LMFDB - Embedded newform 735.2.g.a.734.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [735,2,Mod(734,735)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(735, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("735.734");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 735 = 3 \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	735.g (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:	$5.86900454856$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	16.0.721389578983833600000000.5
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 8x^{14} + 44x^{12} + 128x^{10} + 223x^{8} - 464x^{6} - 724x^{4} + 784x^{2} + 2401 $ x^16 + 8x^14 + 44x^12 + 128x^10 + 223x^8 - 464x^6 - 724x^4 + 784*x^2 + 2401
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{8}\cdot 7^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			734.14
Root			$1.22796 - 0.279124i$ of defining polynomial
Character	$\chi$	$=$	735.734
Dual form			735.2.g.a.734.13

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.45591 q^{2} +1.73205i q^{3} +4.03151 q^{4} +2.23607i q^{5} +4.25377i q^{6} +4.98920 q^{8} -3.00000 q^{9} +5.49159i q^{10} +6.98277i q^{12} -3.87298 q^{15} +4.19003 q^{16} -8.06126i q^{17} -7.36774 q^{18} +5.62561i q^{19} +9.01472i q^{20} +9.58908 q^{23} +8.64155i q^{24} -5.00000 q^{25} -5.19615i q^{27} -9.51171 q^{30} -4.42678i q^{31} +0.311942 q^{32} -19.7977i q^{34} -12.0945 q^{36} +13.8160i q^{38} +11.1562i q^{40} -6.70820i q^{45} +23.5500 q^{46} +1.02391i q^{47} +7.25734i q^{48} -12.2796 q^{50} +13.9625 q^{51} +9.43433 q^{53} -12.7613i q^{54} -9.74384 q^{57} -15.6140 q^{60} +4.08075i q^{61} -10.8718i q^{62} -7.61396 q^{64} -32.4990i q^{68} +16.6088i q^{69} -14.9676 q^{72} -8.66025i q^{75} +22.6797i q^{76} -5.83648 q^{79} +9.36919i q^{80} +9.00000 q^{81} -15.0986i q^{83} +18.0255 q^{85} -16.4748i q^{90} +38.6584 q^{92} +7.66741 q^{93} +2.51464i q^{94} -12.5792 q^{95} +0.540299i q^{96} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 32 q^{4} - 48 q^{9} + 64 q^{16} - 80 q^{25} - 96 q^{36} + 128 q^{64} + 144 q^{81}+O(q^{100}) $ 16 * q + 32 * q^4 - 48 * q^9 + 64 * q^16 - 80 * q^25 - 96 * q^36 + 128 * q^64 + 144 * q^81

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/735\mathbb{Z}\right)^\times$.

$n$	$346$	$442$	$491$
$\chi(n)$	$-1$	$-1$	$-1$

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 $

$2$	2.45591	1.73659	0.868296	−	0.496046i	$-0.165215\pi$
$2$	2.45591	1.73659	0.868296	+	0.496046i	$0.165215\pi$
$3$	1.73205i	1.00000i
$3$	1.73205i	1.00000i
$4$	4.03151	2.01575
$5$	2.23607i	1.00000i
$5$	2.23607i	1.00000i
$6$	4.25377i	1.73659i
$7$	0	0
$7$	0	0
$8$	4.98920	1.76395
$9$	−3.00000	−1.00000
$10$	5.49159i	1.73659i
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	6.98277i	2.01575i
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	−3.87298	−1.00000
$16$	4.19003	1.04751
$17$	− 8.06126i	− 1.95514i	−0.210606	−	0.977571i	$-0.567544\pi$
$17$	− 8.06126i	− 1.95514i	0.210606	−	0.977571i	$-0.432456\pi$
$18$	−7.36774	−1.73659
$19$	5.62561i	1.29060i	0.763928	+	0.645301i	$0.223268\pi$
$19$	5.62561i	1.29060i	−0.763928	+	0.645301i	$0.776732\pi$
$20$	9.01472i	2.01575i
$21$	0	0
$22$	0	0
$23$	9.58908	1.99946	0.999731	−	0.0231881i	$-0.00738165\pi$
$23$	9.58908	1.99946	0.999731	+	0.0231881i	$0.00738165\pi$
$24$	8.64155i	1.76395i
$25$	−5.00000	−1.00000
$26$	0	0
$27$	− 5.19615i	− 1.00000i
$28$	0	0
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	−9.51171	−1.73659
$31$	− 4.42678i	− 0.795074i	−0.917586	−	0.397537i	$-0.869865\pi$
$31$	− 4.42678i	− 0.795074i	0.917586	−	0.397537i	$-0.130135\pi$
$32$	0.311942	0.0551440
$33$	0	0
$34$	− 19.7977i	− 3.39528i
$35$	0	0
$36$	−12.0945	−2.01575
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	13.8160i	2.24125i
$39$	0	0
$40$	11.1562i	1.76395i
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	0	0
$45$	− 6.70820i	− 1.00000i
$46$	23.5500	3.47225
$47$	1.02391i	0.149353i	0.997208	+	0.0746766i	$0.0237924\pi$
$47$	1.02391i	0.149353i	−0.997208	+	0.0746766i	$0.976208\pi$
$48$	7.25734i	1.04751i
$49$	0	0
$50$	−12.2796	−1.73659
$51$	13.9625	1.95514
$52$	0	0
$53$	9.43433	1.29591	0.647953	−	0.761681i	$-0.275625\pi$
$53$	9.43433	1.29591	0.647953	+	0.761681i	$0.275625\pi$
$54$	− 12.7613i	− 1.73659i
$55$	0	0
$56$	0	0
$57$	−9.74384	−1.29060
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	−15.6140	−2.01575
$61$	4.08075i	0.522486i	0.965273	+	0.261243i	$0.0841324\pi$
$61$	4.08075i	0.522486i	−0.965273	+	0.261243i	$0.915868\pi$
$62$	− 10.8718i	− 1.38072i
$63$	0	0
$64$	−7.61396	−0.951744
$65$	0	0
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	− 32.4990i	− 3.94108i
$69$	16.6088i	1.99946i
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	−14.9676	−1.76395
$73$	0	0		−	1.00000i	$-0.5\pi$
$73$	0	0			1.00000i	$0.5\pi$
$74$	0	0
$75$	− 8.66025i	− 1.00000i
$76$	22.6797i	2.60154i
$77$	0	0
$78$	0	0
$79$	−5.83648	−0.656656	−0.328328	−	0.944564i	$-0.606485\pi$
$79$	−5.83648	−0.656656	−0.328328	+	0.944564i	$0.606485\pi$
$80$	9.36919i	1.04751i
$81$	9.00000	1.00000
$82$	0	0
$83$	− 15.0986i	− 1.65729i	−0.559777	−	0.828643i	$-0.689113\pi$
$83$	− 15.0986i	− 1.65729i	0.559777	−	0.828643i	$-0.310887\pi$
$84$	0	0
$85$	18.0255	1.95514
$86$	0	0
$87$	0	0
$88$	0	0
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	− 16.4748i	− 1.73659i
$91$	0	0
$92$	38.6584	4.03042
$93$	7.66741	0.795074
$94$	2.51464i	0.259366i
$95$	−12.5792	−1.29060
$96$	0.540299i	0.0551440i
$97$	0	0		−	1.00000i	$-0.5\pi$
$97$	0	0			1.00000i	$0.5\pi$
$98$	0	0
$99$	0	0
$100$	−20.1575	−2.01575
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	34.2907	3.39528
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	0	0
$105$	0	0
$106$	23.1699	2.25046
$107$	−2.75559	−0.266393	−0.133196	−	0.991090i	$-0.542524\pi$
$107$	−2.75559	−0.266393	−0.133196	+	0.991090i	$0.542524\pi$
$108$	− 20.9483i	− 2.01575i
$109$	−15.4919	−1.48386	−0.741929	−	0.670478i	$-0.766089\pi$
$109$	−15.4919	−1.48386	−0.741929	+	0.670478i	$0.766089\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−17.4911	−1.64542	−0.822710	−	0.568461i	$-0.807539\pi$
$113$	−17.4911	−1.64542	−0.822710	+	0.568461i	$0.807539\pi$
$114$	−23.9300	−2.24125
$115$	21.4418i	1.99946i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	−19.3231	−1.76395
$121$	11.0000	1.00000
$122$	10.0220i	0.907345i
$123$	0	0
$124$	− 17.8466i	− 1.60267i
$125$	− 11.1803i	− 1.00000i
$126$	0	0
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	−19.3231	−1.70794
$129$	0	0
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	11.6190	1.00000
$136$	− 40.2192i	− 3.44877i
$137$	−2.15624	−0.184220	−0.0921099	−	0.995749i	$-0.529361\pi$
$137$	−2.15624	−0.184220	−0.0921099	+	0.995749i	$0.529361\pi$
$138$	40.7897i	3.47225i
$139$	− 12.5883i	− 1.06772i	−0.845572	−	0.533862i	$-0.820740\pi$
$139$	− 12.5883i	− 1.06772i	0.845572	−	0.533862i	$-0.179260\pi$
$140$	0	0
$141$	−1.77347	−0.149353
$142$	0	0
$143$	0	0
$144$	−12.5701	−1.04751
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	− 21.2688i	− 1.73659i
$151$	−22.0885	−1.79754	−0.898770	−	0.438421i	$-0.855538\pi$
$151$	−22.0885	−1.79754	−0.898770	+	0.438421i	$0.855538\pi$
$152$	28.0673i	2.27656i
$153$	24.1838i	1.95514i
$154$	0	0
$155$	9.89859	0.795074
$156$	0	0
$157$	0	0		−	1.00000i	$-0.5\pi$
$157$	0	0			1.00000i	$0.5\pi$
$158$	−14.3339	−1.14034
$159$	16.3407i	1.29591i
$160$	0.697523i	0.0551440i
$161$	0	0
$162$	22.1032	1.73659
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	0	0
$165$	0	0
$166$	− 37.0808i	− 2.87803i
$167$	8.94427i	0.692129i	0.938211	+	0.346064i	$0.112482\pi$
$167$	8.94427i	0.692129i	−0.938211	+	0.346064i	$0.887518\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	44.2691	3.39528
$171$	− 16.8768i	− 1.29060i
$172$	0	0
$173$	6.01343i	0.457193i	0.973521	+	0.228596i	$0.0734136\pi$
$173$	6.01343i	0.457193i	−0.973521	+	0.228596i	$0.926586\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	− 27.0442i	− 2.01575i
$181$	5.89364i	0.438071i	0.975717	+	0.219036i	$0.0702911\pi$
$181$	5.89364i	0.438071i	−0.975717	+	0.219036i	$0.929709\pi$
$182$	0	0
$183$	−7.06806	−0.522486
$184$	47.8419	3.52695
$185$	0	0
$186$	18.8305	1.38072
$187$	0	0
$188$	4.12792i	0.301059i
$189$	0	0
$190$	−30.8935	−2.24125
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	− 13.1878i	− 0.951744i
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	27.3147	1.94609	0.973046	−	0.230610i	$-0.0740720\pi$
$197$	27.3147	1.94609	0.973046	+	0.230610i	$0.0740720\pi$
$198$	0	0
$199$	− 27.5920i	− 1.95594i	−0.208741	−	0.977971i	$-0.566937\pi$
$199$	− 27.5920i	− 1.95594i	0.208741	−	0.977971i	$-0.433063\pi$
$200$	−24.9460	−1.76395
$201$	0	0
$202$	0	0
$203$	0	0
$204$	56.2899	3.94108
$205$	0	0
$206$	0	0
$207$	−28.7673	−1.99946
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−7.74597	−0.533254	−0.266627	−	0.963800i	$-0.585909\pi$
$211$	−7.74597	−0.533254	−0.266627	+	0.963800i	$0.585909\pi$
$212$	38.0346	2.61222
$213$	0	0
$214$	−6.76748	−0.462616
$215$	0	0
$216$	− 25.9246i	− 1.76395i
$217$	0	0
$218$	−38.0468	−2.57686
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	0	0		−	1.00000i	$-0.5\pi$
$223$	0	0			1.00000i	$0.5\pi$
$224$	0	0
$225$	15.0000	1.00000
$226$	−42.9565	−2.85743
$227$	17.8885i	1.18730i	0.804722	+	0.593652i	$0.202314\pi$
$227$	17.8885i	1.18730i	−0.804722	+	0.593652i	$0.797686\pi$
$228$	−39.2823	−2.60154
$229$	29.9494i	1.97911i	0.144155	+	0.989555i	$0.453954\pi$
$229$	29.9494i	1.97911i	−0.144155	+	0.989555i	$0.546046\pi$
$230$	52.6593i	3.47225i
$231$	0	0
$232$	0	0
$233$	−28.6125	−1.87447	−0.937234	−	0.348702i	$-0.886622\pi$
$233$	−28.6125	−1.87447	−0.937234	+	0.348702i	$0.886622\pi$
$234$	0	0
$235$	−2.28954	−0.149353
$236$	0	0
$237$	− 10.1091i	− 0.656656i
$238$	0	0
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	−16.2279	−1.04751
$241$	27.8600i	1.79462i	0.441401	+	0.897310i	$0.354482\pi$
$241$	27.8600i	1.79462i	−0.441401	+	0.897310i	$0.645518\pi$
$242$	27.0150	1.73659
$243$	15.5885i	1.00000i
$244$	16.4516i	1.05320i
$245$	0	0
$246$	0	0
$247$	0	0
$248$	− 22.0861i	− 1.40247i
$249$	26.1515	1.65729
$250$	− 27.4579i	− 1.73659i
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	31.2211i	1.95514i
$256$	−32.2279	−2.01424
$257$	6.92820i	0.432169i	0.976375	+	0.216085i	$0.0693287\pi$
$257$	6.92820i	0.432169i	−0.976375	+	0.216085i	$0.930671\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	32.2265	1.98717	0.993587	−	0.113073i	$-0.0360693\pi$
$263$	32.2265	1.98717	0.993587	+	0.113073i	$0.0360693\pi$
$264$	0	0
$265$	21.0958i	1.29591i
$266$	0	0
$267$	0	0
$268$	0	0
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	28.5351	1.73659
$271$	− 5.08953i	− 0.309167i	−0.987980	−	0.154583i	$-0.950596\pi$
$271$	− 5.08953i	− 0.309167i	0.987980	−	0.154583i	$-0.0494035\pi$
$272$	− 33.7769i	− 2.04803i
$273$	0	0
$274$	−5.29553	−0.319915
$275$	0	0
$276$	66.9584i	4.03042i
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	− 30.9157i	− 1.85420i
$279$	13.2803i	0.795074i
$280$	0	0
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	−4.35549	−0.259366
$283$	0	0		−	1.00000i	$-0.5\pi$
$283$	0	0			1.00000i	$0.5\pi$
$284$	0	0
$285$	− 21.7879i	− 1.29060i
$286$	0	0
$287$	0	0
$288$	−0.935825	−0.0551440
$289$	−47.9839	−2.82258
$290$	0	0
$291$	0	0
$292$	0	0
$293$	13.8564i	0.809500i	0.914427	+	0.404750i	$0.132641\pi$
$293$	13.8564i	0.809500i	−0.914427	+	0.404750i	$0.867359\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	− 34.9139i	− 2.01575i
$301$	0	0
$302$	−54.2475	−3.12159
$303$	0	0
$304$	23.5715i	1.35192i
$305$	−9.12483	−0.522486
$306$	59.3932i	3.39528i
$307$	0	0		−	1.00000i	$-0.5\pi$
$307$	0	0			1.00000i	$0.5\pi$
$308$	0	0
$309$	0	0
$310$	24.3101	1.38072
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	0	0		−	1.00000i	$-0.5\pi$
$313$	0	0			1.00000i	$0.5\pi$
$314$	0	0
$315$	0	0
$316$	−23.5298	−1.32366
$317$	−10.0533	−0.564652	−0.282326	−	0.959319i	$-0.591106\pi$
$317$	−10.0533	−0.564652	−0.282326	+	0.959319i	$0.591106\pi$
$318$	40.1314i	2.25046i
$319$	0	0
$320$	− 17.0253i	− 0.951744i
$321$	− 4.77282i	− 0.266393i
$322$	0	0
$323$	45.3495	2.52331
$324$	36.2836	2.01575
$325$	0	0
$326$	0	0
$327$	− 26.8328i	− 1.48386i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	23.2379	1.27727	0.638635	−	0.769510i	$-0.279499\pi$
$331$	23.2379	1.27727	0.638635	+	0.769510i	$0.279499\pi$
$332$	− 60.8701i	− 3.34068i
$333$	0	0
$334$	21.9663i	1.20195i
$335$	0	0
$336$	0	0
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	−31.9269	−1.73659
$339$	− 30.2954i	− 1.64542i
$340$	72.6700	3.94108
$341$	0	0
$342$	− 41.4480i	− 2.24125i
$343$	0	0
$344$	0	0
$345$	−37.1384	−1.99946
$346$	14.7685i	0.793957i
$347$	29.0768	1.56092	0.780461	−	0.625204i	$-0.214984\pi$
$347$	29.0768	1.56092	0.780461	+	0.625204i	$0.214984\pi$
$348$	0	0
$349$	27.3239i	1.46262i	0.682048	+	0.731308i	$0.261090\pi$
$349$	27.3239i	1.46262i	−0.682048	+	0.731308i	$0.738910\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 20.7846i	− 1.10625i	−0.833097	−	0.553127i	$-0.813435\pi$
$353$	− 20.7846i	− 1.10625i	0.833097	−	0.553127i	$-0.186565\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	− 33.4686i	− 1.76395i
$361$	−12.6474	−0.665655
$362$	14.4743i	0.760751i
$363$	19.0526i	1.00000i
$364$	0	0
$365$	0	0
$366$	−17.3585	−0.907345
$367$	0	0		−	1.00000i	$-0.5\pi$
$367$	0	0			1.00000i	$0.5\pi$
$368$	40.1785	2.09445
$369$	0	0
$370$	0	0
$371$	0	0
$372$	30.9112	1.60267
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	19.3649	1.00000
$376$	5.10851i	0.263451i
$377$	0	0
$378$	0	0
$379$	−30.2146	−1.55202	−0.776009	−	0.630722i	$-0.782759\pi$
$379$	−30.2146	−1.55202	−0.776009	+	0.630722i	$0.782759\pi$
$380$	−50.7133	−2.60154
$381$	0	0
$382$	0	0
$383$	− 33.2689i	− 1.69996i	−0.526812	−	0.849982i	$-0.676613\pi$
$383$	− 33.2689i	− 1.69996i	0.526812	−	0.849982i	$-0.323387\pi$
$384$	− 33.4686i	− 1.70794i
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	− 77.3001i	− 3.90923i
$392$	0	0
$393$	0	0
$394$	67.0825	3.37957
$395$	− 13.0508i	− 0.656656i
$396$	0	0
$397$	0	0		−	1.00000i	$-0.5\pi$
$397$	0	0			1.00000i	$0.5\pi$
$398$	− 67.7634i	− 3.39667i
$399$	0	0
$400$	−20.9501	−1.04751
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	20.1246i	1.00000i
$406$	0	0
$407$	0	0
$408$	69.6618	3.44877
$409$	38.5751i	1.90742i	0.300732	+	0.953709i	$0.402769\pi$
$409$	38.5751i	1.90742i	−0.300732	+	0.953709i	$0.597231\pi$
$410$	0	0
$411$	− 3.73471i	− 0.184220i
$412$	0	0
$413$	0	0
$414$	−70.6499	−3.47225
$415$	33.7615	1.65729
$416$	0	0
$417$	21.8035	1.06772
$418$	0	0
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	37.8245	1.84345	0.921727	−	0.387840i	$-0.126779\pi$
$421$	37.8245	1.84345	0.921727	+	0.387840i	$0.126779\pi$
$422$	−19.0234	−0.926045
$423$	− 3.07174i	− 0.149353i
$424$	47.0698	2.28591
$425$	40.3063i	1.95514i
$426$	0	0
$427$	0	0
$428$	−11.1092	−0.536982
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	− 21.7720i	− 1.04751i
$433$	0	0		−	1.00000i	$-0.5\pi$
$433$	0	0			1.00000i	$0.5\pi$
$434$	0	0
$435$	0	0
$436$	−62.4558	−2.99109
$437$	53.9444i	2.58051i
$438$	0	0
$439$	− 29.6033i	− 1.41289i	−0.707768	−	0.706445i	$-0.750298\pi$
$439$	− 29.6033i	− 1.41289i	0.707768	−	0.706445i	$-0.249702\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	42.0502	1.99786	0.998932	−	0.0461968i	$-0.0147101\pi$
$443$	42.0502	1.99786	0.998932	+	0.0461968i	$0.0147101\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	36.8387	1.73659
$451$	0	0
$452$	−70.5153	−3.31676
$453$	− 38.2585i	− 1.79754i
$454$	43.9327i	2.06186i
$455$	0	0
$456$	−48.6140	−2.27656
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	73.5530i	3.43691i
$459$	−41.8875	−1.95514
$460$	86.4429i	4.03042i
$461$	0	0		−	1.00000i	$-0.5\pi$
$461$	0	0			1.00000i	$0.5\pi$
$462$	0	0
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	0	0
$465$	17.1449i	0.795074i
$466$	−70.2698	−3.25519
$467$	35.7771i	1.65557i	0.561048	+	0.827783i	$0.310398\pi$
$467$	35.7771i	1.65557i	−0.561048	+	0.827783i	$0.689602\pi$
$468$	0	0
$469$	0	0
$470$	−5.62291	−0.259366
$471$	0	0
$472$	0	0
$473$	0	0
$474$	− 24.8270i	− 1.14034i
$475$	− 28.1280i	− 1.29060i
$476$	0	0
$477$	−28.3030	−1.29591
$478$	0	0
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	−1.20815	−0.0551440
$481$	0	0
$482$	68.4217i	3.11652i
$483$	0	0
$484$	44.3466	2.01575
$485$	0	0
$486$	38.2839i	1.73659i
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	20.3597i	0.921639i
$489$	0	0
$490$	0	0
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	− 18.5483i	− 0.832845i
$497$	0	0
$498$	64.2259	2.87803
$499$	−25.6355	−1.14760	−0.573801	−	0.818995i	$-0.694532\pi$
$499$	−25.6355	−1.14760	−0.573801	+	0.818995i	$0.694532\pi$
$500$	− 45.0736i	− 2.01575i
$501$	−15.4919	−0.692129
$502$	0	0
$503$	29.1733i	1.30077i	0.759604	+	0.650386i	$0.225393\pi$
$503$	29.1733i	1.30077i	−0.759604	+	0.650386i	$0.774607\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 22.5167i	− 1.00000i
$508$	0	0
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	76.6763i	3.39528i
$511$	0	0
$512$	−40.5027	−1.78999
$513$	29.2315	1.29060
$514$	17.0151i	0.750502i
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−10.4156	−0.457193
$520$	0	0
$521$	0	0		−	1.00000i	$-0.5\pi$
$521$	0	0			1.00000i	$0.5\pi$
$522$	0	0
$523$	0	0		−	1.00000i	$-0.5\pi$
$523$	0	0			1.00000i	$0.5\pi$
$524$	0	0
$525$	0	0
$526$	79.1456	3.45091
$527$	−35.6854	−1.55448
$528$	0	0
$529$	68.9505	2.99785
$530$	51.8095i	2.25046i
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	− 6.16168i	− 0.266393i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	46.8419	2.01575
$541$	−34.2776	−1.47371	−0.736854	−	0.676052i	$-0.763689\pi$
$541$	−34.2776	−1.47371	−0.736854	+	0.676052i	$0.763689\pi$
$542$	− 12.4994i	− 0.536897i
$543$	−10.2081	−0.438071
$544$	− 2.51464i	− 0.107814i
$545$	− 34.6410i	− 1.48386i
$546$	0	0
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	−8.69288	−0.371341
$549$	− 12.2422i	− 0.522486i
$550$	0	0
$551$	0	0
$552$	82.8645i	3.52695i
$553$	0	0
$554$	0	0
$555$	0	0
$556$	− 50.7497i	− 2.15227i
$557$	−46.9620	−1.98984	−0.994922	−	0.100647i	$-0.967909\pi$
$557$	−46.9620	−1.98984	−0.994922	+	0.100647i	$0.967909\pi$
$558$	32.6154i	1.38072i
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 47.3436i	− 1.99530i	−0.0685449	−	0.997648i	$-0.521836\pi$
$563$	− 47.3436i	− 1.99530i	0.0685449	−	0.997648i	$-0.478164\pi$
$564$	−7.14976	−0.301059
$565$	− 39.1112i	− 1.64542i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	0	0	1.00000			$0$
$569$	0	0	−1.00000			$\pi$
$570$	− 53.5091i	− 2.24125i
$571$	38.7298	1.62079	0.810397	−	0.585882i	$-0.199252\pi$
$571$	38.7298	1.62079	0.810397	+	0.585882i	$0.199252\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−47.9454	−1.99946
$576$	22.8419	0.951744
$577$	0	0		−	1.00000i	$-0.5\pi$
$577$	0	0			1.00000i	$0.5\pi$
$578$	−117.844	−4.90167
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	34.0301i	1.40577i
$587$	19.1943i	0.792232i	0.918201	+	0.396116i	$0.129642\pi$
$587$	19.1943i	0.792232i	−0.918201	+	0.396116i	$0.870358\pi$
$588$	0	0
$589$	24.9033	1.02612
$590$	0	0
$591$	47.3105i	1.94609i
$592$	0	0
$593$	4.47214i	0.183649i	0.995775	+	0.0918243i	$0.0292698\pi$
$593$	4.47214i	0.183649i	−0.995775	+	0.0918243i	$0.970730\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	47.7907	1.95594
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	− 43.2077i	− 1.76395i
$601$	46.9644i	1.91572i	0.287237	+	0.957860i	$0.407263\pi$
$601$	46.9644i	1.91572i	−0.287237	+	0.957860i	$0.592737\pi$
$602$	0	0
$603$	0	0
$604$	−89.0500	−3.62339
$605$	24.5967i	1.00000i
$606$	0	0
$607$	0	0		−	1.00000i	$-0.5\pi$
$607$	0	0			1.00000i	$0.5\pi$
$608$	1.75486i	0.0711690i
$609$	0	0
$610$	−22.4098	−0.907345
$611$	0	0
$612$	97.4970i	3.94108i
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	48.1002	1.93644	0.968220	−	0.250100i	$-0.0804635\pi$
$617$	48.1002	1.93644	0.968220	+	0.250100i	$0.0804635\pi$
$618$	0	0
$619$	− 38.3071i	− 1.53969i	−0.638230	−	0.769846i	$-0.720333\pi$
$619$	− 38.3071i	− 1.53969i	0.638230	−	0.769846i	$-0.279667\pi$
$620$	39.9062	1.60267
$621$	− 49.8263i	− 1.99946i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	25.0000	1.00000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−50.0135	−1.99101	−0.995504	−	0.0947206i	$-0.969804\pi$
$631$	−50.0135	−1.99101	−0.995504	+	0.0947206i	$0.969804\pi$
$632$	−29.1194	−1.15831
$633$	− 13.4164i	− 0.533254i
$634$	−24.6901	−0.980570
$635$	0	0
$636$	65.8778i	2.61222i
$637$	0	0
$638$	0	0
$639$	0	0
$640$	− 43.2077i	− 1.70794i
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	− 11.7216i	− 0.462616i
$643$	0	0		−	1.00000i	$-0.5\pi$
$643$	0	0			1.00000i	$0.5\pi$
$644$	0	0
$645$	0	0
$646$	111.374	4.38196
$647$	− 44.7214i	− 1.75818i	−0.476658	−	0.879089i	$-0.658152\pi$
$647$	− 44.7214i	− 1.75818i	0.476658	−	0.879089i	$-0.341848\pi$
$648$	44.9028	1.76395
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	31.6272	1.23767	0.618834	−	0.785522i	$-0.287605\pi$
$653$	31.6272	1.23767	0.618834	+	0.785522i	$0.287605\pi$
$654$	− 65.8990i	− 2.57686i
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	0	0	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	0	0
$661$	− 38.1109i	− 1.48234i	−0.671317	−	0.741170i	$-0.734271\pi$
$661$	− 38.1109i	− 1.48234i	0.671317	−	0.741170i	$-0.265729\pi$
$662$	57.0702	2.21810
$663$	0	0
$664$	− 75.3300i	− 2.92337i
$665$	0	0
$666$	0	0
$667$	0	0
$668$	36.0589i	1.39516i
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	0	0
$675$	25.9808i	1.00000i
$676$	−52.4096	−2.01575
$677$	− 41.5692i	− 1.59763i	−0.601574	−	0.798817i	$-0.705459\pi$
$677$	− 41.5692i	− 1.59763i	0.601574	−	0.798817i	$-0.294541\pi$
$678$	− 74.4029i	− 2.85743i
$679$	0	0
$680$	89.9329	3.44877
$681$	−30.9839	−1.18730
$682$	0	0
$683$	47.6359	1.82274	0.911369	−	0.411591i	$-0.135027\pi$
$683$	47.6359	1.82274	0.911369	+	0.411591i	$0.135027\pi$
$684$	− 68.0390i	− 2.60154i
$685$	− 4.82149i	− 0.184220i
$686$	0	0
$687$	−51.8738	−1.97911
$688$	0	0
$689$	0	0
$690$	−91.2086	−3.47225
$691$	27.0559i	1.02925i	0.857414	+	0.514627i	$0.172070\pi$
$691$	27.0559i	1.02925i	−0.857414	+	0.514627i	$0.827930\pi$
$692$	24.2432i	0.921587i
$693$	0	0
$694$	71.4100	2.71069
$695$	28.1482	1.06772
$696$	0	0
$697$	0	0
$698$	67.1051i	2.53997i
$699$	− 49.5583i	− 1.87447i
$700$	0	0
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	− 3.96560i	− 0.149353i
$706$	− 51.0452i	− 1.92111i
$707$	0	0
$708$	0	0
$709$	−14.4786	−0.543754	−0.271877	−	0.962332i	$-0.587644\pi$
$709$	−14.4786	−0.543754	−0.271877	+	0.962332i	$0.587644\pi$
$710$	0	0
$711$	17.5094	0.656656
$712$	0	0
$713$	− 42.4488i	− 1.58972i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	− 28.1076i	− 1.04751i
$721$	0	0
$722$	−31.0610	−1.15597
$723$	−48.2549	−1.79462
$724$	23.7603i	0.883043i
$725$	0	0
$726$	46.7914i	1.73659i
$727$	0	0		−	1.00000i	$-0.5\pi$
$727$	0	0			1.00000i	$0.5\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	0	0
$731$	0	0
$732$	−28.4949	−1.05320
$733$	0	0		−	1.00000i	$-0.5\pi$
$733$	0	0			1.00000i	$0.5\pi$
$734$	0	0
$735$	0	0
$736$	2.99124	0.110258
$737$	0	0
$738$	0	0
$739$	4.00000	0.147142	0.0735712	−	0.997290i	$-0.476560\pi$
$739$	4.00000	0.147142	0.0735712	+	0.997290i	$0.476560\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−29.3863	−1.07808	−0.539039	−	0.842281i	$-0.681212\pi$
$743$	−29.3863	−1.07808	−0.539039	+	0.842281i	$0.681212\pi$
$744$	38.2543	1.40247
$745$	0	0
$746$	0	0
$747$	45.2958i	1.65729i
$748$	0	0
$749$	0	0
$750$	47.5585	1.73659
$751$	8.00000	0.291924	0.145962	−	0.989290i	$-0.453372\pi$
$751$	8.00000	0.291924	0.145962	+	0.989290i	$0.453372\pi$
$752$	4.29023i	0.156449i
$753$	0	0
$754$	0	0
$755$	− 49.3915i	− 1.79754i
$756$	0	0
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	−74.2043	−2.69522
$759$	0	0
$760$	−62.7603	−2.27656
$761$	0	0		−	1.00000i	$-0.5\pi$
$761$	0	0			1.00000i	$0.5\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−54.0766	−1.95514
$766$	− 81.7056i	− 2.95214i
$767$	0	0
$768$	− 55.8204i	− 2.01424i
$769$	− 30.6414i	− 1.10496i	−0.833527	−	0.552479i	$-0.813682\pi$
$769$	− 30.6414i	− 1.10496i	0.833527	−	0.552479i	$-0.186318\pi$
$770$	0	0
$771$	−12.0000	−0.432169
$772$	0	0
$773$	− 42.3541i	− 1.52337i	−0.647947	−	0.761686i	$-0.724372\pi$
$773$	− 42.3541i	− 1.52337i	0.647947	−	0.761686i	$-0.275628\pi$
$774$	0	0
$775$	22.1339i	0.795074i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	− 189.842i	− 6.78874i
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	0	0		−	1.00000i	$-0.5\pi$
$787$	0	0			1.00000i	$0.5\pi$
$788$	110.119	3.92284
$789$	55.8180i	1.98717i
$790$	− 32.0516i	− 1.14034i
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	−36.5390	−1.29591
$796$	− 111.237i	− 3.94270i
$797$	− 54.3810i	− 1.92627i	−0.269013	−	0.963136i	$-0.586698\pi$
$797$	− 54.3810i	− 1.92627i	0.269013	−	0.963136i	$-0.413302\pi$
$798$	0	0
$799$	8.25403	0.292007
$800$	−1.55971	−0.0551440
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	0	0	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	49.4243i	1.73659i
$811$	39.1490i	1.37471i	0.726323	+	0.687353i	$0.241228\pi$
$811$	39.1490i	1.37471i	−0.726323	+	0.687353i	$0.758772\pi$
$812$	0	0
$813$	8.81532	0.309167
$814$	0	0
$815$	0	0
$816$	58.5033	2.04803
$817$	0	0
$818$	94.7371i	3.31241i
$819$	0	0
$820$	0	0
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	− 9.17212i	− 0.319915i
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−27.9141	−0.970667	−0.485334	−	0.874329i	$-0.661302\pi$
$827$	−27.9141	−0.970667	−0.485334	+	0.874329i	$0.661302\pi$
$828$	−115.975	−4.03042
$829$	13.6264i	0.473264i	0.971599	+	0.236632i	$0.0760436\pi$
$829$	13.6264i	0.473264i	−0.971599	+	0.236632i	$0.923956\pi$
$830$	82.9153	2.87803
$831$	0	0
$832$	0	0
$833$	0	0
$834$	53.5476	1.85420
$835$	−20.0000	−0.692129
$836$	0	0
$837$	−23.0022	−0.795074
$838$	0	0
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	29.0000	1.00000
$842$	92.8937	3.20133
$843$	0	0
$844$	−31.2279	−1.07491
$845$	− 29.0689i	− 1.00000i
$846$	− 7.54393i	− 0.259366i
$847$	0	0
$848$	39.5301	1.35747
$849$	0	0
$850$	98.9887i	3.39528i
$851$	0	0
$852$	0	0
$853$	0	0		−	1.00000i	$-0.5\pi$
$853$	0	0			1.00000i	$0.5\pi$
$854$	0	0
$855$	37.7377	1.29060
$856$	−13.7482	−0.469903
$857$	36.2106i	1.23693i	0.785812	+	0.618466i	$0.212245\pi$
$857$	36.2106i	1.23693i	−0.785812	+	0.618466i	$0.787755\pi$
$858$	0	0
$859$	− 55.4720i	− 1.89268i	−0.323174	−	0.946340i	$-0.604750\pi$
$859$	− 55.4720i	− 1.89268i	0.323174	−	0.946340i	$-0.395250\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	18.0904	0.615805	0.307902	−	0.951418i	$-0.400373\pi$
$863$	18.0904	0.615805	0.307902	+	0.951418i	$0.400373\pi$
$864$	− 1.62090i	− 0.0551440i
$865$	−13.4464	−0.457193
$866$	0	0
$867$	− 83.1105i	− 2.82258i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	−77.2924	−2.61745
$873$	0	0
$874$	132.483i	4.48130i
$875$	0	0
$876$	0	0
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	− 72.7032i	− 2.45361i
$879$	−24.0000	−0.809500
$880$	0	0
$881$	0	0		−	1.00000i	$-0.5\pi$
$881$	0	0			1.00000i	$0.5\pi$
$882$	0	0
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	0	0
$885$	0	0
$886$	103.272	3.46948
$887$	35.3168i	1.18582i	0.805268	+	0.592911i	$0.202021\pi$
$887$	35.3168i	1.18582i	−0.805268	+	0.592911i	$0.797979\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−5.76014	−0.192756
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	60.4726	2.01575
$901$	− 76.0526i	− 2.53368i
$902$	0	0
$903$	0	0
$904$	−87.2664	−2.90244
$905$	−13.1786	−0.438071
$906$	− 93.9594i	− 3.12159i
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	72.1178i	2.39331i
$909$	0	0
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	−40.8270	−1.35192
$913$	0	0
$914$	0	0
$915$	− 15.8047i	− 0.522486i
$916$	120.741i	3.98940i
$917$	0	0
$918$	−102.872	−3.39528
$919$	23.2379	0.766548	0.383274	−	0.923635i	$-0.374797\pi$
$919$	23.2379	0.766548	0.383274	+	0.923635i	$0.374797\pi$
$920$	106.978i	3.52695i
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	0	0		−	1.00000i	$-0.5\pi$
$929$	0	0			1.00000i	$0.5\pi$
$930$	42.1063i	1.38072i
$931$	0	0
$932$	−115.351	−3.77846
$933$	0	0
$934$	87.8654i	2.87504i
$935$	0	0
$936$	0	0
$937$	0	0		−	1.00000i	$-0.5\pi$
$937$	0	0			1.00000i	$0.5\pi$
$938$	0	0
$939$	0	0
$940$	−9.23030	−0.301059
$941$	0	0		−	1.00000i	$-0.5\pi$
$941$	0	0			1.00000i	$0.5\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−47.5614	−1.54554	−0.772768	−	0.634688i	$-0.781129\pi$
$947$	−47.5614	−1.54554	−0.772768	+	0.634688i	$0.781129\pi$
$948$	− 40.7548i	− 1.32366i
$949$	0	0
$950$	− 69.0800i	− 2.24125i
$951$	− 17.4129i	− 0.564652i
$952$	0	0
$953$	−41.4508	−1.34272	−0.671362	−	0.741129i	$-0.734290\pi$
$953$	−41.4508	−1.34272	−0.671362	+	0.741129i	$0.734290\pi$
$954$	−69.5097	−2.25046
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	29.4887	0.951744
$961$	11.4036	0.367858
$962$	0	0
$963$	8.26677	0.266393
$964$	112.318i	3.61751i
$965$	0	0
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	54.8812	1.76395
$969$	78.5476i	2.52331i
$970$	0	0
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	62.8450i	2.01575i
$973$	0	0
$974$	0	0
$975$	0	0
$976$	17.0984i	0.547308i
$977$	27.9935	0.895591	0.447796	−	0.894136i	$-0.352209\pi$
$977$	27.9935	0.895591	0.447796	+	0.894136i	$0.352209\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	46.4758	1.48386
$982$	0	0
$983$	3.46410i	0.110488i	0.998473	+	0.0552438i	$0.0175936\pi$
$983$	3.46410i	0.110488i	−0.998473	+	0.0552438i	$0.982406\pi$
$984$	0	0
$985$	61.0776i	1.94609i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	32.0000	1.01651	0.508257	−	0.861206i	$-0.330290\pi$
$991$	32.0000	1.01651	0.508257	+	0.861206i	$0.330290\pi$
$992$	− 1.38090i	− 0.0438436i
$993$	40.2492i	1.27727i
$994$	0	0
$995$	61.6975	1.95594
$996$	105.430	3.34068
$997$	0	0		−	1.00000i	$-0.5\pi$
$997$	0	0			1.00000i	$0.5\pi$
$998$	−62.9585	−1.99292
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	735.2.g.a.734.14	yes	16
3.2	odd	2	inner	735.2.g.a.734.3	✓	16
5.4	even	2	inner	735.2.g.a.734.3	✓	16
7.2	even	3		735.2.p.d.374.2		16
7.3	odd	6		735.2.p.d.509.2		16
7.4	even	3		735.2.p.e.509.2		16
7.5	odd	6		735.2.p.e.374.2		16
7.6	odd	2	inner	735.2.g.a.734.13	yes	16
15.14	odd	2	CM	735.2.g.a.734.14	yes	16
21.2	odd	6		735.2.p.e.374.7		16
21.5	even	6		735.2.p.d.374.7		16
21.11	odd	6		735.2.p.d.509.7		16
21.17	even	6		735.2.p.e.509.7		16
21.20	even	2	inner	735.2.g.a.734.4	yes	16
35.4	even	6		735.2.p.d.509.7		16
35.9	even	6		735.2.p.e.374.7		16
35.19	odd	6		735.2.p.d.374.7		16
35.24	odd	6		735.2.p.e.509.7		16
35.34	odd	2	inner	735.2.g.a.734.4	yes	16
105.44	odd	6		735.2.p.d.374.2		16
105.59	even	6		735.2.p.d.509.2		16
105.74	odd	6		735.2.p.e.509.2		16
105.89	even	6		735.2.p.e.374.2		16
105.104	even	2	inner	735.2.g.a.734.13	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
735.2.g.a.734.3	✓	16	3.2	odd	2	inner
735.2.g.a.734.3	✓	16	5.4	even	2	inner
735.2.g.a.734.4	yes	16	21.20	even	2	inner
735.2.g.a.734.4	yes	16	35.34	odd	2	inner
735.2.g.a.734.13	yes	16	7.6	odd	2	inner
735.2.g.a.734.13	yes	16	105.104	even	2	inner
735.2.g.a.734.14	yes	16	1.1	even	1	trivial
735.2.g.a.734.14	yes	16	15.14	odd	2	CM
735.2.p.d.374.2		16	7.2	even	3
735.2.p.d.374.2		16	105.44	odd	6
735.2.p.d.374.7		16	21.5	even	6
735.2.p.d.374.7		16	35.19	odd	6
735.2.p.d.509.2		16	7.3	odd	6
735.2.p.d.509.2		16	105.59	even	6
735.2.p.d.509.7		16	21.11	odd	6
735.2.p.d.509.7		16	35.4	even	6
735.2.p.e.374.2		16	7.5	odd	6
735.2.p.e.374.2		16	105.89	even	6
735.2.p.e.374.7		16	21.2	odd	6
735.2.p.e.374.7		16	35.9	even	6
735.2.p.e.509.2		16	7.4	even	3
735.2.p.e.509.2		16	105.74	odd	6
735.2.p.e.509.7		16	21.17	even	6
735.2.p.e.509.7		16	35.24	odd	6

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:	\( N \)	\(=\)	\( 735 = 3 \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	735.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.45591 q^{2} +1.73205i q^{3} +4.03151 q^{4} +2.23607i q^{5} +4.25377i q^{6} +4.98920 q^{8} -3.00000 q^{9} +5.49159i q^{10} +6.98277i q^{12} -3.87298 q^{15} +4.19003 q^{16} -8.06126i q^{17} -7.36774 q^{18} +5.62561i q^{19} +9.01472i q^{20} +9.58908 q^{23} +8.64155i q^{24} -5.00000 q^{25} -5.19615i q^{27} -9.51171 q^{30} -4.42678i q^{31} +0.311942 q^{32} -19.7977i q^{34} -12.0945 q^{36} +13.8160i q^{38} +11.1562i q^{40} -6.70820i q^{45} +23.5500 q^{46} +1.02391i q^{47} +7.25734i q^{48} -12.2796 q^{50} +13.9625 q^{51} +9.43433 q^{53} -12.7613i q^{54} -9.74384 q^{57} -15.6140 q^{60} +4.08075i q^{61} -10.8718i q^{62} -7.61396 q^{64} -32.4990i q^{68} +16.6088i q^{69} -14.9676 q^{72} -8.66025i q^{75} +22.6797i q^{76} -5.83648 q^{79} +9.36919i q^{80} +9.00000 q^{81} -15.0986i q^{83} +18.0255 q^{85} -16.4748i q^{90} +38.6584 q^{92} +7.66741 q^{93} +2.51464i q^{94} -12.5792 q^{95} +0.540299i q^{96} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 32 q^{4} - 48 q^{9} + 64 q^{16} - 80 q^{25} - 96 q^{36} + 128 q^{64} + 144 q^{81}+O(q^{100}) \) 16 * q + 32 * q^4 - 48 * q^9 + 64 * q^16 - 80 * q^25 - 96 * q^36 + 128 * q^64 + 144 * q^81

Introduction

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