LMFDB - Embedded newform 6900.2.f.s.6349.8

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			6349.8
Root			$-0.672035i$ of defining polynomial
Character	$\chi$	$=$	6900.6349
Dual form			6900.2.f.s.6349.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +4.30400i q^{7} -1.00000 q^{9} -0.672035 q^{11} +3.64807i q^{13} +7.56447i q^{17} -5.22040 q^{19} -4.30400 q^{21} -1.00000i q^{23} -1.00000i q^{27} +1.58844 q^{29} +10.5004 q^{31} -0.672035i q^{33} -1.56447i q^{37} -3.64807 q^{39} -4.63196 q^{41} +8.29614i q^{43} +8.22040i q^{47} -11.5244 q^{49} -7.56447 q^{51} +5.52440i q^{53} -5.22040i q^{57} +2.26048 q^{59} +1.89244 q^{61} -4.30400i q^{63} +10.2800i q^{67} +1.00000 q^{69} -6.91641 q^{71} -5.32010i q^{73} -2.89244i q^{77} +10.4168 q^{79} +1.00000 q^{81} -15.1050i q^{83} +1.58844i q^{87} -5.53226 q^{89} -15.7013 q^{91} +10.5004i q^{93} +8.71556i q^{97} +0.672035 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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}+ \cdots + 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

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
$7$	4.30400i	1.62676i	0.581734	+	0.813379i	$0.302375\pi$
$7$	4.30400i	1.62676i	−0.581734	+	0.813379i	$0.697625\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−0.672035	−0.202626	−0.101313	−	0.994855i	$-0.532304\pi$
$11$	−0.672035	−0.202626	−0.101313	+	0.994855i	$0.532304\pi$
$12$	0	0
$13$	3.64807i	1.01179i	0.862594	+	0.505896i	$0.168838\pi$
$13$	3.64807i	1.01179i	−0.862594	+	0.505896i	$0.831162\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	7.56447i	1.83465i	0.398134	+	0.917327i	$0.369658\pi$
$17$	7.56447i	1.83465i	−0.398134	+	0.917327i	$0.630342\pi$
$18$	0	0
$19$	−5.22040	−1.19764	−0.598821	−	0.800883i	$-0.704364\pi$
$19$	−5.22040	−1.19764	−0.598821	+	0.800883i	$0.704364\pi$
$20$	0	0
$21$	−4.30400	−0.939209
$22$	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
$29$	1.58844	0.294966	0.147483	−	0.989065i	$-0.452883\pi$
$29$	1.58844	0.294966	0.147483	+	0.989065i	$0.452883\pi$
$30$	0	0
$31$	10.5004	1.88593	0.942967	−	0.332886i	$-0.108023\pi$
$31$	10.5004	1.88593	0.942967	+	0.332886i	$0.108023\pi$
$32$	0	0
$33$	− 0.672035i	− 0.116986i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 1.56447i	− 0.257198i	−0.991697	−	0.128599i	$-0.958952\pi$
$37$	− 1.56447i	− 0.257198i	0.991697	−	0.128599i	$-0.0410480\pi$
$38$	0	0
$39$	−3.64807	−0.584159
$40$	0	0
$41$	−4.63196	−0.723391	−0.361696	−	0.932296i	$-0.617802\pi$
$41$	−4.63196	−0.723391	−0.361696	+	0.932296i	$0.617802\pi$
$42$	0	0
$43$	8.29614i	1.26515i	0.774499	+	0.632575i	$0.218002\pi$
$43$	8.29614i	1.26515i	−0.774499	+	0.632575i	$0.781998\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	8.22040i	1.19907i	0.800349	+	0.599535i	$0.204648\pi$
$47$	8.22040i	1.19907i	−0.800349	+	0.599535i	$0.795352\pi$
$48$	0	0
$49$	−11.5244	−1.64634
$50$	0	0
$51$	−7.56447	−1.05924
$52$	0	0
$53$	5.52440i	0.758835i	0.925226	+	0.379418i	$0.123876\pi$
$53$	5.52440i	0.758835i	−0.925226	+	0.379418i	$0.876124\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 5.22040i	− 0.691459i
$58$	0	0
$59$	2.26048	0.294289	0.147144	−	0.989115i	$-0.452992\pi$
$59$	2.26048	0.294289	0.147144	+	0.989115i	$0.452992\pi$
$60$	0	0
$61$	1.89244	0.242302	0.121151	−	0.992634i	$-0.461341\pi$
$61$	1.89244	0.242302	0.121151	+	0.992634i	$0.461341\pi$
$62$	0	0
$63$	− 4.30400i	− 0.542253i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	10.2800i	1.25591i	0.778251	+	0.627953i	$0.216107\pi$
$67$	10.2800i	1.25591i	−0.778251	+	0.627953i	$0.783893\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	−6.91641	−0.820826	−0.410413	−	0.911900i	$-0.634616\pi$
$71$	−6.91641	−0.820826	−0.410413	+	0.911900i	$0.634616\pi$
$72$	0	0
$73$	− 5.32010i	− 0.622671i	−0.950300	−	0.311336i	$-0.899224\pi$
$73$	− 5.32010i	− 0.622671i	0.950300	−	0.311336i	$-0.100776\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 2.89244i	− 0.329624i
$78$	0	0
$79$	10.4168	1.17199	0.585993	−	0.810316i	$-0.300705\pi$
$79$	10.4168	1.17199	0.585993	+	0.810316i	$0.300705\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 15.1050i	− 1.65799i	−0.559258	−	0.828994i	$-0.688914\pi$
$83$	− 15.1050i	− 1.65799i	0.559258	−	0.828994i	$-0.311086\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	1.58844i	0.170299i
$88$	0	0
$89$	−5.53226	−0.586419	−0.293209	−	0.956048i	$-0.594723\pi$
$89$	−5.53226	−0.586419	−0.293209	+	0.956048i	$0.594723\pi$
$90$	0	0
$91$	−15.7013	−1.64594
$92$	0	0
$93$	10.5004i	1.08884i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	8.71556i	0.884931i	0.896786	+	0.442465i	$0.145896\pi$
$97$	8.71556i	0.884931i	−0.896786	+	0.442465i	$0.854104\pi$
$98$	0	0
$99$	0.672035	0.0675421
$100$	0	0
$101$	1.13681	0.113117	0.0565584	−	0.998399i	$-0.481987\pi$
$101$	1.13681	0.113117	0.0565584	+	0.998399i	$0.481987\pi$
$102$	0	0
$103$	− 10.8524i	− 1.06932i	−0.845069	−	0.534658i	$-0.820441\pi$
$103$	− 10.8524i	− 1.06932i	0.845069	−	0.534658i	$-0.179559\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 8.22040i	− 0.794696i	−0.917668	−	0.397348i	$-0.869931\pi$
$107$	− 8.22040i	− 0.794696i	0.917668	−	0.397348i	$-0.130069\pi$
$108$	0	0
$109$	11.8044	1.13066	0.565330	−	0.824865i	$-0.308749\pi$
$109$	11.8044	1.13066	0.565330	+	0.824865i	$0.308749\pi$
$110$	0	0
$111$	1.56447	0.148493
$112$	0	0
$113$	− 9.99214i	− 0.939981i	−0.882671	−	0.469991i	$-0.844257\pi$
$113$	− 9.99214i	− 0.939981i	0.882671	−	0.469991i	$-0.155743\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 3.64807i	− 0.337264i
$118$	0	0
$119$	−32.5575	−2.98454
$120$	0	0
$121$	−10.5484	−0.958943
$122$	0	0
$123$	− 4.63196i	− 0.417650i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 3.95207i	− 0.350689i	−0.984507	−	0.175345i	$-0.943896\pi$
$127$	− 3.95207i	− 0.350689i	0.984507	−	0.175345i	$-0.0561040\pi$
$128$	0	0
$129$	−8.29614	−0.730434
$130$	0	0
$131$	−14.2204	−1.24244	−0.621221	−	0.783635i	$-0.713363\pi$
$131$	−14.2204	−1.24244	−0.621221	+	0.783635i	$0.713363\pi$
$132$	0	0
$133$	− 22.4686i	− 1.94828i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 17.0810i	− 1.45933i	−0.683805	−	0.729665i	$-0.739676\pi$
$137$	− 17.0810i	− 1.45933i	0.683805	−	0.729665i	$-0.260324\pi$
$138$	0	0
$139$	0.180331	0.0152955	0.00764776	−	0.999971i	$-0.497566\pi$
$139$	0.180331	0.0152955	0.00764776	+	0.999971i	$0.497566\pi$
$140$	0	0
$141$	−8.22040	−0.692283
$142$	0	0
$143$	− 2.45163i	− 0.205016i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 11.5244i	− 0.950517i
$148$	0	0
$149$	20.7805	1.70240	0.851201	−	0.524840i	$-0.175875\pi$
$149$	20.7805	1.70240	0.851201	+	0.524840i	$0.175875\pi$
$150$	0	0
$151$	14.4926	1.17939	0.589695	−	0.807626i	$-0.299248\pi$
$151$	14.4926	1.17939	0.589695	+	0.807626i	$0.299248\pi$
$152$	0	0
$153$	− 7.56447i	− 0.611552i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	20.7609i	1.65690i	0.560062	+	0.828451i	$0.310777\pi$
$157$	20.7609i	1.65690i	−0.560062	+	0.828451i	$0.689223\pi$
$158$	0	0
$159$	−5.52440	−0.438114
$160$	0	0
$161$	4.30400	0.339203
$162$	0	0
$163$	9.02925i	0.707225i	0.935392	+	0.353613i	$0.115047\pi$
$163$	9.02925i	0.707225i	−0.935392	+	0.353613i	$0.884953\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1.82495i	0.141219i	0.997504	+	0.0706095i	$0.0224944\pi$
$167$	1.82495i	0.141219i	−0.997504	+	0.0706095i	$0.977506\pi$
$168$	0	0
$169$	−0.308409	−0.0237237
$170$	0	0
$171$	5.22040	0.399214
$172$	0	0
$173$	− 4.47263i	− 0.340048i	−0.985440	−	0.170024i	$-0.945615\pi$
$173$	− 4.47263i	− 0.340048i	0.985440	−	0.170024i	$-0.0543845\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	2.26048i	0.169908i
$178$	0	0
$179$	7.52440	0.562400	0.281200	−	0.959649i	$-0.409268\pi$
$179$	7.52440	0.562400	0.281200	+	0.959649i	$0.409268\pi$
$180$	0	0
$181$	9.41684	0.699948	0.349974	−	0.936759i	$-0.386190\pi$
$181$	9.41684	0.699948	0.349974	+	0.936759i	$0.386190\pi$
$182$	0	0
$183$	1.89244i	0.139893i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 5.08359i	− 0.371749i
$188$	0	0
$189$	4.30400	0.313070
$190$	0	0
$191$	−13.9760	−1.01127	−0.505635	−	0.862747i	$-0.668742\pi$
$191$	−13.9760	−1.01127	−0.505635	+	0.862747i	$0.668742\pi$
$192$	0	0
$193$	− 11.1964i	− 0.805937i	−0.915214	−	0.402969i	$-0.867978\pi$
$193$	− 11.1964i	− 0.805937i	0.915214	−	0.402969i	$-0.132022\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	20.0292i	1.42702i	0.700643	+	0.713512i	$0.252897\pi$
$197$	20.0292i	1.42702i	−0.700643	+	0.713512i	$0.747103\pi$
$198$	0	0
$199$	−6.64462	−0.471025	−0.235512	−	0.971871i	$-0.575677\pi$
$199$	−6.64462	−0.471025	−0.235512	+	0.971871i	$0.575677\pi$
$200$	0	0
$201$	−10.2800	−0.725098
$202$	0	0
$203$	6.83665i	0.479839i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	1.00000i	0.0695048i
$208$	0	0
$209$	3.50830	0.242674
$210$	0	0
$211$	−2.35232	−0.161940	−0.0809701	−	0.996717i	$-0.525802\pi$
$211$	−2.35232	−0.161940	−0.0809701	+	0.996717i	$0.525802\pi$
$212$	0	0
$213$	− 6.91641i	− 0.473904i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	45.1939i	3.06796i
$218$	0	0
$219$	5.32010	0.359499
$220$	0	0
$221$	−27.5957	−1.85629
$222$	0	0
$223$	12.0971i	0.810083i	0.914298	+	0.405042i	$0.132743\pi$
$223$	12.0971i	0.810083i	−0.914298	+	0.405042i	$0.867257\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 25.9573i	− 1.72285i	−0.507885	−	0.861425i	$-0.669573\pi$
$227$	− 25.9573i	− 1.72285i	0.507885	−	0.861425i	$-0.330427\pi$
$228$	0	0
$229$	−3.29230	−0.217561	−0.108781	−	0.994066i	$-0.534695\pi$
$229$	−3.29230	−0.217561	−0.108781	+	0.994066i	$0.534695\pi$
$230$	0	0
$231$	2.89244	0.190309
$232$	0	0
$233$	22.1616i	1.45186i	0.687770	+	0.725929i	$0.258590\pi$
$233$	22.1616i	1.45186i	−0.687770	+	0.725929i	$0.741410\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	10.4168i	0.676647i
$238$	0	0
$239$	−3.39200	−0.219410	−0.109705	−	0.993964i	$-0.534991\pi$
$239$	−3.39200	−0.219410	−0.109705	+	0.993964i	$0.534991\pi$
$240$	0	0
$241$	−3.59189	−0.231374	−0.115687	−	0.993286i	$-0.536907\pi$
$241$	−3.59189	−0.231374	−0.115687	+	0.993286i	$0.536907\pi$
$242$	0	0
$243$	1.00000i	0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 19.0444i	− 1.21177i
$248$	0	0
$249$	15.1050	0.957239
$250$	0	0
$251$	31.0462	1.95962	0.979810	−	0.199930i	$-0.0640714\pi$
$251$	31.0462	1.95962	0.979810	+	0.199930i	$0.0640714\pi$
$252$	0	0
$253$	0.672035i	0.0422505i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	4.81526i	0.300368i	0.988658	+	0.150184i	$0.0479865\pi$
$257$	4.81526i	0.300368i	−0.988658	+	0.150184i	$0.952013\pi$
$258$	0	0
$259$	6.73350	0.418399
$260$	0	0
$261$	−1.58844	−0.0983220
$262$	0	0
$263$	− 31.1602i	− 1.92142i	−0.277554	−	0.960710i	$-0.589524\pi$
$263$	− 31.1602i	− 1.92142i	0.277554	−	0.960710i	$-0.410476\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 5.53226i	− 0.338569i
$268$	0	0
$269$	5.18675	0.316241	0.158121	−	0.987420i	$-0.449456\pi$
$269$	5.18675	0.316241	0.158121	+	0.987420i	$0.449456\pi$
$270$	0	0
$271$	−5.65152	−0.343305	−0.171653	−	0.985158i	$-0.554911\pi$
$271$	−5.65152	−0.343305	−0.171653	+	0.985158i	$0.554911\pi$
$272$	0	0
$273$	− 15.7013i	− 0.950285i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	28.6437i	1.72103i	0.509425	+	0.860515i	$0.329858\pi$
$277$	28.6437i	1.72103i	−0.509425	+	0.860515i	$0.670142\pi$
$278$	0	0
$279$	−10.5004	−0.628645
$280$	0	0
$281$	−14.0854	−0.840266	−0.420133	−	0.907463i	$-0.638017\pi$
$281$	−14.0854	−0.840266	−0.420133	+	0.907463i	$0.638017\pi$
$282$	0	0
$283$	− 23.6695i	− 1.40700i	−0.710694	−	0.703502i	$-0.751619\pi$
$283$	− 23.6695i	− 1.40700i	0.710694	−	0.703502i	$-0.248381\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 19.9360i	− 1.17678i
$288$	0	0
$289$	−40.2213	−2.36596
$290$	0	0
$291$	−8.71556	−0.510915
$292$	0	0
$293$	− 18.7962i	− 1.09809i	−0.835794	−	0.549043i	$-0.814992\pi$
$293$	− 18.7962i	− 1.09809i	0.835794	−	0.549043i	$-0.185008\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0.672035i	0.0389954i
$298$	0	0
$299$	3.64807	0.210973
$300$	0	0
$301$	−35.7066	−2.05809
$302$	0	0
$303$	1.13681i	0.0653080i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 8.23306i	− 0.469886i	−0.972009	−	0.234943i	$-0.924510\pi$
$307$	− 8.23306i	− 0.469886i	0.972009	−	0.234943i	$-0.0754903\pi$
$308$	0	0
$309$	10.8524	0.617370
$310$	0	0
$311$	−23.9251	−1.35667	−0.678335	−	0.734753i	$-0.737298\pi$
$311$	−23.9251	−1.35667	−0.678335	+	0.734753i	$0.737298\pi$
$312$	0	0
$313$	8.54051i	0.482738i	0.970433	+	0.241369i	$0.0775964\pi$
$313$	8.54051i	0.482738i	−0.970433	+	0.241369i	$0.922404\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 15.8079i	− 0.887859i	−0.896062	−	0.443930i	$-0.853584\pi$
$317$	− 15.8079i	− 0.887859i	0.896062	−	0.443930i	$-0.146416\pi$
$318$	0	0
$319$	−1.06749	−0.0597679
$320$	0	0
$321$	8.22040	0.458818
$322$	0	0
$323$	− 39.4896i	− 2.19726i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	11.8044i	0.652787i
$328$	0	0
$329$	−35.3806	−1.95060
$330$	0	0
$331$	−10.4652	−0.575217	−0.287609	−	0.957748i	$-0.592860\pi$
$331$	−10.4652	−0.575217	−0.287609	+	0.957748i	$0.592860\pi$
$332$	0	0
$333$	1.56447i	0.0857327i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	10.1867i	0.554907i	0.960739	+	0.277454i	$0.0894905\pi$
$337$	10.1867i	0.554907i	−0.960739	+	0.277454i	$0.910509\pi$
$338$	0	0
$339$	9.99214	0.542699
$340$	0	0
$341$	−7.05666	−0.382140
$342$	0	0
$343$	− 19.4730i	− 1.05144i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	4.52785i	0.243068i	0.992587	+	0.121534i	$0.0387813\pi$
$347$	4.52785i	0.243068i	−0.992587	+	0.121534i	$0.961219\pi$
$348$	0	0
$349$	−16.7512	−0.896672	−0.448336	−	0.893865i	$-0.647983\pi$
$349$	−16.7512	−0.896672	−0.448336	+	0.893865i	$0.647983\pi$
$350$	0	0
$351$	3.64807	0.194720
$352$	0	0
$353$	14.6084i	0.777526i	0.921338	+	0.388763i	$0.127097\pi$
$353$	14.6084i	0.777526i	−0.921338	+	0.388763i	$0.872903\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 32.5575i	− 1.72312i
$358$	0	0
$359$	−10.4326	−0.550610	−0.275305	−	0.961357i	$-0.588779\pi$
$359$	−10.4326	−0.550610	−0.275305	+	0.961357i	$0.588779\pi$
$360$	0	0
$361$	8.25262	0.434348
$362$	0	0
$363$	− 10.5484i	− 0.553646i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 17.4251i	− 0.909582i	−0.890598	−	0.454791i	$-0.849714\pi$
$367$	− 17.4251i	− 0.909582i	0.890598	−	0.454791i	$-0.150286\pi$
$368$	0	0
$369$	4.63196	0.241130
$370$	0	0
$371$	−23.7770	−1.23444
$372$	0	0
$373$	− 13.5308i	− 0.700599i	−0.936638	−	0.350300i	$-0.886080\pi$
$373$	− 13.5308i	− 0.700599i	0.936638	−	0.350300i	$-0.113920\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	5.79474i	0.298444i
$378$	0	0
$379$	−18.1372	−0.931645	−0.465823	−	0.884878i	$-0.654242\pi$
$379$	−18.1372	−0.931645	−0.465823	+	0.884878i	$0.654242\pi$
$380$	0	0
$381$	3.95207	0.202471
$382$	0	0
$383$	− 27.9202i	− 1.42666i	−0.700829	−	0.713329i	$-0.747187\pi$
$383$	− 27.9202i	− 1.42666i	0.700829	−	0.713329i	$-0.252813\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 8.29614i	− 0.421716i
$388$	0	0
$389$	33.3093	1.68885	0.844424	−	0.535676i	$-0.179943\pi$
$389$	33.3093	1.68885	0.844424	+	0.535676i	$0.179943\pi$
$390$	0	0
$391$	7.56447	0.382552
$392$	0	0
$393$	− 14.2204i	− 0.717324i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	24.6739i	1.23835i	0.785255	+	0.619173i	$0.212532\pi$
$397$	24.6739i	1.23835i	−0.785255	+	0.619173i	$0.787468\pi$
$398$	0	0
$399$	22.4686	1.12484
$400$	0	0
$401$	−4.04535	−0.202015	−0.101008	−	0.994886i	$-0.532207\pi$
$401$	−4.04535	−0.202015	−0.101008	+	0.994886i	$0.532207\pi$
$402$	0	0
$403$	38.3063i	1.90817i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1.05138i	0.0521151i
$408$	0	0
$409$	−0.914573	−0.0452227	−0.0226114	−	0.999744i	$-0.507198\pi$
$409$	−0.914573	−0.0452227	−0.0226114	+	0.999744i	$0.507198\pi$
$410$	0	0
$411$	17.0810	0.842544
$412$	0	0
$413$	9.72909i	0.478737i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0.180331i	0.00883087i
$418$	0	0
$419$	15.0117	0.733369	0.366685	−	0.930345i	$-0.380493\pi$
$419$	15.0117	0.733369	0.366685	+	0.930345i	$0.380493\pi$
$420$	0	0
$421$	16.9696	0.827049	0.413524	−	0.910493i	$-0.364298\pi$
$421$	16.9696	0.827049	0.413524	+	0.910493i	$0.364298\pi$
$422$	0	0
$423$	− 8.22040i	− 0.399690i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	8.14506i	0.394167i
$428$	0	0
$429$	2.45163	0.118366
$430$	0	0
$431$	27.1733	1.30889	0.654447	−	0.756108i	$-0.272902\pi$
$431$	27.1733	1.30889	0.654447	+	0.756108i	$0.272902\pi$
$432$	0	0
$433$	− 23.1094i	− 1.11057i	−0.831661	−	0.555283i	$-0.812610\pi$
$433$	− 23.1094i	− 1.11057i	0.831661	−	0.555283i	$-0.187390\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	5.22040i	0.249726i
$438$	0	0
$439$	−31.1485	−1.48664	−0.743319	−	0.668938i	$-0.766749\pi$
$439$	−31.1485	−1.48664	−0.743319	+	0.668938i	$0.766749\pi$
$440$	0	0
$441$	11.5244	0.548781
$442$	0	0
$443$	13.1681i	0.625633i	0.949814	+	0.312817i	$0.101273\pi$
$443$	13.1681i	0.625633i	−0.949814	+	0.312817i	$0.898727\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	20.7805i	0.982882i
$448$	0	0
$449$	−29.6759	−1.40049	−0.700245	−	0.713902i	$-0.746926\pi$
$449$	−29.6759	−1.40049	−0.700245	+	0.713902i	$0.746926\pi$
$450$	0	0
$451$	3.11284	0.146578
$452$	0	0
$453$	14.4926i	0.680921i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	31.7047i	1.48308i	0.670906	+	0.741542i	$0.265905\pi$
$457$	31.7047i	1.48308i	−0.670906	+	0.741542i	$0.734095\pi$
$458$	0	0
$459$	7.56447	0.353079
$460$	0	0
$461$	33.8723	1.57759	0.788795	−	0.614656i	$-0.210705\pi$
$461$	33.8723	1.57759	0.788795	+	0.614656i	$0.210705\pi$
$462$	0	0
$463$	− 0.781429i	− 0.0363161i	−0.999835	−	0.0181580i	$-0.994220\pi$
$463$	− 0.781429i	− 0.0363161i	0.999835	−	0.0181580i	$-0.00578020\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 19.3802i	− 0.896810i	−0.893831	−	0.448405i	$-0.851992\pi$
$467$	− 19.3802i	− 0.896810i	0.893831	−	0.448405i	$-0.148008\pi$
$468$	0	0
$469$	−44.2452	−2.04306
$470$	0	0
$471$	−20.7609	−0.956612
$472$	0	0
$473$	− 5.57530i	− 0.256352i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 5.52440i	− 0.252945i
$478$	0	0
$479$	2.07718	0.0949088	0.0474544	−	0.998873i	$-0.484889\pi$
$479$	2.07718	0.0949088	0.0474544	+	0.998873i	$0.484889\pi$
$480$	0	0
$481$	5.70731	0.260231
$482$	0	0
$483$	4.30400i	0.195839i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 19.5518i	− 0.885977i	−0.896527	−	0.442989i	$-0.853918\pi$
$487$	− 19.5518i	− 0.885977i	0.896527	−	0.442989i	$-0.146082\pi$
$488$	0	0
$489$	−9.02925	−0.408317
$490$	0	0
$491$	−3.54012	−0.159763	−0.0798817	−	0.996804i	$-0.525454\pi$
$491$	−3.54012	−0.159763	−0.0798817	+	0.996804i	$0.525454\pi$
$492$	0	0
$493$	12.0157i	0.541161i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 29.7682i	− 1.33529i
$498$	0	0
$499$	−26.0196	−1.16480	−0.582398	−	0.812904i	$-0.697885\pi$
$499$	−26.0196	−1.16480	−0.582398	+	0.812904i	$0.697885\pi$
$500$	0	0
$501$	−1.82495	−0.0815328
$502$	0	0
$503$	14.4560i	0.644559i	0.946645	+	0.322280i	$0.104449\pi$
$503$	14.4560i	0.644559i	−0.946645	+	0.322280i	$0.895551\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 0.308409i	− 0.0136969i
$508$	0	0
$509$	20.7814	0.921121	0.460560	−	0.887628i	$-0.347649\pi$
$509$	20.7814	0.921121	0.460560	+	0.887628i	$0.347649\pi$
$510$	0	0
$511$	22.8977	1.01294
$512$	0	0
$513$	5.22040i	0.230486i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 5.52440i	− 0.242963i
$518$	0	0
$519$	4.47263	0.196327
$520$	0	0
$521$	36.8885	1.61612	0.808058	−	0.589103i	$-0.200519\pi$
$521$	36.8885	1.61612	0.808058	+	0.589103i	$0.200519\pi$
$522$	0	0
$523$	− 20.4827i	− 0.895646i	−0.894122	−	0.447823i	$-0.852199\pi$
$523$	− 20.4827i	− 0.895646i	0.894122	−	0.447823i	$-0.147801\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	79.4303i	3.46004i
$528$	0	0
$529$	−1.00000	−0.0434783
$530$	0	0
$531$	−2.26048	−0.0980963
$532$	0	0
$533$	− 16.8977i	− 0.731922i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	7.52440i	0.324702i
$538$	0	0
$539$	7.74481	0.333592
$540$	0	0
$541$	−17.1245	−0.736241	−0.368121	−	0.929778i	$-0.619999\pi$
$541$	−17.1245	−0.736241	−0.368121	+	0.929778i	$0.619999\pi$
$542$	0	0
$543$	9.41684i	0.404115i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 33.5557i	− 1.43474i	−0.696694	−	0.717368i	$-0.745347\pi$
$547$	− 33.5557i	− 1.43474i	0.696694	−	0.717368i	$-0.254653\pi$
$548$	0	0
$549$	−1.89244	−0.0807673
$550$	0	0
$551$	−8.29230	−0.353264
$552$	0	0
$553$	44.8341i	1.90654i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	20.4086i	0.864740i	0.901696	+	0.432370i	$0.142323\pi$
$557$	20.4086i	0.864740i	−0.901696	+	0.432370i	$0.857677\pi$
$558$	0	0
$559$	−30.2649	−1.28007
$560$	0	0
$561$	5.08359	0.214630
$562$	0	0
$563$	− 5.17889i	− 0.218264i	−0.994027	−	0.109132i	$-0.965193\pi$
$563$	− 5.17889i	− 0.218264i	0.994027	−	0.109132i	$-0.0348071\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	4.30400i	0.180751i
$568$	0	0
$569$	6.05321	0.253764	0.126882	−	0.991918i	$-0.459503\pi$
$569$	6.05321	0.253764	0.126882	+	0.991918i	$0.459503\pi$
$570$	0	0
$571$	45.3263	1.89684	0.948422	−	0.317009i	$-0.102679\pi$
$571$	45.3263	1.89684	0.948422	+	0.317009i	$0.102679\pi$
$572$	0	0
$573$	− 13.9760i	− 0.583857i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	4.55133i	0.189474i	0.995502	+	0.0947372i	$0.0302011\pi$
$577$	4.55133i	0.189474i	−0.995502	+	0.0947372i	$0.969799\pi$
$578$	0	0
$579$	11.1964	0.465308
$580$	0	0
$581$	65.0118	2.69714
$582$	0	0
$583$	− 3.71259i	− 0.153760i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	46.5148i	1.91987i	0.280218	+	0.959936i	$0.409593\pi$
$587$	46.5148i	1.91987i	−0.280218	+	0.959936i	$0.590407\pi$
$588$	0	0
$589$	−54.8165	−2.25868
$590$	0	0
$591$	−20.0292	−0.823893
$592$	0	0
$593$	7.81574i	0.320954i	0.987040	+	0.160477i	$0.0513033\pi$
$593$	7.81574i	0.320954i	−0.987040	+	0.160477i	$0.948697\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 6.64462i	− 0.271946i
$598$	0	0
$599$	−44.4461	−1.81602	−0.908009	−	0.418951i	$-0.862398\pi$
$599$	−44.4461	−1.81602	−0.908009	+	0.418951i	$0.862398\pi$
$600$	0	0
$601$	−19.3367	−0.788760	−0.394380	−	0.918947i	$-0.629041\pi$
$601$	−19.3367	−0.788760	−0.394380	+	0.918947i	$0.629041\pi$
$602$	0	0
$603$	− 10.2800i	− 0.418635i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 18.4172i	− 0.747532i	−0.927523	−	0.373766i	$-0.878066\pi$
$607$	− 18.4172i	− 0.747532i	0.927523	−	0.373766i	$-0.121934\pi$
$608$	0	0
$609$	−6.83665	−0.277035
$610$	0	0
$611$	−29.9886	−1.21321
$612$	0	0
$613$	− 46.8170i	− 1.89092i	−0.325737	−	0.945460i	$-0.605612\pi$
$613$	− 46.8170i	− 1.89092i	0.325737	−	0.945460i	$-0.394388\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	35.9338i	1.44664i	0.690513	+	0.723320i	$0.257385\pi$
$617$	35.9338i	1.44664i	−0.690513	+	0.723320i	$0.742615\pi$
$618$	0	0
$619$	17.3880	0.698882	0.349441	−	0.936958i	$-0.386372\pi$
$619$	17.3880	0.698882	0.349441	+	0.936958i	$0.386372\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	− 23.8108i	− 0.953961i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	3.50830i	0.140108i
$628$	0	0
$629$	11.8344	0.471870
$630$	0	0
$631$	−8.19002	−0.326040	−0.163020	−	0.986623i	$-0.552123\pi$
$631$	−8.19002	−0.326040	−0.163020	+	0.986623i	$0.552123\pi$
$632$	0	0
$633$	− 2.35232i	− 0.0934962i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 42.0418i	− 1.66576i
$638$	0	0
$639$	6.91641	0.273609
$640$	0	0
$641$	−4.98773	−0.197003	−0.0985017	−	0.995137i	$-0.531405\pi$
$641$	−4.98773	−0.197003	−0.0985017	+	0.995137i	$0.531405\pi$
$642$	0	0
$643$	19.3514i	0.763143i	0.924339	+	0.381572i	$0.124617\pi$
$643$	19.3514i	0.763143i	−0.924339	+	0.381572i	$0.875383\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 40.4964i	− 1.59208i	−0.605245	−	0.796039i	$-0.706925\pi$
$647$	− 40.4964i	− 1.59208i	0.605245	−	0.796039i	$-0.293075\pi$
$648$	0	0
$649$	−1.51912	−0.0596307
$650$	0	0
$651$	−45.1939	−1.77129
$652$	0	0
$653$	− 26.5683i	− 1.03970i	−0.854258	−	0.519849i	$-0.825988\pi$
$653$	− 26.5683i	− 1.03970i	0.854258	−	0.519849i	$-0.174012\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	5.32010i	0.207557i
$658$	0	0
$659$	−6.82879	−0.266012	−0.133006	−	0.991115i	$-0.542463\pi$
$659$	−6.82879	−0.266012	−0.133006	+	0.991115i	$0.542463\pi$
$660$	0	0
$661$	−6.43457	−0.250276	−0.125138	−	0.992139i	$-0.539937\pi$
$661$	−6.43457	−0.250276	−0.125138	+	0.992139i	$0.539937\pi$
$662$	0	0
$663$	− 27.5957i	− 1.07173i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 1.58844i	− 0.0615047i
$668$	0	0
$669$	−12.0971	−0.467702
$670$	0	0
$671$	−1.27179	−0.0490968
$672$	0	0
$673$	30.6058i	1.17977i	0.807488	+	0.589884i	$0.200826\pi$
$673$	30.6058i	1.17977i	−0.807488	+	0.589884i	$0.799174\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	37.1280i	1.42694i	0.700683	+	0.713472i	$0.252879\pi$
$677$	37.1280i	1.42694i	−0.700683	+	0.713472i	$0.747121\pi$
$678$	0	0
$679$	−37.5117	−1.43957
$680$	0	0
$681$	25.9573	0.994688
$682$	0	0
$683$	4.58699i	0.175516i	0.996142	+	0.0877582i	$0.0279703\pi$
$683$	4.58699i	0.175516i	−0.996142	+	0.0877582i	$0.972030\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 3.29230i	− 0.125609i
$688$	0	0
$689$	−20.1534	−0.767783
$690$	0	0
$691$	7.89885	0.300487	0.150243	−	0.988649i	$-0.451994\pi$
$691$	7.89885	0.300487	0.150243	+	0.988649i	$0.451994\pi$
$692$	0	0
$693$	2.89244i	0.109875i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 35.0384i	− 1.32717i
$698$	0	0
$699$	−22.1616	−0.838230
$700$	0	0
$701$	−27.9416	−1.05534	−0.527670	−	0.849449i	$-0.676934\pi$
$701$	−27.9416	−1.05534	−0.527670	+	0.849449i	$0.676934\pi$
$702$	0	0
$703$	8.16719i	0.308031i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	4.89283i	0.184014i
$708$	0	0
$709$	−1.07190	−0.0402560	−0.0201280	−	0.999797i	$-0.506407\pi$
$709$	−1.07190	−0.0402560	−0.0201280	+	0.999797i	$0.506407\pi$
$710$	0	0
$711$	−10.4168	−0.390662
$712$	0	0
$713$	− 10.5004i	− 0.393244i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 3.39200i	− 0.126677i
$718$	0	0
$719$	32.8284	1.22429	0.612146	−	0.790744i	$-0.290306\pi$
$719$	32.8284	1.22429	0.612146	+	0.790744i	$0.290306\pi$
$720$	0	0
$721$	46.7086	1.73952
$722$	0	0
$723$	− 3.59189i	− 0.133584i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 7.51961i	− 0.278887i	−0.990230	−	0.139443i	$-0.955469\pi$
$727$	− 7.51961i	− 0.278887i	0.990230	−	0.139443i	$-0.0445313\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−62.7559	−2.32111
$732$	0	0
$733$	11.0366i	0.407647i	0.979008	+	0.203823i	$0.0653369\pi$
$733$	11.0366i	0.407647i	−0.979008	+	0.203823i	$0.934663\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 6.90855i	− 0.254480i
$738$	0	0
$739$	−3.73952	−0.137561	−0.0687803	−	0.997632i	$-0.521911\pi$
$739$	−3.73952	−0.137561	−0.0687803	+	0.997632i	$0.521911\pi$
$740$	0	0
$741$	19.0444	0.699613
$742$	0	0
$743$	9.88641i	0.362697i	0.983419	+	0.181349i	$0.0580462\pi$
$743$	9.88641i	0.362697i	−0.983419	+	0.181349i	$0.941954\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	15.1050i	0.552662i
$748$	0	0
$749$	35.3806	1.29278
$750$	0	0
$751$	−52.2974	−1.90836	−0.954180	−	0.299235i	$-0.903269\pi$
$751$	−52.2974	−1.90836	−0.954180	+	0.299235i	$0.903269\pi$
$752$	0	0
$753$	31.0462i	1.13139i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	40.6939i	1.47905i	0.673132	+	0.739523i	$0.264949\pi$
$757$	40.6939i	1.47905i	−0.673132	+	0.739523i	$0.735051\pi$
$758$	0	0
$759$	−0.672035	−0.0243933
$760$	0	0
$761$	12.3524	0.447775	0.223887	−	0.974615i	$-0.428125\pi$
$761$	12.3524	0.447775	0.223887	+	0.974615i	$0.428125\pi$
$762$	0	0
$763$	50.8063i	1.83931i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	8.24637i	0.297759i
$768$	0	0
$769$	8.28003	0.298586	0.149293	−	0.988793i	$-0.452300\pi$
$769$	8.28003	0.298586	0.149293	+	0.988793i	$0.452300\pi$
$770$	0	0
$771$	−4.81526	−0.173417
$772$	0	0
$773$	− 11.8171i	− 0.425031i	−0.977158	−	0.212516i	$-0.931834\pi$
$773$	− 11.8171i	− 0.425031i	0.977158	−	0.212516i	$-0.0681656\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	6.73350i	0.241563i
$778$	0	0
$779$	24.1807	0.866364
$780$	0	0
$781$	4.64807	0.166321
$782$	0	0
$783$	− 1.58844i	− 0.0567662i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	24.5609i	0.875503i	0.899096	+	0.437751i	$0.144225\pi$
$787$	24.5609i	0.875503i	−0.899096	+	0.437751i	$0.855775\pi$
$788$	0	0
$789$	31.1602	1.10933
$790$	0	0
$791$	43.0062	1.52912
$792$	0	0
$793$	6.90375i	0.245159i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	14.7970i	0.524135i	0.965050	+	0.262068i	$0.0844043\pi$
$797$	14.7970i	0.524135i	−0.965050	+	0.262068i	$0.915596\pi$
$798$	0	0
$799$	−62.1830	−2.19988
$800$	0	0
$801$	5.53226	0.195473
$802$	0	0
$803$	3.57530i	0.126170i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	5.18675i	0.182582i
$808$	0	0
$809$	27.9438	0.982452	0.491226	−	0.871032i	$-0.336549\pi$
$809$	27.9438	0.982452	0.491226	+	0.871032i	$0.336549\pi$
$810$	0	0
$811$	−37.2940	−1.30957	−0.654786	−	0.755815i	$-0.727241\pi$
$811$	−37.2940	−1.30957	−0.654786	+	0.755815i	$0.727241\pi$
$812$	0	0
$813$	− 5.65152i	− 0.198207i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 43.3092i	− 1.51520i
$818$	0	0
$819$	15.7013	0.548647
$820$	0	0
$821$	−3.38376	−0.118094	−0.0590470	−	0.998255i	$-0.518806\pi$
$821$	−3.38376	−0.118094	−0.0590470	+	0.998255i	$0.518806\pi$
$822$	0	0
$823$	54.5344i	1.90095i	0.310802	+	0.950475i	$0.399402\pi$
$823$	54.5344i	1.90095i	−0.310802	+	0.950475i	$0.600598\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 39.3201i	− 1.36729i	−0.729813	−	0.683647i	$-0.760393\pi$
$827$	− 39.3201i	− 1.36729i	0.729813	−	0.683647i	$-0.239607\pi$
$828$	0	0
$829$	49.4943	1.71901	0.859504	−	0.511128i	$-0.170772\pi$
$829$	49.4943	1.71901	0.859504	+	0.511128i	$0.170772\pi$
$830$	0	0
$831$	−28.6437	−0.993638
$832$	0	0
$833$	− 87.1760i	− 3.02047i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 10.5004i	− 0.362948i
$838$	0	0
$839$	36.0371	1.24414	0.622070	−	0.782962i	$-0.286292\pi$
$839$	36.0371	1.24414	0.622070	+	0.782962i	$0.286292\pi$
$840$	0	0
$841$	−26.4769	−0.912995
$842$	0	0
$843$	− 14.0854i	− 0.485128i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 45.4002i	− 1.55997i
$848$	0	0
$849$	23.6695	0.812334
$850$	0	0
$851$	−1.56447	−0.0536295
$852$	0	0
$853$	− 4.36507i	− 0.149457i	−0.997204	−	0.0747286i	$-0.976191\pi$
$853$	− 4.36507i	− 0.149457i	0.997204	−	0.0747286i	$-0.0238091\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 51.6385i	− 1.76394i	−0.471308	−	0.881969i	$-0.656218\pi$
$857$	− 51.6385i	− 1.76394i	0.471308	−	0.881969i	$-0.343782\pi$
$858$	0	0
$859$	24.8064	0.846385	0.423192	−	0.906040i	$-0.360909\pi$
$859$	24.8064	0.846385	0.423192	+	0.906040i	$0.360909\pi$
$860$	0	0
$861$	19.9360	0.679416
$862$	0	0
$863$	32.9477i	1.12155i	0.827968	+	0.560776i	$0.189497\pi$
$863$	32.9477i	1.12155i	−0.827968	+	0.560776i	$0.810503\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 40.2213i	− 1.36599i
$868$	0	0
$869$	−7.00049	−0.237475
$870$	0	0
$871$	−37.5023	−1.27072
$872$	0	0
$873$	− 8.71556i	− 0.294977i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 23.4691i	− 0.792495i	−0.918144	−	0.396248i	$-0.870312\pi$
$877$	− 23.4691i	− 0.792495i	0.918144	−	0.396248i	$-0.129688\pi$
$878$	0	0
$879$	18.7962	0.633980
$880$	0	0
$881$	20.9824	0.706917	0.353458	−	0.935450i	$-0.385006\pi$
$881$	20.9824	0.706917	0.353458	+	0.935450i	$0.385006\pi$
$882$	0	0
$883$	55.7940i	1.87762i	0.344439	+	0.938809i	$0.388069\pi$
$883$	55.7940i	1.87762i	−0.344439	+	0.938809i	$0.611931\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 55.7607i	− 1.87226i	−0.351655	−	0.936130i	$-0.614381\pi$
$887$	− 55.7607i	− 1.87226i	0.351655	−	0.936130i	$-0.385619\pi$
$888$	0	0
$889$	17.0097	0.570487
$890$	0	0
$891$	−0.672035	−0.0225140
$892$	0	0
$893$	− 42.9138i	− 1.43606i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	3.64807i	0.121805i
$898$	0	0
$899$	16.6793	0.556287
$900$	0	0
$901$	−41.7892	−1.39220
$902$	0	0
$903$	− 35.7066i	− 1.18824i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	46.1719i	1.53311i	0.642177	+	0.766556i	$0.278031\pi$
$907$	46.1719i	1.53311i	−0.642177	+	0.766556i	$0.721969\pi$
$908$	0	0
$909$	−1.13681	−0.0377056
$910$	0	0
$911$	−56.1216	−1.85939	−0.929695	−	0.368329i	$-0.879930\pi$
$911$	−56.1216	−1.85939	−0.929695	+	0.368329i	$0.879930\pi$
$912$	0	0
$913$	10.1511i	0.335952i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 61.2046i	− 2.02115i
$918$	0	0
$919$	33.1860	1.09471	0.547353	−	0.836902i	$-0.315636\pi$
$919$	33.1860	1.09471	0.547353	+	0.836902i	$0.315636\pi$
$920$	0	0
$921$	8.23306	0.271289
$922$	0	0
$923$	− 25.2315i	− 0.830506i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	10.8524i	0.356438i
$928$	0	0
$929$	16.4090	0.538361	0.269181	−	0.963090i	$-0.413247\pi$
$929$	16.4090	0.538361	0.269181	+	0.963090i	$0.413247\pi$
$930$	0	0
$931$	60.1620	1.97173
$932$	0	0
$933$	− 23.9251i	− 0.783274i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	4.27140i	0.139541i	0.997563	+	0.0697703i	$0.0222266\pi$
$937$	4.27140i	0.139541i	−0.997563	+	0.0697703i	$0.977773\pi$
$938$	0	0
$939$	−8.54051	−0.278709
$940$	0	0
$941$	19.8963	0.648600	0.324300	−	0.945954i	$-0.394871\pi$
$941$	19.8963	0.648600	0.324300	+	0.945954i	$0.394871\pi$
$942$	0	0
$943$	4.63196i	0.150837i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 41.9032i	− 1.36167i	−0.732437	−	0.680835i	$-0.761617\pi$
$947$	− 41.9032i	− 1.36167i	0.732437	−	0.680835i	$-0.238383\pi$
$948$	0	0
$949$	19.4081	0.630014
$950$	0	0
$951$	15.8079	0.512606
$952$	0	0
$953$	53.3885i	1.72942i	0.502270	+	0.864711i	$0.332498\pi$
$953$	53.3885i	1.72942i	−0.502270	+	0.864711i	$0.667502\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 1.06749i	− 0.0345070i
$958$	0	0
$959$	73.5167	2.37398
$960$	0	0
$961$	79.2592	2.55675
$962$	0	0
$963$	8.22040i	0.264899i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	58.2152i	1.87208i	0.351899	+	0.936038i	$0.385536\pi$
$967$	58.2152i	1.87208i	−0.351899	+	0.936038i	$0.614464\pi$
$968$	0	0
$969$	39.4896	1.26859
$970$	0	0
$971$	−42.3742	−1.35985	−0.679926	−	0.733280i	$-0.737988\pi$
$971$	−42.3742	−1.35985	−0.679926	+	0.733280i	$0.737988\pi$
$972$	0	0
$973$	0.776146i	0.0248821i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	37.7866i	1.20890i	0.796643	+	0.604451i	$0.206607\pi$
$977$	37.7866i	1.20890i	−0.796643	+	0.604451i	$0.793393\pi$
$978$	0	0
$979$	3.71788	0.118824
$980$	0	0
$981$	−11.8044	−0.376887
$982$	0	0
$983$	38.8498i	1.23912i	0.784951	+	0.619558i	$0.212688\pi$
$983$	38.8498i	1.23912i	−0.784951	+	0.619558i	$0.787312\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 35.3806i	− 1.12618i
$988$	0	0
$989$	8.29614	0.263802
$990$	0	0
$991$	31.7227	1.00770	0.503852	−	0.863790i	$-0.331916\pi$
$991$	31.7227	1.00770	0.503852	+	0.863790i	$0.331916\pi$
$992$	0	0
$993$	− 10.4652i	− 0.332102i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	52.9046i	1.67551i	0.546049	+	0.837753i	$0.316131\pi$
$997$	52.9046i	1.67551i	−0.546049	+	0.837753i	$0.683869\pi$
$998$	0	0
$999$	−1.56447	−0.0494978

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6900.2.f.s.6349.8		8
5.2	odd	4		6900.2.a.bb.1.1	yes	4
5.3	odd	4		6900.2.a.ba.1.4	✓	4
5.4	even	2	inner	6900.2.f.s.6349.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6900.2.a.ba.1.4	✓	4	5.3	odd	4
6900.2.a.bb.1.1	yes	4	5.2	odd	4
6900.2.f.s.6349.1		8	5.4	even	2	inner
6900.2.f.s.6349.8		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)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +4.30400i q^{7} -1.00000 q^{9} -0.672035 q^{11} +3.64807i q^{13} +7.56447i q^{17} -5.22040 q^{19} -4.30400 q^{21} -1.00000i q^{23} -1.00000i q^{27} +1.58844 q^{29} +10.5004 q^{31} -0.672035i q^{33} -1.56447i q^{37} -3.64807 q^{39} -4.63196 q^{41} +8.29614i q^{43} +8.22040i q^{47} -11.5244 q^{49} -7.56447 q^{51} +5.52440i q^{53} -5.22040i q^{57} +2.26048 q^{59} +1.89244 q^{61} -4.30400i q^{63} +10.2800i q^{67} +1.00000 q^{69} -6.91641 q^{71} -5.32010i q^{73} -2.89244i q^{77} +10.4168 q^{79} +1.00000 q^{81} -15.1050i q^{83} +1.58844i q^{87} -5.53226 q^{89} -15.7013 q^{91} +10.5004i q^{93} +8.71556i q^{97} +0.672035 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{9} - 4 q^{11} - 4 q^{19} - 6 q^{21} + 2 q^{29} - 12 q^{31} + 2 q^{39} - 10 q^{41} - 26 q^{49} - 20 q^{51} + 6 q^{59} - 24 q^{61} + 8 q^{69} - 46 q^{71} - 22 q^{79} + 8 q^{81} - 12 q^{89} - 38 q^{91}+ \cdots + 4 q^{99}+O(q^{100}) \) 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