LMFDB - Embedded newform 6900.2.f.s.6349.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6900,2,Mod(6349,6900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6900, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6900.6349");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6900 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6900.f (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:	$55.0967773947$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 20x^{6} + 80x^{4} + 41x^{2} + 4 $ x^8 + 20x^6 + 80x^4 + 41*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			6349.6
Root			$-3.84324i$ of defining polynomial
Character	$\chi$	$=$	6900.6349
Dual form			6900.2.f.s.6349.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} -1.32284i q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} -1.32284i q^{7} -1.00000 q^{9} -3.84324 q^{11} +4.36363i q^{13} +2.75924i q^{17} +5.92724 q^{19} +1.32284 q^{21} -1.00000i q^{23} -1.00000i q^{27} -0.761159 q^{29} -8.72969 q^{31} -3.84324i q^{33} +3.24076i q^{37} -4.36363 q^{39} +4.16608 q^{41} +9.72727i q^{43} -2.92724i q^{47} +5.25008 q^{49} -2.75924 q^{51} -11.2501i q^{53} +5.92724i q^{57} +3.08208 q^{59} -6.08400 q^{61} +1.32284i q^{63} +2.19755i q^{67} +1.00000 q^{69} -1.39560 q^{71} -9.20687i q^{73} +5.08400i q^{77} -14.3341 q^{79} +1.00000 q^{81} -3.03886i q^{83} -0.761159i q^{87} +18.3002 q^{89} +5.77240 q^{91} -8.72969i q^{93} +5.43832i q^{97} +3.84324 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{9}+O(q^{10}) $ 8 * q - 8 * q^9	$ 8 q - 8 q^{9} - 4 q^{11} - 4 q^{19} - 6 q^{21} + 2 q^{29} - 12 q^{31} + 2 q^{39} - 10 q^{41} - 26 q^{49} - 20 q^{51} + 6 q^{59} - 24 q^{61} + 8 q^{69} - 46 q^{71} - 22 q^{79} + 8 q^{81} - 12 q^{89} - 38 q^{91} + 4 q^{99}+O(q^{100}) $ 8 * q - 8 * q^9 - 4 * q^11 - 4 * q^19 - 6 * q^21 + 2 * q^29 - 12 * q^31 + 2 * q^39 - 10 * q^41 - 26 * q^49 - 20 * q^51 + 6 * q^59 - 24 * q^61 + 8 * q^69 - 46 * q^71 - 22 * q^79 + 8 * q^81 - 12 * q^89 - 38 * q^91 + 4 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/6900\mathbb{Z}\right)^\times$.

$n$	$277$	$1201$	$3451$	$4601$
$\chi(n)$	$-1$	$1$	$1$	$1$

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 $

$2$	0	0
$2$	0	0
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 1.32284i	− 0.499988i	−0.968247	−	0.249994i	$-0.919571\pi$
$7$	− 1.32284i	− 0.499988i	0.968247	−	0.249994i	$-0.0804286\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−3.84324	−1.15878	−0.579390	−	0.815050i	$-0.696709\pi$
$11$	−3.84324	−1.15878	−0.579390	+	0.815050i	$0.696709\pi$
$12$	0	0
$13$	4.36363i	1.21025i	0.796129	+	0.605127i	$0.206878\pi$
$13$	4.36363i	1.21025i	−0.796129	+	0.605127i	$0.793122\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.75924i	0.669213i	0.942358	+	0.334606i	$0.108603\pi$
$17$	2.75924i	0.669213i	−0.942358	+	0.334606i	$0.891397\pi$
$18$	0	0
$19$	5.92724	1.35980	0.679901	−	0.733304i	$-0.262023\pi$
$19$	5.92724	1.35980	0.679901	+	0.733304i	$0.262023\pi$
$20$	0	0
$21$	1.32284	0.288668
$22$	0	0
$23$	− 1.00000i	− 0.208514i
$23$	− 1.00000i	− 0.208514i
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 1.00000i	− 0.192450i
$28$	0	0
$29$	−0.761159	−0.141344	−0.0706719	−	0.997500i	$-0.522514\pi$
$29$	−0.761159	−0.141344	−0.0706719	+	0.997500i	$0.522514\pi$
$30$	0	0
$31$	−8.72969	−1.56790	−0.783949	−	0.620825i	$-0.786798\pi$
$31$	−8.72969	−1.56790	−0.783949	+	0.620825i	$0.786798\pi$
$32$	0	0
$33$	− 3.84324i	− 0.669022i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	3.24076i	0.532778i	0.963866	+	0.266389i	$0.0858307\pi$
$37$	3.24076i	0.532778i	−0.963866	+	0.266389i	$0.914169\pi$
$38$	0	0
$39$	−4.36363	−0.698740
$40$	0	0
$41$	4.16608	0.650633	0.325316	−	0.945605i	$-0.394529\pi$
$41$	4.16608	0.650633	0.325316	+	0.945605i	$0.394529\pi$
$42$	0	0
$43$	9.72727i	1.48339i	0.670735	+	0.741697i	$0.265979\pi$
$43$	9.72727i	1.48339i	−0.670735	+	0.741697i	$0.734021\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 2.92724i	− 0.426982i	−0.976945	−	0.213491i	$-0.931517\pi$
$47$	− 2.92724i	− 0.426982i	0.976945	−	0.213491i	$-0.0684834\pi$
$48$	0	0
$49$	5.25008	0.750012
$50$	0	0
$51$	−2.75924	−0.386370
$52$	0	0
$53$	− 11.2501i	− 1.54532i	−0.634821	−	0.772659i	$-0.718926\pi$
$53$	− 11.2501i	− 1.54532i	0.634821	−	0.772659i	$-0.281074\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	5.92724i	0.785082i
$58$	0	0
$59$	3.08208	0.401252	0.200626	−	0.979668i	$-0.435702\pi$
$59$	3.08208	0.401252	0.200626	+	0.979668i	$0.435702\pi$
$60$	0	0
$61$	−6.08400	−0.778977	−0.389488	−	0.921031i	$-0.627348\pi$
$61$	−6.08400	−0.778977	−0.389488	+	0.921031i	$0.627348\pi$
$62$	0	0
$63$	1.32284i	0.166663i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	2.19755i	0.268474i	0.990949	+	0.134237i	$0.0428583\pi$
$67$	2.19755i	0.268474i	−0.990949	+	0.134237i	$0.957142\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	−1.39560	−0.165628	−0.0828138	−	0.996565i	$-0.526391\pi$
$71$	−1.39560	−0.165628	−0.0828138	+	0.996565i	$0.526391\pi$
$72$	0	0
$73$	− 9.20687i	− 1.07758i	−0.842439	−	0.538791i	$-0.818881\pi$
$73$	− 9.20687i	− 1.07758i	0.842439	−	0.538791i	$-0.181119\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	5.08400i	0.579376i
$78$	0	0
$79$	−14.3341	−1.61271	−0.806355	−	0.591431i	$-0.798563\pi$
$79$	−14.3341	−1.61271	−0.806355	+	0.591431i	$0.798563\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 3.03886i	− 0.333559i	−0.985994	−	0.166779i	$-0.946663\pi$
$83$	− 3.03886i	− 0.333559i	0.985994	−	0.166779i	$-0.0533368\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 0.761159i	− 0.0816049i
$88$	0	0
$89$	18.3002	1.93982	0.969908	−	0.243470i	$-0.0782859\pi$
$89$	18.3002	1.93982	0.969908	+	0.243470i	$0.0782859\pi$
$90$	0	0
$91$	5.77240	0.605112
$92$	0	0
$93$	− 8.72969i	− 0.905227i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	5.43832i	0.552177i	0.961132	+	0.276089i	$0.0890383\pi$
$97$	5.43832i	0.552177i	−0.961132	+	0.276089i	$0.910962\pi$
$98$	0	0
$99$	3.84324	0.386260
$100$	0	0
$101$	−15.5316	−1.54546	−0.772728	−	0.634737i	$-0.781108\pi$
$101$	−15.5316	−1.54546	−0.772728	+	0.634737i	$0.781108\pi$
$102$	0	0
$103$	9.09332i	0.895992i	0.894036	+	0.447996i	$0.147862\pi$
$103$	9.09332i	0.895992i	−0.894036	+	0.447996i	$0.852138\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	2.92724i	0.282987i	0.989939	+	0.141494i	$0.0451904\pi$
$107$	2.92724i	0.282987i	−0.989939	+	0.141494i	$0.954810\pi$
$108$	0	0
$109$	−13.0525	−1.25021	−0.625103	−	0.780542i	$-0.714943\pi$
$109$	−13.0525	−1.25021	−0.625103	+	0.780542i	$0.714943\pi$
$110$	0	0
$111$	−3.24076	−0.307600
$112$	0	0
$113$	− 17.0501i	− 1.60394i	−0.597365	−	0.801970i	$-0.703786\pi$
$113$	− 17.0501i	− 1.60394i	0.597365	−	0.801970i	$-0.296214\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 4.36363i	− 0.403418i
$118$	0	0
$119$	3.65004	0.334598
$120$	0	0
$121$	3.77048	0.342771
$122$	0	0
$123$	4.16608i	0.375643i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	0.959211i	0.0851162i	0.999094	+	0.0425581i	$0.0135508\pi$
$127$	0.959211i	0.0851162i	−0.999094	+	0.0425581i	$0.986449\pi$
$128$	0	0
$129$	−9.72727	−0.856438
$130$	0	0
$131$	−3.07276	−0.268468	−0.134234	−	0.990950i	$-0.542857\pi$
$131$	−3.07276	−0.268468	−0.134234	+	0.990950i	$0.542857\pi$
$132$	0	0
$133$	− 7.84081i	− 0.679885i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 2.55926i	− 0.218652i	−0.994006	−	0.109326i	$-0.965131\pi$
$137$	− 2.55926i	− 0.218652i	0.994006	−	0.109326i	$-0.0348693\pi$
$138$	0	0
$139$	−22.9366	−1.94545	−0.972727	−	0.231954i	$-0.925488\pi$
$139$	−22.9366	−1.94545	−0.972727	+	0.231954i	$0.925488\pi$
$140$	0	0
$141$	2.92724	0.246518
$142$	0	0
$143$	− 16.7705i	− 1.40242i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	5.25008i	0.433020i
$148$	0	0
$149$	−6.53214	−0.535134	−0.267567	−	0.963539i	$-0.586220\pi$
$149$	−6.53214	−0.535134	−0.267567	+	0.963539i	$0.586220\pi$
$150$	0	0
$151$	2.32042	0.188833	0.0944165	−	0.995533i	$-0.469901\pi$
$151$	2.32042	0.188833	0.0944165	+	0.995533i	$0.469901\pi$
$152$	0	0
$153$	− 2.75924i	− 0.223071i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	2.35239i	0.187741i	0.995584	+	0.0938705i	$0.0299240\pi$
$157$	2.35239i	0.187741i	−0.995584	+	0.0938705i	$0.970076\pi$
$158$	0	0
$159$	11.2501	0.892190
$160$	0	0
$161$	−1.32284	−0.104255
$162$	0	0
$163$	− 15.6156i	− 1.22311i	−0.791201	−	0.611556i	$-0.790544\pi$
$163$	− 15.6156i	− 1.22311i	0.791201	−	0.611556i	$-0.209456\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 2.15869i	− 0.167044i	−0.996506	−	0.0835221i	$-0.973383\pi$
$167$	− 2.15869i	− 0.167044i	0.996506	−	0.0835221i	$-0.0266169\pi$
$168$	0	0
$169$	−6.04129	−0.464715
$170$	0	0
$171$	−5.92724	−0.453267
$172$	0	0
$173$	22.4250i	1.70494i	0.522776	+	0.852470i	$0.324896\pi$
$173$	22.4250i	1.70494i	−0.522776	+	0.852470i	$0.675104\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	3.08208i	0.231663i
$178$	0	0
$179$	−9.25008	−0.691384	−0.345692	−	0.938348i	$-0.612356\pi$
$179$	−9.25008	−0.691384	−0.345692	+	0.938348i	$0.612356\pi$
$180$	0	0
$181$	−15.3341	−1.13977	−0.569887	−	0.821723i	$-0.693013\pi$
$181$	−15.3341	−1.13977	−0.569887	+	0.821723i	$0.693013\pi$
$182$	0	0
$183$	− 6.08400i	− 0.449742i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 10.6044i	− 0.775470i
$188$	0	0
$189$	−1.32284	−0.0962227
$190$	0	0
$191$	−11.5204	−0.833586	−0.416793	−	0.909001i	$-0.636846\pi$
$191$	−11.5204	−0.833586	−0.416793	+	0.909001i	$0.636846\pi$
$192$	0	0
$193$	2.40685i	0.173249i	0.996241	+	0.0866243i	$0.0276080\pi$
$193$	2.40685i	0.173249i	−0.996241	+	0.0866243i	$0.972392\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 4.61564i	− 0.328851i	−0.986390	−	0.164425i	$-0.947423\pi$
$197$	− 4.61564i	− 0.328851i	0.986390	−	0.164425i	$-0.0525770\pi$
$198$	0	0
$199$	−25.7779	−1.82735	−0.913673	−	0.406451i	$-0.866766\pi$
$199$	−25.7779	−1.82735	−0.913673	+	0.406451i	$0.866766\pi$
$200$	0	0
$201$	−2.19755	−0.155003
$202$	0	0
$203$	1.00689i	0.0706702i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	1.00000i	0.0695048i
$208$	0	0
$209$	−22.7798	−1.57571
$210$	0	0
$211$	−25.2663	−1.73940	−0.869702	−	0.493577i	$-0.835689\pi$
$211$	−25.2663	−1.73940	−0.869702	+	0.493577i	$0.835689\pi$
$212$	0	0
$213$	− 1.39560i	− 0.0956251i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	11.5480i	0.783930i
$218$	0	0
$219$	9.20687	0.622143
$220$	0	0
$221$	−12.0403	−0.809917
$222$	0	0
$223$	7.08897i	0.474713i	0.971423	+	0.237356i	$0.0762809\pi$
$223$	7.08897i	0.474713i	−0.971423	+	0.237356i	$0.923719\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	6.05446i	0.401849i	0.979607	+	0.200924i	$0.0643945\pi$
$227$	6.05446i	0.401849i	−0.979607	+	0.200924i	$0.935605\pi$
$228$	0	0
$229$	0.488425	0.0322760	0.0161380	−	0.999870i	$-0.494863\pi$
$229$	0.488425	0.0322760	0.0161380	+	0.999870i	$0.494863\pi$
$230$	0	0
$231$	−5.08400	−0.334503
$232$	0	0
$233$	− 30.5114i	− 1.99887i	−0.0336230	−	0.999435i	$-0.510705\pi$
$233$	− 30.5114i	− 1.99887i	0.0336230	−	0.999435i	$-0.489295\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 14.3341i	− 0.931099i
$238$	0	0
$239$	−14.6457	−0.947351	−0.473675	−	0.880700i	$-0.657073\pi$
$239$	−14.6457	−0.947351	−0.473675	+	0.880700i	$0.657073\pi$
$240$	0	0
$241$	17.1754	1.10636	0.553182	−	0.833060i	$-0.313413\pi$
$241$	17.1754	1.10636	0.553182	+	0.833060i	$0.313413\pi$
$242$	0	0
$243$	1.00000i	0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	25.8643i	1.64571i
$248$	0	0
$249$	3.03886	0.192580
$250$	0	0
$251$	−22.5453	−1.42305	−0.711524	−	0.702662i	$-0.751995\pi$
$251$	−22.5453	−1.42305	−0.711524	+	0.702662i	$0.751995\pi$
$252$	0	0
$253$	3.84324i	0.241622i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	16.5724i	1.03376i	0.856058	+	0.516880i	$0.172907\pi$
$257$	16.5724i	1.03376i	−0.856058	+	0.516880i	$0.827093\pi$
$258$	0	0
$259$	4.28703	0.266383
$260$	0	0
$261$	0.761159	0.0471146
$262$	0	0
$263$	− 10.7995i	− 0.665927i	−0.942940	−	0.332964i	$-0.891951\pi$
$263$	− 10.7995i	− 0.665927i	0.942940	−	0.332964i	$-0.108049\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	18.3002i	1.11995i
$268$	0	0
$269$	7.32427	0.446568	0.223284	−	0.974753i	$-0.428322\pi$
$269$	7.32427	0.446568	0.223284	+	0.974753i	$0.428322\pi$
$270$	0	0
$271$	12.0506	0.732022	0.366011	−	0.930610i	$-0.380723\pi$
$271$	12.0506	0.732022	0.366011	+	0.930610i	$0.380723\pi$
$272$	0	0
$273$	5.77240i	0.349362i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	17.9995i	1.08149i	0.841188	+	0.540743i	$0.181857\pi$
$277$	17.9995i	1.08149i	−0.841188	+	0.540743i	$0.818143\pi$
$278$	0	0
$279$	8.72969	0.522633
$280$	0	0
$281$	−10.9234	−0.651635	−0.325817	−	0.945433i	$-0.605639\pi$
$281$	−10.9234	−0.651635	−0.325817	+	0.945433i	$0.605639\pi$
$282$	0	0
$283$	− 6.79810i	− 0.404105i	−0.979375	−	0.202053i	$-0.935239\pi$
$283$	− 6.79810i	− 0.404105i	0.979375	−	0.202053i	$-0.0647612\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 5.51107i	− 0.325308i
$288$	0	0
$289$	9.38662	0.552154
$290$	0	0
$291$	−5.43832	−0.318800
$292$	0	0
$293$	22.6324i	1.32220i	0.750299	+	0.661098i	$0.229909\pi$
$293$	22.6324i	1.32220i	−0.750299	+	0.661098i	$0.770091\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	3.84324i	0.223007i
$298$	0	0
$299$	4.36363	0.252355
$300$	0	0
$301$	12.8677	0.741679
$302$	0	0
$303$	− 15.5316i	− 0.892269i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 25.0167i	− 1.42778i	−0.700258	−	0.713890i	$-0.746932\pi$
$307$	− 25.0167i	− 1.42778i	0.700258	−	0.713890i	$-0.253068\pi$
$308$	0	0
$309$	−9.09332	−0.517301
$310$	0	0
$311$	27.1139	1.53749	0.768744	−	0.639557i	$-0.220882\pi$
$311$	27.1139	1.53749	0.768744	+	0.639557i	$0.220882\pi$
$312$	0	0
$313$	1.27963i	0.0723289i	0.999346	+	0.0361645i	$0.0115140\pi$
$313$	1.27963i	0.0723289i	−0.999346	+	0.0361645i	$0.988486\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	27.4668i	1.54269i	0.636419	+	0.771344i	$0.280415\pi$
$317$	27.4668i	1.54269i	−0.636419	+	0.771344i	$0.719585\pi$
$318$	0	0
$319$	2.92532	0.163786
$320$	0	0
$321$	−2.92724	−0.163383
$322$	0	0
$323$	16.3547i	0.909997i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 13.0525i	− 0.721807i
$328$	0	0
$329$	−3.87228	−0.213486
$330$	0	0
$331$	−14.2551	−0.783529	−0.391764	−	0.920066i	$-0.628135\pi$
$331$	−14.2551	−0.783529	−0.391764	+	0.920066i	$0.628135\pi$
$332$	0	0
$333$	− 3.24076i	− 0.177593i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	12.3243i	0.671346i	0.941979	+	0.335673i	$0.108964\pi$
$337$	12.3243i	0.671346i	−0.941979	+	0.335673i	$0.891036\pi$
$338$	0	0
$339$	17.0501	0.926035
$340$	0	0
$341$	33.5503	1.81685
$342$	0	0
$343$	− 16.2049i	− 0.874985i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 30.6643i	− 1.64615i	−0.567935	−	0.823074i	$-0.692257\pi$
$347$	− 30.6643i	− 1.64615i	0.567935	−	0.823074i	$-0.307743\pi$
$348$	0	0
$349$	−14.0835	−0.753873	−0.376936	−	0.926239i	$-0.623022\pi$
$349$	−14.0835	−0.753873	−0.376936	+	0.926239i	$0.623022\pi$
$350$	0	0
$351$	4.36363	0.232913
$352$	0	0
$353$	26.9842i	1.43623i	0.695926	+	0.718113i	$0.254994\pi$
$353$	26.9842i	1.43623i	−0.695926	+	0.718113i	$0.745006\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	3.65004i	0.193180i
$358$	0	0
$359$	28.4343	1.50071	0.750353	−	0.661038i	$-0.229884\pi$
$359$	28.4343	1.50071	0.750353	+	0.661038i	$0.229884\pi$
$360$	0	0
$361$	16.1322	0.849063
$362$	0	0
$363$	3.77048i	0.197899i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 9.24574i	− 0.482623i	−0.970448	−	0.241312i	$-0.922422\pi$
$367$	− 9.24574i	− 0.482623i	0.970448	−	0.241312i	$-0.0775776\pi$
$368$	0	0
$369$	−4.16608	−0.216878
$370$	0	0
$371$	−14.8821	−0.772640
$372$	0	0
$373$	− 22.0107i	− 1.13967i	−0.821758	−	0.569837i	$-0.807007\pi$
$373$	− 22.0107i	− 1.13967i	0.821758	−	0.569837i	$-0.192993\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 3.32142i	− 0.171062i
$378$	0	0
$379$	−25.0983	−1.28921	−0.644606	−	0.764515i	$-0.722979\pi$
$379$	−25.0983	−1.28921	−0.644606	+	0.764515i	$0.722979\pi$
$380$	0	0
$381$	−0.959211	−0.0491419
$382$	0	0
$383$	− 27.6113i	− 1.41087i	−0.708774	−	0.705436i	$-0.750751\pi$
$383$	− 27.6113i	− 1.41087i	0.708774	−	0.705436i	$-0.249249\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 9.72727i	− 0.494465i
$388$	0	0
$389$	0.581909	0.0295040	0.0147520	−	0.999891i	$-0.495304\pi$
$389$	0.581909	0.0295040	0.0147520	+	0.999891i	$0.495304\pi$
$390$	0	0
$391$	2.75924	0.139541
$392$	0	0
$393$	− 3.07276i	− 0.155000i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	19.1622i	0.961725i	0.876796	+	0.480862i	$0.159676\pi$
$397$	19.1622i	0.961725i	−0.876796	+	0.480862i	$0.840324\pi$
$398$	0	0
$399$	7.84081	0.392532
$400$	0	0
$401$	11.0859	0.553605	0.276802	−	0.960927i	$-0.410725\pi$
$401$	11.0859	0.553605	0.276802	+	0.960927i	$0.410725\pi$
$402$	0	0
$403$	− 38.0932i	− 1.89756i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 12.4550i	− 0.617373i
$408$	0	0
$409$	−4.07661	−0.201575	−0.100788	−	0.994908i	$-0.532136\pi$
$409$	−4.07661	−0.201575	−0.100788	+	0.994908i	$0.532136\pi$
$410$	0	0
$411$	2.55926	0.126239
$412$	0	0
$413$	− 4.07711i	− 0.200621i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 22.9366i	− 1.12321i
$418$	0	0
$419$	13.1656	0.643181	0.321590	−	0.946879i	$-0.395783\pi$
$419$	13.1656	0.643181	0.321590	+	0.946879i	$0.395783\pi$
$420$	0	0
$421$	−10.7404	−0.523457	−0.261728	−	0.965142i	$-0.584292\pi$
$421$	−10.7404	−0.523457	−0.261728	+	0.965142i	$0.584292\pi$
$422$	0	0
$423$	2.92724i	0.142327i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	8.04818i	0.389479i
$428$	0	0
$429$	16.7705	0.809686
$430$	0	0
$431$	−27.3458	−1.31720	−0.658601	−	0.752492i	$-0.728852\pi$
$431$	−27.3458	−1.31720	−0.658601	+	0.752492i	$0.728852\pi$
$432$	0	0
$433$	− 22.4030i	− 1.07662i	−0.842747	−	0.538310i	$-0.819063\pi$
$433$	− 22.4030i	− 1.07662i	0.842747	−	0.538310i	$-0.180937\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 5.92724i	− 0.283538i
$438$	0	0
$439$	−12.6339	−0.602985	−0.301493	−	0.953469i	$-0.597485\pi$
$439$	−12.6339	−0.602985	−0.301493	+	0.953469i	$0.597485\pi$
$440$	0	0
$441$	−5.25008	−0.250004
$442$	0	0
$443$	− 14.2506i	− 0.677066i	−0.940955	−	0.338533i	$-0.890069\pi$
$443$	− 14.2506i	− 0.677066i	0.940955	−	0.338533i	$-0.109931\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 6.53214i	− 0.308960i
$448$	0	0
$449$	−38.0589	−1.79611	−0.898056	−	0.439881i	$-0.855021\pi$
$449$	−38.0589	−1.79611	−0.898056	+	0.439881i	$0.855021\pi$
$450$	0	0
$451$	−16.0112	−0.753940
$452$	0	0
$453$	2.32042i	0.109023i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 8.18665i	− 0.382955i	−0.981497	−	0.191478i	$-0.938672\pi$
$457$	− 8.18665i	− 0.382955i	0.981497	−	0.191478i	$-0.0613279\pi$
$458$	0	0
$459$	2.75924	0.128790
$460$	0	0
$461$	28.6521	1.33446	0.667230	−	0.744852i	$-0.267480\pi$
$461$	28.6521	1.33446	0.667230	+	0.744852i	$0.267480\pi$
$462$	0	0
$463$	− 3.24624i	− 0.150865i	−0.997151	−	0.0754327i	$-0.975966\pi$
$463$	− 3.24624i	− 0.150865i	0.997151	−	0.0754327i	$-0.0240338\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	35.7576i	1.65467i	0.561711	+	0.827333i	$0.310143\pi$
$467$	35.7576i	1.65467i	−0.561711	+	0.827333i	$0.689857\pi$
$468$	0	0
$469$	2.90702	0.134234
$470$	0	0
$471$	−2.35239	−0.108392
$472$	0	0
$473$	− 37.3842i	− 1.71893i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	11.2501i	0.515106i
$478$	0	0
$479$	−17.6564	−0.806743	−0.403371	−	0.915036i	$-0.632162\pi$
$479$	−17.6564	−0.806743	−0.403371	+	0.915036i	$0.632162\pi$
$480$	0	0
$481$	−14.1415	−0.644797
$482$	0	0
$483$	− 1.32284i	− 0.0601915i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	13.1847i	0.597457i	0.954338	+	0.298728i	$0.0965625\pi$
$487$	13.1847i	0.597457i	−0.954338	+	0.298728i	$0.903438\pi$
$488$	0	0
$489$	15.6156	0.706164
$490$	0	0
$491$	27.3503	1.23430	0.617151	−	0.786845i	$-0.288287\pi$
$491$	27.3503	1.23430	0.617151	+	0.786845i	$0.288287\pi$
$492$	0	0
$493$	− 2.10022i	− 0.0945891i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1.84616i	0.0828118i
$498$	0	0
$499$	−17.1155	−0.766194	−0.383097	−	0.923708i	$-0.625142\pi$
$499$	−17.1155	−0.766194	−0.383097	+	0.923708i	$0.625142\pi$
$500$	0	0
$501$	2.15869	0.0964430
$502$	0	0
$503$	− 28.1031i	− 1.25306i	−0.779398	−	0.626529i	$-0.784475\pi$
$503$	− 28.1031i	− 1.25306i	0.779398	−	0.626529i	$-0.215525\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 6.04129i	− 0.268303i
$508$	0	0
$509$	23.2462	1.03037	0.515186	−	0.857079i	$-0.327723\pi$
$509$	23.2462	1.03037	0.515186	+	0.857079i	$0.327723\pi$
$510$	0	0
$511$	−12.1793	−0.538778
$512$	0	0
$513$	− 5.92724i	− 0.261694i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	11.2501i	0.494778i
$518$	0	0
$519$	−22.4250	−0.984348
$520$	0	0
$521$	−44.5374	−1.95122	−0.975611	−	0.219509i	$-0.929555\pi$
$521$	−44.5374	−1.95122	−0.975611	+	0.219509i	$0.929555\pi$
$522$	0	0
$523$	− 1.47383i	− 0.0644462i	−0.999481	−	0.0322231i	$-0.989741\pi$
$523$	− 1.47383i	− 0.0644462i	0.999481	−	0.0322231i	$-0.0102587\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 24.0873i	− 1.04926i
$528$	0	0
$529$	−1.00000	−0.0434783
$530$	0	0
$531$	−3.08208	−0.133751
$532$	0	0
$533$	18.1793i	0.787431i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 9.25008i	− 0.399171i
$538$	0	0
$539$	−20.1773	−0.869099
$540$	0	0
$541$	3.84566	0.165338	0.0826690	−	0.996577i	$-0.473656\pi$
$541$	3.84566	0.165338	0.0826690	+	0.996577i	$0.473656\pi$
$542$	0	0
$543$	− 15.3341i	− 0.658049i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 6.03097i	− 0.257866i	−0.991653	−	0.128933i	$-0.958845\pi$
$547$	− 6.03097i	− 0.257866i	0.991653	−	0.128933i	$-0.0411551\pi$
$548$	0	0
$549$	6.08400	0.259659
$550$	0	0
$551$	−4.51158	−0.192200
$552$	0	0
$553$	18.9618i	0.806336i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 20.9139i	− 0.886151i	−0.896484	−	0.443075i	$-0.853887\pi$
$557$	− 20.9139i	− 0.886151i	0.896484	−	0.443075i	$-0.146113\pi$
$558$	0	0
$559$	−42.4462	−1.79528
$560$	0	0
$561$	10.6044	0.447718
$562$	0	0
$563$	− 14.3744i	− 0.605808i	−0.953021	−	0.302904i	$-0.902044\pi$
$563$	− 14.3744i	− 0.605808i	0.953021	−	0.302904i	$-0.0979561\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 1.32284i	− 0.0555542i
$568$	0	0
$569$	−16.1360	−0.676458	−0.338229	−	0.941064i	$-0.609828\pi$
$569$	−16.1360	−0.676458	−0.338229	+	0.941064i	$0.609828\pi$
$570$	0	0
$571$	−16.3478	−0.684132	−0.342066	−	0.939676i	$-0.611127\pi$
$571$	−16.3478	−0.684132	−0.342066	+	0.939676i	$0.611127\pi$
$572$	0	0
$573$	− 11.5204i	− 0.481271i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	33.9046i	1.41147i	0.708478	+	0.705733i	$0.249382\pi$
$577$	33.9046i	1.41147i	−0.708478	+	0.705733i	$0.750618\pi$
$578$	0	0
$579$	−2.40685	−0.100025
$580$	0	0
$581$	−4.01994	−0.166775
$582$	0	0
$583$	43.2368i	1.79068i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 21.7045i	− 0.895840i	−0.894074	−	0.447920i	$-0.852165\pi$
$587$	− 21.7045i	− 0.895840i	0.894074	−	0.447920i	$-0.147835\pi$
$588$	0	0
$589$	−51.7430	−2.13203
$590$	0	0
$591$	4.61564	0.189862
$592$	0	0
$593$	− 42.5169i	− 1.74596i	−0.487757	−	0.872980i	$-0.662185\pi$
$593$	− 42.5169i	− 1.74596i	0.487757	−	0.872980i	$-0.337815\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 25.7779i	− 1.05502i
$598$	0	0
$599$	4.94973	0.202240	0.101120	−	0.994874i	$-0.467757\pi$
$599$	4.94973	0.202240	0.101120	+	0.994874i	$0.467757\pi$
$600$	0	0
$601$	29.3527	1.19732	0.598661	−	0.801002i	$-0.295700\pi$
$601$	29.3527	1.19732	0.598661	+	0.801002i	$0.295700\pi$
$602$	0	0
$603$	− 2.19755i	− 0.0894912i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 17.2958i	− 0.702017i	−0.936372	−	0.351008i	$-0.885839\pi$
$607$	− 17.2958i	− 0.702017i	0.936372	−	0.351008i	$-0.114161\pi$
$608$	0	0
$609$	−1.00689	−0.0408014
$610$	0	0
$611$	12.7734	0.516757
$612$	0	0
$613$	18.3463i	0.741001i	0.928832	+	0.370501i	$0.120814\pi$
$613$	18.3463i	0.741001i	−0.928832	+	0.370501i	$0.879186\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	25.0959i	1.01032i	0.863025	+	0.505161i	$0.168567\pi$
$617$	25.0959i	1.01032i	−0.863025	+	0.505161i	$0.831433\pi$
$618$	0	0
$619$	40.9115	1.64437	0.822186	−	0.569219i	$-0.192754\pi$
$619$	40.9115	1.64437	0.822186	+	0.569219i	$0.192754\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	− 24.2083i	− 0.969885i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 22.7798i	− 0.909738i
$628$	0	0
$629$	−8.94203	−0.356542
$630$	0	0
$631$	30.6677	1.22086	0.610430	−	0.792070i	$-0.290996\pi$
$631$	30.6677	1.22086	0.610430	+	0.792070i	$0.290996\pi$
$632$	0	0
$633$	− 25.2663i	− 1.00425i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	22.9094i	0.907705i
$638$	0	0
$639$	1.39560	0.0552092
$640$	0	0
$641$	−0.685976	−0.0270944	−0.0135472	−	0.999908i	$-0.504312\pi$
$641$	−0.685976	−0.0270944	−0.0135472	+	0.999908i	$0.504312\pi$
$642$	0	0
$643$	12.4879i	0.492476i	0.969209	+	0.246238i	$0.0791944\pi$
$643$	12.4879i	0.492476i	−0.969209	+	0.246238i	$0.920806\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 33.5361i	− 1.31844i	−0.751950	−	0.659220i	$-0.770886\pi$
$647$	− 33.5361i	− 1.31844i	0.751950	−	0.659220i	$-0.229114\pi$
$648$	0	0
$649$	−11.8452	−0.464963
$650$	0	0
$651$	−11.5480	−0.452602
$652$	0	0
$653$	− 26.9749i	− 1.05561i	−0.849365	−	0.527805i	$-0.823015\pi$
$653$	− 26.9749i	− 1.05561i	0.849365	−	0.527805i	$-0.176985\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	9.20687i	0.359194i
$658$	0	0
$659$	−8.05700	−0.313856	−0.156928	−	0.987610i	$-0.550159\pi$
$659$	−8.05700	−0.313856	−0.156928	+	0.987610i	$0.550159\pi$
$660$	0	0
$661$	18.5376	0.721029	0.360515	−	0.932754i	$-0.382601\pi$
$661$	18.5376	0.721029	0.360515	+	0.932754i	$0.382601\pi$
$662$	0	0
$663$	− 12.0403i	− 0.467606i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0.761159i	0.0294722i
$668$	0	0
$669$	−7.08897	−0.274076
$670$	0	0
$671$	23.3823	0.902663
$672$	0	0
$673$	22.9391i	0.884238i	0.896957	+	0.442119i	$0.145773\pi$
$673$	22.9391i	0.884238i	−0.896957	+	0.442119i	$0.854227\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 2.25991i	− 0.0868552i	−0.999057	−	0.0434276i	$-0.986172\pi$
$677$	− 2.25991i	− 0.0868552i	0.999057	−	0.0434276i	$-0.0138278\pi$
$678$	0	0
$679$	7.19404	0.276082
$680$	0	0
$681$	−6.05446	−0.232007
$682$	0	0
$683$	34.5498i	1.32201i	0.750381	+	0.661005i	$0.229870\pi$
$683$	34.5498i	1.32201i	−0.750381	+	0.661005i	$0.770130\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0.488425i	0.0186346i
$688$	0	0
$689$	49.0912	1.87023
$690$	0	0
$691$	25.1768	0.957772	0.478886	−	0.877877i	$-0.341041\pi$
$691$	25.1768	0.957772	0.478886	+	0.877877i	$0.341041\pi$
$692$	0	0
$693$	− 5.08400i	− 0.193125i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	11.4952i	0.435412i
$698$	0	0
$699$	30.5114	1.15405
$700$	0	0
$701$	−10.0458	−0.379423	−0.189712	−	0.981840i	$-0.560755\pi$
$701$	−10.0458	−0.379423	−0.189712	+	0.981840i	$0.560755\pi$
$702$	0	0
$703$	19.2088i	0.724473i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	20.5459i	0.772709i
$708$	0	0
$709$	−8.43882	−0.316926	−0.158463	−	0.987365i	$-0.550654\pi$
$709$	−8.43882	−0.316926	−0.158463	+	0.987365i	$0.550654\pi$
$710$	0	0
$711$	14.3341	0.537570
$712$	0	0
$713$	8.72969i	0.326929i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 14.6457i	− 0.546953i
$718$	0	0
$719$	10.4271	0.388864	0.194432	−	0.980916i	$-0.437714\pi$
$719$	10.4271	0.388864	0.194432	+	0.980916i	$0.437714\pi$
$720$	0	0
$721$	12.0290	0.447985
$722$	0	0
$723$	17.1754i	0.638760i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	44.2442i	1.64092i	0.571701	+	0.820462i	$0.306284\pi$
$727$	44.2442i	1.64092i	−0.571701	+	0.820462i	$0.693716\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−26.8398	−0.992706
$732$	0	0
$733$	41.4236i	1.53001i	0.644022	+	0.765007i	$0.277265\pi$
$733$	41.4236i	1.53001i	−0.644022	+	0.765007i	$0.722735\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 8.44571i	− 0.311102i
$738$	0	0
$739$	−2.91792	−0.107337	−0.0536687	−	0.998559i	$-0.517092\pi$
$739$	−2.91792	−0.107337	−0.0536687	+	0.998559i	$0.517092\pi$
$740$	0	0
$741$	−25.8643	−0.950149
$742$	0	0
$743$	0.285101i	0.0104593i	0.999986	+	0.00522967i	$0.00166466\pi$
$743$	0.285101i	0.0104593i	−0.999986	+	0.00522967i	$0.998335\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	3.03886i	0.111186i
$748$	0	0
$749$	3.87228	0.141490
$750$	0	0
$751$	−38.8978	−1.41940	−0.709701	−	0.704503i	$-0.751170\pi$
$751$	−38.8978	−1.41940	−0.709701	+	0.704503i	$0.751170\pi$
$752$	0	0
$753$	− 22.5453i	− 0.821597i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 35.8116i	− 1.30160i	−0.759251	−	0.650798i	$-0.774435\pi$
$757$	− 35.8116i	− 1.30160i	0.759251	−	0.650798i	$-0.225565\pi$
$758$	0	0
$759$	−3.84324	−0.139501
$760$	0	0
$761$	−50.4529	−1.82892	−0.914459	−	0.404679i	$-0.867383\pi$
$761$	−50.4529	−1.82892	−0.914459	+	0.404679i	$0.867383\pi$
$762$	0	0
$763$	17.2665i	0.625088i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	13.4491i	0.485617i
$768$	0	0
$769$	0.197551	0.00712387	0.00356193	−	0.999994i	$-0.498866\pi$
$769$	0.197551	0.00712387	0.00356193	+	0.999994i	$0.498866\pi$
$770$	0	0
$771$	−16.5724	−0.596841
$772$	0	0
$773$	− 14.8914i	− 0.535607i	−0.963474	−	0.267804i	$-0.913702\pi$
$773$	− 14.8914i	− 0.535607i	0.963474	−	0.267804i	$-0.0862978\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	4.28703i	0.153796i
$778$	0	0
$779$	24.6934	0.884732
$780$	0	0
$781$	5.36363	0.191926
$782$	0	0
$783$	0.761159i	0.0272016i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 30.0643i	− 1.07168i	−0.844321	−	0.535838i	$-0.819996\pi$
$787$	− 30.0643i	− 1.07168i	0.844321	−	0.535838i	$-0.180004\pi$
$788$	0	0
$789$	10.7995	0.384473
$790$	0	0
$791$	−22.5546	−0.801950
$792$	0	0
$793$	− 26.5484i	− 0.942760i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	20.6275i	0.730664i	0.930877	+	0.365332i	$0.119044\pi$
$797$	20.6275i	0.730664i	−0.930877	+	0.365332i	$0.880956\pi$
$798$	0	0
$799$	8.07695	0.285742
$800$	0	0
$801$	−18.3002	−0.646606
$802$	0	0
$803$	35.3842i	1.24868i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	7.32427i	0.257826i
$808$	0	0
$809$	6.46097	0.227155	0.113578	−	0.993529i	$-0.463769\pi$
$809$	6.46097	0.227155	0.113578	+	0.993529i	$0.463769\pi$
$810$	0	0
$811$	43.4072	1.52423	0.762116	−	0.647440i	$-0.224160\pi$
$811$	43.4072	1.52423	0.762116	+	0.647440i	$0.224160\pi$
$812$	0	0
$813$	12.0506i	0.422633i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	57.6558i	2.01712i
$818$	0	0
$819$	−5.77240	−0.201704
$820$	0	0
$821$	1.93414	0.0675018	0.0337509	−	0.999430i	$-0.489255\pi$
$821$	1.93414	0.0675018	0.0337509	+	0.999430i	$0.489255\pi$
$822$	0	0
$823$	− 22.5890i	− 0.787404i	−0.919238	−	0.393702i	$-0.871194\pi$
$823$	− 22.5890i	− 0.787404i	0.919238	−	0.393702i	$-0.128806\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 43.2069i	− 1.50245i	−0.660046	−	0.751225i	$-0.729463\pi$
$827$	− 43.2069i	− 1.50245i	0.660046	−	0.751225i	$-0.270537\pi$
$828$	0	0
$829$	−39.5983	−1.37531	−0.687654	−	0.726039i	$-0.741359\pi$
$829$	−39.5983	−1.37531	−0.687654	+	0.726039i	$0.741359\pi$
$830$	0	0
$831$	−17.9995	−0.624396
$832$	0	0
$833$	14.4862i	0.501918i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	8.72969i	0.301742i
$838$	0	0
$839$	4.33425	0.149635	0.0748175	−	0.997197i	$-0.476163\pi$
$839$	4.33425	0.149635	0.0748175	+	0.997197i	$0.476163\pi$
$840$	0	0
$841$	−28.4206	−0.980022
$842$	0	0
$843$	− 10.9234i	− 0.376222i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 4.98775i	− 0.171381i
$848$	0	0
$849$	6.79810	0.233310
$850$	0	0
$851$	3.24076	0.111092
$852$	0	0
$853$	30.5090i	1.04461i	0.852759	+	0.522304i	$0.174927\pi$
$853$	30.5090i	1.04461i	−0.852759	+	0.522304i	$0.825073\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 0.909223i	− 0.0310585i	−0.999879	−	0.0155292i	$-0.995057\pi$
$857$	− 0.909223i	− 0.0310585i	0.999879	−	0.0155292i	$-0.00494331\pi$
$858$	0	0
$859$	13.8442	0.472357	0.236178	−	0.971710i	$-0.424105\pi$
$859$	13.8442	0.472357	0.236178	+	0.971710i	$0.424105\pi$
$860$	0	0
$861$	5.51107	0.187817
$862$	0	0
$863$	16.6767i	0.567680i	0.958872	+	0.283840i	$0.0916085\pi$
$863$	16.6767i	0.567680i	−0.958872	+	0.283840i	$0.908392\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	9.38662i	0.318786i
$868$	0	0
$869$	55.0893	1.86878
$870$	0	0
$871$	−9.58930	−0.324921
$872$	0	0
$873$	− 5.43832i	− 0.184059i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	53.2485i	1.79807i	0.437873	+	0.899037i	$0.355732\pi$
$877$	53.2485i	1.79807i	−0.437873	+	0.899037i	$0.644268\pi$
$878$	0	0
$879$	−22.6324	−0.763370
$880$	0	0
$881$	43.7812	1.47503	0.737513	−	0.675332i	$-0.236000\pi$
$881$	43.7812	1.47503	0.737513	+	0.675332i	$0.236000\pi$
$882$	0	0
$883$	17.9524i	0.604148i	0.953285	+	0.302074i	$0.0976789\pi$
$883$	17.9524i	0.604148i	−0.953285	+	0.302074i	$0.902321\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	53.4631i	1.79511i	0.440898	+	0.897557i	$0.354660\pi$
$887$	53.4631i	1.79511i	−0.440898	+	0.897557i	$0.645340\pi$
$888$	0	0
$889$	1.26889	0.0425571
$890$	0	0
$891$	−3.84324	−0.128753
$892$	0	0
$893$	− 17.3505i	− 0.580611i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	4.36363i	0.145697i
$898$	0	0
$899$	6.64469	0.221613
$900$	0	0
$901$	31.0416	1.03415
$902$	0	0
$903$	12.8677i	0.428209i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	23.9651i	0.795748i	0.917440	+	0.397874i	$0.130252\pi$
$907$	23.9651i	0.795748i	−0.917440	+	0.397874i	$0.869748\pi$
$908$	0	0
$909$	15.5316	0.515152
$910$	0	0
$911$	8.52073	0.282304	0.141152	−	0.989988i	$-0.454919\pi$
$911$	8.52073	0.282304	0.141152	+	0.989988i	$0.454919\pi$
$912$	0	0
$913$	11.6791i	0.386521i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	4.06478i	0.134231i
$918$	0	0
$919$	6.59812	0.217652	0.108826	−	0.994061i	$-0.465291\pi$
$919$	6.59812	0.217652	0.108826	+	0.994061i	$0.465291\pi$
$920$	0	0
$921$	25.0167	0.824329
$922$	0	0
$923$	− 6.08990i	− 0.200451i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 9.09332i	− 0.298664i
$928$	0	0
$929$	−1.28398	−0.0421260	−0.0210630	−	0.999778i	$-0.506705\pi$
$929$	−1.28398	−0.0421260	−0.0210630	+	0.999778i	$0.506705\pi$
$930$	0	0
$931$	31.1185	1.01987
$932$	0	0
$933$	27.1139i	0.887669i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 44.0122i	− 1.43782i	−0.695105	−	0.718908i	$-0.744642\pi$
$937$	− 44.0122i	− 1.43782i	0.695105	−	0.718908i	$-0.255358\pi$
$938$	0	0
$939$	−1.27963	−0.0417591
$940$	0	0
$941$	17.1317	0.558477	0.279239	−	0.960222i	$-0.409918\pi$
$941$	17.1317	0.558477	0.279239	+	0.960222i	$0.409918\pi$
$942$	0	0
$943$	− 4.16608i	− 0.135666i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 2.30320i	− 0.0748440i	−0.999300	−	0.0374220i	$-0.988085\pi$
$947$	− 2.30320i	− 0.0748440i	0.999300	−	0.0374220i	$-0.0119146\pi$
$948$	0	0
$949$	40.1754	1.30415
$950$	0	0
$951$	−27.4668	−0.890671
$952$	0	0
$953$	14.8222i	0.480137i	0.970756	+	0.240069i	$0.0771700\pi$
$953$	14.8222i	0.480137i	−0.970756	+	0.240069i	$0.922830\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	2.92532i	0.0945621i
$958$	0	0
$959$	−3.38550	−0.109323
$960$	0	0
$961$	45.2075	1.45831
$962$	0	0
$963$	− 2.92724i	− 0.0943290i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	6.98248i	0.224541i	0.993678	+	0.112271i	$0.0358124\pi$
$967$	6.98248i	0.224541i	−0.993678	+	0.112271i	$0.964188\pi$
$968$	0	0
$969$	−16.3547	−0.525387
$970$	0	0
$971$	14.3885	0.461750	0.230875	−	0.972983i	$-0.425841\pi$
$971$	14.3885	0.461750	0.230875	+	0.972983i	$0.425841\pi$
$972$	0	0
$973$	30.3415i	0.972703i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 55.0868i	− 1.76238i	−0.472761	−	0.881191i	$-0.656742\pi$
$977$	− 55.0868i	− 1.76238i	0.472761	−	0.881191i	$-0.343258\pi$
$978$	0	0
$979$	−70.3320	−2.24782
$980$	0	0
$981$	13.0525	0.416735
$982$	0	0
$983$	− 1.13846i	− 0.0363113i	−0.999835	−	0.0181556i	$-0.994221\pi$
$983$	− 1.13846i	− 0.0363113i	0.999835	−	0.0181556i	$-0.00577944\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 3.87228i	− 0.123256i
$988$	0	0
$989$	9.72727	0.309309
$990$	0	0
$991$	−7.33794	−0.233097	−0.116549	−	0.993185i	$-0.537183\pi$
$991$	−7.33794	−0.233097	−0.116549	+	0.993185i	$0.537183\pi$
$992$	0	0
$993$	− 14.2551i	− 0.452371i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 19.0077i	− 0.601981i	−0.953627	−	0.300990i	$-0.902683\pi$
$997$	− 19.0077i	− 0.601981i	0.953627	−	0.300990i	$-0.0973172\pi$
$998$	0	0
$999$	3.24076	0.102533

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6900.2.f.s.6349.6		8
5.2	odd	4		6900.2.a.bb.1.3	yes	4
5.3	odd	4		6900.2.a.ba.1.2	✓	4
5.4	even	2	inner	6900.2.f.s.6349.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6900.2.a.ba.1.2	✓	4	5.3	odd	4
6900.2.a.bb.1.3	yes	4	5.2	odd	4
6900.2.f.s.6349.3		8	5.4	even	2	inner
6900.2.f.s.6349.6		8	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 6900 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6900.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} -1.32284i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} -1.32284i q^{7} -1.00000 q^{9} -3.84324 q^{11} +4.36363i q^{13} +2.75924i q^{17} +5.92724 q^{19} +1.32284 q^{21} -1.00000i q^{23} -1.00000i q^{27} -0.761159 q^{29} -8.72969 q^{31} -3.84324i q^{33} +3.24076i q^{37} -4.36363 q^{39} +4.16608 q^{41} +9.72727i q^{43} -2.92724i q^{47} +5.25008 q^{49} -2.75924 q^{51} -11.2501i q^{53} +5.92724i q^{57} +3.08208 q^{59} -6.08400 q^{61} +1.32284i q^{63} +2.19755i q^{67} +1.00000 q^{69} -1.39560 q^{71} -9.20687i q^{73} +5.08400i q^{77} -14.3341 q^{79} +1.00000 q^{81} -3.03886i q^{83} -0.761159i q^{87} +18.3002 q^{89} +5.77240 q^{91} -8.72969i q^{93} +5.43832i q^{97} +3.84324 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{9}+O(q^{10}) \) 8 * q - 8 * q^9	\( 8 q - 8 q^{9} - 4 q^{11} - 4 q^{19} - 6 q^{21} + 2 q^{29} - 12 q^{31} + 2 q^{39} - 10 q^{41} - 26 q^{49} - 20 q^{51} + 6 q^{59} - 24 q^{61} + 8 q^{69} - 46 q^{71} - 22 q^{79} + 8 q^{81} - 12 q^{89} - 38 q^{91} + 4 q^{99}+O(q^{100}) \) 8 * q - 8 * q^9 - 4 * q^11 - 4 * q^19 - 6 * q^21 + 2 * q^29 - 12 * q^31 + 2 * q^39 - 10 * q^41 - 26 * q^49 - 20 * q^51 + 6 * q^59 - 24 * q^61 + 8 * q^69 - 46 * q^71 - 22 * q^79 + 8 * q^81 - 12 * q^89 - 38 * q^91 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more