LMFDB - Embedded newform 2925.2.c.q.2224.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2925,2,Mod(2224,2925)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage: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")

magma://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:traces = [4,0,0,-4,0,0,0,0,0,0,-20,0,0,12,0,12,0,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$23.3562425912$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 325)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2224.1
Root			$0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	2925.2224
Dual form			2925.2.c.q.2224.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.41421i q^{2} -3.82843 q^{4} +2.41421i q^{7} +4.41421i q^{8} -6.41421 q^{11} -1.00000i q^{13} +5.82843 q^{14} +3.00000 q^{16} +3.82843i q^{17} +3.65685 q^{19} +15.4853i q^{22} -3.17157i q^{23} -2.41421 q^{26} -9.24264i q^{28} -0.171573 q^{29} +1.24264 q^{31} +1.58579i q^{32} +9.24264 q^{34} -6.00000i q^{37} -8.82843i q^{38} +5.65685 q^{41} -0.828427i q^{43} +24.5563 q^{44} -7.65685 q^{46} -3.58579i q^{47} +1.17157 q^{49} +3.82843i q^{52} +3.00000i q^{53} -10.6569 q^{56} +0.414214i q^{58} +13.2426 q^{59} +1.00000 q^{61} -3.00000i q^{62} +9.82843 q^{64} -2.75736i q^{67} -14.6569i q^{68} +9.31371 q^{71} +6.00000i q^{73} -14.4853 q^{74} -14.0000 q^{76} -15.4853i q^{77} +6.00000 q^{79} -13.6569i q^{82} +6.89949i q^{83} -2.00000 q^{86} -28.3137i q^{88} -8.48528 q^{89} +2.41421 q^{91} +12.1421i q^{92} -8.65685 q^{94} -0.828427i q^{97} -2.82843i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4} - 20 q^{11} + 12 q^{14} + 12 q^{16} - 8 q^{19} - 4 q^{26} - 12 q^{29} - 12 q^{31} + 20 q^{34} + 36 q^{44} - 8 q^{46} + 16 q^{49} - 20 q^{56} + 36 q^{59} + 4 q^{61} + 28 q^{64} - 8 q^{71} - 24 q^{74}+ \cdots - 12 q^{94}+O(q^{100}) $ 4 * q - 4 * q^4 - 20 * q^11 + 12 * q^14 + 12 * q^16 - 8 * q^19 - 4 * q^26 - 12 * q^29 - 12 * q^31 + 20 * q^34 + 36 * q^44 - 8 * q^46 + 16 * q^49 - 20 * q^56 + 36 * q^59 + 4 * q^61 + 28 * q^64 - 8 * q^71 - 24 * q^74 - 56 * q^76 + 24 * q^79 - 8 * q^86 + 4 * q^91 - 12 * q^94

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 $

$2$	− 2.41421i	− 1.70711i	−0.521005	−	0.853553i	$-0.674443\pi$
$2$	− 2.41421i	− 1.70711i	0.521005	−	0.853553i	$-0.325557\pi$
$3$	0	0
$3$	0	0
$4$	−3.82843	−1.91421
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.41421i	0.912487i	0.889855	+	0.456243i	$0.150805\pi$
$7$	2.41421i	0.912487i	−0.889855	+	0.456243i	$0.849195\pi$
$8$	4.41421i	1.56066i
$9$	0	0
$10$	0	0
$11$	−6.41421	−1.93396	−0.966979	−	0.254856i	$-0.917972\pi$
$11$	−6.41421	−1.93396	−0.966979	+	0.254856i	$0.917972\pi$
$12$	0	0
$13$	− 1.00000i	− 0.277350i
$13$	− 1.00000i	− 0.277350i
$14$	5.82843	1.55771
$15$	0	0
$16$	3.00000	0.750000
$17$	3.82843i	0.928530i	0.885696	+	0.464265i	$0.153681\pi$
$17$	3.82843i	0.928530i	−0.885696	+	0.464265i	$0.846319\pi$
$18$	0	0
$19$	3.65685	0.838940	0.419470	−	0.907769i	$-0.362216\pi$
$19$	3.65685	0.838940	0.419470	+	0.907769i	$0.362216\pi$
$20$	0	0
$21$	0	0
$22$	15.4853i	3.30147i
$23$	− 3.17157i	− 0.661319i	−0.943750	−	0.330659i	$-0.892729\pi$
$23$	− 3.17157i	− 0.661319i	0.943750	−	0.330659i	$-0.107271\pi$
$24$	0	0
$25$	0	0
$26$	−2.41421	−0.473466
$27$	0	0
$28$	− 9.24264i	− 1.74669i
$29$	−0.171573	−0.0318603	−0.0159301	−	0.999873i	$-0.505071\pi$
$29$	−0.171573	−0.0318603	−0.0159301	+	0.999873i	$0.505071\pi$
$30$	0	0
$31$	1.24264	0.223185	0.111592	−	0.993754i	$-0.464405\pi$
$31$	1.24264	0.223185	0.111592	+	0.993754i	$0.464405\pi$
$32$	1.58579i	0.280330i
$33$	0	0
$34$	9.24264	1.58510
$35$	0	0
$36$	0	0
$37$	− 6.00000i	− 0.986394i	−0.869918	−	0.493197i	$-0.835828\pi$
$37$	− 6.00000i	− 0.986394i	0.869918	−	0.493197i	$-0.164172\pi$
$38$	− 8.82843i	− 1.43216i
$39$	0	0
$40$	0	0
$41$	5.65685	0.883452	0.441726	−	0.897150i	$-0.354366\pi$
$41$	5.65685	0.883452	0.441726	+	0.897150i	$0.354366\pi$
$42$	0	0
$43$	− 0.828427i	− 0.126334i	−0.998003	−	0.0631670i	$-0.979880\pi$
$43$	− 0.828427i	− 0.126334i	0.998003	−	0.0631670i	$-0.0201201\pi$
$44$	24.5563	3.70201
$45$	0	0
$46$	−7.65685	−1.12894
$47$	− 3.58579i	− 0.523041i	−0.965198	−	0.261520i	$-0.915776\pi$
$47$	− 3.58579i	− 0.523041i	0.965198	−	0.261520i	$-0.0842239\pi$
$48$	0	0
$49$	1.17157	0.167368
$50$	0	0
$51$	0	0
$52$	3.82843i	0.530907i
$53$	3.00000i	0.412082i	0.978543	+	0.206041i	$0.0660580\pi$
$53$	3.00000i	0.412082i	−0.978543	+	0.206041i	$0.933942\pi$
$54$	0	0
$55$	0	0
$56$	−10.6569	−1.42408
$57$	0	0
$58$	0.414214i	0.0543889i
$59$	13.2426	1.72404	0.862022	−	0.506870i	$-0.169198\pi$
$59$	13.2426	1.72404	0.862022	+	0.506870i	$0.169198\pi$
$60$	0	0
$61$	1.00000	0.128037	0.0640184	−	0.997949i	$-0.479608\pi$
$61$	1.00000	0.128037	0.0640184	+	0.997949i	$0.479608\pi$
$62$	− 3.00000i	− 0.381000i
$63$	0	0
$64$	9.82843	1.22855
$65$	0	0
$66$	0	0
$67$	− 2.75736i	− 0.336865i	−0.985713	−	0.168433i	$-0.946129\pi$
$67$	− 2.75736i	− 0.336865i	0.985713	−	0.168433i	$-0.0538705\pi$
$68$	− 14.6569i	− 1.77740i
$69$	0	0
$70$	0	0
$71$	9.31371	1.10533	0.552667	−	0.833402i	$-0.313610\pi$
$71$	9.31371	1.10533	0.552667	+	0.833402i	$0.313610\pi$
$72$	0	0
$73$	6.00000i	0.702247i	0.936329	+	0.351123i	$0.114200\pi$
$73$	6.00000i	0.702247i	−0.936329	+	0.351123i	$0.885800\pi$
$74$	−14.4853	−1.68388
$75$	0	0
$76$	−14.0000	−1.60591
$77$	− 15.4853i	− 1.76471i
$78$	0	0
$79$	6.00000	0.675053	0.337526	−	0.941316i	$-0.390410\pi$
$79$	6.00000	0.675053	0.337526	+	0.941316i	$0.390410\pi$
$80$	0	0
$81$	0	0
$82$	− 13.6569i	− 1.50815i
$83$	6.89949i	0.757318i	0.925536	+	0.378659i	$0.123615\pi$
$83$	6.89949i	0.757318i	−0.925536	+	0.378659i	$0.876385\pi$
$84$	0	0
$85$	0	0
$86$	−2.00000	−0.215666
$87$	0	0
$88$	− 28.3137i	− 3.01825i
$89$	−8.48528	−0.899438	−0.449719	−	0.893170i	$-0.648476\pi$
$89$	−8.48528	−0.899438	−0.449719	+	0.893170i	$0.648476\pi$
$90$	0	0
$91$	2.41421	0.253078
$92$	12.1421i	1.26591i
$93$	0	0
$94$	−8.65685	−0.892886
$95$	0	0
$96$	0	0
$97$	− 0.828427i	− 0.0841140i	−0.999115	−	0.0420570i	$-0.986609\pi$
$97$	− 0.828427i	− 0.0841140i	0.999115	−	0.0420570i	$-0.0133911\pi$
$98$	− 2.82843i	− 0.285714i
$99$	0	0
$100$	0	0
$101$	6.65685	0.662382	0.331191	−	0.943564i	$-0.392550\pi$
$101$	6.65685	0.662382	0.331191	+	0.943564i	$0.392550\pi$
$102$	0	0
$103$	− 13.6569i	− 1.34565i	−0.739802	−	0.672825i	$-0.765081\pi$
$103$	− 13.6569i	− 1.34565i	0.739802	−	0.672825i	$-0.234919\pi$
$104$	4.41421	0.432849
$105$	0	0
$106$	7.24264	0.703467
$107$	− 16.1421i	− 1.56052i	−0.625456	−	0.780260i	$-0.715087\pi$
$107$	− 16.1421i	− 1.56052i	0.625456	−	0.780260i	$-0.284913\pi$
$108$	0	0
$109$	16.4853	1.57900	0.789502	−	0.613748i	$-0.210339\pi$
$109$	16.4853	1.57900	0.789502	+	0.613748i	$0.210339\pi$
$110$	0	0
$111$	0	0
$112$	7.24264i	0.684365i
$113$	− 11.6569i	− 1.09658i	−0.836287	−	0.548292i	$-0.815278\pi$
$113$	− 11.6569i	− 1.09658i	0.836287	−	0.548292i	$-0.184722\pi$
$114$	0	0
$115$	0	0
$116$	0.656854	0.0609874
$117$	0	0
$118$	− 31.9706i	− 2.94313i
$119$	−9.24264	−0.847271
$120$	0	0
$121$	30.1421	2.74019
$122$	− 2.41421i	− 0.218573i
$123$	0	0
$124$	−4.75736	−0.427223
$125$	0	0
$126$	0	0
$127$	− 3.65685i	− 0.324493i	−0.986750	−	0.162247i	$-0.948126\pi$
$127$	− 3.65685i	− 0.324493i	0.986750	−	0.162247i	$-0.0518740\pi$
$128$	− 20.5563i	− 1.81694i
$129$	0	0
$130$	0	0
$131$	−8.48528	−0.741362	−0.370681	−	0.928760i	$-0.620876\pi$
$131$	−8.48528	−0.741362	−0.370681	+	0.928760i	$0.620876\pi$
$132$	0	0
$133$	8.82843i	0.765522i
$134$	−6.65685	−0.575065
$135$	0	0
$136$	−16.8995	−1.44912
$137$	− 2.82843i	− 0.241649i	−0.992674	−	0.120824i	$-0.961446\pi$
$137$	− 2.82843i	− 0.241649i	0.992674	−	0.120824i	$-0.0385538\pi$
$138$	0	0
$139$	18.4853	1.56790	0.783951	−	0.620823i	$-0.213202\pi$
$139$	18.4853	1.56790	0.783951	+	0.620823i	$0.213202\pi$
$140$	0	0
$141$	0	0
$142$	− 22.4853i	− 1.88692i
$143$	6.41421i	0.536383i
$144$	0	0
$145$	0	0
$146$	14.4853	1.19881
$147$	0	0
$148$	22.9706i	1.88817i
$149$	8.82843	0.723253	0.361626	−	0.932323i	$-0.382222\pi$
$149$	8.82843	0.723253	0.361626	+	0.932323i	$0.382222\pi$
$150$	0	0
$151$	−8.75736	−0.712664	−0.356332	−	0.934359i	$-0.615973\pi$
$151$	−8.75736	−0.712664	−0.356332	+	0.934359i	$0.615973\pi$
$152$	16.1421i	1.30930i
$153$	0	0
$154$	−37.3848	−3.01255
$155$	0	0
$156$	0	0
$157$	23.4853i	1.87433i	0.348887	+	0.937165i	$0.386560\pi$
$157$	23.4853i	1.87433i	−0.348887	+	0.937165i	$0.613440\pi$
$158$	− 14.4853i	− 1.15239i
$159$	0	0
$160$	0	0
$161$	7.65685	0.603445
$162$	0	0
$163$	− 10.0000i	− 0.783260i	−0.920123	−	0.391630i	$-0.871911\pi$
$163$	− 10.0000i	− 0.783260i	0.920123	−	0.391630i	$-0.128089\pi$
$164$	−21.6569	−1.69112
$165$	0	0
$166$	16.6569	1.29282
$167$	5.31371i	0.411187i	0.978637	+	0.205594i	$0.0659125\pi$
$167$	5.31371i	0.411187i	−0.978637	+	0.205594i	$0.934088\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	0	0
$172$	3.17157i	0.241830i
$173$	18.6569i	1.41845i	0.704980	+	0.709227i	$0.250956\pi$
$173$	18.6569i	1.41845i	−0.704980	+	0.709227i	$0.749044\pi$
$174$	0	0
$175$	0	0
$176$	−19.2426	−1.45047
$177$	0	0
$178$	20.4853i	1.53544i
$179$	0.686292	0.0512958	0.0256479	−	0.999671i	$-0.491835\pi$
$179$	0.686292	0.0512958	0.0256479	+	0.999671i	$0.491835\pi$
$180$	0	0
$181$	−17.4853	−1.29967	−0.649835	−	0.760075i	$-0.725162\pi$
$181$	−17.4853	−1.29967	−0.649835	+	0.760075i	$0.725162\pi$
$182$	− 5.82843i	− 0.432032i
$183$	0	0
$184$	14.0000	1.03209
$185$	0	0
$186$	0	0
$187$	− 24.5563i	− 1.79574i
$188$	13.7279i	1.00121i
$189$	0	0
$190$	0	0
$191$	−25.3137	−1.83164	−0.915818	−	0.401594i	$-0.868456\pi$
$191$	−25.3137	−1.83164	−0.915818	+	0.401594i	$0.868456\pi$
$192$	0	0
$193$	− 1.65685i	− 0.119263i	−0.998220	−	0.0596315i	$-0.981007\pi$
$193$	− 1.65685i	− 0.119263i	0.998220	−	0.0596315i	$-0.0189926\pi$
$194$	−2.00000	−0.143592
$195$	0	0
$196$	−4.48528	−0.320377
$197$	12.0000i	0.854965i	0.904024	+	0.427482i	$0.140599\pi$
$197$	12.0000i	0.854965i	−0.904024	+	0.427482i	$0.859401\pi$
$198$	0	0
$199$	−10.0000	−0.708881	−0.354441	−	0.935079i	$-0.615329\pi$
$199$	−10.0000	−0.708881	−0.354441	+	0.935079i	$0.615329\pi$
$200$	0	0
$201$	0	0
$202$	− 16.0711i	− 1.13076i
$203$	− 0.414214i	− 0.0290721i
$204$	0	0
$205$	0	0
$206$	−32.9706	−2.29717
$207$	0	0
$208$	− 3.00000i	− 0.208013i
$209$	−23.4558	−1.62247
$210$	0	0
$211$	17.7990	1.22533	0.612666	−	0.790342i	$-0.290097\pi$
$211$	17.7990	1.22533	0.612666	+	0.790342i	$0.290097\pi$
$212$	− 11.4853i	− 0.788812i
$213$	0	0
$214$	−38.9706	−2.66397
$215$	0	0
$216$	0	0
$217$	3.00000i	0.203653i
$218$	− 39.7990i	− 2.69553i
$219$	0	0
$220$	0	0
$221$	3.82843	0.257528
$222$	0	0
$223$	10.9706i	0.734643i	0.930094	+	0.367322i	$0.119725\pi$
$223$	10.9706i	0.734643i	−0.930094	+	0.367322i	$0.880275\pi$
$224$	−3.82843	−0.255798
$225$	0	0
$226$	−28.1421	−1.87199
$227$	8.89949i	0.590680i	0.955392	+	0.295340i	$0.0954330\pi$
$227$	8.89949i	0.590680i	−0.955392	+	0.295340i	$0.904567\pi$
$228$	0	0
$229$	4.82843	0.319071	0.159536	−	0.987192i	$-0.449000\pi$
$229$	4.82843	0.319071	0.159536	+	0.987192i	$0.449000\pi$
$230$	0	0
$231$	0	0
$232$	− 0.757359i	− 0.0497231i
$233$	20.6274i	1.35135i	0.737201	+	0.675674i	$0.236147\pi$
$233$	20.6274i	1.35135i	−0.737201	+	0.675674i	$0.763853\pi$
$234$	0	0
$235$	0	0
$236$	−50.6985	−3.30019
$237$	0	0
$238$	22.3137i	1.44638i
$239$	23.5858	1.52564	0.762819	−	0.646612i	$-0.223815\pi$
$239$	23.5858	1.52564	0.762819	+	0.646612i	$0.223815\pi$
$240$	0	0
$241$	4.97056	0.320182	0.160091	−	0.987102i	$-0.448821\pi$
$241$	4.97056	0.320182	0.160091	+	0.987102i	$0.448821\pi$
$242$	− 72.7696i	− 4.67780i
$243$	0	0
$244$	−3.82843	−0.245090
$245$	0	0
$246$	0	0
$247$	− 3.65685i	− 0.232680i
$248$	5.48528i	0.348316i
$249$	0	0
$250$	0	0
$251$	5.65685	0.357057	0.178529	−	0.983935i	$-0.442866\pi$
$251$	5.65685	0.357057	0.178529	+	0.983935i	$0.442866\pi$
$252$	0	0
$253$	20.3431i	1.27896i
$254$	−8.82843	−0.553945
$255$	0	0
$256$	−29.9706	−1.87316
$257$	6.65685i	0.415243i	0.978209	+	0.207622i	$0.0665723\pi$
$257$	6.65685i	0.415243i	−0.978209	+	0.207622i	$0.933428\pi$
$258$	0	0
$259$	14.4853	0.900072
$260$	0	0
$261$	0	0
$262$	20.4853i	1.26558i
$263$	− 26.1421i	− 1.61199i	−0.591920	−	0.805997i	$-0.701630\pi$
$263$	− 26.1421i	− 1.61199i	0.591920	−	0.805997i	$-0.298370\pi$
$264$	0	0
$265$	0	0
$266$	21.3137	1.30683
$267$	0	0
$268$	10.5563i	0.644832i
$269$	15.1421	0.923232	0.461616	−	0.887080i	$-0.347270\pi$
$269$	15.1421	0.923232	0.461616	+	0.887080i	$0.347270\pi$
$270$	0	0
$271$	−25.7279	−1.56286	−0.781430	−	0.623993i	$-0.785509\pi$
$271$	−25.7279	−1.56286	−0.781430	+	0.623993i	$0.785509\pi$
$272$	11.4853i	0.696397i
$273$	0	0
$274$	−6.82843	−0.412520
$275$	0	0
$276$	0	0
$277$	9.31371i	0.559607i	0.960057	+	0.279803i	$0.0902692\pi$
$277$	9.31371i	0.559607i	−0.960057	+	0.279803i	$0.909731\pi$
$278$	− 44.6274i	− 2.67657i
$279$	0	0
$280$	0	0
$281$	−4.82843	−0.288040	−0.144020	−	0.989575i	$-0.546003\pi$
$281$	−4.82843	−0.288040	−0.144020	+	0.989575i	$0.546003\pi$
$282$	0	0
$283$	22.4853i	1.33661i	0.743887	+	0.668306i	$0.232980\pi$
$283$	22.4853i	1.33661i	−0.743887	+	0.668306i	$0.767020\pi$
$284$	−35.6569	−2.11585
$285$	0	0
$286$	15.4853	0.915664
$287$	13.6569i	0.806139i
$288$	0	0
$289$	2.34315	0.137832
$290$	0	0
$291$	0	0
$292$	− 22.9706i	− 1.34425i
$293$	8.82843i	0.515762i	0.966177	+	0.257881i	$0.0830243\pi$
$293$	8.82843i	0.515762i	−0.966177	+	0.257881i	$0.916976\pi$
$294$	0	0
$295$	0	0
$296$	26.4853	1.53943
$297$	0	0
$298$	− 21.3137i	− 1.23467i
$299$	−3.17157	−0.183417
$300$	0	0
$301$	2.00000	0.115278
$302$	21.1421i	1.21659i
$303$	0	0
$304$	10.9706	0.629205
$305$	0	0
$306$	0	0
$307$	− 9.31371i	− 0.531561i	−0.964034	−	0.265781i	$-0.914370\pi$
$307$	− 9.31371i	− 0.531561i	0.964034	−	0.265781i	$-0.0856297\pi$
$308$	59.2843i	3.37803i
$309$	0	0
$310$	0	0
$311$	−9.51472	−0.539530	−0.269765	−	0.962926i	$-0.586946\pi$
$311$	−9.51472	−0.539530	−0.269765	+	0.962926i	$0.586946\pi$
$312$	0	0
$313$	− 2.85786i	− 0.161536i	−0.996733	−	0.0807680i	$-0.974263\pi$
$313$	− 2.85786i	− 0.161536i	0.996733	−	0.0807680i	$-0.0257373\pi$
$314$	56.6985	3.19968
$315$	0	0
$316$	−22.9706	−1.29220
$317$	− 20.1421i	− 1.13130i	−0.824647	−	0.565648i	$-0.808626\pi$
$317$	− 20.1421i	− 1.13130i	0.824647	−	0.565648i	$-0.191374\pi$
$318$	0	0
$319$	1.10051	0.0616165
$320$	0	0
$321$	0	0
$322$	− 18.4853i	− 1.03014i
$323$	14.0000i	0.778981i
$324$	0	0
$325$	0	0
$326$	−24.1421	−1.33711
$327$	0	0
$328$	24.9706i	1.37877i
$329$	8.65685	0.477268
$330$	0	0
$331$	27.6569	1.52016	0.760079	−	0.649831i	$-0.225160\pi$
$331$	27.6569	1.52016	0.760079	+	0.649831i	$0.225160\pi$
$332$	− 26.4142i	− 1.44967i
$333$	0	0
$334$	12.8284	0.701940
$335$	0	0
$336$	0	0
$337$	− 20.7990i	− 1.13299i	−0.824064	−	0.566497i	$-0.808298\pi$
$337$	− 20.7990i	− 1.13299i	0.824064	−	0.566497i	$-0.191702\pi$
$338$	2.41421i	0.131316i
$339$	0	0
$340$	0	0
$341$	−7.97056	−0.431630
$342$	0	0
$343$	19.7279i	1.06521i
$344$	3.65685	0.197164
$345$	0	0
$346$	45.0416	2.42145
$347$	19.4558i	1.04444i	0.852809	+	0.522222i	$0.174897\pi$
$347$	19.4558i	1.04444i	−0.852809	+	0.522222i	$0.825103\pi$
$348$	0	0
$349$	−35.4558	−1.89791	−0.948954	−	0.315415i	$-0.897856\pi$
$349$	−35.4558	−1.89791	−0.948954	+	0.315415i	$0.897856\pi$
$350$	0	0
$351$	0	0
$352$	− 10.1716i	− 0.542147i
$353$	26.1421i	1.39141i	0.718330	+	0.695703i	$0.244907\pi$
$353$	26.1421i	1.39141i	−0.718330	+	0.695703i	$0.755093\pi$
$354$	0	0
$355$	0	0
$356$	32.4853	1.72172
$357$	0	0
$358$	− 1.65685i	− 0.0875675i
$359$	21.3848	1.12865	0.564323	−	0.825554i	$-0.309137\pi$
$359$	21.3848	1.12865	0.564323	+	0.825554i	$0.309137\pi$
$360$	0	0
$361$	−5.62742	−0.296180
$362$	42.2132i	2.21868i
$363$	0	0
$364$	−9.24264	−0.484446
$365$	0	0
$366$	0	0
$367$	− 34.8284i	− 1.81803i	−0.416764	−	0.909015i	$-0.636836\pi$
$367$	− 34.8284i	− 1.81803i	0.416764	−	0.909015i	$-0.363164\pi$
$368$	− 9.51472i	− 0.495989i
$369$	0	0
$370$	0	0
$371$	−7.24264	−0.376019
$372$	0	0
$373$	− 1.14214i	− 0.0591375i	−0.999563	−	0.0295688i	$-0.990587\pi$
$373$	− 1.14214i	− 0.0591375i	0.999563	−	0.0295688i	$-0.00941340\pi$
$374$	−59.2843	−3.06552
$375$	0	0
$376$	15.8284	0.816289
$377$	0.171573i	0.00883645i
$378$	0	0
$379$	31.8701	1.63705	0.818527	−	0.574467i	$-0.194791\pi$
$379$	31.8701	1.63705	0.818527	+	0.574467i	$0.194791\pi$
$380$	0	0
$381$	0	0
$382$	61.1127i	3.12680i
$383$	27.6569i	1.41320i	0.707614	+	0.706600i	$0.249772\pi$
$383$	27.6569i	1.41320i	−0.707614	+	0.706600i	$0.750228\pi$
$384$	0	0
$385$	0	0
$386$	−4.00000	−0.203595
$387$	0	0
$388$	3.17157i	0.161012i
$389$	28.6274	1.45147	0.725734	−	0.687976i	$-0.241500\pi$
$389$	28.6274	1.45147	0.725734	+	0.687976i	$0.241500\pi$
$390$	0	0
$391$	12.1421	0.614054
$392$	5.17157i	0.261204i
$393$	0	0
$394$	28.9706	1.45952
$395$	0	0
$396$	0	0
$397$	7.65685i	0.384286i	0.981367	+	0.192143i	$0.0615438\pi$
$397$	7.65685i	0.384286i	−0.981367	+	0.192143i	$0.938456\pi$
$398$	24.1421i	1.21014i
$399$	0	0
$400$	0	0
$401$	3.85786	0.192653	0.0963263	−	0.995350i	$-0.469291\pi$
$401$	3.85786	0.192653	0.0963263	+	0.995350i	$0.469291\pi$
$402$	0	0
$403$	− 1.24264i	− 0.0619003i
$404$	−25.4853	−1.26794
$405$	0	0
$406$	−1.00000	−0.0496292
$407$	38.4853i	1.90764i
$408$	0	0
$409$	10.8284	0.535431	0.267716	−	0.963498i	$-0.413731\pi$
$409$	10.8284	0.535431	0.267716	+	0.963498i	$0.413731\pi$
$410$	0	0
$411$	0	0
$412$	52.2843i	2.57586i
$413$	31.9706i	1.57317i
$414$	0	0
$415$	0	0
$416$	1.58579	0.0777496
$417$	0	0
$418$	56.6274i	2.76974i
$419$	22.8284	1.11524	0.557621	−	0.830096i	$-0.311714\pi$
$419$	22.8284	1.11524	0.557621	+	0.830096i	$0.311714\pi$
$420$	0	0
$421$	−16.9706	−0.827095	−0.413547	−	0.910483i	$-0.635710\pi$
$421$	−16.9706	−0.827095	−0.413547	+	0.910483i	$0.635710\pi$
$422$	− 42.9706i	− 2.09177i
$423$	0	0
$424$	−13.2426	−0.643119
$425$	0	0
$426$	0	0
$427$	2.41421i	0.116832i
$428$	61.7990i	2.98717i
$429$	0	0
$430$	0	0
$431$	8.34315	0.401875	0.200938	−	0.979604i	$-0.435601\pi$
$431$	8.34315	0.401875	0.200938	+	0.979604i	$0.435601\pi$
$432$	0	0
$433$	− 16.3431i	− 0.785401i	−0.919666	−	0.392701i	$-0.871541\pi$
$433$	− 16.3431i	− 0.785401i	0.919666	−	0.392701i	$-0.128459\pi$
$434$	7.24264	0.347658
$435$	0	0
$436$	−63.1127	−3.02255
$437$	− 11.5980i	− 0.554807i
$438$	0	0
$439$	−14.0000	−0.668184	−0.334092	−	0.942541i	$-0.608430\pi$
$439$	−14.0000	−0.668184	−0.334092	+	0.942541i	$0.608430\pi$
$440$	0	0
$441$	0	0
$442$	− 9.24264i	− 0.439628i
$443$	− 29.7990i	− 1.41579i	−0.706316	−	0.707896i	$-0.749644\pi$
$443$	− 29.7990i	− 1.41579i	0.706316	−	0.707896i	$-0.250356\pi$
$444$	0	0
$445$	0	0
$446$	26.4853	1.25411
$447$	0	0
$448$	23.7279i	1.12104i
$449$	−5.17157	−0.244062	−0.122031	−	0.992526i	$-0.538941\pi$
$449$	−5.17157	−0.244062	−0.122031	+	0.992526i	$0.538941\pi$
$450$	0	0
$451$	−36.2843	−1.70856
$452$	44.6274i	2.09910i
$453$	0	0
$454$	21.4853	1.00835
$455$	0	0
$456$	0	0
$457$	8.48528i	0.396925i	0.980109	+	0.198462i	$0.0635948\pi$
$457$	8.48528i	0.396925i	−0.980109	+	0.198462i	$0.936405\pi$
$458$	− 11.6569i	− 0.544689i
$459$	0	0
$460$	0	0
$461$	19.4558	0.906149	0.453074	−	0.891473i	$-0.350327\pi$
$461$	19.4558	0.906149	0.453074	+	0.891473i	$0.350327\pi$
$462$	0	0
$463$	− 28.5563i	− 1.32713i	−0.748120	−	0.663563i	$-0.769043\pi$
$463$	− 28.5563i	− 1.32713i	0.748120	−	0.663563i	$-0.230957\pi$
$464$	−0.514719	−0.0238952
$465$	0	0
$466$	49.7990	2.30689
$467$	− 31.1127i	− 1.43972i	−0.694117	−	0.719862i	$-0.744205\pi$
$467$	− 31.1127i	− 1.43972i	0.694117	−	0.719862i	$-0.255795\pi$
$468$	0	0
$469$	6.65685	0.307385
$470$	0	0
$471$	0	0
$472$	58.4558i	2.69065i
$473$	5.31371i	0.244325i
$474$	0	0
$475$	0	0
$476$	35.3848	1.62186
$477$	0	0
$478$	− 56.9411i	− 2.60443i
$479$	14.2721	0.652108	0.326054	−	0.945351i	$-0.394281\pi$
$479$	14.2721	0.652108	0.326054	+	0.945351i	$0.394281\pi$
$480$	0	0
$481$	−6.00000	−0.273576
$482$	− 12.0000i	− 0.546585i
$483$	0	0
$484$	−115.397	−5.24532
$485$	0	0
$486$	0	0
$487$	− 42.2132i	− 1.91286i	−0.291957	−	0.956431i	$-0.594306\pi$
$487$	− 42.2132i	− 1.91286i	0.291957	−	0.956431i	$-0.405694\pi$
$488$	4.41421i	0.199822i
$489$	0	0
$490$	0	0
$491$	−11.1716	−0.504166	−0.252083	−	0.967706i	$-0.581116\pi$
$491$	−11.1716	−0.504166	−0.252083	+	0.967706i	$0.581116\pi$
$492$	0	0
$493$	− 0.656854i	− 0.0295832i
$494$	−8.82843	−0.397210
$495$	0	0
$496$	3.72792	0.167389
$497$	22.4853i	1.00860i
$498$	0	0
$499$	−2.55635	−0.114438	−0.0572190	−	0.998362i	$-0.518223\pi$
$499$	−2.55635	−0.114438	−0.0572190	+	0.998362i	$0.518223\pi$
$500$	0	0
$501$	0	0
$502$	− 13.6569i	− 0.609535i
$503$	− 14.6274i	− 0.652204i	−0.945335	−	0.326102i	$-0.894265\pi$
$503$	− 14.6274i	− 0.652204i	0.945335	−	0.326102i	$-0.105735\pi$
$504$	0	0
$505$	0	0
$506$	49.1127	2.18333
$507$	0	0
$508$	14.0000i	0.621150i
$509$	14.6274	0.648349	0.324174	−	0.945997i	$-0.394914\pi$
$509$	14.6274	0.648349	0.324174	+	0.945997i	$0.394914\pi$
$510$	0	0
$511$	−14.4853	−0.640791
$512$	31.2426i	1.38074i
$513$	0	0
$514$	16.0711	0.708864
$515$	0	0
$516$	0	0
$517$	23.0000i	1.01154i
$518$	− 34.9706i	− 1.53652i
$519$	0	0
$520$	0	0
$521$	−0.343146	−0.0150335	−0.00751674	−	0.999972i	$-0.502393\pi$
$521$	−0.343146	−0.0150335	−0.00751674	+	0.999972i	$0.502393\pi$
$522$	0	0
$523$	− 6.00000i	− 0.262362i	−0.991358	−	0.131181i	$-0.958123\pi$
$523$	− 6.00000i	− 0.262362i	0.991358	−	0.131181i	$-0.0418769\pi$
$524$	32.4853	1.41913
$525$	0	0
$526$	−63.1127	−2.75184
$527$	4.75736i	0.207234i
$528$	0	0
$529$	12.9411	0.562658
$530$	0	0
$531$	0	0
$532$	− 33.7990i	− 1.46537i
$533$	− 5.65685i	− 0.245026i
$534$	0	0
$535$	0	0
$536$	12.1716	0.525732
$537$	0	0
$538$	− 36.5563i	− 1.57606i
$539$	−7.51472	−0.323682
$540$	0	0
$541$	33.7990	1.45313	0.726566	−	0.687097i	$-0.241115\pi$
$541$	33.7990	1.45313	0.726566	+	0.687097i	$0.241115\pi$
$542$	62.1127i	2.66797i
$543$	0	0
$544$	−6.07107	−0.260295
$545$	0	0
$546$	0	0
$547$	28.4853i	1.21794i	0.793192	+	0.608971i	$0.208418\pi$
$547$	28.4853i	1.21794i	−0.793192	+	0.608971i	$0.791582\pi$
$548$	10.8284i	0.462567i
$549$	0	0
$550$	0	0
$551$	−0.627417	−0.0267289
$552$	0	0
$553$	14.4853i	0.615977i
$554$	22.4853	0.955308
$555$	0	0
$556$	−70.7696	−3.00130
$557$	− 27.3137i	− 1.15732i	−0.815569	−	0.578659i	$-0.803576\pi$
$557$	− 27.3137i	− 1.15732i	0.815569	−	0.578659i	$-0.196424\pi$
$558$	0	0
$559$	−0.828427	−0.0350387
$560$	0	0
$561$	0	0
$562$	11.6569i	0.491715i
$563$	− 18.0000i	− 0.758610i	−0.925272	−	0.379305i	$-0.876163\pi$
$563$	− 18.0000i	− 0.758610i	0.925272	−	0.379305i	$-0.123837\pi$
$564$	0	0
$565$	0	0
$566$	54.2843	2.28174
$567$	0	0
$568$	41.1127i	1.72505i
$569$	−35.2843	−1.47919	−0.739597	−	0.673050i	$-0.764984\pi$
$569$	−35.2843	−1.47919	−0.739597	+	0.673050i	$0.764984\pi$
$570$	0	0
$571$	−2.00000	−0.0836974	−0.0418487	−	0.999124i	$-0.513325\pi$
$571$	−2.00000	−0.0836974	−0.0418487	+	0.999124i	$0.513325\pi$
$572$	− 24.5563i	− 1.02675i
$573$	0	0
$574$	32.9706	1.37616
$575$	0	0
$576$	0	0
$577$	− 7.17157i	− 0.298556i	−0.988795	−	0.149278i	$-0.952305\pi$
$577$	− 7.17157i	− 0.298556i	0.988795	−	0.149278i	$-0.0476950\pi$
$578$	− 5.65685i	− 0.235294i
$579$	0	0
$580$	0	0
$581$	−16.6569	−0.691043
$582$	0	0
$583$	− 19.2426i	− 0.796949i
$584$	−26.4853	−1.09597
$585$	0	0
$586$	21.3137	0.880461
$587$	− 14.4142i	− 0.594938i	−0.954731	−	0.297469i	$-0.903857\pi$
$587$	− 14.4142i	− 0.594938i	0.954731	−	0.297469i	$-0.0961425\pi$
$588$	0	0
$589$	4.54416	0.187239
$590$	0	0
$591$	0	0
$592$	− 18.0000i	− 0.739795i
$593$	19.6569i	0.807210i	0.914933	+	0.403605i	$0.132243\pi$
$593$	19.6569i	0.807210i	−0.914933	+	0.403605i	$0.867757\pi$
$594$	0	0
$595$	0	0
$596$	−33.7990	−1.38446
$597$	0	0
$598$	7.65685i	0.313112i
$599$	−3.51472	−0.143608	−0.0718038	−	0.997419i	$-0.522876\pi$
$599$	−3.51472	−0.143608	−0.0718038	+	0.997419i	$0.522876\pi$
$600$	0	0
$601$	13.8284	0.564073	0.282037	−	0.959404i	$-0.408990\pi$
$601$	13.8284	0.564073	0.282037	+	0.959404i	$0.408990\pi$
$602$	− 4.82843i	− 0.196792i
$603$	0	0
$604$	33.5269	1.36419
$605$	0	0
$606$	0	0
$607$	− 21.5147i	− 0.873255i	−0.899642	−	0.436628i	$-0.856173\pi$
$607$	− 21.5147i	− 0.873255i	0.899642	−	0.436628i	$-0.143827\pi$
$608$	5.79899i	0.235180i
$609$	0	0
$610$	0	0
$611$	−3.58579	−0.145065
$612$	0	0
$613$	12.8284i	0.518135i	0.965859	+	0.259068i	$0.0834153\pi$
$613$	12.8284i	0.518135i	−0.965859	+	0.259068i	$0.916585\pi$
$614$	−22.4853	−0.907432
$615$	0	0
$616$	68.3553	2.75412
$617$	− 6.00000i	− 0.241551i	−0.992680	−	0.120775i	$-0.961462\pi$
$617$	− 6.00000i	− 0.241551i	0.992680	−	0.120775i	$-0.0385381\pi$
$618$	0	0
$619$	−22.9706	−0.923265	−0.461632	−	0.887071i	$-0.652736\pi$
$619$	−22.9706	−0.923265	−0.461632	+	0.887071i	$0.652736\pi$
$620$	0	0
$621$	0	0
$622$	22.9706i	0.921036i
$623$	− 20.4853i	− 0.820725i
$624$	0	0
$625$	0	0
$626$	−6.89949	−0.275759
$627$	0	0
$628$	− 89.9117i	− 3.58787i
$629$	22.9706	0.915896
$630$	0	0
$631$	34.0000	1.35352	0.676759	−	0.736204i	$-0.263384\pi$
$631$	34.0000	1.35352	0.676759	+	0.736204i	$0.263384\pi$
$632$	26.4853i	1.05353i
$633$	0	0
$634$	−48.6274	−1.93124
$635$	0	0
$636$	0	0
$637$	− 1.17157i	− 0.0464194i
$638$	− 2.65685i	− 0.105186i
$639$	0	0
$640$	0	0
$641$	−28.1127	−1.11038	−0.555192	−	0.831722i	$-0.687355\pi$
$641$	−28.1127	−1.11038	−0.555192	+	0.831722i	$0.687355\pi$
$642$	0	0
$643$	1.02944i	0.0405970i	0.999794	+	0.0202985i	$0.00646166\pi$
$643$	1.02944i	0.0405970i	−0.999794	+	0.0202985i	$0.993538\pi$
$644$	−29.3137	−1.15512
$645$	0	0
$646$	33.7990	1.32980
$647$	− 18.8284i	− 0.740222i	−0.928988	−	0.370111i	$-0.879320\pi$
$647$	− 18.8284i	− 0.740222i	0.928988	−	0.370111i	$-0.120680\pi$
$648$	0	0
$649$	−84.9411	−3.33423
$650$	0	0
$651$	0	0
$652$	38.2843i	1.49933i
$653$	− 22.4558i	− 0.878765i	−0.898300	−	0.439383i	$-0.855197\pi$
$653$	− 22.4558i	− 0.878765i	0.898300	−	0.439383i	$-0.144803\pi$
$654$	0	0
$655$	0	0
$656$	16.9706	0.662589
$657$	0	0
$658$	− 20.8995i	− 0.814747i
$659$	−8.62742	−0.336076	−0.168038	−	0.985780i	$-0.553743\pi$
$659$	−8.62742	−0.336076	−0.168038	+	0.985780i	$0.553743\pi$
$660$	0	0
$661$	−6.82843	−0.265595	−0.132798	−	0.991143i	$-0.542396\pi$
$661$	−6.82843	−0.265595	−0.132798	+	0.991143i	$0.542396\pi$
$662$	− 66.7696i	− 2.59507i
$663$	0	0
$664$	−30.4558	−1.18192
$665$	0	0
$666$	0	0
$667$	0.544156i	0.0210698i
$668$	− 20.3431i	− 0.787100i
$669$	0	0
$670$	0	0
$671$	−6.41421	−0.247618
$672$	0	0
$673$	− 24.4558i	− 0.942704i	−0.881945	−	0.471352i	$-0.843766\pi$
$673$	− 24.4558i	− 0.942704i	0.881945	−	0.471352i	$-0.156234\pi$
$674$	−50.2132	−1.93414
$675$	0	0
$676$	3.82843	0.147247
$677$	41.3137i	1.58781i	0.608039	+	0.793907i	$0.291957\pi$
$677$	41.3137i	1.58781i	−0.608039	+	0.793907i	$0.708043\pi$
$678$	0	0
$679$	2.00000	0.0767530
$680$	0	0
$681$	0	0
$682$	19.2426i	0.736839i
$683$	21.5858i	0.825957i	0.910741	+	0.412979i	$0.135512\pi$
$683$	21.5858i	0.825957i	−0.910741	+	0.412979i	$0.864488\pi$
$684$	0	0
$685$	0	0
$686$	47.6274	1.81842
$687$	0	0
$688$	− 2.48528i	− 0.0947505i
$689$	3.00000	0.114291
$690$	0	0
$691$	8.89949	0.338553	0.169276	−	0.985569i	$-0.445857\pi$
$691$	8.89949	0.338553	0.169276	+	0.985569i	$0.445857\pi$
$692$	− 71.4264i	− 2.71522i
$693$	0	0
$694$	46.9706	1.78298
$695$	0	0
$696$	0	0
$697$	21.6569i	0.820312i
$698$	85.5980i	3.23993i
$699$	0	0
$700$	0	0
$701$	15.8284	0.597831	0.298916	−	0.954280i	$-0.403375\pi$
$701$	15.8284	0.597831	0.298916	+	0.954280i	$0.403375\pi$
$702$	0	0
$703$	− 21.9411i	− 0.827525i
$704$	−63.0416	−2.37597
$705$	0	0
$706$	63.1127	2.37528
$707$	16.0711i	0.604415i
$708$	0	0
$709$	49.2548	1.84980	0.924902	−	0.380205i	$-0.124147\pi$
$709$	49.2548	1.84980	0.924902	+	0.380205i	$0.124147\pi$
$710$	0	0
$711$	0	0
$712$	− 37.4558i	− 1.40372i
$713$	− 3.94113i	− 0.147596i
$714$	0	0
$715$	0	0
$716$	−2.62742	−0.0981912
$717$	0	0
$718$	− 51.6274i	− 1.92672i
$719$	−49.4558	−1.84439	−0.922196	−	0.386723i	$-0.873607\pi$
$719$	−49.4558	−1.84439	−0.922196	+	0.386723i	$0.873607\pi$
$720$	0	0
$721$	32.9706	1.22789
$722$	13.5858i	0.505611i
$723$	0	0
$724$	66.9411	2.48785
$725$	0	0
$726$	0	0
$727$	16.6863i	0.618860i	0.950922	+	0.309430i	$0.100138\pi$
$727$	16.6863i	0.618860i	−0.950922	+	0.309430i	$0.899862\pi$
$728$	10.6569i	0.394969i
$729$	0	0
$730$	0	0
$731$	3.17157	0.117305
$732$	0	0
$733$	35.7990i	1.32227i	0.750269	+	0.661133i	$0.229924\pi$
$733$	35.7990i	1.32227i	−0.750269	+	0.661133i	$0.770076\pi$
$734$	−84.0833	−3.10357
$735$	0	0
$736$	5.02944	0.185388
$737$	17.6863i	0.651483i
$738$	0	0
$739$	27.7279	1.01999	0.509994	−	0.860178i	$-0.329648\pi$
$739$	27.7279	1.01999	0.509994	+	0.860178i	$0.329648\pi$
$740$	0	0
$741$	0	0
$742$	17.4853i	0.641905i
$743$	− 14.6985i	− 0.539235i	−0.962967	−	0.269618i	$-0.913103\pi$
$743$	− 14.6985i	− 0.539235i	0.962967	−	0.269618i	$-0.0868973\pi$
$744$	0	0
$745$	0	0
$746$	−2.75736	−0.100954
$747$	0	0
$748$	94.0122i	3.43743i
$749$	38.9706	1.42395
$750$	0	0
$751$	31.4558	1.14784	0.573920	−	0.818911i	$-0.305422\pi$
$751$	31.4558	1.14784	0.573920	+	0.818911i	$0.305422\pi$
$752$	− 10.7574i	− 0.392281i
$753$	0	0
$754$	0.414214	0.0150848
$755$	0	0
$756$	0	0
$757$	− 16.3137i	− 0.592932i	−0.955043	−	0.296466i	$-0.904192\pi$
$757$	− 16.3137i	− 0.592932i	0.955043	−	0.296466i	$-0.0958081\pi$
$758$	− 76.9411i	− 2.79463i
$759$	0	0
$760$	0	0
$761$	21.7990	0.790213	0.395106	−	0.918635i	$-0.370708\pi$
$761$	21.7990	0.790213	0.395106	+	0.918635i	$0.370708\pi$
$762$	0	0
$763$	39.7990i	1.44082i
$764$	96.9117	3.50614
$765$	0	0
$766$	66.7696	2.41248
$767$	− 13.2426i	− 0.478164i
$768$	0	0
$769$	−4.97056	−0.179243	−0.0896215	−	0.995976i	$-0.528566\pi$
$769$	−4.97056	−0.179243	−0.0896215	+	0.995976i	$0.528566\pi$
$770$	0	0
$771$	0	0
$772$	6.34315i	0.228295i
$773$	− 5.79899i	− 0.208575i	−0.994547	−	0.104288i	$-0.966744\pi$
$773$	− 5.79899i	− 0.208575i	0.994547	−	0.104288i	$-0.0332562\pi$
$774$	0	0
$775$	0	0
$776$	3.65685	0.131273
$777$	0	0
$778$	− 69.1127i	− 2.47781i
$779$	20.6863	0.741163
$780$	0	0
$781$	−59.7401	−2.13767
$782$	− 29.3137i	− 1.04826i
$783$	0	0
$784$	3.51472	0.125526
$785$	0	0
$786$	0	0
$787$	38.7574i	1.38155i	0.723069	+	0.690775i	$0.242731\pi$
$787$	38.7574i	1.38155i	−0.723069	+	0.690775i	$0.757269\pi$
$788$	− 45.9411i	− 1.63658i
$789$	0	0
$790$	0	0
$791$	28.1421	1.00062
$792$	0	0
$793$	− 1.00000i	− 0.0355110i
$794$	18.4853	0.656018
$795$	0	0
$796$	38.2843	1.35695
$797$	11.4853i	0.406830i	0.979093	+	0.203415i	$0.0652040\pi$
$797$	11.4853i	0.406830i	−0.979093	+	0.203415i	$0.934796\pi$
$798$	0	0
$799$	13.7279	0.485659
$800$	0	0
$801$	0	0
$802$	− 9.31371i	− 0.328878i
$803$	− 38.4853i	− 1.35812i
$804$	0	0
$805$	0	0
$806$	−3.00000	−0.105670
$807$	0	0
$808$	29.3848i	1.03375i
$809$	17.3137	0.608718	0.304359	−	0.952557i	$-0.401558\pi$
$809$	17.3137	0.608718	0.304359	+	0.952557i	$0.401558\pi$
$810$	0	0
$811$	9.58579	0.336602	0.168301	−	0.985736i	$-0.446172\pi$
$811$	9.58579	0.336602	0.168301	+	0.985736i	$0.446172\pi$
$812$	1.58579i	0.0556502i
$813$	0	0
$814$	92.9117	3.25655
$815$	0	0
$816$	0	0
$817$	− 3.02944i	− 0.105987i
$818$	− 26.1421i	− 0.914038i
$819$	0	0
$820$	0	0
$821$	20.8284	0.726917	0.363459	−	0.931610i	$-0.381596\pi$
$821$	20.8284	0.726917	0.363459	+	0.931610i	$0.381596\pi$
$822$	0	0
$823$	20.2843i	0.707065i	0.935422	+	0.353533i	$0.115020\pi$
$823$	20.2843i	0.707065i	−0.935422	+	0.353533i	$0.884980\pi$
$824$	60.2843	2.10010
$825$	0	0
$826$	77.1838	2.68557
$827$	32.3553i	1.12511i	0.826761	+	0.562553i	$0.190181\pi$
$827$	32.3553i	1.12511i	−0.826761	+	0.562553i	$0.809819\pi$
$828$	0	0
$829$	−7.97056	−0.276829	−0.138415	−	0.990374i	$-0.544201\pi$
$829$	−7.97056	−0.276829	−0.138415	+	0.990374i	$0.544201\pi$
$830$	0	0
$831$	0	0
$832$	− 9.82843i	− 0.340739i
$833$	4.48528i	0.155406i
$834$	0	0
$835$	0	0
$836$	89.7990	3.10576
$837$	0	0
$838$	− 55.1127i	− 1.90384i
$839$	−1.02944	−0.0355401	−0.0177701	−	0.999842i	$-0.505657\pi$
$839$	−1.02944	−0.0355401	−0.0177701	+	0.999842i	$0.505657\pi$
$840$	0	0
$841$	−28.9706	−0.998985
$842$	40.9706i	1.41194i
$843$	0	0
$844$	−68.1421	−2.34555
$845$	0	0
$846$	0	0
$847$	72.7696i	2.50039i
$848$	9.00000i	0.309061i
$849$	0	0
$850$	0	0
$851$	−19.0294	−0.652321
$852$	0	0
$853$	42.4264i	1.45265i	0.687350	+	0.726326i	$0.258774\pi$
$853$	42.4264i	1.45265i	−0.687350	+	0.726326i	$0.741226\pi$
$854$	5.82843	0.199445
$855$	0	0
$856$	71.2548	2.43544
$857$	4.62742i	0.158070i	0.996872	+	0.0790348i	$0.0251838\pi$
$857$	4.62742i	0.158070i	−0.996872	+	0.0790348i	$0.974816\pi$
$858$	0	0
$859$	−24.1421	−0.823719	−0.411860	−	0.911247i	$-0.635121\pi$
$859$	−24.1421	−0.823719	−0.411860	+	0.911247i	$0.635121\pi$
$860$	0	0
$861$	0	0
$862$	− 20.1421i	− 0.686044i
$863$	29.1838i	0.993427i	0.867915	+	0.496713i	$0.165460\pi$
$863$	29.1838i	0.993427i	−0.867915	+	0.496713i	$0.834540\pi$
$864$	0	0
$865$	0	0
$866$	−39.4558	−1.34076
$867$	0	0
$868$	− 11.4853i	− 0.389836i
$869$	−38.4853	−1.30552
$870$	0	0
$871$	−2.75736	−0.0934296
$872$	72.7696i	2.46429i
$873$	0	0
$874$	−28.0000	−0.947114
$875$	0	0
$876$	0	0
$877$	− 15.7990i	− 0.533494i	−0.963767	−	0.266747i	$-0.914051\pi$
$877$	− 15.7990i	− 0.533494i	0.963767	−	0.266747i	$-0.0859488\pi$
$878$	33.7990i	1.14066i
$879$	0	0
$880$	0	0
$881$	15.0000	0.505363	0.252681	−	0.967550i	$-0.418688\pi$
$881$	15.0000	0.505363	0.252681	+	0.967550i	$0.418688\pi$
$882$	0	0
$883$	− 32.9706i	− 1.10955i	−0.832001	−	0.554774i	$-0.812805\pi$
$883$	− 32.9706i	− 1.10955i	0.832001	−	0.554774i	$-0.187195\pi$
$884$	−14.6569	−0.492963
$885$	0	0
$886$	−71.9411	−2.41691
$887$	− 8.62742i	− 0.289680i	−0.989455	−	0.144840i	$-0.953733\pi$
$887$	− 8.62742i	− 0.289680i	0.989455	−	0.144840i	$-0.0462668\pi$
$888$	0	0
$889$	8.82843	0.296096
$890$	0	0
$891$	0	0
$892$	− 42.0000i	− 1.40626i
$893$	− 13.1127i	− 0.438800i
$894$	0	0
$895$	0	0
$896$	49.6274	1.65794
$897$	0	0
$898$	12.4853i	0.416639i
$899$	−0.213203	−0.00711073
$900$	0	0
$901$	−11.4853	−0.382630
$902$	87.5980i	2.91669i
$903$	0	0
$904$	51.4558	1.71140
$905$	0	0
$906$	0	0
$907$	− 24.0000i	− 0.796907i	−0.917189	−	0.398453i	$-0.869547\pi$
$907$	− 24.0000i	− 0.796907i	0.917189	−	0.398453i	$-0.130453\pi$
$908$	− 34.0711i	− 1.13069i
$909$	0	0
$910$	0	0
$911$	−6.00000	−0.198789	−0.0993944	−	0.995048i	$-0.531691\pi$
$911$	−6.00000	−0.198789	−0.0993944	+	0.995048i	$0.531691\pi$
$912$	0	0
$913$	− 44.2548i	− 1.46462i
$914$	20.4853	0.677593
$915$	0	0
$916$	−18.4853	−0.610771
$917$	− 20.4853i	− 0.676484i
$918$	0	0
$919$	18.9706	0.625781	0.312891	−	0.949789i	$-0.398703\pi$
$919$	18.9706	0.625781	0.312891	+	0.949789i	$0.398703\pi$
$920$	0	0
$921$	0	0
$922$	− 46.9706i	− 1.54689i
$923$	− 9.31371i	− 0.306564i
$924$	0	0
$925$	0	0
$926$	−68.9411	−2.26555
$927$	0	0
$928$	− 0.272078i	− 0.00893140i
$929$	30.2843	0.993595	0.496797	−	0.867867i	$-0.334509\pi$
$929$	30.2843	0.993595	0.496797	+	0.867867i	$0.334509\pi$
$930$	0	0
$931$	4.28427	0.140411
$932$	− 78.9706i	− 2.58677i
$933$	0	0
$934$	−75.1127	−2.45776
$935$	0	0
$936$	0	0
$937$	1.97056i	0.0643755i	0.999482	+	0.0321877i	$0.0102474\pi$
$937$	1.97056i	0.0643755i	−0.999482	+	0.0321877i	$0.989753\pi$
$938$	− 16.0711i	− 0.524739i
$939$	0	0
$940$	0	0
$941$	35.6569	1.16238	0.581190	−	0.813768i	$-0.302587\pi$
$941$	35.6569	1.16238	0.581190	+	0.813768i	$0.302587\pi$
$942$	0	0
$943$	− 17.9411i	− 0.584243i
$944$	39.7279	1.29303
$945$	0	0
$946$	12.8284	0.417088
$947$	26.4142i	0.858347i	0.903222	+	0.429173i	$0.141195\pi$
$947$	26.4142i	0.858347i	−0.903222	+	0.429173i	$0.858805\pi$
$948$	0	0
$949$	6.00000	0.194768
$950$	0	0
$951$	0	0
$952$	− 40.7990i	− 1.32230i
$953$	− 47.3431i	− 1.53359i	−0.641889	−	0.766797i	$-0.721849\pi$
$953$	− 47.3431i	− 1.53359i	0.641889	−	0.766797i	$-0.278151\pi$
$954$	0	0
$955$	0	0
$956$	−90.2965	−2.92040
$957$	0	0
$958$	− 34.4558i	− 1.11322i
$959$	6.82843	0.220501
$960$	0	0
$961$	−29.4558	−0.950189
$962$	14.4853i	0.467024i
$963$	0	0
$964$	−19.0294	−0.612897
$965$	0	0
$966$	0	0
$967$	43.7279i	1.40620i	0.711093	+	0.703098i	$0.248200\pi$
$967$	43.7279i	1.40620i	−0.711093	+	0.703098i	$0.751800\pi$
$968$	133.054i	4.27651i
$969$	0	0
$970$	0	0
$971$	−36.8284	−1.18188	−0.590940	−	0.806715i	$-0.701243\pi$
$971$	−36.8284	−1.18188	−0.590940	+	0.806715i	$0.701243\pi$
$972$	0	0
$973$	44.6274i	1.43069i
$974$	−101.912	−3.26546
$975$	0	0
$976$	3.00000	0.0960277
$977$	8.48528i	0.271468i	0.990745	+	0.135734i	$0.0433393\pi$
$977$	8.48528i	0.271468i	−0.990745	+	0.135734i	$0.956661\pi$
$978$	0	0
$979$	54.4264	1.73948
$980$	0	0
$981$	0	0
$982$	26.9706i	0.860665i
$983$	26.6985i	0.851549i	0.904829	+	0.425775i	$0.139998\pi$
$983$	26.6985i	0.851549i	−0.904829	+	0.425775i	$0.860002\pi$
$984$	0	0
$985$	0	0
$986$	−1.58579	−0.0505017
$987$	0	0
$988$	14.0000i	0.445399i
$989$	−2.62742	−0.0835470
$990$	0	0
$991$	−13.5147	−0.429309	−0.214655	−	0.976690i	$-0.568863\pi$
$991$	−13.5147	−0.429309	−0.214655	+	0.976690i	$0.568863\pi$
$992$	1.97056i	0.0625654i
$993$	0	0
$994$	54.2843	1.72179
$995$	0	0
$996$	0	0
$997$	12.6569i	0.400847i	0.979709	+	0.200423i	$0.0642317\pi$
$997$	12.6569i	0.400847i	−0.979709	+	0.200423i	$0.935768\pi$
$998$	6.17157i	0.195358i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2925.2.c.q.2224.1		4
3.2	odd	2		325.2.b.d.274.4		4
5.2	odd	4		2925.2.a.bd.1.2		2
5.3	odd	4		2925.2.a.w.1.1		2
5.4	even	2	inner	2925.2.c.q.2224.4		4
15.2	even	4		325.2.a.f.1.1	✓	2
15.8	even	4		325.2.a.h.1.2	yes	2
15.14	odd	2		325.2.b.d.274.1		4
60.23	odd	4		5200.2.a.br.1.2		2
60.47	odd	4		5200.2.a.bt.1.1		2
195.38	even	4		4225.2.a.s.1.1		2
195.77	even	4		4225.2.a.z.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
325.2.a.f.1.1	✓	2	15.2	even	4
325.2.a.h.1.2	yes	2	15.8	even	4
325.2.b.d.274.1		4	15.14	odd	2
325.2.b.d.274.4		4	3.2	odd	2
2925.2.a.w.1.1		2	5.3	odd	4
2925.2.a.bd.1.2		2	5.2	odd	4
2925.2.c.q.2224.1		4	1.1	even	1	trivial
2925.2.c.q.2224.4		4	5.4	even	2	inner
4225.2.a.s.1.1		2	195.38	even	4
4225.2.a.z.1.2		2	195.77	even	4
5200.2.a.br.1.2		2	60.23	odd	4
5200.2.a.bt.1.1		2	60.47	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 $\F_{q}$
Belyi maps

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)

\(f(q)\)	\(=\)	\(q-2.41421i q^{2} -3.82843 q^{4} +2.41421i q^{7} +4.41421i q^{8} -6.41421 q^{11} -1.00000i q^{13} +5.82843 q^{14} +3.00000 q^{16} +3.82843i q^{17} +3.65685 q^{19} +15.4853i q^{22} -3.17157i q^{23} -2.41421 q^{26} -9.24264i q^{28} -0.171573 q^{29} +1.24264 q^{31} +1.58579i q^{32} +9.24264 q^{34} -6.00000i q^{37} -8.82843i q^{38} +5.65685 q^{41} -0.828427i q^{43} +24.5563 q^{44} -7.65685 q^{46} -3.58579i q^{47} +1.17157 q^{49} +3.82843i q^{52} +3.00000i q^{53} -10.6569 q^{56} +0.414214i q^{58} +13.2426 q^{59} +1.00000 q^{61} -3.00000i q^{62} +9.82843 q^{64} -2.75736i q^{67} -14.6569i q^{68} +9.31371 q^{71} +6.00000i q^{73} -14.4853 q^{74} -14.0000 q^{76} -15.4853i q^{77} +6.00000 q^{79} -13.6569i q^{82} +6.89949i q^{83} -2.00000 q^{86} -28.3137i q^{88} -8.48528 q^{89} +2.41421 q^{91} +12.1421i q^{92} -8.65685 q^{94} -0.828427i q^{97} -2.82843i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4} - 20 q^{11} + 12 q^{14} + 12 q^{16} - 8 q^{19} - 4 q^{26} - 12 q^{29} - 12 q^{31} + 20 q^{34} + 36 q^{44} - 8 q^{46} + 16 q^{49} - 20 q^{56} + 36 q^{59} + 4 q^{61} + 28 q^{64} - 8 q^{71} - 24 q^{74}+ \cdots - 12 q^{94}+O(q^{100}) \) 4 * q - 4 * q^4 - 20 * q^11 + 12 * q^14 + 12 * q^16 - 8 * q^19 - 4 * q^26 - 12 * q^29 - 12 * q^31 + 20 * q^34 + 36 * q^44 - 8 * q^46 + 16 * q^49 - 20 * q^56 + 36 * q^59 + 4 * q^61 + 28 * q^64 - 8 * q^71 - 24 * q^74 - 56 * q^76 + 24 * q^79 - 8 * q^86 + 4 * q^91 - 12 * q^94

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