LMFDB - Embedded newform 6900.2.f.o.6349.2

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:	$4$
Coefficient field:	$\Q(i, \sqrt{15})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 7x^{2} + 16 $ x^4 - 7*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 1380)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			6349.2
Root			$1.93649 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	6900.6349
Dual form			6900.2.f.o.6349.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +3.00000i q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} +3.00000i q^{7} -1.00000 q^{9} +4.87298 q^{11} +2.87298i q^{13} +3.87298i q^{17} +2.87298 q^{19} +3.00000 q^{21} +1.00000i q^{23} +1.00000i q^{27} +5.87298 q^{29} +3.00000 q^{31} -4.87298i q^{33} +1.00000i q^{37} +2.87298 q^{39} -5.87298 q^{41} -3.74597i q^{43} -6.87298i q^{47} -2.00000 q^{49} +3.87298 q^{51} +3.87298i q^{53} -2.87298i q^{57} -5.87298 q^{59} +8.87298 q^{61} -3.00000i q^{63} +10.7460i q^{67} +1.00000 q^{69} -13.6190 q^{71} +2.87298i q^{73} +14.6190i q^{77} -4.00000 q^{79} +1.00000 q^{81} +0.127017i q^{83} -5.87298i q^{87} -1.74597 q^{89} -8.61895 q^{91} -3.00000i q^{93} +8.00000i q^{97} -4.87298 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^9	$ 4 q - 4 q^{9} + 4 q^{11} - 4 q^{19} + 12 q^{21} + 8 q^{29} + 12 q^{31} - 4 q^{39} - 8 q^{41} - 8 q^{49} - 8 q^{59} + 20 q^{61} + 4 q^{69} - 8 q^{71} - 16 q^{79} + 4 q^{81} + 24 q^{89} + 12 q^{91} - 4 q^{99}+O(q^{100}) $ 4 * q - 4 * q^9 + 4 * q^11 - 4 * q^19 + 12 * q^21 + 8 * q^29 + 12 * q^31 - 4 * q^39 - 8 * q^41 - 8 * q^49 - 8 * q^59 + 20 * q^61 + 4 * q^69 - 8 * q^71 - 16 * q^79 + 4 * q^81 + 24 * q^89 + 12 * 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
$7$	3.00000i	1.13389i	0.823754	+	0.566947i	$0.191875\pi$
$7$	3.00000i	1.13389i	−0.823754	+	0.566947i	$0.808125\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	4.87298	1.46926	0.734630	−	0.678468i	$-0.237356\pi$
$11$	4.87298	1.46926	0.734630	+	0.678468i	$0.237356\pi$
$12$	0	0
$13$	2.87298i	0.796822i	0.917207	+	0.398411i	$0.130438\pi$
$13$	2.87298i	0.796822i	−0.917207	+	0.398411i	$0.869562\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.87298i	0.939336i	0.882843	+	0.469668i	$0.155626\pi$
$17$	3.87298i	0.939336i	−0.882843	+	0.469668i	$0.844374\pi$
$18$	0	0
$19$	2.87298	0.659108	0.329554	−	0.944137i	$-0.393102\pi$
$19$	2.87298	0.659108	0.329554	+	0.944137i	$0.393102\pi$
$20$	0	0
$21$	3.00000	0.654654
$22$	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
$29$	5.87298	1.09059	0.545293	−	0.838246i	$-0.316418\pi$
$29$	5.87298	1.09059	0.545293	+	0.838246i	$0.316418\pi$
$30$	0	0
$31$	3.00000	0.538816	0.269408	−	0.963026i	$-0.413172\pi$
$31$	3.00000	0.538816	0.269408	+	0.963026i	$0.413172\pi$
$32$	0	0
$33$	− 4.87298i	− 0.848278i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	1.00000i	0.164399i	0.996616	+	0.0821995i	$0.0261945\pi$
$37$	1.00000i	0.164399i	−0.996616	+	0.0821995i	$0.973806\pi$
$38$	0	0
$39$	2.87298	0.460046
$40$	0	0
$41$	−5.87298	−0.917206	−0.458603	−	0.888641i	$-0.651650\pi$
$41$	−5.87298	−0.917206	−0.458603	+	0.888641i	$0.651650\pi$
$42$	0	0
$43$	− 3.74597i	− 0.571255i	−0.958341	−	0.285627i	$-0.907798\pi$
$43$	− 3.74597i	− 0.571255i	0.958341	−	0.285627i	$-0.0922019\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 6.87298i	− 1.00253i	−0.865295	−	0.501264i	$-0.832869\pi$
$47$	− 6.87298i	− 1.00253i	0.865295	−	0.501264i	$-0.167131\pi$
$48$	0	0
$49$	−2.00000	−0.285714
$50$	0	0
$51$	3.87298	0.542326
$52$	0	0
$53$	3.87298i	0.531995i	0.963974	+	0.265998i	$0.0857013\pi$
$53$	3.87298i	0.531995i	−0.963974	+	0.265998i	$0.914299\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 2.87298i	− 0.380536i
$58$	0	0
$59$	−5.87298	−0.764597	−0.382299	−	0.924039i	$-0.624867\pi$
$59$	−5.87298	−0.764597	−0.382299	+	0.924039i	$0.624867\pi$
$60$	0	0
$61$	8.87298	1.13607	0.568035	−	0.823005i	$-0.307704\pi$
$61$	8.87298	1.13607	0.568035	+	0.823005i	$0.307704\pi$
$62$	0	0
$63$	− 3.00000i	− 0.377964i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	10.7460i	1.31283i	0.754401	+	0.656414i	$0.227928\pi$
$67$	10.7460i	1.31283i	−0.754401	+	0.656414i	$0.772072\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	−13.6190	−1.61627	−0.808136	−	0.588996i	$-0.799523\pi$
$71$	−13.6190	−1.61627	−0.808136	+	0.588996i	$0.799523\pi$
$72$	0	0
$73$	2.87298i	0.336257i	0.985765	+	0.168129i	$0.0537724\pi$
$73$	2.87298i	0.336257i	−0.985765	+	0.168129i	$0.946228\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	14.6190i	1.66598i
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	0.127017i	0.0139419i	0.999976	+	0.00697094i	$0.00221894\pi$
$83$	0.127017i	0.0139419i	−0.999976	+	0.00697094i	$0.997781\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 5.87298i	− 0.629650i
$88$	0	0
$89$	−1.74597	−0.185072	−0.0925360	−	0.995709i	$-0.529497\pi$
$89$	−1.74597	−0.185072	−0.0925360	+	0.995709i	$0.529497\pi$
$90$	0	0
$91$	−8.61895	−0.903511
$92$	0	0
$93$	− 3.00000i	− 0.311086i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	8.00000i	0.812277i	0.913812	+	0.406138i	$0.133125\pi$
$97$	8.00000i	0.812277i	−0.913812	+	0.406138i	$0.866875\pi$
$98$	0	0
$99$	−4.87298	−0.489753
$100$	0	0
$101$	−3.87298	−0.385376	−0.192688	−	0.981260i	$-0.561721\pi$
$101$	−3.87298	−0.385376	−0.192688	+	0.981260i	$0.561721\pi$
$102$	0	0
$103$	6.00000i	0.591198i	0.955312	+	0.295599i	$0.0955191\pi$
$103$	6.00000i	0.591198i	−0.955312	+	0.295599i	$0.904481\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 5.61895i	− 0.543204i	−0.962410	−	0.271602i	$-0.912447\pi$
$107$	− 5.61895i	− 0.543204i	0.962410	−	0.271602i	$-0.0875535\pi$
$108$	0	0
$109$	−16.6190	−1.59181	−0.795903	−	0.605424i	$-0.793004\pi$
$109$	−16.6190	−1.59181	−0.795903	+	0.605424i	$0.793004\pi$
$110$	0	0
$111$	1.00000	0.0949158
$112$	0	0
$113$	3.87298i	0.364340i	0.983267	+	0.182170i	$0.0583121\pi$
$113$	3.87298i	0.364340i	−0.983267	+	0.182170i	$0.941688\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 2.87298i	− 0.265607i
$118$	0	0
$119$	−11.6190	−1.06511
$120$	0	0
$121$	12.7460	1.15872
$122$	0	0
$123$	5.87298i	0.529549i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 6.61895i	− 0.587337i	−0.955907	−	0.293668i	$-0.905124\pi$
$127$	− 6.61895i	− 0.587337i	0.955907	−	0.293668i	$-0.0948762\pi$
$128$	0	0
$129$	−3.74597	−0.329814
$130$	0	0
$131$	11.4919	1.00405	0.502027	−	0.864852i	$-0.332588\pi$
$131$	11.4919	1.00405	0.502027	+	0.864852i	$0.332588\pi$
$132$	0	0
$133$	8.61895i	0.747358i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	8.00000i	0.683486i	0.939793	+	0.341743i	$0.111017\pi$
$137$	8.00000i	0.683486i	−0.939793	+	0.341743i	$0.888983\pi$
$138$	0	0
$139$	20.4919	1.73810	0.869052	−	0.494722i	$-0.164730\pi$
$139$	20.4919	1.73810	0.869052	+	0.494722i	$0.164730\pi$
$140$	0	0
$141$	−6.87298	−0.578810
$142$	0	0
$143$	14.0000i	1.17074i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	2.00000i	0.164957i
$148$	0	0
$149$	10.8730	0.890750	0.445375	−	0.895344i	$-0.353070\pi$
$149$	10.8730	0.890750	0.445375	+	0.895344i	$0.353070\pi$
$150$	0	0
$151$	3.74597	0.304842	0.152421	−	0.988316i	$-0.451293\pi$
$151$	3.74597	0.304842	0.152421	+	0.988316i	$0.451293\pi$
$152$	0	0
$153$	− 3.87298i	− 0.313112i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	17.0000i	1.35675i	0.734717	+	0.678374i	$0.237315\pi$
$157$	17.0000i	1.35675i	−0.734717	+	0.678374i	$0.762685\pi$
$158$	0	0
$159$	3.87298	0.307148
$160$	0	0
$161$	−3.00000	−0.236433
$162$	0	0
$163$	22.0000i	1.72317i	0.507611	+	0.861586i	$0.330529\pi$
$163$	22.0000i	1.72317i	−0.507611	+	0.861586i	$0.669471\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 9.12702i	− 0.706270i	−0.935572	−	0.353135i	$-0.885116\pi$
$167$	− 9.12702i	− 0.706270i	0.935572	−	0.353135i	$-0.114884\pi$
$168$	0	0
$169$	4.74597	0.365074
$170$	0	0
$171$	−2.87298	−0.219703
$172$	0	0
$173$	− 23.4919i	− 1.78606i	−0.449999	−	0.893029i	$-0.648575\pi$
$173$	− 23.4919i	− 1.78606i	0.449999	−	0.893029i	$-0.351425\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	5.87298i	0.441440i
$178$	0	0
$179$	15.7460	1.17691	0.588454	−	0.808530i	$-0.299737\pi$
$179$	15.7460	1.17691	0.588454	+	0.808530i	$0.299737\pi$
$180$	0	0
$181$	−21.7460	−1.61636	−0.808182	−	0.588932i	$-0.799549\pi$
$181$	−21.7460	−1.61636	−0.808182	+	0.588932i	$0.799549\pi$
$182$	0	0
$183$	− 8.87298i	− 0.655910i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	18.8730i	1.38013i
$188$	0	0
$189$	−3.00000	−0.218218
$190$	0	0
$191$	−18.8730	−1.36560	−0.682801	−	0.730605i	$-0.739238\pi$
$191$	−18.8730	−1.36560	−0.682801	+	0.730605i	$0.739238\pi$
$192$	0	0
$193$	− 9.74597i	− 0.701530i	−0.936464	−	0.350765i	$-0.885922\pi$
$193$	− 9.74597i	− 0.701530i	0.936464	−	0.350765i	$-0.114078\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	4.25403i	0.303087i	0.988451	+	0.151544i	$0.0484244\pi$
$197$	4.25403i	0.303087i	−0.988451	+	0.151544i	$0.951576\pi$
$198$	0	0
$199$	−19.2379	−1.36374	−0.681869	−	0.731474i	$-0.738833\pi$
$199$	−19.2379	−1.36374	−0.681869	+	0.731474i	$0.738833\pi$
$200$	0	0
$201$	10.7460	0.757962
$202$	0	0
$203$	17.6190i	1.23661i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 1.00000i	− 0.0695048i
$208$	0	0
$209$	14.0000	0.968400
$210$	0	0
$211$	1.00000	0.0688428	0.0344214	−	0.999407i	$-0.489041\pi$
$211$	1.00000	0.0688428	0.0344214	+	0.999407i	$0.489041\pi$
$212$	0	0
$213$	13.6190i	0.933155i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	9.00000i	0.610960i
$218$	0	0
$219$	2.87298	0.194138
$220$	0	0
$221$	−11.1270	−0.748484
$222$	0	0
$223$	2.00000i	0.133930i	0.997755	+	0.0669650i	$0.0213316\pi$
$223$	2.00000i	0.133930i	−0.997755	+	0.0669650i	$0.978668\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	7.49193i	0.497257i	0.968599	+	0.248629i	$0.0799798\pi$
$227$	7.49193i	0.497257i	−0.968599	+	0.248629i	$0.920020\pi$
$228$	0	0
$229$	5.74597	0.379704	0.189852	−	0.981813i	$-0.439199\pi$
$229$	5.74597	0.379704	0.189852	+	0.981813i	$0.439199\pi$
$230$	0	0
$231$	14.6190	0.961856
$232$	0	0
$233$	− 15.4919i	− 1.01491i	−0.861678	−	0.507455i	$-0.830586\pi$
$233$	− 15.4919i	− 1.01491i	0.861678	−	0.507455i	$-0.169414\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	4.00000i	0.259828i
$238$	0	0
$239$	17.8730	1.15611	0.578054	−	0.815999i	$-0.303812\pi$
$239$	17.8730	1.15611	0.578054	+	0.815999i	$0.303812\pi$
$240$	0	0
$241$	−5.12702	−0.330260	−0.165130	−	0.986272i	$-0.552804\pi$
$241$	−5.12702	−0.330260	−0.165130	+	0.986272i	$0.552804\pi$
$242$	0	0
$243$	− 1.00000i	− 0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	8.25403i	0.525192i
$248$	0	0
$249$	0.127017	0.00804935
$250$	0	0
$251$	2.00000	0.126239	0.0631194	−	0.998006i	$-0.479895\pi$
$251$	2.00000	0.126239	0.0631194	+	0.998006i	$0.479895\pi$
$252$	0	0
$253$	4.87298i	0.306362i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 16.6190i	− 1.03666i	−0.855180	−	0.518331i	$-0.826554\pi$
$257$	− 16.6190i	− 1.03666i	0.855180	−	0.518331i	$-0.173446\pi$
$258$	0	0
$259$	−3.00000	−0.186411
$260$	0	0
$261$	−5.87298	−0.363529
$262$	0	0
$263$	21.8730i	1.34875i	0.738391	+	0.674373i	$0.235586\pi$
$263$	21.8730i	1.34875i	−0.738391	+	0.674373i	$0.764414\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	1.74597i	0.106851i
$268$	0	0
$269$	23.3649	1.42458	0.712292	−	0.701883i	$-0.247657\pi$
$269$	23.3649	1.42458	0.712292	+	0.701883i	$0.247657\pi$
$270$	0	0
$271$	−7.25403	−0.440651	−0.220326	−	0.975426i	$-0.570712\pi$
$271$	−7.25403	−0.440651	−0.220326	+	0.975426i	$0.570712\pi$
$272$	0	0
$273$	8.61895i	0.521643i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	14.0000i	0.841178i	0.907251	+	0.420589i	$0.138177\pi$
$277$	14.0000i	0.841178i	−0.907251	+	0.420589i	$0.861823\pi$
$278$	0	0
$279$	−3.00000	−0.179605
$280$	0	0
$281$	−16.3649	−0.976249	−0.488125	−	0.872774i	$-0.662319\pi$
$281$	−16.3649	−0.976249	−0.488125	+	0.872774i	$0.662319\pi$
$282$	0	0
$283$	− 22.2379i	− 1.32191i	−0.750427	−	0.660953i	$-0.770152\pi$
$283$	− 22.2379i	− 1.32191i	0.750427	−	0.660953i	$-0.229848\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 17.6190i	− 1.04001i
$288$	0	0
$289$	2.00000	0.117647
$290$	0	0
$291$	8.00000	0.468968
$292$	0	0
$293$	13.6190i	0.795628i	0.917466	+	0.397814i	$0.130231\pi$
$293$	13.6190i	0.795628i	−0.917466	+	0.397814i	$0.869769\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	4.87298i	0.282759i
$298$	0	0
$299$	−2.87298	−0.166149
$300$	0	0
$301$	11.2379	0.647742
$302$	0	0
$303$	3.87298i	0.222497i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	9.12702i	0.520906i	0.965486	+	0.260453i	$0.0838720\pi$
$307$	9.12702i	0.520906i	−0.965486	+	0.260453i	$0.916128\pi$
$308$	0	0
$309$	6.00000	0.341328
$310$	0	0
$311$	−4.00000	−0.226819	−0.113410	−	0.993548i	$-0.536177\pi$
$311$	−4.00000	−0.226819	−0.113410	+	0.993548i	$0.536177\pi$
$312$	0	0
$313$	− 7.00000i	− 0.395663i	−0.980236	−	0.197832i	$-0.936610\pi$
$313$	− 7.00000i	− 0.395663i	0.980236	−	0.197832i	$-0.0633900\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	22.3649i	1.25614i	0.778157	+	0.628069i	$0.216155\pi$
$317$	22.3649i	1.25614i	−0.778157	+	0.628069i	$0.783845\pi$
$318$	0	0
$319$	28.6190	1.60235
$320$	0	0
$321$	−5.61895	−0.313619
$322$	0	0
$323$	11.1270i	0.619124i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	16.6190i	0.919030i
$328$	0	0
$329$	20.6190	1.13676
$330$	0	0
$331$	−22.2379	−1.22231	−0.611153	−	0.791513i	$-0.709294\pi$
$331$	−22.2379	−1.22231	−0.611153	+	0.791513i	$0.709294\pi$
$332$	0	0
$333$	− 1.00000i	− 0.0547997i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 1.49193i	− 0.0812708i	−0.999174	−	0.0406354i	$-0.987062\pi$
$337$	− 1.49193i	− 0.0812708i	0.999174	−	0.0406354i	$-0.0129382\pi$
$338$	0	0
$339$	3.87298	0.210352
$340$	0	0
$341$	14.6190	0.791661
$342$	0	0
$343$	15.0000i	0.809924i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 25.7460i	− 1.38212i	−0.722799	−	0.691058i	$-0.757145\pi$
$347$	− 25.7460i	− 1.38212i	0.722799	−	0.691058i	$-0.242855\pi$
$348$	0	0
$349$	14.7460	0.789334	0.394667	−	0.918824i	$-0.370860\pi$
$349$	14.7460	0.789334	0.394667	+	0.918824i	$0.370860\pi$
$350$	0	0
$351$	−2.87298	−0.153349
$352$	0	0
$353$	1.12702i	0.0599850i	0.999550	+	0.0299925i	$0.00954835\pi$
$353$	1.12702i	0.0599850i	−0.999550	+	0.0299925i	$0.990452\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	11.6190i	0.614940i
$358$	0	0
$359$	21.1270	1.11504	0.557521	−	0.830163i	$-0.311753\pi$
$359$	21.1270	1.11504	0.557521	+	0.830163i	$0.311753\pi$
$360$	0	0
$361$	−10.7460	−0.565577
$362$	0	0
$363$	− 12.7460i	− 0.668990i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	25.9839i	1.35635i	0.734902	+	0.678173i	$0.237228\pi$
$367$	25.9839i	1.35635i	−0.734902	+	0.678173i	$0.762772\pi$
$368$	0	0
$369$	5.87298	0.305735
$370$	0	0
$371$	−11.6190	−0.603226
$372$	0	0
$373$	− 1.74597i	− 0.0904027i	−0.998978	−	0.0452014i	$-0.985607\pi$
$373$	− 1.74597i	− 0.0904027i	0.998978	−	0.0452014i	$-0.0143929\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	16.8730i	0.869003i
$378$	0	0
$379$	12.0000	0.616399	0.308199	−	0.951322i	$-0.400274\pi$
$379$	12.0000	0.616399	0.308199	+	0.951322i	$0.400274\pi$
$380$	0	0
$381$	−6.61895	−0.339099
$382$	0	0
$383$	25.6190i	1.30907i	0.756033	+	0.654534i	$0.227135\pi$
$383$	25.6190i	1.30907i	−0.756033	+	0.654534i	$0.772865\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	3.74597i	0.190418i
$388$	0	0
$389$	−3.74597	−0.189928	−0.0949640	−	0.995481i	$-0.530274\pi$
$389$	−3.74597	−0.189928	−0.0949640	+	0.995481i	$0.530274\pi$
$390$	0	0
$391$	−3.87298	−0.195865
$392$	0	0
$393$	− 11.4919i	− 0.579691i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	19.4919i	0.978272i	0.872208	+	0.489136i	$0.162688\pi$
$397$	19.4919i	0.978272i	−0.872208	+	0.489136i	$0.837312\pi$
$398$	0	0
$399$	8.61895	0.431487
$400$	0	0
$401$	27.2379	1.36020	0.680098	−	0.733121i	$-0.261937\pi$
$401$	27.2379	1.36020	0.680098	+	0.733121i	$0.261937\pi$
$402$	0	0
$403$	8.61895i	0.429340i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	4.87298i	0.241545i
$408$	0	0
$409$	−22.2379	−1.09959	−0.549797	−	0.835299i	$-0.685295\pi$
$409$	−22.2379	−1.09959	−0.549797	+	0.835299i	$0.685295\pi$
$410$	0	0
$411$	8.00000	0.394611
$412$	0	0
$413$	− 17.6190i	− 0.866972i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 20.4919i	− 1.00349i
$418$	0	0
$419$	−2.61895	−0.127944	−0.0639720	−	0.997952i	$-0.520377\pi$
$419$	−2.61895	−0.127944	−0.0639720	+	0.997952i	$0.520377\pi$
$420$	0	0
$421$	−38.8730	−1.89455	−0.947277	−	0.320417i	$-0.896177\pi$
$421$	−38.8730	−1.89455	−0.947277	+	0.320417i	$0.896177\pi$
$422$	0	0
$423$	6.87298i	0.334176i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	26.6190i	1.28818i
$428$	0	0
$429$	14.0000	0.675926
$430$	0	0
$431$	5.74597	0.276773	0.138387	−	0.990378i	$-0.455808\pi$
$431$	5.74597	0.276773	0.138387	+	0.990378i	$0.455808\pi$
$432$	0	0
$433$	− 19.0000i	− 0.913082i	−0.889702	−	0.456541i	$-0.849088\pi$
$433$	− 19.0000i	− 0.913082i	0.889702	−	0.456541i	$-0.150912\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	2.87298i	0.137433i
$438$	0	0
$439$	26.0000	1.24091	0.620456	−	0.784241i	$-0.286947\pi$
$439$	26.0000	1.24091	0.620456	+	0.784241i	$0.286947\pi$
$440$	0	0
$441$	2.00000	0.0952381
$442$	0	0
$443$	24.6190i	1.16968i	0.811148	+	0.584841i	$0.198843\pi$
$443$	24.6190i	1.16968i	−0.811148	+	0.584841i	$0.801157\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 10.8730i	− 0.514274i
$448$	0	0
$449$	21.8730	1.03225	0.516125	−	0.856513i	$-0.327374\pi$
$449$	21.8730	1.03225	0.516125	+	0.856513i	$0.327374\pi$
$450$	0	0
$451$	−28.6190	−1.34761
$452$	0	0
$453$	− 3.74597i	− 0.176001i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 17.0000i	− 0.795226i	−0.917553	−	0.397613i	$-0.869839\pi$
$457$	− 17.0000i	− 0.795226i	0.917553	−	0.397613i	$-0.130161\pi$
$458$	0	0
$459$	−3.87298	−0.180775
$460$	0	0
$461$	31.7460	1.47856	0.739279	−	0.673400i	$-0.235167\pi$
$461$	31.7460	1.47856	0.739279	+	0.673400i	$0.235167\pi$
$462$	0	0
$463$	− 34.8730i	− 1.62068i	−0.585957	−	0.810342i	$-0.699281\pi$
$463$	− 34.8730i	− 1.62068i	0.585957	−	0.810342i	$-0.300719\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 5.61895i	− 0.260014i	−0.991513	−	0.130007i	$-0.958500\pi$
$467$	− 5.61895i	− 0.260014i	0.991513	−	0.130007i	$-0.0415000\pi$
$468$	0	0
$469$	−32.2379	−1.48861
$470$	0	0
$471$	17.0000	0.783319
$472$	0	0
$473$	− 18.2540i	− 0.839321i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 3.87298i	− 0.177332i
$478$	0	0
$479$	34.8730	1.59339	0.796694	−	0.604383i	$-0.206580\pi$
$479$	34.8730	1.59339	0.796694	+	0.604383i	$0.206580\pi$
$480$	0	0
$481$	−2.87298	−0.130997
$482$	0	0
$483$	3.00000i	0.136505i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 26.1109i	− 1.18320i	−0.806233	−	0.591599i	$-0.798497\pi$
$487$	− 26.1109i	− 1.18320i	0.806233	−	0.591599i	$-0.201503\pi$
$488$	0	0
$489$	22.0000	0.994874
$490$	0	0
$491$	28.1270	1.26935	0.634677	−	0.772777i	$-0.281133\pi$
$491$	28.1270	1.26935	0.634677	+	0.772777i	$0.281133\pi$
$492$	0	0
$493$	22.7460i	1.02443i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 40.8569i	− 1.83268i
$498$	0	0
$499$	−3.25403	−0.145671	−0.0728353	−	0.997344i	$-0.523205\pi$
$499$	−3.25403	−0.145671	−0.0728353	+	0.997344i	$0.523205\pi$
$500$	0	0
$501$	−9.12702	−0.407765
$502$	0	0
$503$	37.1109i	1.65469i	0.561692	+	0.827346i	$0.310151\pi$
$503$	37.1109i	1.65469i	−0.561692	+	0.827346i	$0.689849\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 4.74597i	− 0.210776i
$508$	0	0
$509$	−20.0000	−0.886484	−0.443242	−	0.896402i	$-0.646172\pi$
$509$	−20.0000	−0.886484	−0.443242	+	0.896402i	$0.646172\pi$
$510$	0	0
$511$	−8.61895	−0.381280
$512$	0	0
$513$	2.87298i	0.126845i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 33.4919i	− 1.47297i
$518$	0	0
$519$	−23.4919	−1.03118
$520$	0	0
$521$	40.8730	1.79068	0.895339	−	0.445385i	$-0.146933\pi$
$521$	40.8730	1.79068	0.895339	+	0.445385i	$0.146933\pi$
$522$	0	0
$523$	18.9839i	0.830107i	0.909797	+	0.415053i	$0.136237\pi$
$523$	18.9839i	0.830107i	−0.909797	+	0.415053i	$0.863763\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	11.6190i	0.506129i
$528$	0	0
$529$	−1.00000	−0.0434783
$530$	0	0
$531$	5.87298	0.254866
$532$	0	0
$533$	− 16.8730i	− 0.730850i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 15.7460i	− 0.679489i
$538$	0	0
$539$	−9.74597	−0.419789
$540$	0	0
$541$	−40.0000	−1.71973	−0.859867	−	0.510518i	$-0.829454\pi$
$541$	−40.0000	−1.71973	−0.859867	+	0.510518i	$0.829454\pi$
$542$	0	0
$543$	21.7460i	0.933209i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	16.0000i	0.684111i	0.939680	+	0.342055i	$0.111123\pi$
$547$	16.0000i	0.684111i	−0.939680	+	0.342055i	$0.888877\pi$
$548$	0	0
$549$	−8.87298	−0.378690
$550$	0	0
$551$	16.8730	0.718813
$552$	0	0
$553$	− 12.0000i	− 0.510292i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	28.1270i	1.19178i	0.803066	+	0.595890i	$0.203201\pi$
$557$	28.1270i	1.19178i	−0.803066	+	0.595890i	$0.796799\pi$
$558$	0	0
$559$	10.7621	0.455188
$560$	0	0
$561$	18.8730	0.796818
$562$	0	0
$563$	− 33.1109i	− 1.39546i	−0.716362	−	0.697729i	$-0.754194\pi$
$563$	− 33.1109i	− 1.39546i	0.716362	−	0.697729i	$-0.245806\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	3.00000i	0.125988i
$568$	0	0
$569$	−4.25403	−0.178338	−0.0891692	−	0.996016i	$-0.528421\pi$
$569$	−4.25403	−0.178338	−0.0891692	+	0.996016i	$0.528421\pi$
$570$	0	0
$571$	36.3649	1.52182	0.760912	−	0.648855i	$-0.224752\pi$
$571$	36.3649	1.52182	0.760912	+	0.648855i	$0.224752\pi$
$572$	0	0
$573$	18.8730i	0.788430i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	8.00000i	0.333044i	0.986038	+	0.166522i	$0.0532537\pi$
$577$	8.00000i	0.333044i	−0.986038	+	0.166522i	$0.946746\pi$
$578$	0	0
$579$	−9.74597	−0.405029
$580$	0	0
$581$	−0.381050	−0.0158086
$582$	0	0
$583$	18.8730i	0.781639i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	13.4919i	0.556872i	0.960455	+	0.278436i	$0.0898160\pi$
$587$	13.4919i	0.556872i	−0.960455	+	0.278436i	$0.910184\pi$
$588$	0	0
$589$	8.61895	0.355138
$590$	0	0
$591$	4.25403	0.174988
$592$	0	0
$593$	− 44.6190i	− 1.83228i	−0.400858	−	0.916140i	$-0.631288\pi$
$593$	− 44.6190i	− 1.83228i	0.400858	−	0.916140i	$-0.368712\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	19.2379i	0.787355i
$598$	0	0
$599$	−21.7460	−0.888516	−0.444258	−	0.895899i	$-0.646533\pi$
$599$	−21.7460	−0.888516	−0.444258	+	0.895899i	$0.646533\pi$
$600$	0	0
$601$	−12.7460	−0.519919	−0.259959	−	0.965620i	$-0.583709\pi$
$601$	−12.7460	−0.519919	−0.259959	+	0.965620i	$0.583709\pi$
$602$	0	0
$603$	− 10.7460i	− 0.437610i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	31.1270i	1.26341i	0.775210	+	0.631703i	$0.217644\pi$
$607$	31.1270i	1.26341i	−0.775210	+	0.631703i	$0.782356\pi$
$608$	0	0
$609$	17.6190	0.713956
$610$	0	0
$611$	19.7460	0.798836
$612$	0	0
$613$	− 48.9839i	− 1.97844i	−0.146438	−	0.989220i	$-0.546781\pi$
$613$	− 48.9839i	− 1.97844i	0.146438	−	0.989220i	$-0.453219\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 17.6190i	− 0.709312i	−0.934997	−	0.354656i	$-0.884598\pi$
$617$	− 17.6190i	− 0.709312i	0.934997	−	0.354656i	$-0.115402\pi$
$618$	0	0
$619$	16.0000	0.643094	0.321547	−	0.946894i	$-0.395797\pi$
$619$	16.0000	0.643094	0.321547	+	0.946894i	$0.395797\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	− 5.23790i	− 0.209852i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 14.0000i	− 0.559106i
$628$	0	0
$629$	−3.87298	−0.154426
$630$	0	0
$631$	32.3649	1.28843	0.644213	−	0.764846i	$-0.277185\pi$
$631$	32.3649	1.28843	0.644213	+	0.764846i	$0.277185\pi$
$632$	0	0
$633$	− 1.00000i	− 0.0397464i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 5.74597i	− 0.227663i
$638$	0	0
$639$	13.6190	0.538757
$640$	0	0
$641$	−10.1109	−0.399356	−0.199678	−	0.979862i	$-0.563990\pi$
$641$	−10.1109	−0.399356	−0.199678	+	0.979862i	$0.563990\pi$
$642$	0	0
$643$	22.4919i	0.886995i	0.896276	+	0.443498i	$0.146263\pi$
$643$	22.4919i	0.886995i	−0.896276	+	0.443498i	$0.853737\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	27.8569i	1.09517i	0.836751	+	0.547583i	$0.184452\pi$
$647$	27.8569i	1.09517i	−0.836751	+	0.547583i	$0.815548\pi$
$648$	0	0
$649$	−28.6190	−1.12339
$650$	0	0
$651$	9.00000	0.352738
$652$	0	0
$653$	10.1109i	0.395669i	0.980235	+	0.197835i	$0.0633909\pi$
$653$	10.1109i	0.395669i	−0.980235	+	0.197835i	$0.936609\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 2.87298i	− 0.112086i
$658$	0	0
$659$	−28.8730	−1.12473	−0.562366	−	0.826889i	$-0.690109\pi$
$659$	−28.8730	−1.12473	−0.562366	+	0.826889i	$0.690109\pi$
$660$	0	0
$661$	−43.2379	−1.68176	−0.840880	−	0.541222i	$-0.817962\pi$
$661$	−43.2379	−1.68176	−0.840880	+	0.541222i	$0.817962\pi$
$662$	0	0
$663$	11.1270i	0.432138i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	5.87298i	0.227403i
$668$	0	0
$669$	2.00000	0.0773245
$670$	0	0
$671$	43.2379	1.66918
$672$	0	0
$673$	− 19.1270i	− 0.737292i	−0.929570	−	0.368646i	$-0.879821\pi$
$673$	− 19.1270i	− 0.737292i	0.929570	−	0.368646i	$-0.120179\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 7.11088i	− 0.273293i	−0.990620	−	0.136647i	$-0.956367\pi$
$677$	− 7.11088i	− 0.273293i	0.990620	−	0.136647i	$-0.0436325\pi$
$678$	0	0
$679$	−24.0000	−0.921035
$680$	0	0
$681$	7.49193	0.287092
$682$	0	0
$683$	− 17.1270i	− 0.655347i	−0.944791	−	0.327674i	$-0.893735\pi$
$683$	− 17.1270i	− 0.655347i	0.944791	−	0.327674i	$-0.106265\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 5.74597i	− 0.219222i
$688$	0	0
$689$	−11.1270	−0.423906
$690$	0	0
$691$	46.7298	1.77769	0.888843	−	0.458211i	$-0.151510\pi$
$691$	46.7298	1.77769	0.888843	+	0.458211i	$0.151510\pi$
$692$	0	0
$693$	− 14.6190i	− 0.555328i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 22.7460i	− 0.861565i
$698$	0	0
$699$	−15.4919	−0.585959
$700$	0	0
$701$	29.8569	1.12768	0.563839	−	0.825885i	$-0.309324\pi$
$701$	29.8569	1.12768	0.563839	+	0.825885i	$0.309324\pi$
$702$	0	0
$703$	2.87298i	0.108357i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 11.6190i	− 0.436976i
$708$	0	0
$709$	6.61895	0.248580	0.124290	−	0.992246i	$-0.460335\pi$
$709$	6.61895	0.248580	0.124290	+	0.992246i	$0.460335\pi$
$710$	0	0
$711$	4.00000	0.150012
$712$	0	0
$713$	3.00000i	0.112351i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 17.8730i	− 0.667479i
$718$	0	0
$719$	39.3649	1.46806	0.734032	−	0.679115i	$-0.237636\pi$
$719$	39.3649	1.46806	0.734032	+	0.679115i	$0.237636\pi$
$720$	0	0
$721$	−18.0000	−0.670355
$722$	0	0
$723$	5.12702i	0.190676i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	11.2540i	0.417389i	0.977981	+	0.208694i	$0.0669214\pi$
$727$	11.2540i	0.417389i	−0.977981	+	0.208694i	$0.933079\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	14.5081	0.536600
$732$	0	0
$733$	− 22.7460i	− 0.840141i	−0.907491	−	0.420071i	$-0.862005\pi$
$733$	− 22.7460i	− 0.840141i	0.907491	−	0.420071i	$-0.137995\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	52.3649i	1.92889i
$738$	0	0
$739$	−13.2540	−0.487557	−0.243779	−	0.969831i	$-0.578387\pi$
$739$	−13.2540	−0.487557	−0.243779	+	0.969831i	$0.578387\pi$
$740$	0	0
$741$	8.25403	0.303219
$742$	0	0
$743$	36.0000i	1.32071i	0.750953	+	0.660356i	$0.229595\pi$
$743$	36.0000i	1.32071i	−0.750953	+	0.660356i	$0.770405\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 0.127017i	− 0.00464730i
$748$	0	0
$749$	16.8569	0.615936
$750$	0	0
$751$	−29.8569	−1.08949	−0.544746	−	0.838601i	$-0.683374\pi$
$751$	−29.8569	−1.08949	−0.544746	+	0.838601i	$0.683374\pi$
$752$	0	0
$753$	− 2.00000i	− 0.0728841i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 4.74597i	− 0.172495i	−0.996274	−	0.0862475i	$-0.972512\pi$
$757$	− 4.74597i	− 0.172495i	0.996274	−	0.0862475i	$-0.0274876\pi$
$758$	0	0
$759$	4.87298	0.176878
$760$	0	0
$761$	−14.8569	−0.538560	−0.269280	−	0.963062i	$-0.586786\pi$
$761$	−14.8569	−0.538560	−0.269280	+	0.963062i	$0.586786\pi$
$762$	0	0
$763$	− 49.8569i	− 1.80494i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 16.8730i	− 0.609248i
$768$	0	0
$769$	−28.8730	−1.04119	−0.520593	−	0.853805i	$-0.674289\pi$
$769$	−28.8730	−1.04119	−0.520593	+	0.853805i	$0.674289\pi$
$770$	0	0
$771$	−16.6190	−0.598517
$772$	0	0
$773$	− 4.25403i	− 0.153007i	−0.997069	−	0.0765035i	$-0.975624\pi$
$773$	− 4.25403i	− 0.153007i	0.997069	−	0.0765035i	$-0.0243756\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	3.00000i	0.107624i
$778$	0	0
$779$	−16.8730	−0.604537
$780$	0	0
$781$	−66.3649	−2.37472
$782$	0	0
$783$	5.87298i	0.209883i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	25.0000i	0.891154i	0.895244	+	0.445577i	$0.147001\pi$
$787$	25.0000i	0.891154i	−0.895244	+	0.445577i	$0.852999\pi$
$788$	0	0
$789$	21.8730	0.778699
$790$	0	0
$791$	−11.6190	−0.413122
$792$	0	0
$793$	25.4919i	0.905245i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	11.8730i	0.420563i	0.977641	+	0.210281i	$0.0674380\pi$
$797$	11.8730i	0.420563i	−0.977641	+	0.210281i	$0.932562\pi$
$798$	0	0
$799$	26.6190	0.941711
$800$	0	0
$801$	1.74597	0.0616907
$802$	0	0
$803$	14.0000i	0.494049i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 23.3649i	− 0.822484i
$808$	0	0
$809$	−24.8569	−0.873920	−0.436960	−	0.899481i	$-0.643945\pi$
$809$	−24.8569	−0.873920	−0.436960	+	0.899481i	$0.643945\pi$
$810$	0	0
$811$	28.4919	1.00049	0.500244	−	0.865885i	$-0.333244\pi$
$811$	28.4919	1.00049	0.500244	+	0.865885i	$0.333244\pi$
$812$	0	0
$813$	7.25403i	0.254410i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 10.7621i	− 0.376518i
$818$	0	0
$819$	8.61895	0.301170
$820$	0	0
$821$	−3.49193	−0.121869	−0.0609347	−	0.998142i	$-0.519408\pi$
$821$	−3.49193	−0.121869	−0.0609347	+	0.998142i	$0.519408\pi$
$822$	0	0
$823$	− 44.0000i	− 1.53374i	−0.641800	−	0.766872i	$-0.721812\pi$
$823$	− 44.0000i	− 1.53374i	0.641800	−	0.766872i	$-0.278188\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 15.1109i	− 0.525457i	−0.964870	−	0.262728i	$-0.915378\pi$
$827$	− 15.1109i	− 0.525457i	0.964870	−	0.262728i	$-0.0846223\pi$
$828$	0	0
$829$	−3.50807	−0.121840	−0.0609201	−	0.998143i	$-0.519403\pi$
$829$	−3.50807	−0.121840	−0.0609201	+	0.998143i	$0.519403\pi$
$830$	0	0
$831$	14.0000	0.485655
$832$	0	0
$833$	− 7.74597i	− 0.268382i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	3.00000i	0.103695i
$838$	0	0
$839$	−18.0000	−0.621429	−0.310715	−	0.950503i	$-0.600568\pi$
$839$	−18.0000	−0.621429	−0.310715	+	0.950503i	$0.600568\pi$
$840$	0	0
$841$	5.49193	0.189377
$842$	0	0
$843$	16.3649i	0.563638i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	38.2379i	1.31387i
$848$	0	0
$849$	−22.2379	−0.763203
$850$	0	0
$851$	−1.00000	−0.0342796
$852$	0	0
$853$	43.7460i	1.49783i	0.662664	+	0.748917i	$0.269426\pi$
$853$	43.7460i	1.49783i	−0.662664	+	0.748917i	$0.730574\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 14.5081i	− 0.495586i	−0.968813	−	0.247793i	$-0.920295\pi$
$857$	− 14.5081i	− 0.495586i	0.968813	−	0.247793i	$-0.0797053\pi$
$858$	0	0
$859$	−29.0000	−0.989467	−0.494734	−	0.869045i	$-0.664734\pi$
$859$	−29.0000	−0.989467	−0.494734	+	0.869045i	$0.664734\pi$
$860$	0	0
$861$	−17.6190	−0.600452
$862$	0	0
$863$	− 8.25403i	− 0.280971i	−0.990083	−	0.140485i	$-0.955134\pi$
$863$	− 8.25403i	− 0.280971i	0.990083	−	0.140485i	$-0.0448663\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 2.00000i	− 0.0679236i
$868$	0	0
$869$	−19.4919	−0.661219
$870$	0	0
$871$	−30.8730	−1.04609
$872$	0	0
$873$	− 8.00000i	− 0.270759i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	54.7298i	1.84810i	0.382277	+	0.924048i	$0.375140\pi$
$877$	54.7298i	1.84810i	−0.382277	+	0.924048i	$0.624860\pi$
$878$	0	0
$879$	13.6190	0.459356
$880$	0	0
$881$	4.11088	0.138499	0.0692496	−	0.997599i	$-0.477940\pi$
$881$	4.11088	0.138499	0.0692496	+	0.997599i	$0.477940\pi$
$882$	0	0
$883$	− 55.3488i	− 1.86263i	−0.364209	−	0.931317i	$-0.618660\pi$
$883$	− 55.3488i	− 1.86263i	0.364209	−	0.931317i	$-0.381340\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	19.7460i	0.663005i	0.943454	+	0.331502i	$0.107555\pi$
$887$	19.7460i	0.663005i	−0.943454	+	0.331502i	$0.892445\pi$
$888$	0	0
$889$	19.8569	0.665977
$890$	0	0
$891$	4.87298	0.163251
$892$	0	0
$893$	− 19.7460i	− 0.660774i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	2.87298i	0.0959261i
$898$	0	0
$899$	17.6190	0.587625
$900$	0	0
$901$	−15.0000	−0.499722
$902$	0	0
$903$	− 11.2379i	− 0.373974i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	30.7460i	1.02090i	0.859907	+	0.510452i	$0.170522\pi$
$907$	30.7460i	1.02090i	−0.859907	+	0.510452i	$0.829478\pi$
$908$	0	0
$909$	3.87298	0.128459
$910$	0	0
$911$	−16.0000	−0.530104	−0.265052	−	0.964234i	$-0.585389\pi$
$911$	−16.0000	−0.530104	−0.265052	+	0.964234i	$0.585389\pi$
$912$	0	0
$913$	0.618950i	0.0204843i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	34.4758i	1.13849i
$918$	0	0
$919$	44.0000	1.45143	0.725713	−	0.687998i	$-0.241510\pi$
$919$	44.0000	1.45143	0.725713	+	0.687998i	$0.241510\pi$
$920$	0	0
$921$	9.12702	0.300745
$922$	0	0
$923$	− 39.1270i	− 1.28788i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 6.00000i	− 0.197066i
$928$	0	0
$929$	−31.3649	−1.02905	−0.514525	−	0.857476i	$-0.672032\pi$
$929$	−31.3649	−1.02905	−0.514525	+	0.857476i	$0.672032\pi$
$930$	0	0
$931$	−5.74597	−0.188316
$932$	0	0
$933$	4.00000i	0.130954i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	54.4758i	1.77965i	0.456305	+	0.889823i	$0.349173\pi$
$937$	54.4758i	1.77965i	−0.456305	+	0.889823i	$0.650827\pi$
$938$	0	0
$939$	−7.00000	−0.228436
$940$	0	0
$941$	−16.1109	−0.525200	−0.262600	−	0.964905i	$-0.584580\pi$
$941$	−16.1109	−0.525200	−0.262600	+	0.964905i	$0.584580\pi$
$942$	0	0
$943$	− 5.87298i	− 0.191251i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 25.2379i	− 0.820122i	−0.912058	−	0.410061i	$-0.865507\pi$
$947$	− 25.2379i	− 0.820122i	0.912058	−	0.410061i	$-0.134493\pi$
$948$	0	0
$949$	−8.25403	−0.267937
$950$	0	0
$951$	22.3649	0.725232
$952$	0	0
$953$	− 8.98387i	− 0.291016i	−0.989357	−	0.145508i	$-0.953518\pi$
$953$	− 8.98387i	− 0.291016i	0.989357	−	0.145508i	$-0.0464817\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 28.6190i	− 0.925119i
$958$	0	0
$959$	−24.0000	−0.775000
$960$	0	0
$961$	−22.0000	−0.709677
$962$	0	0
$963$	5.61895i	0.181068i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	21.3810i	0.687568i	0.939049	+	0.343784i	$0.111709\pi$
$967$	21.3810i	0.687568i	−0.939049	+	0.343784i	$0.888291\pi$
$968$	0	0
$969$	11.1270	0.357451
$970$	0	0
$971$	53.7460	1.72479	0.862395	−	0.506236i	$-0.168963\pi$
$971$	53.7460	1.72479	0.862395	+	0.506236i	$0.168963\pi$
$972$	0	0
$973$	61.4758i	1.97082i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 51.8730i	− 1.65956i	−0.558088	−	0.829782i	$-0.688465\pi$
$977$	− 51.8730i	− 1.65956i	0.558088	−	0.829782i	$-0.311535\pi$
$978$	0	0
$979$	−8.50807	−0.271919
$980$	0	0
$981$	16.6190	0.530602
$982$	0	0
$983$	− 7.36492i	− 0.234904i	−0.993079	−	0.117452i	$-0.962527\pi$
$983$	− 7.36492i	− 0.234904i	0.993079	−	0.117452i	$-0.0374727\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 20.6190i	− 0.656308i
$988$	0	0
$989$	3.74597	0.119115
$990$	0	0
$991$	−11.5081	−0.365566	−0.182783	−	0.983153i	$-0.558511\pi$
$991$	−11.5081	−0.365566	−0.182783	+	0.983153i	$0.558511\pi$
$992$	0	0
$993$	22.2379i	0.705698i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 41.7460i	− 1.32211i	−0.750338	−	0.661054i	$-0.770109\pi$
$997$	− 41.7460i	− 1.32211i	0.750338	−	0.661054i	$-0.229891\pi$
$998$	0	0
$999$	−1.00000	−0.0316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6900.2.f.o.6349.2		4
5.2	odd	4		6900.2.a.j.1.2		2
5.3	odd	4		1380.2.a.i.1.2	✓	2
5.4	even	2	inner	6900.2.f.o.6349.4		4
15.8	even	4		4140.2.a.p.1.1		2
20.3	even	4		5520.2.a.bj.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1380.2.a.i.1.2	✓	2	5.3	odd	4
4140.2.a.p.1.1		2	15.8	even	4
5520.2.a.bj.1.1		2	20.3	even	4
6900.2.a.j.1.2		2	5.2	odd	4
6900.2.f.o.6349.2		4	1.1	even	1	trivial
6900.2.f.o.6349.4		4	5.4	even	2	inner

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)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +3.00000i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} +3.00000i q^{7} -1.00000 q^{9} +4.87298 q^{11} +2.87298i q^{13} +3.87298i q^{17} +2.87298 q^{19} +3.00000 q^{21} +1.00000i q^{23} +1.00000i q^{27} +5.87298 q^{29} +3.00000 q^{31} -4.87298i q^{33} +1.00000i q^{37} +2.87298 q^{39} -5.87298 q^{41} -3.74597i q^{43} -6.87298i q^{47} -2.00000 q^{49} +3.87298 q^{51} +3.87298i q^{53} -2.87298i q^{57} -5.87298 q^{59} +8.87298 q^{61} -3.00000i q^{63} +10.7460i q^{67} +1.00000 q^{69} -13.6190 q^{71} +2.87298i q^{73} +14.6190i q^{77} -4.00000 q^{79} +1.00000 q^{81} +0.127017i q^{83} -5.87298i q^{87} -1.74597 q^{89} -8.61895 q^{91} -3.00000i q^{93} +8.00000i q^{97} -4.87298 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^9	\( 4 q - 4 q^{9} + 4 q^{11} - 4 q^{19} + 12 q^{21} + 8 q^{29} + 12 q^{31} - 4 q^{39} - 8 q^{41} - 8 q^{49} - 8 q^{59} + 20 q^{61} + 4 q^{69} - 8 q^{71} - 16 q^{79} + 4 q^{81} + 24 q^{89} + 12 q^{91} - 4 q^{99}+O(q^{100}) \) 4 * q - 4 * q^9 + 4 * q^11 - 4 * q^19 + 12 * q^21 + 8 * q^29 + 12 * q^31 - 4 * q^39 - 8 * q^41 - 8 * q^49 - 8 * q^59 + 20 * q^61 + 4 * q^69 - 8 * q^71 - 16 * q^79 + 4 * q^81 + 24 * q^89 + 12 * q^91 - 4 * q^99

Introduction

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