423e8i	− 1.08921i	−0.838694	−	0.544603i	$-0.816680\pi$
$797$	− 5.51423e8i	− 1.08921i	0.838694	−	0.544603i	$-0.183320\pi$
$798$	0	0
$799$	−1.19963e9	−2.35184
$800$	0	0
$801$	0	0
$802$	4.07009e8	0.789008
$803$	− 3.83532e8i	− 0.740722i
$804$	0	0
$805$	0	0
$806$	1.37268e7i	0.0262158i
$807$	0	0
$808$	−2.19082e8	−0.415310
$809$	3.95120e7i	0.0746249i	0.999304	+	0.0373125i	$0.0118797\pi$
$809$	3.95120e7i	0.0746249i	−0.999304	+	0.0373125i	$0.988120\pi$
$810$	0	0
$811$	−8.11405e8	−1.52116	−0.760580	−	0.649244i	$-0.775085\pi$
$811$	−8.11405e8	−1.52116	−0.760580	+	0.649244i	$0.775085\pi$
$812$	1.29308e7i	0.0241522i
$813$	0	0
$814$	7.55819e8	1.40134
$815$	0	0
$816$	0	0
$817$	1.15089e9	2.11042
$818$	5.62553e8i	1.02779i
$819$	0	0
$820$	0	0
$821$	− 1.22832e8i	− 0.221963i	−0.993822	−	0.110982i	$-0.964601\pi$
$821$	− 1.22832e8i	− 0.221963i	0.993822	−	0.110982i	$-0.0353994\pi$
$822$	0	0
$823$	8.46296e8	1.51818	0.759089	−	0.650987i	$-0.225645\pi$
$823$	8.46296e8	1.51818	0.759089	+	0.650987i	$0.225645\pi$
$824$	5.08357e7i	0.0908630i
$825$	0	0
$826$	−7.23514e8	−1.28383
$827$	5.80388e8i	1.02613i	0.858350	+	0.513065i	$0.171490\pi$
$827$	5.80388e8i	1.02613i	−0.858350	+	0.513065i	$0.828510\pi$
$828$	0	0
$829$	6.30126e8	1.10602	0.553011	−	0.833174i	$-0.313479\pi$
$829$	6.30126e8	1.10602	0.553011	+	0.833174i	$0.313479\pi$
$830$	0	0
$831$	0	0
$832$	1.22121e8	0.212041
$833$	2.94231e8i	0.509041i
$834$	0	0
$835$	0	0
$836$	− 6.67811e7i	− 0.114297i
$837$	0	0
$838$	−3.40863e8	−0.579225
$839$	2.51051e7i	0.0425085i	0.999774	+	0.0212542i	$0.00676595\pi$
$839$	2.51051e7i	0.0425085i	−0.999774	+	0.0212542i	$0.993234\pi$
$840$	0	0
$841$	5.35040e8	0.899494
$842$	− 8.30748e8i	− 1.39166i
$843$	0	0
$844$	8.16992e6	0.0135891
$845$	0	0
$846$	0	0
$847$	9.78978e7	0.161110
$848$	− 1.41700e8i	− 0.232371i
$849$	0	0
$850$	0	0
$851$	6.78079e8i	1.10025i
$852$	0	0
$853$	−3.53908e8	−0.570221	−0.285111	−	0.958495i	$-0.592030\pi$
$853$	−3.53908e8	−0.570221	−0.285111	+	0.958495i	$0.592030\pi$
$854$	8.13800e8i	1.30660i
$855$	0	0
$856$	3.18252e8	0.507400
$857$	− 8.06395e8i	− 1.28117i	−0.767889	−	0.640583i	$-0.778693\pi$
$857$	− 8.06395e8i	− 1.28117i	0.767889	−	0.640583i	$-0.221307\pi$
$858$	0	0
$859$	9.26709e8	1.46206	0.731028	−	0.682348i	$-0.239041\pi$
$859$	9.26709e8	1.46206	0.731028	+	0.682348i	$0.239041\pi$
$860$	0	0
$861$	0	0
$862$	−1.98656e8	−0.310155
$863$	6.89660e8i	1.07301i	0.843898	+	0.536504i	$0.180255\pi$
$863$	6.89660e8i	1.07301i	−0.843898	+	0.536504i	$0.819745\pi$
$864$	0	0
$865$	0	0
$866$	5.10771e8i	0.786452i
$867$	0	0
$868$	−5.20827e6	−0.00796405
$869$	7.83499e7i	0.119393i
$870$	0	0
$871$	−2.56414e8	−0.388050
$872$	8.02327e8i	1.21005i
$873$	0	0
$874$	6.79365e8	1.01758
$875$	0	0
$876$	0	0
$877$	2.94275e8	0.436269	0.218134	−	0.975919i	$-0.430003\pi$
$877$	2.94275e8	0.436269	0.218134	+	0.975919i	$0.430003\pi$
$878$	1.28360e8i	0.189647i
$879$	0	0
$880$	0	0
$881$	− 6.79591e7i	− 0.0993847i	−0.998765	−	0.0496924i	$-0.984176\pi$
$881$	− 6.79591e7i	− 0.0993847i	0.998765	−	0.0496924i	$-0.0158241\pi$
$882$	0	0
$883$	−9.49523e8	−1.37919	−0.689594	−	0.724196i	$-0.742211\pi$
$883$	−9.49523e8	−1.37919	−0.689594	+	0.724196i	$0.742211\pi$
$884$	− 2.14579e7i	− 0.0310620i
$885$	0	0
$886$	9.39340e8	1.35058
$887$	− 1.87598e8i	− 0.268817i	−0.990926	−	0.134409i	$-0.957086\pi$
$887$	− 1.87598e8i	− 0.268817i	0.990926	−	0.134409i	$-0.0429135\pi$
$888$	0	0
$889$	1.01566e8	0.144558
$890$	0	0
$891$	0	0
$892$	8.25207e6	0.0116270
$893$	− 1.65462e9i	− 2.32351i
$894$	0	0
$895$	0	0
$896$	6.34642e8i	0.882276i
$897$	0	0
$898$	1.06583e9	1.47183
$899$	− 2.40796e7i	− 0.0331413i
$900$	0	0
$901$	2.09633e8	0.286607
$902$	− 5.64330e8i	− 0.768977i
$903$	0	0
$904$	−4.97024e8	−0.672778
$905$	0	0
$906$	0	0
$907$	1.30184e9	1.74476	0.872378	−	0.488832i	$-0.162577\pi$
$907$	1.30184e9	1.74476	0.872378	+	0.488832i	$0.162577\pi$
$908$	− 4.93480e7i	− 0.0659192i
$909$	0	0
$910$	0	0
$911$	− 1.25808e9i	− 1.66400i	−0.554775	−	0.832000i	$-0.687196\pi$
$911$	− 1.25808e9i	− 1.66400i	0.554775	−	0.832000i	$-0.312804\pi$
$912$	0	0
$913$	−8.78846e6	−0.0115478
$914$	6.48797e8i	0.849709i
$915$	0	0
$916$	−6.92759e7	−0.0901354
$917$	− 3.85646e8i	− 0.500127i
$918$	0	0
$919$	2.32755e8	0.299883	0.149942	−	0.988695i	$-0.452091\pi$
$919$	2.32755e8	0.299883	0.149942	+	0.988695i	$0.452091\pi$
$920$	0	0
$921$	0	0
$922$	1.97422e8	0.251886
$923$	− 1.95020e8i	− 0.248013i
$924$	0	0
$925$	0	0
$926$	3.91651e8i	0.493250i
$927$	0	0
$928$	4.88427e7	0.0611161
$929$	1.02421e9i	1.27745i	0.769436	+	0.638724i	$0.220538\pi$
$929$	1.02421e9i	1.27745i	−0.769436	+	0.638724i	$0.779462\pi$
$930$	0	0
$931$	−4.05825e8	−0.502910
$932$	8.28485e7i	0.102338i
$933$	0	0
$934$	−8.72917e8	−1.07135
$935$	0	0
$936$	0	0
$937$	−5.87353e8	−0.713971	−0.356985	−	0.934110i	$-0.616195\pi$
$937$	−5.87353e8	−0.713971	−0.356985	+	0.934110i	$0.616195\pi$
$938$	− 1.10320e9i	− 1.33673i
$939$	0	0
$940$	0	0
$941$	1.44532e8i	0.173458i	0.996232	+	0.0867292i	$0.0276415\pi$
$941$	1.44532e8i	0.173458i	−0.996232	+	0.0867292i	$0.972359\pi$
$942$	0	0
$943$	5.06285e8	0.603755
$944$	1.42363e9i	1.69232i
$945$	0	0
$946$	1.25946e9	1.48768
$947$	2.40595e8i	0.283294i	0.989917	+	0.141647i	$0.0452398\pi$
$947$	2.40595e8i	0.283294i	−0.989917	+	0.141647i	$0.954760\pi$
$948$	0	0
$949$	1.69975e8	0.198878
$950$	0	0
$951$	0	0
$952$	−8.62208e8	−0.999313
$953$	1.59931e9i	1.84779i	0.382644	+	0.923896i	$0.375014\pi$
$953$	1.59931e9i	1.84779i	−0.382644	+	0.923896i	$0.624986\pi$
$954$	0	0
$955$	0	0
$956$	1.96260e7i	0.0224625i
$957$	0	0
$958$	8.34129e8	0.948717
$959$	− 7.46440e8i	− 0.846328i
$960$	0	0
$961$	−8.77805e8	−0.989072
$962$	3.34967e8i	0.376250i
$963$	0	0
$964$	−2.08181e7	−0.0232386
$965$	0	0
$966$	0	0
$967$	−3.53519e8	−0.390960	−0.195480	−	0.980708i	$-0.562627\pi$
$967$	−3.53519e8	−0.390960	−0.195480	+	0.980708i	$0.562627\pi$
$968$	− 1.75496e8i	− 0.193482i
$969$	0	0
$970$	0	0
$971$	− 1.33953e8i	− 0.146317i	−0.997320	−	0.0731583i	$-0.976692\pi$
$971$	− 1.33953e8i	− 0.146317i	0.997320	−	0.0731583i	$-0.0233078\pi$
$972$	0	0
$973$	6.30969e8	0.684967
$974$	1.59636e9i	1.72764i
$975$	0	0
$976$	1.60129e9	1.72234
$977$	− 2.99781e8i	− 0.321455i	−0.986999	−	0.160728i	$-0.948616\pi$
$977$	− 2.99781e8i	− 0.321455i	0.986999	−	0.160728i	$-0.0513840\pi$
$978$	0	0
$979$	1.17694e9	1.25431
$980$	0	0
$981$	0	0
$982$	9.08685e8	0.959575
$983$	1.04294e9i	1.09799i	0.835825	+	0.548996i	$0.184990\pi$
$983$	1.04294e9i	1.09799i	−0.835825	+	0.548996i	$0.815010\pi$
$984$	0	0
$985$	0	0
$986$	4.26829e8i	0.445269i
$987$	0	0
$988$	2.95963e7	0.0306879
$989$	1.12992e9i	1.16804i
$990$	0	0
$991$	3.39676e8	0.349014	0.174507	−	0.984656i	$-0.444167\pi$
$991$	3.39676e8	0.349014	0.174507	+	0.984656i	$0.444167\pi$
$992$	1.96729e7i	0.0201527i
$993$	0	0
$994$	8.39054e8	0.854340
$995$	0	0
$996$	0	0
$997$	−1.79345e9	−1.80969	−0.904846	−	0.425739i	$-0.860014\pi$
$997$	−1.79345e9	−1.80969	−0.904846	+	0.425739i	$0.860014\pi$
$998$	− 1.55761e9i	− 1.56700i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.7.c.e.26.4		12
3.2	odd	2	inner	225.7.c.e.26.10		12
5.2	odd	4		45.7.d.a.44.10	yes	12
5.3	odd	4		45.7.d.a.44.4	yes	12
5.4	even	2	inner	225.7.c.e.26.9		12
15.2	even	4		45.7.d.a.44.3	✓	12
15.8	even	4		45.7.d.a.44.9	yes	12
15.14	odd	2	inner	225.7.c.e.26.3		12
20.3	even	4		720.7.c.a.449.10		12
20.7	even	4		720.7.c.a.449.4		12
60.23	odd	4		720.7.c.a.449.3		12
60.47	odd	4		720.7.c.a.449.9		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.7.d.a.44.3	✓	12	15.2	even	4
45.7.d.a.44.4	yes	12	5.3	odd	4
45.7.d.a.44.9	yes	12	15.8	even	4
45.7.d.a.44.10	yes	12	5.2	odd	4
225.7.c.e.26.3		12	15.14	odd	2	inner
225.7.c.e.26.4		12	1.1	even	1	trivial
225.7.c.e.26.9		12	5.4	even	2	inner
225.7.c.e.26.10		12	3.2	odd	2	inner
720.7.c.a.449.3		12	60.23	odd	4
720.7.c.a.449.4		12	20.7	even	4
720.7.c.a.449.9		12	60.47	odd	4
720.7.c.a.449.10		12	20.3	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	225.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-8.37795i q^{2} -6.18997 q^{4} +270.176 q^{7} -484.329i q^{8} +1187.10i q^{11} -526.105 q^{13} -2263.52i q^{14} -4453.84 q^{16} -6589.09i q^{17} +9088.18 q^{19} +9945.48 q^{22} +8922.54i q^{23} +4407.68i q^{26} -1672.38 q^{28} -7731.97i q^{29} +3114.29 q^{31} +6316.98i q^{32} -55203.1 q^{34} +75996.2 q^{37} -76140.3i q^{38} -56742.3i q^{41} +126636. q^{43} -7348.13i q^{44} +74752.5 q^{46} -182063. i q^{47} -44654.2 q^{49} +3256.57 q^{52} +31815.2i q^{53} -130854. i q^{56} -64778.1 q^{58} -319642. i q^{59} -359529. q^{61} -26091.3i q^{62} -232123. q^{64} +487383. q^{67} +40786.3i q^{68} +370686. i q^{71} -323083. q^{73} -636692. i q^{74} -56255.6 q^{76} +320726. i q^{77} +66000.9 q^{79} -475384. q^{82} +7403.28i q^{83} -1.06095e6i q^{86} +574948. q^{88} -991440. i q^{89} -142141. q^{91} -55230.3i q^{92} -1.52532e6 q^{94} -1.52751e6 q^{97} +374110. i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 516 q^{4} + 36372 q^{16} - 4320 q^{19} + 60192 q^{31} - 106296 q^{34} - 1078968 q^{46} - 711516 q^{49} - 449784 q^{61} - 3964572 q^{64} - 584400 q^{76} - 4324608 q^{79} + 631152 q^{91} - 5793408 q^{94}+O(q^{100}) \) 12 * q - 516 * q^4 + 36372 * q^16 - 4320 * q^19 + 60192 * q^31 - 106296 * q^34 - 1078968 * q^46 - 711516 * q^49 - 449784 * q^61 - 3964572 * q^64 - 584400 * q^76 - 4324608 * q^79 + 631152 * q^91 - 5793408 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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