LMFDB - Embedded newform 2925.2.c.s.2224.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2925,2,Mod(2224,2925)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2925, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2925.2224");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2925 = 3^{2} \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2925.c (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:	$23.3562425912$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 117)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2224.4
Root			$0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	2925.2224
Dual form			2925.2.c.s.2224.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205i q^{2} -1.00000 q^{4} +2.00000i q^{7} +1.73205i q^{8} +O(q^{10})$	$q+1.73205i q^{2} -1.00000 q^{4} +2.00000i q^{7} +1.73205i q^{8} -3.46410 q^{11} -1.00000i q^{13} -3.46410 q^{14} -5.00000 q^{16} -6.92820i q^{17} -2.00000 q^{19} -6.00000i q^{22} -6.92820i q^{23} +1.73205 q^{26} -2.00000i q^{28} -6.92820 q^{29} +2.00000 q^{31} -5.19615i q^{32} +12.0000 q^{34} +2.00000i q^{37} -3.46410i q^{38} +6.92820 q^{41} -8.00000i q^{43} +3.46410 q^{44} +12.0000 q^{46} -10.3923i q^{47} +3.00000 q^{49} +1.00000i q^{52} -3.46410 q^{56} -12.0000i q^{58} -3.46410 q^{59} -10.0000 q^{61} +3.46410i q^{62} -1.00000 q^{64} +14.0000i q^{67} +6.92820i q^{68} +3.46410 q^{71} +10.0000i q^{73} -3.46410 q^{74} +2.00000 q^{76} -6.92820i q^{77} +4.00000 q^{79} +12.0000i q^{82} -10.3923i q^{83} +13.8564 q^{86} -6.00000i q^{88} -6.92820 q^{89} +2.00000 q^{91} +6.92820i q^{92} +18.0000 q^{94} -10.0000i q^{97} +5.19615i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4}+O(q^{10}) $ 4 * q - 4 * q^4	$ 4 q - 4 q^{4} - 20 q^{16} - 8 q^{19} + 8 q^{31} + 48 q^{34} + 48 q^{46} + 12 q^{49} - 40 q^{61} - 4 q^{64} + 8 q^{76} + 16 q^{79} + 8 q^{91} + 72 q^{94}+O(q^{100}) $ 4 * q - 4 * q^4 - 20 * q^16 - 8 * q^19 + 8 * q^31 + 48 * q^34 + 48 * q^46 + 12 * q^49 - 40 * q^61 - 4 * q^64 + 8 * q^76 + 16 * q^79 + 8 * q^91 + 72 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2925\mathbb{Z}\right)^\times$.

$n$	$326$	$352$	$2251$
$\chi(n)$	$1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.73205i	1.22474i	0.790569	+	0.612372i	$0.209785\pi$
$2$	1.73205i	1.22474i	−0.790569	+	0.612372i	$0.790215\pi$
$3$	0	0
$3$	0	0
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.00000i	0.755929i	0.925820	+	0.377964i	$0.123376\pi$
$7$	2.00000i	0.755929i	−0.925820	+	0.377964i	$0.876624\pi$
$8$	1.73205i	0.612372i
$9$	0	0
$10$	0	0
$11$	−3.46410	−1.04447	−0.522233	−	0.852803i	$-0.674901\pi$
$11$	−3.46410	−1.04447	−0.522233	+	0.852803i	$0.674901\pi$
$12$	0	0
$13$	− 1.00000i	− 0.277350i
$13$	− 1.00000i	− 0.277350i
$14$	−3.46410	−0.925820
$15$	0	0
$16$	−5.00000	−1.25000
$17$	− 6.92820i	− 1.68034i	−0.542326	−	0.840168i	$-0.682456\pi$
$17$	− 6.92820i	− 1.68034i	0.542326	−	0.840168i	$-0.317544\pi$
$18$	0	0
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	0	0
$21$	0	0
$22$	− 6.00000i	− 1.27920i
$23$	− 6.92820i	− 1.44463i	−0.691564	−	0.722315i	$-0.743078\pi$
$23$	− 6.92820i	− 1.44463i	0.691564	−	0.722315i	$-0.256922\pi$
$24$	0	0
$25$	0	0
$26$	1.73205	0.339683
$27$	0	0
$28$	− 2.00000i	− 0.377964i
$29$	−6.92820	−1.28654	−0.643268	−	0.765641i	$-0.722422\pi$
$29$	−6.92820	−1.28654	−0.643268	+	0.765641i	$0.722422\pi$
$30$	0	0
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	− 5.19615i	− 0.918559i
$33$	0	0
$34$	12.0000	2.05798
$35$	0	0
$36$	0	0
$37$	2.00000i	0.328798i	0.986394	+	0.164399i	$0.0525685\pi$
$37$	2.00000i	0.328798i	−0.986394	+	0.164399i	$0.947432\pi$
$38$	− 3.46410i	− 0.561951i
$39$	0	0
$40$	0	0
$41$	6.92820	1.08200	0.541002	−	0.841021i	$-0.318045\pi$
$41$	6.92820	1.08200	0.541002	+	0.841021i	$0.318045\pi$
$42$	0	0
$43$	− 8.00000i	− 1.21999i	−0.792406	−	0.609994i	$-0.791172\pi$
$43$	− 8.00000i	− 1.21999i	0.792406	−	0.609994i	$-0.208828\pi$
$44$	3.46410	0.522233
$45$	0	0
$46$	12.0000	1.76930
$47$	− 10.3923i	− 1.51587i	−0.652328	−	0.757937i	$-0.726208\pi$
$47$	− 10.3923i	− 1.51587i	0.652328	−	0.757937i	$-0.273792\pi$
$48$	0	0
$49$	3.00000	0.428571
$50$	0	0
$51$	0	0
$52$	1.00000i	0.138675i
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	0	0
$55$	0	0
$56$	−3.46410	−0.462910
$57$	0	0
$58$	− 12.0000i	− 1.57568i
$59$	−3.46410	−0.450988	−0.225494	−	0.974245i	$-0.572400\pi$
$59$	−3.46410	−0.450988	−0.225494	+	0.974245i	$0.572400\pi$
$60$	0	0
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	3.46410i	0.439941i
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	14.0000i	1.71037i	0.518321	+	0.855186i	$0.326557\pi$
$67$	14.0000i	1.71037i	−0.518321	+	0.855186i	$0.673443\pi$
$68$	6.92820i	0.840168i
$69$	0	0
$70$	0	0
$71$	3.46410	0.411113	0.205557	−	0.978645i	$-0.434100\pi$
$71$	3.46410	0.411113	0.205557	+	0.978645i	$0.434100\pi$
$72$	0	0
$73$	10.0000i	1.17041i	0.810885	+	0.585206i	$0.198986\pi$
$73$	10.0000i	1.17041i	−0.810885	+	0.585206i	$0.801014\pi$
$74$	−3.46410	−0.402694
$75$	0	0
$76$	2.00000	0.229416
$77$	− 6.92820i	− 0.789542i
$78$	0	0
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	0	0
$81$	0	0
$82$	12.0000i	1.32518i
$83$	− 10.3923i	− 1.14070i	−0.821401	−	0.570352i	$-0.806807\pi$
$83$	− 10.3923i	− 1.14070i	0.821401	−	0.570352i	$-0.193193\pi$
$84$	0	0
$85$	0	0
$86$	13.8564	1.49417
$87$	0	0
$88$	− 6.00000i	− 0.639602i
$89$	−6.92820	−0.734388	−0.367194	−	0.930144i	$-0.619682\pi$
$89$	−6.92820	−0.734388	−0.367194	+	0.930144i	$0.619682\pi$
$90$	0	0
$91$	2.00000	0.209657
$92$	6.92820i	0.722315i
$93$	0	0
$94$	18.0000	1.85656
$95$	0	0
$96$	0	0
$97$	− 10.0000i	− 1.01535i	−0.861550	−	0.507673i	$-0.830506\pi$
$97$	− 10.0000i	− 1.01535i	0.861550	−	0.507673i	$-0.169494\pi$
$98$	5.19615i	0.524891i
$99$	0	0
$100$	0	0
$101$	−13.8564	−1.37876	−0.689382	−	0.724398i	$-0.742118\pi$
$101$	−13.8564	−1.37876	−0.689382	+	0.724398i	$0.742118\pi$
$102$	0	0
$103$	4.00000i	0.394132i	0.980390	+	0.197066i	$0.0631413\pi$
$103$	4.00000i	0.394132i	−0.980390	+	0.197066i	$0.936859\pi$
$104$	1.73205	0.169842
$105$	0	0
$106$	0	0
$107$	6.92820i	0.669775i	0.942258	+	0.334887i	$0.108698\pi$
$107$	6.92820i	0.669775i	−0.942258	+	0.334887i	$0.891302\pi$
$108$	0	0
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	0	0
$111$	0	0
$112$	− 10.0000i	− 0.944911i
$113$	6.92820i	0.651751i	0.945413	+	0.325875i	$0.105659\pi$
$113$	6.92820i	0.651751i	−0.945413	+	0.325875i	$0.894341\pi$
$114$	0	0
$115$	0	0
$116$	6.92820	0.643268
$117$	0	0
$118$	− 6.00000i	− 0.552345i
$119$	13.8564	1.27021
$120$	0	0
$121$	1.00000	0.0909091
$122$	− 17.3205i	− 1.56813i
$123$	0	0
$124$	−2.00000	−0.179605
$125$	0	0
$126$	0	0
$127$	− 16.0000i	− 1.41977i	−0.704317	−	0.709885i	$-0.748747\pi$
$127$	− 16.0000i	− 1.41977i	0.704317	−	0.709885i	$-0.251253\pi$
$128$	− 12.1244i	− 1.07165i
$129$	0	0
$130$	0	0
$131$	−20.7846	−1.81596	−0.907980	−	0.419014i	$-0.862376\pi$
$131$	−20.7846	−1.81596	−0.907980	+	0.419014i	$0.862376\pi$
$132$	0	0
$133$	− 4.00000i	− 0.346844i
$134$	−24.2487	−2.09477
$135$	0	0
$136$	12.0000	1.02899
$137$	− 6.92820i	− 0.591916i	−0.955201	−	0.295958i	$-0.904361\pi$
$137$	− 6.92820i	− 0.591916i	0.955201	−	0.295958i	$-0.0956389\pi$
$138$	0	0
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	0	0
$141$	0	0
$142$	6.00000i	0.503509i
$143$	3.46410i	0.289683i
$144$	0	0
$145$	0	0
$146$	−17.3205	−1.43346
$147$	0	0
$148$	− 2.00000i	− 0.164399i
$149$	−13.8564	−1.13516	−0.567581	−	0.823318i	$-0.692120\pi$
$149$	−13.8564	−1.13516	−0.567581	+	0.823318i	$0.692120\pi$
$150$	0	0
$151$	14.0000	1.13930	0.569652	−	0.821886i	$-0.307078\pi$
$151$	14.0000	1.13930	0.569652	+	0.821886i	$0.307078\pi$
$152$	− 3.46410i	− 0.280976i
$153$	0	0
$154$	12.0000	0.966988
$155$	0	0
$156$	0	0
$157$	2.00000i	0.159617i	0.996810	+	0.0798087i	$0.0254309\pi$
$157$	2.00000i	0.159617i	−0.996810	+	0.0798087i	$0.974569\pi$
$158$	6.92820i	0.551178i
$159$	0	0
$160$	0	0
$161$	13.8564	1.09204
$162$	0	0
$163$	− 14.0000i	− 1.09656i	−0.836293	−	0.548282i	$-0.815282\pi$
$163$	− 14.0000i	− 1.09656i	0.836293	−	0.548282i	$-0.184718\pi$
$164$	−6.92820	−0.541002
$165$	0	0
$166$	18.0000	1.39707
$167$	17.3205i	1.34030i	0.742225	+	0.670151i	$0.233770\pi$
$167$	17.3205i	1.34030i	−0.742225	+	0.670151i	$0.766230\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	0	0
$172$	8.00000i	0.609994i
$173$	− 13.8564i	− 1.05348i	−0.850026	−	0.526742i	$-0.823414\pi$
$173$	− 13.8564i	− 1.05348i	0.850026	−	0.526742i	$-0.176586\pi$
$174$	0	0
$175$	0	0
$176$	17.3205	1.30558
$177$	0	0
$178$	− 12.0000i	− 0.899438i
$179$	13.8564	1.03568	0.517838	−	0.855479i	$-0.326737\pi$
$179$	13.8564	1.03568	0.517838	+	0.855479i	$0.326737\pi$
$180$	0	0
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	3.46410i	0.256776i
$183$	0	0
$184$	12.0000	0.884652
$185$	0	0
$186$	0	0
$187$	24.0000i	1.75505i
$188$	10.3923i	0.757937i
$189$	0	0
$190$	0	0
$191$	13.8564	1.00261	0.501307	−	0.865269i	$-0.332853\pi$
$191$	13.8564	1.00261	0.501307	+	0.865269i	$0.332853\pi$
$192$	0	0
$193$	− 26.0000i	− 1.87152i	−0.352636	−	0.935760i	$-0.614715\pi$
$193$	− 26.0000i	− 1.87152i	0.352636	−	0.935760i	$-0.385285\pi$
$194$	17.3205	1.24354
$195$	0	0
$196$	−3.00000	−0.214286
$197$	− 13.8564i	− 0.987228i	−0.869681	−	0.493614i	$-0.835676\pi$
$197$	− 13.8564i	− 0.987228i	0.869681	−	0.493614i	$-0.164324\pi$
$198$	0	0
$199$	16.0000	1.13421	0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000	1.13421	0.567105	+	0.823646i	$0.308063\pi$
$200$	0	0
$201$	0	0
$202$	− 24.0000i	− 1.68863i
$203$	− 13.8564i	− 0.972529i
$204$	0	0
$205$	0	0
$206$	−6.92820	−0.482711
$207$	0	0
$208$	5.00000i	0.346688i
$209$	6.92820	0.479234
$210$	0	0
$211$	8.00000	0.550743	0.275371	−	0.961338i	$-0.411199\pi$
$211$	8.00000	0.550743	0.275371	+	0.961338i	$0.411199\pi$
$212$	0	0
$213$	0	0
$214$	−12.0000	−0.820303
$215$	0	0
$216$	0	0
$217$	4.00000i	0.271538i
$218$	17.3205i	1.17309i
$219$	0	0
$220$	0	0
$221$	−6.92820	−0.466041
$222$	0	0
$223$	− 2.00000i	− 0.133930i	−0.997755	−	0.0669650i	$-0.978668\pi$
$223$	− 2.00000i	− 0.133930i	0.997755	−	0.0669650i	$-0.0213316\pi$
$224$	10.3923	0.694365
$225$	0	0
$226$	−12.0000	−0.798228
$227$	3.46410i	0.229920i	0.993370	+	0.114960i	$0.0366741\pi$
$227$	3.46410i	0.229920i	−0.993370	+	0.114960i	$0.963326\pi$
$228$	0	0
$229$	−14.0000	−0.925146	−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000	−0.925146	−0.462573	+	0.886581i	$0.653074\pi$
$230$	0	0
$231$	0	0
$232$	− 12.0000i	− 0.787839i
$233$	0	0	1.00000			$0$
$233$	0	0	−1.00000			$\pi$
$234$	0	0
$235$	0	0
$236$	3.46410	0.225494
$237$	0	0
$238$	24.0000i	1.55569i
$239$	10.3923	0.672222	0.336111	−	0.941822i	$-0.390888\pi$
$239$	10.3923	0.672222	0.336111	+	0.941822i	$0.390888\pi$
$240$	0	0
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	1.73205i	0.111340i
$243$	0	0
$244$	10.0000	0.640184
$245$	0	0
$246$	0	0
$247$	2.00000i	0.127257i
$248$	3.46410i	0.219971i
$249$	0	0
$250$	0	0
$251$	−6.92820	−0.437304	−0.218652	−	0.975803i	$-0.570166\pi$
$251$	−6.92820	−0.437304	−0.218652	+	0.975803i	$0.570166\pi$
$252$	0	0
$253$	24.0000i	1.50887i
$254$	27.7128	1.73886
$255$	0	0
$256$	19.0000	1.18750
$257$	27.7128i	1.72868i	0.502910	+	0.864339i	$0.332263\pi$
$257$	27.7128i	1.72868i	−0.502910	+	0.864339i	$0.667737\pi$
$258$	0	0
$259$	−4.00000	−0.248548
$260$	0	0
$261$	0	0
$262$	− 36.0000i	− 2.22409i
$263$	− 6.92820i	− 0.427211i	−0.976920	−	0.213606i	$-0.931479\pi$
$263$	− 6.92820i	− 0.427211i	0.976920	−	0.213606i	$-0.0685208\pi$
$264$	0	0
$265$	0	0
$266$	6.92820	0.424795
$267$	0	0
$268$	− 14.0000i	− 0.855186i
$269$	6.92820	0.422420	0.211210	−	0.977441i	$-0.432260\pi$
$269$	6.92820	0.422420	0.211210	+	0.977441i	$0.432260\pi$
$270$	0	0
$271$	−22.0000	−1.33640	−0.668202	−	0.743980i	$-0.732936\pi$
$271$	−22.0000	−1.33640	−0.668202	+	0.743980i	$0.732936\pi$
$272$	34.6410i	2.10042i
$273$	0	0
$274$	12.0000	0.724947
$275$	0	0
$276$	0	0
$277$	2.00000i	0.120168i	0.998193	+	0.0600842i	$0.0191369\pi$
$277$	2.00000i	0.120168i	−0.998193	+	0.0600842i	$0.980863\pi$
$278$	6.92820i	0.415526i
$279$	0	0
$280$	0	0
$281$	−20.7846	−1.23991	−0.619953	−	0.784639i	$-0.712848\pi$
$281$	−20.7846	−1.23991	−0.619953	+	0.784639i	$0.712848\pi$
$282$	0	0
$283$	4.00000i	0.237775i	0.992908	+	0.118888i	$0.0379328\pi$
$283$	4.00000i	0.237775i	−0.992908	+	0.118888i	$0.962067\pi$
$284$	−3.46410	−0.205557
$285$	0	0
$286$	−6.00000	−0.354787
$287$	13.8564i	0.817918i
$288$	0	0
$289$	−31.0000	−1.82353
$290$	0	0
$291$	0	0
$292$	− 10.0000i	− 0.585206i
$293$	− 13.8564i	− 0.809500i	−0.914427	−	0.404750i	$-0.867359\pi$
$293$	− 13.8564i	− 0.809500i	0.914427	−	0.404750i	$-0.132641\pi$
$294$	0	0
$295$	0	0
$296$	−3.46410	−0.201347
$297$	0	0
$298$	− 24.0000i	− 1.39028i
$299$	−6.92820	−0.400668
$300$	0	0
$301$	16.0000	0.922225
$302$	24.2487i	1.39536i
$303$	0	0
$304$	10.0000	0.573539
$305$	0	0
$306$	0	0
$307$	26.0000i	1.48390i	0.670456	+	0.741949i	$0.266098\pi$
$307$	26.0000i	1.48390i	−0.670456	+	0.741949i	$0.733902\pi$
$308$	6.92820i	0.394771i
$309$	0	0
$310$	0	0
$311$	−20.7846	−1.17859	−0.589294	−	0.807919i	$-0.700594\pi$
$311$	−20.7846	−1.17859	−0.589294	+	0.807919i	$0.700594\pi$
$312$	0	0
$313$	− 14.0000i	− 0.791327i	−0.918396	−	0.395663i	$-0.870515\pi$
$313$	− 14.0000i	− 0.791327i	0.918396	−	0.395663i	$-0.129485\pi$
$314$	−3.46410	−0.195491
$315$	0	0
$316$	−4.00000	−0.225018
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	0	0
$319$	24.0000	1.34374
$320$	0	0
$321$	0	0
$322$	24.0000i	1.33747i
$323$	13.8564i	0.770991i
$324$	0	0
$325$	0	0
$326$	24.2487	1.34301
$327$	0	0
$328$	12.0000i	0.662589i
$329$	20.7846	1.14589
$330$	0	0
$331$	−10.0000	−0.549650	−0.274825	−	0.961494i	$-0.588620\pi$
$331$	−10.0000	−0.549650	−0.274825	+	0.961494i	$0.588620\pi$
$332$	10.3923i	0.570352i
$333$	0	0
$334$	−30.0000	−1.64153
$335$	0	0
$336$	0	0
$337$	2.00000i	0.108947i	0.998515	+	0.0544735i	$0.0173480\pi$
$337$	2.00000i	0.108947i	−0.998515	+	0.0544735i	$0.982652\pi$
$338$	− 1.73205i	− 0.0942111i
$339$	0	0
$340$	0	0
$341$	−6.92820	−0.375183
$342$	0	0
$343$	20.0000i	1.07990i
$344$	13.8564	0.747087
$345$	0	0
$346$	24.0000	1.29025
$347$	− 27.7128i	− 1.48770i	−0.668346	−	0.743851i	$-0.732997\pi$
$347$	− 27.7128i	− 1.48770i	0.668346	−	0.743851i	$-0.267003\pi$
$348$	0	0
$349$	−26.0000	−1.39175	−0.695874	−	0.718164i	$-0.744983\pi$
$349$	−26.0000	−1.39175	−0.695874	+	0.718164i	$0.744983\pi$
$350$	0	0
$351$	0	0
$352$	18.0000i	0.959403i
$353$	− 6.92820i	− 0.368751i	−0.982856	−	0.184376i	$-0.940974\pi$
$353$	− 6.92820i	− 0.368751i	0.982856	−	0.184376i	$-0.0590263\pi$
$354$	0	0
$355$	0	0
$356$	6.92820	0.367194
$357$	0	0
$358$	24.0000i	1.26844i
$359$	10.3923	0.548485	0.274242	−	0.961661i	$-0.411573\pi$
$359$	10.3923	0.548485	0.274242	+	0.961661i	$0.411573\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	− 17.3205i	− 0.910346i
$363$	0	0
$364$	−2.00000	−0.104828
$365$	0	0
$366$	0	0
$367$	8.00000i	0.417597i	0.977959	+	0.208798i	$0.0669552\pi$
$367$	8.00000i	0.417597i	−0.977959	+	0.208798i	$0.933045\pi$
$368$	34.6410i	1.80579i
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	− 26.0000i	− 1.34623i	−0.739538	−	0.673114i	$-0.764956\pi$
$373$	− 26.0000i	− 1.34623i	0.739538	−	0.673114i	$-0.235044\pi$
$374$	−41.5692	−2.14949
$375$	0	0
$376$	18.0000	0.928279
$377$	6.92820i	0.356821i
$378$	0	0
$379$	−2.00000	−0.102733	−0.0513665	−	0.998680i	$-0.516358\pi$
$379$	−2.00000	−0.102733	−0.0513665	+	0.998680i	$0.516358\pi$
$380$	0	0
$381$	0	0
$382$	24.0000i	1.22795i
$383$	17.3205i	0.885037i	0.896759	+	0.442518i	$0.145915\pi$
$383$	17.3205i	0.885037i	−0.896759	+	0.442518i	$0.854085\pi$
$384$	0	0
$385$	0	0
$386$	45.0333	2.29214
$387$	0	0
$388$	10.0000i	0.507673i
$389$	−20.7846	−1.05382	−0.526911	−	0.849921i	$-0.676650\pi$
$389$	−20.7846	−1.05382	−0.526911	+	0.849921i	$0.676650\pi$
$390$	0	0
$391$	−48.0000	−2.42746
$392$	5.19615i	0.262445i
$393$	0	0
$394$	24.0000	1.20910
$395$	0	0
$396$	0	0
$397$	2.00000i	0.100377i	0.998740	+	0.0501886i	$0.0159822\pi$
$397$	2.00000i	0.100377i	−0.998740	+	0.0501886i	$0.984018\pi$
$398$	27.7128i	1.38912i
$399$	0	0
$400$	0	0
$401$	−34.6410	−1.72989	−0.864945	−	0.501867i	$-0.832647\pi$
$401$	−34.6410	−1.72989	−0.864945	+	0.501867i	$0.832647\pi$
$402$	0	0
$403$	− 2.00000i	− 0.0996271i
$404$	13.8564	0.689382
$405$	0	0
$406$	24.0000	1.19110
$407$	− 6.92820i	− 0.343418i
$408$	0	0
$409$	10.0000	0.494468	0.247234	−	0.968956i	$-0.420478\pi$
$409$	10.0000	0.494468	0.247234	+	0.968956i	$0.420478\pi$
$410$	0	0
$411$	0	0
$412$	− 4.00000i	− 0.197066i
$413$	− 6.92820i	− 0.340915i
$414$	0	0
$415$	0	0
$416$	−5.19615	−0.254762
$417$	0	0
$418$	12.0000i	0.586939i
$419$	−6.92820	−0.338465	−0.169232	−	0.985576i	$-0.554129\pi$
$419$	−6.92820	−0.338465	−0.169232	+	0.985576i	$0.554129\pi$
$420$	0	0
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	13.8564i	0.674519i
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 20.0000i	− 0.967868i
$428$	− 6.92820i	− 0.334887i
$429$	0	0
$430$	0	0
$431$	38.1051	1.83546	0.917729	−	0.397206i	$-0.130020\pi$
$431$	38.1051	1.83546	0.917729	+	0.397206i	$0.130020\pi$
$432$	0	0
$433$	− 2.00000i	− 0.0961139i	−0.998845	−	0.0480569i	$-0.984697\pi$
$433$	− 2.00000i	− 0.0961139i	0.998845	−	0.0480569i	$-0.0153029\pi$
$434$	−6.92820	−0.332564
$435$	0	0
$436$	−10.0000	−0.478913
$437$	13.8564i	0.662842i
$438$	0	0
$439$	28.0000	1.33637	0.668184	−	0.743996i	$-0.267072\pi$
$439$	28.0000	1.33637	0.668184	+	0.743996i	$0.267072\pi$
$440$	0	0
$441$	0	0
$442$	− 12.0000i	− 0.570782i
$443$	− 20.7846i	− 0.987507i	−0.869602	−	0.493753i	$-0.835625\pi$
$443$	− 20.7846i	− 0.987507i	0.869602	−	0.493753i	$-0.164375\pi$
$444$	0	0
$445$	0	0
$446$	3.46410	0.164030
$447$	0	0
$448$	− 2.00000i	− 0.0944911i
$449$	6.92820	0.326962	0.163481	−	0.986546i	$-0.447728\pi$
$449$	6.92820	0.326962	0.163481	+	0.986546i	$0.447728\pi$
$450$	0	0
$451$	−24.0000	−1.13012
$452$	− 6.92820i	− 0.325875i
$453$	0	0
$454$	−6.00000	−0.281594
$455$	0	0
$456$	0	0
$457$	− 10.0000i	− 0.467780i	−0.972263	−	0.233890i	$-0.924854\pi$
$457$	− 10.0000i	− 0.467780i	0.972263	−	0.233890i	$-0.0751456\pi$
$458$	− 24.2487i	− 1.13307i
$459$	0	0
$460$	0	0
$461$	−27.7128	−1.29071	−0.645357	−	0.763881i	$-0.723291\pi$
$461$	−27.7128	−1.29071	−0.645357	+	0.763881i	$0.723291\pi$
$462$	0	0
$463$	− 2.00000i	− 0.0929479i	−0.998920	−	0.0464739i	$-0.985202\pi$
$463$	− 2.00000i	− 0.0929479i	0.998920	−	0.0464739i	$-0.0147984\pi$
$464$	34.6410	1.60817
$465$	0	0
$466$	0	0
$467$	20.7846i	0.961797i	0.876776	+	0.480899i	$0.159689\pi$
$467$	20.7846i	0.961797i	−0.876776	+	0.480899i	$0.840311\pi$
$468$	0	0
$469$	−28.0000	−1.29292
$470$	0	0
$471$	0	0
$472$	− 6.00000i	− 0.276172i
$473$	27.7128i	1.27424i
$474$	0	0
$475$	0	0
$476$	−13.8564	−0.635107
$477$	0	0
$478$	18.0000i	0.823301i
$479$	3.46410	0.158279	0.0791394	−	0.996864i	$-0.474783\pi$
$479$	3.46410	0.158279	0.0791394	+	0.996864i	$0.474783\pi$
$480$	0	0
$481$	2.00000	0.0911922
$482$	− 17.3205i	− 0.788928i
$483$	0	0
$484$	−1.00000	−0.0454545
$485$	0	0
$486$	0	0
$487$	2.00000i	0.0906287i	0.998973	+	0.0453143i	$0.0144289\pi$
$487$	2.00000i	0.0906287i	−0.998973	+	0.0453143i	$0.985571\pi$
$488$	− 17.3205i	− 0.784063i
$489$	0	0
$490$	0	0
$491$	6.92820	0.312665	0.156333	−	0.987704i	$-0.450033\pi$
$491$	6.92820	0.312665	0.156333	+	0.987704i	$0.450033\pi$
$492$	0	0
$493$	48.0000i	2.16181i
$494$	−3.46410	−0.155857
$495$	0	0
$496$	−10.0000	−0.449013
$497$	6.92820i	0.310772i
$498$	0	0
$499$	−14.0000	−0.626726	−0.313363	−	0.949633i	$-0.601456\pi$
$499$	−14.0000	−0.626726	−0.313363	+	0.949633i	$0.601456\pi$
$500$	0	0
$501$	0	0
$502$	− 12.0000i	− 0.535586i
$503$	6.92820i	0.308913i	0.988000	+	0.154457i	$0.0493627\pi$
$503$	6.92820i	0.308913i	−0.988000	+	0.154457i	$0.950637\pi$
$504$	0	0
$505$	0	0
$506$	−41.5692	−1.84798
$507$	0	0
$508$	16.0000i	0.709885i
$509$	13.8564	0.614174	0.307087	−	0.951681i	$-0.400646\pi$
$509$	13.8564	0.614174	0.307087	+	0.951681i	$0.400646\pi$
$510$	0	0
$511$	−20.0000	−0.884748
$512$	8.66025i	0.382733i
$513$	0	0
$514$	−48.0000	−2.11719
$515$	0	0
$516$	0	0
$517$	36.0000i	1.58328i
$518$	− 6.92820i	− 0.304408i
$519$	0	0
$520$	0	0
$521$	20.7846	0.910590	0.455295	−	0.890341i	$-0.349534\pi$
$521$	20.7846	0.910590	0.455295	+	0.890341i	$0.349534\pi$
$522$	0	0
$523$	− 20.0000i	− 0.874539i	−0.899331	−	0.437269i	$-0.855946\pi$
$523$	− 20.0000i	− 0.874539i	0.899331	−	0.437269i	$-0.144054\pi$
$524$	20.7846	0.907980
$525$	0	0
$526$	12.0000	0.523225
$527$	− 13.8564i	− 0.603595i
$528$	0	0
$529$	−25.0000	−1.08696
$530$	0	0
$531$	0	0
$532$	4.00000i	0.173422i
$533$	− 6.92820i	− 0.300094i
$534$	0	0
$535$	0	0
$536$	−24.2487	−1.04738
$537$	0	0
$538$	12.0000i	0.517357i
$539$	−10.3923	−0.447628
$540$	0	0
$541$	−10.0000	−0.429934	−0.214967	−	0.976621i	$-0.568964\pi$
$541$	−10.0000	−0.429934	−0.214967	+	0.976621i	$0.568964\pi$
$542$	− 38.1051i	− 1.63675i
$543$	0	0
$544$	−36.0000	−1.54349
$545$	0	0
$546$	0	0
$547$	− 4.00000i	− 0.171028i	−0.996337	−	0.0855138i	$-0.972747\pi$
$547$	− 4.00000i	− 0.171028i	0.996337	−	0.0855138i	$-0.0272532\pi$
$548$	6.92820i	0.295958i
$549$	0	0
$550$	0	0
$551$	13.8564	0.590303
$552$	0	0
$553$	8.00000i	0.340195i
$554$	−3.46410	−0.147176
$555$	0	0
$556$	−4.00000	−0.169638
$557$	27.7128i	1.17423i	0.809504	+	0.587115i	$0.199736\pi$
$557$	27.7128i	1.17423i	−0.809504	+	0.587115i	$0.800264\pi$
$558$	0	0
$559$	−8.00000	−0.338364
$560$	0	0
$561$	0	0
$562$	− 36.0000i	− 1.51857i
$563$	− 13.8564i	− 0.583978i	−0.956422	−	0.291989i	$-0.905683\pi$
$563$	− 13.8564i	− 0.583978i	0.956422	−	0.291989i	$-0.0943171\pi$
$564$	0	0
$565$	0	0
$566$	−6.92820	−0.291214
$567$	0	0
$568$	6.00000i	0.251754i
$569$	13.8564	0.580891	0.290445	−	0.956892i	$-0.406197\pi$
$569$	13.8564	0.580891	0.290445	+	0.956892i	$0.406197\pi$
$570$	0	0
$571$	20.0000	0.836974	0.418487	−	0.908223i	$-0.362561\pi$
$571$	20.0000	0.836974	0.418487	+	0.908223i	$0.362561\pi$
$572$	− 3.46410i	− 0.144841i
$573$	0	0
$574$	−24.0000	−1.00174
$575$	0	0
$576$	0	0
$577$	26.0000i	1.08239i	0.840896	+	0.541197i	$0.182029\pi$
$577$	26.0000i	1.08239i	−0.840896	+	0.541197i	$0.817971\pi$
$578$	− 53.6936i	− 2.23336i
$579$	0	0
$580$	0	0
$581$	20.7846	0.862291
$582$	0	0
$583$	0	0
$584$	−17.3205	−0.716728
$585$	0	0
$586$	24.0000	0.991431
$587$	− 45.0333i	− 1.85872i	−0.369170	−	0.929362i	$-0.620358\pi$
$587$	− 45.0333i	− 1.85872i	0.369170	−	0.929362i	$-0.379642\pi$
$588$	0	0
$589$	−4.00000	−0.164817
$590$	0	0
$591$	0	0
$592$	− 10.0000i	− 0.410997i
$593$	− 20.7846i	− 0.853522i	−0.904365	−	0.426761i	$-0.859655\pi$
$593$	− 20.7846i	− 0.853522i	0.904365	−	0.426761i	$-0.140345\pi$
$594$	0	0
$595$	0	0
$596$	13.8564	0.567581
$597$	0	0
$598$	− 12.0000i	− 0.490716i
$599$	−20.7846	−0.849236	−0.424618	−	0.905373i	$-0.639592\pi$
$599$	−20.7846	−0.849236	−0.424618	+	0.905373i	$0.639592\pi$
$600$	0	0
$601$	−34.0000	−1.38689	−0.693444	−	0.720510i	$-0.743908\pi$
$601$	−34.0000	−1.38689	−0.693444	+	0.720510i	$0.743908\pi$
$602$	27.7128i	1.12949i
$603$	0	0
$604$	−14.0000	−0.569652
$605$	0	0
$606$	0	0
$607$	− 40.0000i	− 1.62355i	−0.583970	−	0.811775i	$-0.698502\pi$
$607$	− 40.0000i	− 1.62355i	0.583970	−	0.811775i	$-0.301498\pi$
$608$	10.3923i	0.421464i
$609$	0	0
$610$	0	0
$611$	−10.3923	−0.420428
$612$	0	0
$613$	− 2.00000i	− 0.0807792i	−0.999184	−	0.0403896i	$-0.987140\pi$
$613$	− 2.00000i	− 0.0807792i	0.999184	−	0.0403896i	$-0.0128599\pi$
$614$	−45.0333	−1.81740
$615$	0	0
$616$	12.0000	0.483494
$617$	− 6.92820i	− 0.278919i	−0.990228	−	0.139459i	$-0.955464\pi$
$617$	− 6.92820i	− 0.278919i	0.990228	−	0.139459i	$-0.0445365\pi$
$618$	0	0
$619$	−26.0000	−1.04503	−0.522514	−	0.852631i	$-0.675006\pi$
$619$	−26.0000	−1.04503	−0.522514	+	0.852631i	$0.675006\pi$
$620$	0	0
$621$	0	0
$622$	− 36.0000i	− 1.44347i
$623$	− 13.8564i	− 0.555145i
$624$	0	0
$625$	0	0
$626$	24.2487	0.969173
$627$	0	0
$628$	− 2.00000i	− 0.0798087i
$629$	13.8564	0.552491
$630$	0	0
$631$	38.0000	1.51276	0.756378	−	0.654135i	$-0.226967\pi$
$631$	38.0000	1.51276	0.756378	+	0.654135i	$0.226967\pi$
$632$	6.92820i	0.275589i
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 3.00000i	− 0.118864i
$638$	41.5692i	1.64574i
$639$	0	0
$640$	0	0
$641$	34.6410	1.36824	0.684119	−	0.729370i	$-0.260187\pi$
$641$	34.6410	1.36824	0.684119	+	0.729370i	$0.260187\pi$
$642$	0	0
$643$	− 14.0000i	− 0.552106i	−0.961142	−	0.276053i	$-0.910973\pi$
$643$	− 14.0000i	− 0.552106i	0.961142	−	0.276053i	$-0.0890266\pi$
$644$	−13.8564	−0.546019
$645$	0	0
$646$	−24.0000	−0.944267
$647$	− 13.8564i	− 0.544752i	−0.962191	−	0.272376i	$-0.912191\pi$
$647$	− 13.8564i	− 0.544752i	0.962191	−	0.272376i	$-0.0878094\pi$
$648$	0	0
$649$	12.0000	0.471041
$650$	0	0
$651$	0	0
$652$	14.0000i	0.548282i
$653$	34.6410i	1.35561i	0.735243	+	0.677804i	$0.237068\pi$
$653$	34.6410i	1.35561i	−0.735243	+	0.677804i	$0.762932\pi$
$654$	0	0
$655$	0	0
$656$	−34.6410	−1.35250
$657$	0	0
$658$	36.0000i	1.40343i
$659$	−13.8564	−0.539769	−0.269884	−	0.962893i	$-0.586986\pi$
$659$	−13.8564	−0.539769	−0.269884	+	0.962893i	$0.586986\pi$
$660$	0	0
$661$	26.0000	1.01128	0.505641	−	0.862744i	$-0.331256\pi$
$661$	26.0000	1.01128	0.505641	+	0.862744i	$0.331256\pi$
$662$	− 17.3205i	− 0.673181i
$663$	0	0
$664$	18.0000	0.698535
$665$	0	0
$666$	0	0
$667$	48.0000i	1.85857i
$668$	− 17.3205i	− 0.670151i
$669$	0	0
$670$	0	0
$671$	34.6410	1.33730
$672$	0	0
$673$	10.0000i	0.385472i	0.981251	+	0.192736i	$0.0617360\pi$
$673$	10.0000i	0.385472i	−0.981251	+	0.192736i	$0.938264\pi$
$674$	−3.46410	−0.133432
$675$	0	0
$676$	1.00000	0.0384615
$677$	− 41.5692i	− 1.59763i	−0.601574	−	0.798817i	$-0.705459\pi$
$677$	− 41.5692i	− 1.59763i	0.601574	−	0.798817i	$-0.294541\pi$
$678$	0	0
$679$	20.0000	0.767530
$680$	0	0
$681$	0	0
$682$	− 12.0000i	− 0.459504i
$683$	17.3205i	0.662751i	0.943499	+	0.331375i	$0.107513\pi$
$683$	17.3205i	0.662751i	−0.943499	+	0.331375i	$0.892487\pi$
$684$	0	0
$685$	0	0
$686$	−34.6410	−1.32260
$687$	0	0
$688$	40.0000i	1.52499i
$689$	0	0
$690$	0	0
$691$	−10.0000	−0.380418	−0.190209	−	0.981744i	$-0.560917\pi$
$691$	−10.0000	−0.380418	−0.190209	+	0.981744i	$0.560917\pi$
$692$	13.8564i	0.526742i
$693$	0	0
$694$	48.0000	1.82206
$695$	0	0
$696$	0	0
$697$	− 48.0000i	− 1.81813i
$698$	− 45.0333i	− 1.70454i
$699$	0	0
$700$	0	0
$701$	−20.7846	−0.785024	−0.392512	−	0.919747i	$-0.628394\pi$
$701$	−20.7846	−0.785024	−0.392512	+	0.919747i	$0.628394\pi$
$702$	0	0
$703$	− 4.00000i	− 0.150863i
$704$	3.46410	0.130558
$705$	0	0
$706$	12.0000	0.451626
$707$	− 27.7128i	− 1.04225i
$708$	0	0
$709$	−38.0000	−1.42712	−0.713560	−	0.700594i	$-0.752918\pi$
$709$	−38.0000	−1.42712	−0.713560	+	0.700594i	$0.752918\pi$
$710$	0	0
$711$	0	0
$712$	− 12.0000i	− 0.449719i
$713$	− 13.8564i	− 0.518927i
$714$	0	0
$715$	0	0
$716$	−13.8564	−0.517838
$717$	0	0
$718$	18.0000i	0.671754i
$719$	−34.6410	−1.29189	−0.645946	−	0.763383i	$-0.723537\pi$
$719$	−34.6410	−1.29189	−0.645946	+	0.763383i	$0.723537\pi$
$720$	0	0
$721$	−8.00000	−0.297936
$722$	− 25.9808i	− 0.966904i
$723$	0	0
$724$	10.0000	0.371647
$725$	0	0
$726$	0	0
$727$	32.0000i	1.18681i	0.804902	+	0.593407i	$0.202218\pi$
$727$	32.0000i	1.18681i	−0.804902	+	0.593407i	$0.797782\pi$
$728$	3.46410i	0.128388i
$729$	0	0
$730$	0	0
$731$	−55.4256	−2.04999
$732$	0	0
$733$	34.0000i	1.25582i	0.778287	+	0.627909i	$0.216089\pi$
$733$	34.0000i	1.25582i	−0.778287	+	0.627909i	$0.783911\pi$
$734$	−13.8564	−0.511449
$735$	0	0
$736$	−36.0000	−1.32698
$737$	− 48.4974i	− 1.78643i
$738$	0	0
$739$	22.0000	0.809283	0.404642	−	0.914475i	$-0.367396\pi$
$739$	22.0000	0.809283	0.404642	+	0.914475i	$0.367396\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	24.2487i	0.889599i	0.895630	+	0.444799i	$0.146725\pi$
$743$	24.2487i	0.889599i	−0.895630	+	0.444799i	$0.853275\pi$
$744$	0	0
$745$	0	0
$746$	45.0333	1.64879
$747$	0	0
$748$	− 24.0000i	− 0.877527i
$749$	−13.8564	−0.506302
$750$	0	0
$751$	−4.00000	−0.145962	−0.0729810	−	0.997333i	$-0.523251\pi$
$751$	−4.00000	−0.145962	−0.0729810	+	0.997333i	$0.523251\pi$
$752$	51.9615i	1.89484i
$753$	0	0
$754$	−12.0000	−0.437014
$755$	0	0
$756$	0	0
$757$	2.00000i	0.0726912i	0.999339	+	0.0363456i	$0.0115717\pi$
$757$	2.00000i	0.0726912i	−0.999339	+	0.0363456i	$0.988428\pi$
$758$	− 3.46410i	− 0.125822i
$759$	0	0
$760$	0	0
$761$	34.6410	1.25574	0.627868	−	0.778320i	$-0.283928\pi$
$761$	34.6410	1.25574	0.627868	+	0.778320i	$0.283928\pi$
$762$	0	0
$763$	20.0000i	0.724049i
$764$	−13.8564	−0.501307
$765$	0	0
$766$	−30.0000	−1.08394
$767$	3.46410i	0.125081i
$768$	0	0
$769$	−50.0000	−1.80305	−0.901523	−	0.432731i	$-0.857550\pi$
$769$	−50.0000	−1.80305	−0.901523	+	0.432731i	$0.857550\pi$
$770$	0	0
$771$	0	0
$772$	26.0000i	0.935760i
$773$	− 13.8564i	− 0.498380i	−0.968455	−	0.249190i	$-0.919836\pi$
$773$	− 13.8564i	− 0.498380i	0.968455	−	0.249190i	$-0.0801644\pi$
$774$	0	0
$775$	0	0
$776$	17.3205	0.621770
$777$	0	0
$778$	− 36.0000i	− 1.29066i
$779$	−13.8564	−0.496457
$780$	0	0
$781$	−12.0000	−0.429394
$782$	− 83.1384i	− 2.97302i
$783$	0	0
$784$	−15.0000	−0.535714
$785$	0	0
$786$	0	0
$787$	− 22.0000i	− 0.784215i	−0.919919	−	0.392108i	$-0.871746\pi$
$787$	− 22.0000i	− 0.784215i	0.919919	−	0.392108i	$-0.128254\pi$
$788$	13.8564i	0.493614i
$789$	0	0
$790$	0	0
$791$	−13.8564	−0.492677
$792$	0	0
$793$	10.0000i	0.355110i
$794$	−3.46410	−0.122936
$795$	0	0
$796$	−16.0000	−0.567105
$797$	34.6410i	1.22705i	0.789676	+	0.613524i	$0.210249\pi$
$797$	34.6410i	1.22705i	−0.789676	+	0.613524i	$0.789751\pi$
$798$	0	0
$799$	−72.0000	−2.54718
$800$	0	0
$801$	0	0
$802$	− 60.0000i	− 2.11867i
$803$	− 34.6410i	− 1.22245i
$804$	0	0
$805$	0	0
$806$	3.46410	0.122018
$807$	0	0
$808$	− 24.0000i	− 0.844317i
$809$	−6.92820	−0.243583	−0.121791	−	0.992556i	$-0.538864\pi$
$809$	−6.92820	−0.243583	−0.121791	+	0.992556i	$0.538864\pi$
$810$	0	0
$811$	26.0000	0.912983	0.456492	−	0.889728i	$-0.349106\pi$
$811$	26.0000	0.912983	0.456492	+	0.889728i	$0.349106\pi$
$812$	13.8564i	0.486265i
$813$	0	0
$814$	12.0000	0.420600
$815$	0	0
$816$	0	0
$817$	16.0000i	0.559769i
$818$	17.3205i	0.605597i
$819$	0	0
$820$	0	0
$821$	27.7128	0.967184	0.483592	−	0.875294i	$-0.339332\pi$
$821$	27.7128	0.967184	0.483592	+	0.875294i	$0.339332\pi$
$822$	0	0
$823$	4.00000i	0.139431i	0.997567	+	0.0697156i	$0.0222092\pi$
$823$	4.00000i	0.139431i	−0.997567	+	0.0697156i	$0.977791\pi$
$824$	−6.92820	−0.241355
$825$	0	0
$826$	12.0000	0.417533
$827$	10.3923i	0.361376i	0.983540	+	0.180688i	$0.0578324\pi$
$827$	10.3923i	0.361376i	−0.983540	+	0.180688i	$0.942168\pi$
$828$	0	0
$829$	−2.00000	−0.0694629	−0.0347314	−	0.999397i	$-0.511058\pi$
$829$	−2.00000	−0.0694629	−0.0347314	+	0.999397i	$0.511058\pi$
$830$	0	0
$831$	0	0
$832$	1.00000i	0.0346688i
$833$	− 20.7846i	− 0.720144i
$834$	0	0
$835$	0	0
$836$	−6.92820	−0.239617
$837$	0	0
$838$	− 12.0000i	− 0.414533i
$839$	−24.2487	−0.837158	−0.418579	−	0.908180i	$-0.637472\pi$
$839$	−24.2487	−0.837158	−0.418579	+	0.908180i	$0.637472\pi$
$840$	0	0
$841$	19.0000	0.655172
$842$	− 17.3205i	− 0.596904i
$843$	0	0
$844$	−8.00000	−0.275371
$845$	0	0
$846$	0	0
$847$	2.00000i	0.0687208i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	13.8564	0.474991
$852$	0	0
$853$	46.0000i	1.57501i	0.616308	+	0.787505i	$0.288628\pi$
$853$	46.0000i	1.57501i	−0.616308	+	0.787505i	$0.711372\pi$
$854$	34.6410	1.18539
$855$	0	0
$856$	−12.0000	−0.410152
$857$	0	0	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	0	0
$859$	40.0000	1.36478	0.682391	−	0.730987i	$-0.260940\pi$
$859$	40.0000	1.36478	0.682391	+	0.730987i	$0.260940\pi$
$860$	0	0
$861$	0	0
$862$	66.0000i	2.24797i
$863$	31.1769i	1.06127i	0.847599	+	0.530637i	$0.178047\pi$
$863$	31.1769i	1.06127i	−0.847599	+	0.530637i	$0.821953\pi$
$864$	0	0
$865$	0	0
$866$	3.46410	0.117715
$867$	0	0
$868$	− 4.00000i	− 0.135769i
$869$	−13.8564	−0.470046
$870$	0	0
$871$	14.0000	0.474372
$872$	17.3205i	0.586546i
$873$	0	0
$874$	−24.0000	−0.811812
$875$	0	0
$876$	0	0
$877$	− 22.0000i	− 0.742887i	−0.928456	−	0.371444i	$-0.878863\pi$
$877$	− 22.0000i	− 0.742887i	0.928456	−	0.371444i	$-0.121137\pi$
$878$	48.4974i	1.63671i
$879$	0	0
$880$	0	0
$881$	−13.8564	−0.466834	−0.233417	−	0.972377i	$-0.574991\pi$
$881$	−13.8564	−0.466834	−0.233417	+	0.972377i	$0.574991\pi$
$882$	0	0
$883$	− 44.0000i	− 1.48072i	−0.672212	−	0.740359i	$-0.734656\pi$
$883$	− 44.0000i	− 1.48072i	0.672212	−	0.740359i	$-0.265344\pi$
$884$	6.92820	0.233021
$885$	0	0
$886$	36.0000	1.20944
$887$	27.7128i	0.930505i	0.885178	+	0.465253i	$0.154037\pi$
$887$	27.7128i	0.930505i	−0.885178	+	0.465253i	$0.845963\pi$
$888$	0	0
$889$	32.0000	1.07325
$890$	0	0
$891$	0	0
$892$	2.00000i	0.0669650i
$893$	20.7846i	0.695530i
$894$	0	0
$895$	0	0
$896$	24.2487	0.810093
$897$	0	0
$898$	12.0000i	0.400445i
$899$	−13.8564	−0.462137
$900$	0	0
$901$	0	0
$902$	− 41.5692i	− 1.38410i
$903$	0	0
$904$	−12.0000	−0.399114
$905$	0	0
$906$	0	0
$907$	44.0000i	1.46100i	0.682915	+	0.730498i	$0.260712\pi$
$907$	44.0000i	1.46100i	−0.682915	+	0.730498i	$0.739288\pi$
$908$	− 3.46410i	− 0.114960i
$909$	0	0
$910$	0	0
$911$	−41.5692	−1.37725	−0.688625	−	0.725118i	$-0.741785\pi$
$911$	−41.5692	−1.37725	−0.688625	+	0.725118i	$0.741785\pi$
$912$	0	0
$913$	36.0000i	1.19143i
$914$	17.3205	0.572911
$915$	0	0
$916$	14.0000	0.462573
$917$	− 41.5692i	− 1.37274i
$918$	0	0
$919$	4.00000	0.131948	0.0659739	−	0.997821i	$-0.478985\pi$
$919$	4.00000	0.131948	0.0659739	+	0.997821i	$0.478985\pi$
$920$	0	0
$921$	0	0
$922$	− 48.0000i	− 1.58080i
$923$	− 3.46410i	− 0.114022i
$924$	0	0
$925$	0	0
$926$	3.46410	0.113837
$927$	0	0
$928$	36.0000i	1.18176i
$929$	48.4974	1.59115	0.795574	−	0.605856i	$-0.207169\pi$
$929$	48.4974	1.59115	0.795574	+	0.605856i	$0.207169\pi$
$930$	0	0
$931$	−6.00000	−0.196642
$932$	0	0
$933$	0	0
$934$	−36.0000	−1.17796
$935$	0	0
$936$	0	0
$937$	26.0000i	0.849383i	0.905338	+	0.424691i	$0.139617\pi$
$937$	26.0000i	0.849383i	−0.905338	+	0.424691i	$0.860383\pi$
$938$	− 48.4974i	− 1.58350i
$939$	0	0
$940$	0	0
$941$	41.5692	1.35512	0.677559	−	0.735469i	$-0.263038\pi$
$941$	41.5692	1.35512	0.677559	+	0.735469i	$0.263038\pi$
$942$	0	0
$943$	− 48.0000i	− 1.56310i
$944$	17.3205	0.563735
$945$	0	0
$946$	−48.0000	−1.56061
$947$	17.3205i	0.562841i	0.959585	+	0.281420i	$0.0908056\pi$
$947$	17.3205i	0.562841i	−0.959585	+	0.281420i	$0.909194\pi$
$948$	0	0
$949$	10.0000	0.324614
$950$	0	0
$951$	0	0
$952$	24.0000i	0.777844i
$953$	27.7128i	0.897706i	0.893606	+	0.448853i	$0.148167\pi$
$953$	27.7128i	0.897706i	−0.893606	+	0.448853i	$0.851833\pi$
$954$	0	0
$955$	0	0
$956$	−10.3923	−0.336111
$957$	0	0
$958$	6.00000i	0.193851i
$959$	13.8564	0.447447
$960$	0	0
$961$	−27.0000	−0.870968
$962$	3.46410i	0.111687i
$963$	0	0
$964$	10.0000	0.322078
$965$	0	0
$966$	0	0
$967$	2.00000i	0.0643157i	0.999483	+	0.0321578i	$0.0102379\pi$
$967$	2.00000i	0.0643157i	−0.999483	+	0.0321578i	$0.989762\pi$
$968$	1.73205i	0.0556702i
$969$	0	0
$970$	0	0
$971$	55.4256	1.77869	0.889346	−	0.457234i	$-0.151160\pi$
$971$	55.4256	1.77869	0.889346	+	0.457234i	$0.151160\pi$
$972$	0	0
$973$	8.00000i	0.256468i
$974$	−3.46410	−0.110997
$975$	0	0
$976$	50.0000	1.60046
$977$	6.92820i	0.221653i	0.993840	+	0.110826i	$0.0353498\pi$
$977$	6.92820i	0.221653i	−0.993840	+	0.110826i	$0.964650\pi$
$978$	0	0
$979$	24.0000	0.767043
$980$	0	0
$981$	0	0
$982$	12.0000i	0.382935i
$983$	− 10.3923i	− 0.331463i	−0.986171	−	0.165732i	$-0.947001\pi$
$983$	− 10.3923i	− 0.331463i	0.986171	−	0.165732i	$-0.0529985\pi$
$984$	0	0
$985$	0	0
$986$	−83.1384	−2.64767
$987$	0	0
$988$	− 2.00000i	− 0.0636285i
$989$	−55.4256	−1.76243
$990$	0	0
$991$	−16.0000	−0.508257	−0.254128	−	0.967170i	$-0.581789\pi$
$991$	−16.0000	−0.508257	−0.254128	+	0.967170i	$0.581789\pi$
$992$	− 10.3923i	− 0.329956i
$993$	0	0
$994$	−12.0000	−0.380617
$995$	0	0
$996$	0	0
$997$	− 10.0000i	− 0.316703i	−0.987383	−	0.158352i	$-0.949382\pi$
$997$	− 10.0000i	− 0.316703i	0.987383	−	0.158352i	$-0.0506179\pi$
$998$	− 24.2487i	− 0.767580i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2925.2.c.s.2224.4		4
3.2	odd	2	inner	2925.2.c.s.2224.2		4
5.2	odd	4		2925.2.a.y.1.1		2
5.3	odd	4		117.2.a.b.1.2	yes	2
5.4	even	2	inner	2925.2.c.s.2224.1		4
15.2	even	4		2925.2.a.y.1.2		2
15.8	even	4		117.2.a.b.1.1	✓	2
15.14	odd	2	inner	2925.2.c.s.2224.3		4
20.3	even	4		1872.2.a.v.1.2		2
35.13	even	4		5733.2.a.t.1.2		2
40.3	even	4		7488.2.a.cj.1.1		2
40.13	odd	4		7488.2.a.cq.1.2		2
45.13	odd	12		1053.2.e.i.703.1		4
45.23	even	12		1053.2.e.i.703.2		4
45.38	even	12		1053.2.e.i.352.2		4
45.43	odd	12		1053.2.e.i.352.1		4
60.23	odd	4		1872.2.a.v.1.1		2
65.8	even	4		1521.2.b.i.1351.3		4
65.18	even	4		1521.2.b.i.1351.2		4
65.38	odd	4		1521.2.a.j.1.1		2
105.83	odd	4		5733.2.a.t.1.1		2
120.53	even	4		7488.2.a.cq.1.1		2
120.83	odd	4		7488.2.a.cj.1.2		2
195.8	odd	4		1521.2.b.i.1351.1		4
195.38	even	4		1521.2.a.j.1.2		2
195.83	odd	4		1521.2.b.i.1351.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
117.2.a.b.1.1	✓	2	15.8	even	4
117.2.a.b.1.2	yes	2	5.3	odd	4
1053.2.e.i.352.1		4	45.43	odd	12
1053.2.e.i.352.2		4	45.38	even	12
1053.2.e.i.703.1		4	45.13	odd	12
1053.2.e.i.703.2		4	45.23	even	12
1521.2.a.j.1.1		2	65.38	odd	4
1521.2.a.j.1.2		2	195.38	even	4
1521.2.b.i.1351.1		4	195.8	odd	4
1521.2.b.i.1351.2		4	65.18	even	4
1521.2.b.i.1351.3		4	65.8	even	4
1521.2.b.i.1351.4		4	195.83	odd	4
1872.2.a.v.1.1		2	60.23	odd	4
1872.2.a.v.1.2		2	20.3	even	4
2925.2.a.y.1.1		2	5.2	odd	4
2925.2.a.y.1.2		2	15.2	even	4
2925.2.c.s.2224.1		4	5.4	even	2	inner
2925.2.c.s.2224.2		4	3.2	odd	2	inner
2925.2.c.s.2224.3		4	15.14	odd	2	inner
2925.2.c.s.2224.4		4	1.1	even	1	trivial
5733.2.a.t.1.1		2	105.83	odd	4
5733.2.a.t.1.2		2	35.13	even	4
7488.2.a.cj.1.1		2	40.3	even	4
7488.2.a.cj.1.2		2	120.83	odd	4
7488.2.a.cq.1.1		2	120.53	even	4
7488.2.a.cq.1.2		2	40.13	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2925 = 3^{2} \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2925.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.73205i q^{2} -1.00000 q^{4} +2.00000i q^{7} +1.73205i q^{8} +O(q^{10})\)	\(q+1.73205i q^{2} -1.00000 q^{4} +2.00000i q^{7} +1.73205i q^{8} -3.46410 q^{11} -1.00000i q^{13} -3.46410 q^{14} -5.00000 q^{16} -6.92820i q^{17} -2.00000 q^{19} -6.00000i q^{22} -6.92820i q^{23} +1.73205 q^{26} -2.00000i q^{28} -6.92820 q^{29} +2.00000 q^{31} -5.19615i q^{32} +12.0000 q^{34} +2.00000i q^{37} -3.46410i q^{38} +6.92820 q^{41} -8.00000i q^{43} +3.46410 q^{44} +12.0000 q^{46} -10.3923i q^{47} +3.00000 q^{49} +1.00000i q^{52} -3.46410 q^{56} -12.0000i q^{58} -3.46410 q^{59} -10.0000 q^{61} +3.46410i q^{62} -1.00000 q^{64} +14.0000i q^{67} +6.92820i q^{68} +3.46410 q^{71} +10.0000i q^{73} -3.46410 q^{74} +2.00000 q^{76} -6.92820i q^{77} +4.00000 q^{79} +12.0000i q^{82} -10.3923i q^{83} +13.8564 q^{86} -6.00000i q^{88} -6.92820 q^{89} +2.00000 q^{91} +6.92820i q^{92} +18.0000 q^{94} -10.0000i q^{97} +5.19615i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4}+O(q^{10}) \) 4 * q - 4 * q^4	\( 4 q - 4 q^{4} - 20 q^{16} - 8 q^{19} + 8 q^{31} + 48 q^{34} + 48 q^{46} + 12 q^{49} - 40 q^{61} - 4 q^{64} + 8 q^{76} + 16 q^{79} + 8 q^{91} + 72 q^{94}+O(q^{100}) \) 4 * q - 4 * q^4 - 20 * q^16 - 8 * q^19 + 8 * q^31 + 48 * q^34 + 48 * q^46 + 12 * q^49 - 40 * q^61 - 4 * q^64 + 8 * q^76 + 16 * q^79 + 8 * q^91 + 72 * q^94

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