LMFDB - Embedded newform 7605.2.a.bw.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7605,2,Mod(1,7605)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7605, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("7605.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7605 = 3^{2} \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7605.a (trivial)

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:	yes
Analytic conductor:	$60.7262307372$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.756.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 6x - 2 $ x^3 - 6*x - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 195)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.26180$ of defining polynomial
Character	$\chi$	$=$	7605.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.26180 q^{2} +3.11575 q^{4} +1.00000 q^{5} +1.26180 q^{7} -2.52360 q^{8} +O(q^{10})$	$q-2.26180 q^{2} +3.11575 q^{4} +1.00000 q^{5} +1.26180 q^{7} -2.52360 q^{8} -2.26180 q^{10} +4.52360 q^{11} -2.85395 q^{14} -0.523604 q^{16} +4.49330 q^{17} -5.11575 q^{19} +3.11575 q^{20} -10.2315 q^{22} -2.23150 q^{23} +1.00000 q^{25} +3.93146 q^{28} +1.37755 q^{29} -8.87085 q^{31} +6.23150 q^{32} -10.1630 q^{34} +1.26180 q^{35} +0.231499 q^{37} +11.5708 q^{38} -2.52360 q^{40} +1.14605 q^{41} +6.37755 q^{43} +14.0944 q^{44} +5.04721 q^{46} -10.7854 q^{47} -5.40786 q^{49} -2.26180 q^{50} -4.52360 q^{53} +4.52360 q^{55} -3.18429 q^{56} -3.11575 q^{58} -0.853947 q^{59} +4.63935 q^{61} +20.0641 q^{62} -13.0472 q^{64} -13.1327 q^{67} +14.0000 q^{68} -2.85395 q^{70} -9.60905 q^{71} -13.7854 q^{73} -0.523604 q^{74} -15.9394 q^{76} +5.70789 q^{77} -8.87085 q^{79} -0.523604 q^{80} -2.59214 q^{82} -8.23150 q^{83} +4.49330 q^{85} -14.4248 q^{86} -11.4158 q^{88} +6.62245 q^{89} -6.95279 q^{92} +24.3945 q^{94} -5.11575 q^{95} -10.6697 q^{97} +12.2315 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 6 q^{4} + 3 q^{5} - 3 q^{7} + 6 q^{8}+O(q^{10}) $ 3 * q + 6 * q^4 + 3 * q^5 - 3 * q^7 + 6 * q^8	$ 3 q + 6 q^{4} + 3 q^{5} - 3 q^{7} + 6 q^{8} - 12 q^{14} + 12 q^{16} - 12 q^{19} + 6 q^{20} - 24 q^{22} + 3 q^{25} - 12 q^{28} - 6 q^{29} - 3 q^{31} + 12 q^{32} - 3 q^{35} - 6 q^{37} - 6 q^{38} + 6 q^{40} + 9 q^{43} - 12 q^{44} - 12 q^{46} - 12 q^{47} - 6 q^{49} - 30 q^{56} - 6 q^{58} - 6 q^{59} - 3 q^{61} + 6 q^{62} - 12 q^{64} - 9 q^{67} + 42 q^{68} - 12 q^{70} - 12 q^{71} - 21 q^{73} + 12 q^{74} - 48 q^{76} + 24 q^{77} - 3 q^{79} + 12 q^{80} - 18 q^{82} - 18 q^{83} - 6 q^{86} - 48 q^{88} + 30 q^{89} - 48 q^{92} + 36 q^{94} - 12 q^{95} - 15 q^{97} + 30 q^{98}+O(q^{100}) $ 3 * q + 6 * q^4 + 3 * q^5 - 3 * q^7 + 6 * q^8 - 12 * q^14 + 12 * q^16 - 12 * q^19 + 6 * q^20 - 24 * q^22 + 3 * q^25 - 12 * q^28 - 6 * q^29 - 3 * q^31 + 12 * q^32 - 3 * q^35 - 6 * q^37 - 6 * q^38 + 6 * q^40 + 9 * q^43 - 12 * q^44 - 12 * q^46 - 12 * q^47 - 6 * q^49 - 30 * q^56 - 6 * q^58 - 6 * q^59 - 3 * q^61 + 6 * q^62 - 12 * q^64 - 9 * q^67 + 42 * q^68 - 12 * q^70 - 12 * q^71 - 21 * q^73 + 12 * q^74 - 48 * q^76 + 24 * q^77 - 3 * q^79 + 12 * q^80 - 18 * q^82 - 18 * q^83 - 6 * q^86 - 48 * q^88 + 30 * q^89 - 48 * q^92 + 36 * q^94 - 12 * q^95 - 15 * q^97 + 30 * q^98

Display 100 coefficients 10 coefficients

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 $

$2$	−2.26180	−1.59934	−0.799668	−	0.600443i	$-0.794991\pi$
$2$	−2.26180	−1.59934	−0.799668	+	0.600443i	$0.794991\pi$
$3$	0	0
$3$	0	0
$4$	3.11575	1.55787
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	1.26180	0.476916	0.238458	−	0.971153i	$-0.423358\pi$
$7$	1.26180	0.476916	0.238458	+	0.971153i	$0.423358\pi$
$8$	−2.52360	−0.892229
$9$	0	0
$10$	−2.26180	−0.715245
$11$	4.52360	1.36392	0.681959	−	0.731390i	$-0.261128\pi$
$11$	4.52360	1.36392	0.681959	+	0.731390i	$0.261128\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	−2.85395	−0.762749
$15$	0	0
$16$	−0.523604	−0.130901
$17$	4.49330	1.08979	0.544893	−	0.838506i	$-0.316570\pi$
$17$	4.49330	1.08979	0.544893	+	0.838506i	$0.316570\pi$
$18$	0	0
$19$	−5.11575	−1.17363	−0.586817	−	0.809720i	$-0.699619\pi$
$19$	−5.11575	−1.17363	−0.586817	+	0.809720i	$0.699619\pi$
$20$	3.11575	0.696703
$21$	0	0
$22$	−10.2315	−2.18136
$23$	−2.23150	−0.465300	−0.232650	−	0.972561i	$-0.574740\pi$
$23$	−2.23150	−0.465300	−0.232650	+	0.972561i	$0.574740\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	3.93146	0.742976
$29$	1.37755	0.255805	0.127902	−	0.991787i	$-0.459176\pi$
$29$	1.37755	0.255805	0.127902	+	0.991787i	$0.459176\pi$
$30$	0	0
$31$	−8.87085	−1.59325	−0.796626	−	0.604472i	$-0.793384\pi$
$31$	−8.87085	−1.59325	−0.796626	+	0.604472i	$0.793384\pi$
$32$	6.23150	1.10158
$33$	0	0
$34$	−10.1630	−1.74293
$35$	1.26180	0.213284
$36$	0	0
$37$	0.231499	0.0380582	0.0190291	−	0.999819i	$-0.493942\pi$
$37$	0.231499	0.0380582	0.0190291	+	0.999819i	$0.493942\pi$
$38$	11.5708	1.87703
$39$	0	0
$40$	−2.52360	−0.399017
$41$	1.14605	0.178983	0.0894917	−	0.995988i	$-0.471476\pi$
$41$	1.14605	0.178983	0.0894917	+	0.995988i	$0.471476\pi$
$42$	0	0
$43$	6.37755	0.972568	0.486284	−	0.873801i	$-0.338352\pi$
$43$	6.37755	0.972568	0.486284	+	0.873801i	$0.338352\pi$
$44$	14.0944	2.12481
$45$	0	0
$46$	5.04721	0.744170
$47$	−10.7854	−1.57321	−0.786607	−	0.617454i	$-0.788164\pi$
$47$	−10.7854	−1.57321	−0.786607	+	0.617454i	$0.788164\pi$
$48$	0	0
$49$	−5.40786	−0.772551
$50$	−2.26180	−0.319867
$51$	0	0
$52$	0	0
$53$	−4.52360	−0.621365	−0.310682	−	0.950514i	$-0.600558\pi$
$53$	−4.52360	−0.621365	−0.310682	+	0.950514i	$0.600558\pi$
$54$	0	0
$55$	4.52360	0.609963
$56$	−3.18429	−0.425519
$57$	0	0
$58$	−3.11575	−0.409118
$59$	−0.853947	−0.111174	−0.0555872	−	0.998454i	$-0.517703\pi$
$59$	−0.853947	−0.111174	−0.0555872	+	0.998454i	$0.517703\pi$
$60$	0	0
$61$	4.63935	0.594008	0.297004	−	0.954876i	$-0.404012\pi$
$61$	4.63935	0.594008	0.297004	+	0.954876i	$0.404012\pi$
$62$	20.0641	2.54815
$63$	0	0
$64$	−13.0472	−1.63090
$65$	0	0
$66$	0	0
$67$	−13.1327	−1.60441	−0.802205	−	0.597049i	$-0.796340\pi$
$67$	−13.1327	−1.60441	−0.802205	+	0.597049i	$0.796340\pi$
$68$	14.0000	1.69775
$69$	0	0
$70$	−2.85395	−0.341112
$71$	−9.60905	−1.14038	−0.570192	−	0.821511i	$-0.693131\pi$
$71$	−9.60905	−1.14038	−0.570192	+	0.821511i	$0.693131\pi$
$72$	0	0
$73$	−13.7854	−1.61346	−0.806730	−	0.590920i	$-0.798765\pi$
$73$	−13.7854	−1.61346	−0.806730	+	0.590920i	$0.798765\pi$
$74$	−0.523604	−0.0608678
$75$	0	0
$76$	−15.9394	−1.82837
$77$	5.70789	0.650475
$78$	0	0
$79$	−8.87085	−0.998049	−0.499024	−	0.866588i	$-0.666308\pi$
$79$	−8.87085	−0.998049	−0.499024	+	0.866588i	$0.666308\pi$
$80$	−0.523604	−0.0585408
$81$	0	0
$82$	−2.59214	−0.286255
$83$	−8.23150	−0.903524	−0.451762	−	0.892138i	$-0.649204\pi$
$83$	−8.23150	−0.903524	−0.451762	+	0.892138i	$0.649204\pi$
$84$	0	0
$85$	4.49330	0.487367
$86$	−14.4248	−1.55546
$87$	0	0
$88$	−11.4158	−1.21693
$89$	6.62245	0.701978	0.350989	−	0.936380i	$-0.385845\pi$
$89$	6.62245	0.701978	0.350989	+	0.936380i	$0.385845\pi$
$90$	0	0
$91$	0	0
$92$	−6.95279	−0.724879
$93$	0	0
$94$	24.3945	2.51610
$95$	−5.11575	−0.524865
$96$	0	0
$97$	−10.6697	−1.08334	−0.541670	−	0.840591i	$-0.682208\pi$
$97$	−10.6697	−1.08334	−0.541670	+	0.840591i	$0.682208\pi$
$98$	12.2315	1.23557
$99$	0	0
$100$	3.11575	0.311575
$101$	17.0472	1.69626	0.848130	−	0.529788i	$-0.177728\pi$
$101$	17.0472	1.69626	0.848130	+	0.529788i	$0.177728\pi$
$102$	0	0
$103$	−13.9484	−1.37437	−0.687187	−	0.726481i	$-0.741155\pi$
$103$	−13.9484	−1.37437	−0.687187	+	0.726481i	$0.741155\pi$
$104$	0	0
$105$	0	0
$106$	10.2315	0.993771
$107$	−12.4933	−1.20777	−0.603886	−	0.797070i	$-0.706382\pi$
$107$	−12.4933	−1.20777	−0.603886	+	0.797070i	$0.706382\pi$
$108$	0	0
$109$	2.27871	0.218261	0.109130	−	0.994027i	$-0.465193\pi$
$109$	2.27871	0.218261	0.109130	+	0.994027i	$0.465193\pi$
$110$	−10.2315	−0.975535
$111$	0	0
$112$	−0.660685	−0.0624289
$113$	1.47640	0.138888	0.0694438	−	0.997586i	$-0.477878\pi$
$113$	1.47640	0.138888	0.0694438	+	0.997586i	$0.477878\pi$
$114$	0	0
$115$	−2.23150	−0.208088
$116$	4.29211	0.398512
$117$	0	0
$118$	1.93146	0.177805
$119$	5.66966	0.519737
$120$	0	0
$121$	9.46300	0.860273
$122$	−10.4933	−0.950019
$123$	0	0
$124$	−27.6394	−2.48209
$125$	1.00000	0.0894427
$126$	0	0
$127$	−17.7854	−1.57820	−0.789100	−	0.614265i	$-0.789453\pi$
$127$	−17.7854	−1.57820	−0.789100	+	0.614265i	$0.789453\pi$
$128$	17.0472	1.50677
$129$	0	0
$130$	0	0
$131$	−11.6697	−1.01958	−0.509791	−	0.860298i	$-0.670277\pi$
$131$	−11.6697	−1.01958	−0.509791	+	0.860298i	$0.670277\pi$
$132$	0	0
$133$	−6.45506	−0.559725
$134$	29.7035	2.56599
$135$	0	0
$136$	−11.3393	−0.972338
$137$	−20.0641	−1.71419	−0.857096	−	0.515156i	$-0.827734\pi$
$137$	−20.0641	−1.71419	−0.857096	+	0.515156i	$0.827734\pi$
$138$	0	0
$139$	16.3393	1.38588	0.692941	−	0.720994i	$-0.256314\pi$
$139$	16.3393	1.38588	0.692941	+	0.720994i	$0.256314\pi$
$140$	3.93146	0.332269
$141$	0	0
$142$	21.7338	1.82386
$143$	0	0
$144$	0	0
$145$	1.37755	0.114399
$146$	31.1799	2.58046
$147$	0	0
$148$	0.721292	0.0592899
$149$	−3.24490	−0.265832	−0.132916	−	0.991127i	$-0.542434\pi$
$149$	−3.24490	−0.265832	−0.132916	+	0.991127i	$0.542434\pi$
$150$	0	0
$151$	−5.69996	−0.463856	−0.231928	−	0.972733i	$-0.574503\pi$
$151$	−5.69996	−0.463856	−0.231928	+	0.972733i	$0.574503\pi$
$152$	12.9101	1.04715
$153$	0	0
$154$	−12.9101	−1.04033
$155$	−8.87085	−0.712524
$156$	0	0
$157$	−3.85395	−0.307578	−0.153789	−	0.988104i	$-0.549148\pi$
$157$	−3.85395	−0.307578	−0.153789	+	0.988104i	$0.549148\pi$
$158$	20.0641	1.59622
$159$	0	0
$160$	6.23150	0.492643
$161$	−2.81571	−0.221909
$162$	0	0
$163$	9.72480	0.761705	0.380853	−	0.924636i	$-0.375631\pi$
$163$	9.72480	0.761705	0.380853	+	0.924636i	$0.375631\pi$
$164$	3.57081	0.278834
$165$	0	0
$166$	18.6180	1.44504
$167$	−19.2787	−1.49183	−0.745916	−	0.666040i	$-0.767988\pi$
$167$	−19.2787	−1.49183	−0.745916	+	0.666040i	$0.767988\pi$
$168$	0	0
$169$	0	0
$170$	−10.1630	−0.779463
$171$	0	0
$172$	19.8709	1.51514
$173$	3.01691	0.229371	0.114686	−	0.993402i	$-0.463414\pi$
$173$	3.01691	0.229371	0.114686	+	0.993402i	$0.463414\pi$
$174$	0	0
$175$	1.26180	0.0953833
$176$	−2.36858	−0.178538
$177$	0	0
$178$	−14.9787	−1.12270
$179$	5.31694	0.397407	0.198704	−	0.980060i	$-0.436327\pi$
$179$	5.31694	0.397407	0.198704	+	0.980060i	$0.436327\pi$
$180$	0	0
$181$	25.8709	1.92297	0.961483	−	0.274866i	$-0.0886333\pi$
$181$	25.8709	1.92297	0.961483	+	0.274866i	$0.0886333\pi$
$182$	0	0
$183$	0	0
$184$	5.63142	0.415154
$185$	0.231499	0.0170201
$186$	0	0
$187$	20.3259	1.48638
$188$	−33.6046	−2.45087
$189$	0	0
$190$	11.5708	0.839435
$191$	−10.6563	−0.771060	−0.385530	−	0.922695i	$-0.625981\pi$
$191$	−10.6563	−0.771060	−0.385530	+	0.922695i	$0.625981\pi$
$192$	0	0
$193$	−7.98309	−0.574636	−0.287318	−	0.957835i	$-0.592764\pi$
$193$	−7.98309	−0.574636	−0.287318	+	0.957835i	$0.592764\pi$
$194$	24.1327	1.73262
$195$	0	0
$196$	−16.8495	−1.20354
$197$	17.3393	1.23538	0.617688	−	0.786424i	$-0.288070\pi$
$197$	17.3393	1.23538	0.617688	+	0.786424i	$0.288070\pi$
$198$	0	0
$199$	−1.23150	−0.0872986	−0.0436493	−	0.999047i	$-0.513898\pi$
$199$	−1.23150	−0.0872986	−0.0436493	+	0.999047i	$0.513898\pi$
$200$	−2.52360	−0.178446
$201$	0	0
$202$	−38.5574	−2.71289
$203$	1.73820	0.121998
$204$	0	0
$205$	1.14605	0.0800438
$206$	31.5484	2.19808
$207$	0	0
$208$	0	0
$209$	−23.1416	−1.60074
$210$	0	0
$211$	10.3393	0.711788	0.355894	−	0.934526i	$-0.384176\pi$
$211$	10.3393	0.711788	0.355894	+	0.934526i	$0.384176\pi$
$212$	−14.0944	−0.968009
$213$	0	0
$214$	28.2574	1.93163
$215$	6.37755	0.434945
$216$	0	0
$217$	−11.1933	−0.759848
$218$	−5.15399	−0.349072
$219$	0	0
$220$	14.0944	0.950245
$221$	0	0
$222$	0	0
$223$	−21.5708	−1.44449	−0.722244	−	0.691638i	$-0.756889\pi$
$223$	−21.5708	−1.44449	−0.722244	+	0.691638i	$0.756889\pi$
$224$	7.86292	0.525363
$225$	0	0
$226$	−3.33931	−0.222128
$227$	5.44609	0.361470	0.180735	−	0.983532i	$-0.442152\pi$
$227$	5.44609	0.361470	0.180735	+	0.983532i	$0.442152\pi$
$228$	0	0
$229$	−21.9315	−1.44927	−0.724636	−	0.689132i	$-0.757992\pi$
$229$	−21.9315	−1.44927	−0.724636	+	0.689132i	$0.757992\pi$
$230$	5.04721	0.332803
$231$	0	0
$232$	−3.47640	−0.228237
$233$	−25.3125	−1.65828	−0.829139	−	0.559042i	$-0.811169\pi$
$233$	−25.3125	−1.65828	−0.829139	+	0.559042i	$0.811169\pi$
$234$	0	0
$235$	−10.7854	−0.703562
$236$	−2.66069	−0.173196
$237$	0	0
$238$	−12.8236	−0.831233
$239$	22.5236	1.45693	0.728465	−	0.685083i	$-0.240234\pi$
$239$	22.5236	1.45693	0.728465	+	0.685083i	$0.240234\pi$
$240$	0	0
$241$	−11.1157	−0.716028	−0.358014	−	0.933716i	$-0.616546\pi$
$241$	−11.1157	−0.716028	−0.358014	+	0.933716i	$0.616546\pi$
$242$	−21.4034	−1.37586
$243$	0	0
$244$	14.4551	0.925391
$245$	−5.40786	−0.345495
$246$	0	0
$247$	0	0
$248$	22.3865	1.42155
$249$	0	0
$250$	−2.26180	−0.143049
$251$	−11.0472	−0.697294	−0.348647	−	0.937254i	$-0.613359\pi$
$251$	−11.0472	−0.697294	−0.348647	+	0.937254i	$0.613359\pi$
$252$	0	0
$253$	−10.0944	−0.634631
$254$	40.2271	2.52407
$255$	0	0
$256$	−12.4630	−0.778937
$257$	2.03030	0.126647	0.0633234	−	0.997993i	$-0.479830\pi$
$257$	2.03030	0.126647	0.0633234	+	0.997993i	$0.479830\pi$
$258$	0	0
$259$	0.292106	0.0181506
$260$	0	0
$261$	0	0
$262$	26.3945	1.63066
$263$	−3.24840	−0.200305	−0.100153	−	0.994972i	$-0.531933\pi$
$263$	−3.24840	−0.200305	−0.100153	+	0.994972i	$0.531933\pi$
$264$	0	0
$265$	−4.52360	−0.277883
$266$	14.6001	0.895188
$267$	0	0
$268$	−40.9181	−2.49947
$269$	−26.9484	−1.64307	−0.821535	−	0.570157i	$-0.806882\pi$
$269$	−26.9484	−1.64307	−0.821535	+	0.570157i	$0.806882\pi$
$270$	0	0
$271$	19.8495	1.20577	0.602886	−	0.797827i	$-0.294017\pi$
$271$	19.8495	1.20577	0.602886	+	0.797827i	$0.294017\pi$
$272$	−2.35271	−0.142654
$273$	0	0
$274$	45.3811	2.74157
$275$	4.52360	0.272784
$276$	0	0
$277$	16.0944	0.967020	0.483510	−	0.875339i	$-0.339362\pi$
$277$	16.0944	0.967020	0.483510	+	0.875339i	$0.339362\pi$
$278$	−36.9563	−2.21649
$279$	0	0
$280$	−3.18429	−0.190298
$281$	−5.37755	−0.320798	−0.160399	−	0.987052i	$-0.551278\pi$
$281$	−5.37755	−0.320798	−0.160399	+	0.987052i	$0.551278\pi$
$282$	0	0
$283$	26.8664	1.59704	0.798522	−	0.601966i	$-0.205616\pi$
$283$	26.8664	1.59704	0.798522	+	0.601966i	$0.205616\pi$
$284$	−29.9394	−1.77658
$285$	0	0
$286$	0	0
$287$	1.44609	0.0853601
$288$	0	0
$289$	3.18975	0.187633
$290$	−3.11575	−0.182963
$291$	0	0
$292$	−42.9519	−2.51357
$293$	21.4193	1.25133	0.625664	−	0.780092i	$-0.284828\pi$
$293$	21.4193	1.25133	0.625664	+	0.780092i	$0.284828\pi$
$294$	0	0
$295$	−0.853947	−0.0497187
$296$	−0.584211	−0.0339566
$297$	0	0
$298$	7.33931	0.425155
$299$	0	0
$300$	0	0
$301$	8.04721	0.463833
$302$	12.8922	0.741862
$303$	0	0
$304$	2.67863	0.153630
$305$	4.63935	0.265649
$306$	0	0
$307$	7.85395	0.448248	0.224124	−	0.974561i	$-0.428048\pi$
$307$	7.85395	0.448248	0.224124	+	0.974561i	$0.428048\pi$
$308$	17.7844	1.01336
$309$	0	0
$310$	20.0641	1.13957
$311$	27.8744	1.58061	0.790305	−	0.612714i	$-0.209922\pi$
$311$	27.8744	1.58061	0.790305	+	0.612714i	$0.209922\pi$
$312$	0	0
$313$	7.88776	0.445842	0.222921	−	0.974836i	$-0.428441\pi$
$313$	7.88776	0.445842	0.222921	+	0.974836i	$0.428441\pi$
$314$	8.71687	0.491921
$315$	0	0
$316$	−27.6394	−1.55484
$317$	13.2181	0.742403	0.371201	−	0.928552i	$-0.378946\pi$
$317$	13.2181	0.742403	0.371201	+	0.928552i	$0.378946\pi$
$318$	0	0
$319$	6.23150	0.348897
$320$	−13.0472	−0.729361
$321$	0	0
$322$	6.36858	0.354907
$323$	−22.9866	−1.27901
$324$	0	0
$325$	0	0
$326$	−21.9956	−1.21822
$327$	0	0
$328$	−2.89218	−0.159694
$329$	−13.6091	−0.750291
$330$	0	0
$331$	23.8495	1.31089	0.655444	−	0.755244i	$-0.272481\pi$
$331$	23.8495	1.31089	0.655444	+	0.755244i	$0.272481\pi$
$332$	−25.6473	−1.40758
$333$	0	0
$334$	43.6046	2.38594
$335$	−13.1327	−0.717514
$336$	0	0
$337$	5.68306	0.309576	0.154788	−	0.987948i	$-0.450531\pi$
$337$	5.68306	0.309576	0.154788	+	0.987948i	$0.450531\pi$
$338$	0	0
$339$	0	0
$340$	14.0000	0.759257
$341$	−40.1282	−2.17307
$342$	0	0
$343$	−15.6563	−0.845359
$344$	−16.0944	−0.867753
$345$	0	0
$346$	−6.82364	−0.366841
$347$	21.4496	1.15147	0.575737	−	0.817635i	$-0.304715\pi$
$347$	21.4496	1.15147	0.575737	+	0.817635i	$0.304715\pi$
$348$	0	0
$349$	−6.87085	−0.367788	−0.183894	−	0.982946i	$-0.558870\pi$
$349$	−6.87085	−0.367788	−0.183894	+	0.982946i	$0.558870\pi$
$350$	−2.85395	−0.152550
$351$	0	0
$352$	28.1888	1.50247
$353$	34.4968	1.83608	0.918040	−	0.396488i	$-0.129771\pi$
$353$	34.4968	1.83608	0.918040	+	0.396488i	$0.129771\pi$
$354$	0	0
$355$	−9.60905	−0.509995
$356$	20.6339	1.09359
$357$	0	0
$358$	−12.0259	−0.635587
$359$	−8.15945	−0.430639	−0.215320	−	0.976544i	$-0.569079\pi$
$359$	−8.15945	−0.430639	−0.215320	+	0.976544i	$0.569079\pi$
$360$	0	0
$361$	7.17089	0.377415
$362$	−58.5148	−3.07547
$363$	0	0
$364$	0	0
$365$	−13.7854	−0.721561
$366$	0	0
$367$	−6.84055	−0.357074	−0.178537	−	0.983933i	$-0.557136\pi$
$367$	−6.84055	−0.357074	−0.178537	+	0.983933i	$0.557136\pi$
$368$	1.16842	0.0609082
$369$	0	0
$370$	−0.523604	−0.0272209
$371$	−5.70789	−0.296339
$372$	0	0
$373$	13.8192	0.715532	0.357766	−	0.933811i	$-0.383539\pi$
$373$	13.8192	0.715532	0.357766	+	0.933811i	$0.383539\pi$
$374$	−45.9732	−2.37722
$375$	0	0
$376$	27.2181	1.40367
$377$	0	0
$378$	0	0
$379$	−11.5157	−0.591520	−0.295760	−	0.955262i	$-0.595573\pi$
$379$	−11.5157	−0.591520	−0.295760	+	0.955262i	$0.595573\pi$
$380$	−15.9394	−0.817674
$381$	0	0
$382$	24.1024	1.23318
$383$	22.8798	1.16910	0.584552	−	0.811356i	$-0.301270\pi$
$383$	22.8798	1.16910	0.584552	+	0.811356i	$0.301270\pi$
$384$	0	0
$385$	5.70789	0.290901
$386$	18.0562	0.919035
$387$	0	0
$388$	−33.2440	−1.68771
$389$	4.35271	0.220691	0.110346	−	0.993893i	$-0.464804\pi$
$389$	4.35271	0.220691	0.110346	+	0.993893i	$0.464804\pi$
$390$	0	0
$391$	−10.0268	−0.507077
$392$	13.6473	0.689292
$393$	0	0
$394$	−39.2181	−1.97578
$395$	−8.87085	−0.446341
$396$	0	0
$397$	−7.62245	−0.382560	−0.191280	−	0.981536i	$-0.561264\pi$
$397$	−7.62245	−0.382560	−0.191280	+	0.981536i	$0.561264\pi$
$398$	2.78541	0.139620
$399$	0	0
$400$	−0.523604	−0.0261802
$401$	20.8495	1.04118	0.520588	−	0.853808i	$-0.325713\pi$
$401$	20.8495	1.04118	0.520588	+	0.853808i	$0.325713\pi$
$402$	0	0
$403$	0	0
$404$	53.1148	2.64256
$405$	0	0
$406$	−3.93146	−0.195115
$407$	1.04721	0.0519082
$408$	0	0
$409$	16.9653	0.838879	0.419439	−	0.907783i	$-0.362227\pi$
$409$	16.9653	0.838879	0.419439	+	0.907783i	$0.362227\pi$
$410$	−2.59214	−0.128017
$411$	0	0
$412$	−43.4596	−2.14110
$413$	−1.07751	−0.0530209
$414$	0	0
$415$	−8.23150	−0.404068
$416$	0	0
$417$	0	0
$418$	52.3418	2.56012
$419$	−2.09884	−0.102535	−0.0512676	−	0.998685i	$-0.516326\pi$
$419$	−2.09884	−0.102535	−0.0512676	+	0.998685i	$0.516326\pi$
$420$	0	0
$421$	39.6598	1.93290	0.966449	−	0.256857i	$-0.0826870\pi$
$421$	39.6598	1.93290	0.966449	+	0.256857i	$0.0826870\pi$
$422$	−23.3855	−1.13839
$423$	0	0
$424$	11.4158	0.554400
$425$	4.49330	0.217957
$426$	0	0
$427$	5.85395	0.283292
$428$	−38.9260	−1.88156
$429$	0	0
$430$	−14.4248	−0.695624
$431$	−22.1282	−1.06588	−0.532940	−	0.846153i	$-0.678913\pi$
$431$	−22.1282	−1.06588	−0.532940	+	0.846153i	$0.678913\pi$
$432$	0	0
$433$	−31.3562	−1.50688	−0.753442	−	0.657515i	$-0.771608\pi$
$433$	−31.3562	−1.50688	−0.753442	+	0.657515i	$0.771608\pi$
$434$	25.3169	1.21525
$435$	0	0
$436$	7.09988	0.340023
$437$	11.4158	0.546091
$438$	0	0
$439$	4.31344	0.205869	0.102935	−	0.994688i	$-0.467177\pi$
$439$	4.31344	0.205869	0.102935	+	0.994688i	$0.467177\pi$
$440$	−11.4158	−0.544226
$441$	0	0
$442$	0	0
$443$	−0.493301	−0.0234374	−0.0117187	−	0.999931i	$-0.503730\pi$
$443$	−0.493301	−0.0234374	−0.0117187	+	0.999931i	$0.503730\pi$
$444$	0	0
$445$	6.62245	0.313934
$446$	48.7889	2.31022
$447$	0	0
$448$	−16.4630	−0.777804
$449$	−23.3125	−1.10019	−0.550093	−	0.835103i	$-0.685408\pi$
$449$	−23.3125	−1.10019	−0.550093	+	0.835103i	$0.685408\pi$
$450$	0	0
$451$	5.18429	0.244119
$452$	4.60008	0.216369
$453$	0	0
$454$	−12.3180	−0.578112
$455$	0	0
$456$	0	0
$457$	−6.66966	−0.311993	−0.155997	−	0.987758i	$-0.549859\pi$
$457$	−6.66966	−0.311993	−0.155997	+	0.987758i	$0.549859\pi$
$458$	49.6046	2.31787
$459$	0	0
$460$	−6.95279	−0.324176
$461$	29.1193	1.35622	0.678110	−	0.734961i	$-0.262800\pi$
$461$	29.1193	1.35622	0.678110	+	0.734961i	$0.262800\pi$
$462$	0	0
$463$	1.79334	0.0833436	0.0416718	−	0.999131i	$-0.486732\pi$
$463$	1.79334	0.0833436	0.0416718	+	0.999131i	$0.486732\pi$
$464$	−0.721292	−0.0334852
$465$	0	0
$466$	57.2519	2.65214
$467$	−16.8192	−0.778301	−0.389150	−	0.921174i	$-0.627231\pi$
$467$	−16.8192	−0.778301	−0.389150	+	0.921174i	$0.627231\pi$
$468$	0	0
$469$	−16.5708	−0.765169
$470$	24.3945	1.12523
$471$	0	0
$472$	2.15502	0.0991931
$473$	28.8495	1.32650
$474$	0	0
$475$	−5.11575	−0.234727
$476$	17.6652	0.809685
$477$	0	0
$478$	−50.9439	−2.33012
$479$	35.1531	1.60618	0.803092	−	0.595855i	$-0.203187\pi$
$479$	35.1531	1.60618	0.803092	+	0.595855i	$0.203187\pi$
$480$	0	0
$481$	0	0
$482$	25.1416	1.14517
$483$	0	0
$484$	29.4843	1.34020
$485$	−10.6697	−0.484484
$486$	0	0
$487$	8.62596	0.390879	0.195440	−	0.980716i	$-0.437387\pi$
$487$	8.62596	0.390879	0.195440	+	0.980716i	$0.437387\pi$
$488$	−11.7079	−0.529991
$489$	0	0
$490$	12.2315	0.552563
$491$	23.2137	1.04762	0.523809	−	0.851836i	$-0.324510\pi$
$491$	23.2137	1.04762	0.523809	+	0.851836i	$0.324510\pi$
$492$	0	0
$493$	6.18975	0.278773
$494$	0	0
$495$	0	0
$496$	4.64482	0.208558
$497$	−12.1247	−0.543868
$498$	0	0
$499$	−29.0393	−1.29998	−0.649988	−	0.759944i	$-0.725226\pi$
$499$	−29.0393	−1.29998	−0.649988	+	0.759944i	$0.725226\pi$
$500$	3.11575	0.139341
$501$	0	0
$502$	24.9866	1.11521
$503$	17.3999	0.775824	0.387912	−	0.921696i	$-0.373196\pi$
$503$	17.3999	0.775824	0.387912	+	0.921696i	$0.373196\pi$
$504$	0	0
$505$	17.0472	0.758591
$506$	22.8316	1.01499
$507$	0	0
$508$	−55.4149	−2.45864
$509$	−22.8763	−1.01397	−0.506987	−	0.861953i	$-0.669241\pi$
$509$	−22.8763	−1.01397	−0.506987	+	0.861953i	$0.669241\pi$
$510$	0	0
$511$	−17.3945	−0.769485
$512$	−5.90558	−0.260992
$513$	0	0
$514$	−4.59214	−0.202551
$515$	−13.9484	−0.614638
$516$	0	0
$517$	−48.7889	−2.14573
$518$	−0.660685	−0.0290288
$519$	0	0
$520$	0	0
$521$	26.3642	1.15503	0.577517	−	0.816378i	$-0.304022\pi$
$521$	26.3642	1.15503	0.577517	+	0.816378i	$0.304022\pi$
$522$	0	0
$523$	−26.2921	−1.14967	−0.574837	−	0.818268i	$-0.694934\pi$
$523$	−26.2921	−1.14967	−0.574837	+	0.818268i	$0.694934\pi$
$524$	−36.3597	−1.58838
$525$	0	0
$526$	7.34725	0.320355
$527$	−39.8594	−1.73630
$528$	0	0
$529$	−18.0204	−0.783496
$530$	10.2315	0.444428
$531$	0	0
$532$	−20.1124	−0.871981
$533$	0	0
$534$	0	0
$535$	−12.4933	−0.540133
$536$	33.1416	1.43150
$537$	0	0
$538$	60.9519	2.62782
$539$	−24.4630	−1.05370
$540$	0	0
$541$	0.107816	0.00463537	0.00231768	−	0.999997i	$-0.499262\pi$
$541$	0.107816	0.00463537	0.00231768	+	0.999997i	$0.499262\pi$
$542$	−44.8957	−1.92844
$543$	0	0
$544$	28.0000	1.20049
$545$	2.27871	0.0976091
$546$	0	0
$547$	12.1193	0.518182	0.259091	−	0.965853i	$-0.416577\pi$
$547$	12.1193	0.518182	0.259091	+	0.965853i	$0.416577\pi$
$548$	−62.5148	−2.67050
$549$	0	0
$550$	−10.2315	−0.436273
$551$	−7.04721	−0.300221
$552$	0	0
$553$	−11.1933	−0.475986
$554$	−36.4024	−1.54659
$555$	0	0
$556$	50.9092	2.15903
$557$	38.0070	1.61041	0.805204	−	0.592997i	$-0.202056\pi$
$557$	38.0070	1.61041	0.805204	+	0.592997i	$0.202056\pi$
$558$	0	0
$559$	0	0
$560$	−0.660685	−0.0279191
$561$	0	0
$562$	12.1630	0.513063
$563$	−15.5102	−0.653677	−0.326839	−	0.945080i	$-0.605983\pi$
$563$	−15.5102	−0.653677	−0.326839	+	0.945080i	$0.605983\pi$
$564$	0	0
$565$	1.47640	0.0621124
$566$	−60.7665	−2.55421
$567$	0	0
$568$	24.2494	1.01748
$569$	−19.2405	−0.806602	−0.403301	−	0.915067i	$-0.632137\pi$
$569$	−19.2405	−0.806602	−0.403301	+	0.915067i	$0.632137\pi$
$570$	0	0
$571$	27.7338	1.16062	0.580311	−	0.814395i	$-0.302931\pi$
$571$	27.7338	1.16062	0.580311	+	0.814395i	$0.302931\pi$
$572$	0	0
$573$	0	0
$574$	−3.27077	−0.136519
$575$	−2.23150	−0.0930599
$576$	0	0
$577$	−26.9866	−1.12347	−0.561733	−	0.827318i	$-0.689865\pi$
$577$	−26.9866	−1.12347	−0.561733	+	0.827318i	$0.689865\pi$
$578$	−7.21459	−0.300088
$579$	0	0
$580$	4.29211	0.178220
$581$	−10.3865	−0.430906
$582$	0	0
$583$	−20.4630	−0.847491
$584$	34.7889	1.43958
$585$	0	0
$586$	−48.4462	−2.00129
$587$	−30.8530	−1.27344	−0.636720	−	0.771095i	$-0.719709\pi$
$587$	−30.8530	−1.27344	−0.636720	+	0.771095i	$0.719709\pi$
$588$	0	0
$589$	45.3811	1.86989
$590$	1.93146	0.0795169
$591$	0	0
$592$	−0.121214	−0.00498186
$593$	4.29211	0.176256	0.0881278	−	0.996109i	$-0.471912\pi$
$593$	4.29211	0.176256	0.0881278	+	0.996109i	$0.471912\pi$
$594$	0	0
$595$	5.66966	0.232433
$596$	−10.1103	−0.414133
$597$	0	0
$598$	0	0
$599$	24.0224	0.981527	0.490764	−	0.871293i	$-0.336718\pi$
$599$	24.0224	0.981527	0.490764	+	0.871293i	$0.336718\pi$
$600$	0	0
$601$	2.53154	0.103264	0.0516318	−	0.998666i	$-0.483558\pi$
$601$	2.53154	0.103264	0.0516318	+	0.998666i	$0.483558\pi$
$602$	−18.2012	−0.741825
$603$	0	0
$604$	−17.7596	−0.722630
$605$	9.46300	0.384726
$606$	0	0
$607$	11.4417	0.464403	0.232201	−	0.972668i	$-0.425407\pi$
$607$	11.4417	0.464403	0.232201	+	0.972668i	$0.425407\pi$
$608$	−31.8788	−1.29286
$609$	0	0
$610$	−10.4933	−0.424861
$611$	0	0
$612$	0	0
$613$	7.19326	0.290533	0.145267	−	0.989393i	$-0.453596\pi$
$613$	7.19326	0.290533	0.145267	+	0.989393i	$0.453596\pi$
$614$	−17.7641	−0.716900
$615$	0	0
$616$	−14.4045	−0.580373
$617$	25.5708	1.02944	0.514721	−	0.857358i	$-0.327895\pi$
$617$	25.5708	1.02944	0.514721	+	0.857358i	$0.327895\pi$
$618$	0	0
$619$	−5.98660	−0.240622	−0.120311	−	0.992736i	$-0.538389\pi$
$619$	−5.98660	−0.240622	−0.120311	+	0.992736i	$0.538389\pi$
$620$	−27.6394	−1.11002
$621$	0	0
$622$	−63.0463	−2.52793
$623$	8.35622	0.334785
$624$	0	0
$625$	1.00000	0.0400000
$626$	−17.8405	−0.713052
$627$	0	0
$628$	−12.0079	−0.479169
$629$	1.04019	0.0414752
$630$	0	0
$631$	43.0259	1.71283	0.856417	−	0.516285i	$-0.172686\pi$
$631$	43.0259	1.71283	0.856417	+	0.516285i	$0.172686\pi$
$632$	22.3865	0.890488
$633$	0	0
$634$	−29.8967	−1.18735
$635$	−17.7854	−0.705792
$636$	0	0
$637$	0	0
$638$	−14.0944	−0.558003
$639$	0	0
$640$	17.0472	0.673850
$641$	6.69450	0.264417	0.132208	−	0.991222i	$-0.457793\pi$
$641$	6.69450	0.264417	0.132208	+	0.991222i	$0.457793\pi$
$642$	0	0
$643$	−8.18078	−0.322619	−0.161309	−	0.986904i	$-0.551572\pi$
$643$	−8.18078	−0.322619	−0.161309	+	0.986904i	$0.551572\pi$
$644$	−8.77305	−0.345706
$645$	0	0
$646$	51.9911	2.04556
$647$	11.2787	0.443412	0.221706	−	0.975114i	$-0.428838\pi$
$647$	11.2787	0.443412	0.221706	+	0.975114i	$0.428838\pi$
$648$	0	0
$649$	−3.86292	−0.151633
$650$	0	0
$651$	0	0
$652$	30.3000	1.18664
$653$	−27.2484	−1.06631	−0.533156	−	0.846017i	$-0.678994\pi$
$653$	−27.2484	−1.06631	−0.533156	+	0.846017i	$0.678994\pi$
$654$	0	0
$655$	−11.6697	−0.455971
$656$	−0.600078	−0.0234291
$657$	0	0
$658$	30.7810	1.19997
$659$	−6.75510	−0.263141	−0.131571	−	0.991307i	$-0.542002\pi$
$659$	−6.75510	−0.263141	−0.131571	+	0.991307i	$0.542002\pi$
$660$	0	0
$661$	26.5102	1.03113	0.515564	−	0.856851i	$-0.327583\pi$
$661$	26.5102	1.03113	0.515564	+	0.856851i	$0.327583\pi$
$662$	−53.9429	−2.09655
$663$	0	0
$664$	20.7730	0.806151
$665$	−6.45506	−0.250317
$666$	0	0
$667$	−3.07400	−0.119026
$668$	−60.0676	−2.32409
$669$	0	0
$670$	29.7035	1.14755
$671$	20.9866	0.810179
$672$	0	0
$673$	−0.677591	−0.0261192	−0.0130596	−	0.999915i	$-0.504157\pi$
$673$	−0.677591	−0.0261192	−0.0130596	+	0.999915i	$0.504157\pi$
$674$	−12.8539	−0.495116
$675$	0	0
$676$	0	0
$677$	10.4630	0.402126	0.201063	−	0.979578i	$-0.435560\pi$
$677$	10.4630	0.402126	0.201063	+	0.979578i	$0.435560\pi$
$678$	0	0
$679$	−13.4630	−0.516662
$680$	−11.3393	−0.434843
$681$	0	0
$682$	90.7621	3.47546
$683$	−18.7516	−0.717510	−0.358755	−	0.933432i	$-0.616799\pi$
$683$	−18.7516	−0.717510	−0.358755	+	0.933432i	$0.616799\pi$
$684$	0	0
$685$	−20.0641	−0.766610
$686$	35.4114	1.35201
$687$	0	0
$688$	−3.33931	−0.127310
$689$	0	0
$690$	0	0
$691$	−43.5653	−1.65730	−0.828652	−	0.559764i	$-0.810892\pi$
$691$	−43.5653	−1.65730	−0.828652	+	0.559764i	$0.810892\pi$
$692$	9.39992	0.357331
$693$	0	0
$694$	−48.5148	−1.84159
$695$	16.3393	0.619786
$696$	0	0
$697$	5.14956	0.195054
$698$	15.5405	0.588217
$699$	0	0
$700$	3.93146	0.148595
$701$	19.0810	0.720680	0.360340	−	0.932821i	$-0.382661\pi$
$701$	19.0810	0.720680	0.360340	+	0.932821i	$0.382661\pi$
$702$	0	0
$703$	−1.18429	−0.0446663
$704$	−59.0204	−2.22442
$705$	0	0
$706$	−78.0250	−2.93651
$707$	21.5102	0.808975
$708$	0	0
$709$	−44.6046	−1.67516	−0.837581	−	0.546313i	$-0.816031\pi$
$709$	−44.6046	−1.67516	−0.837581	+	0.546313i	$0.816031\pi$
$710$	21.7338	0.815654
$711$	0	0
$712$	−16.7124	−0.626325
$713$	19.7953	0.741340
$714$	0	0
$715$	0	0
$716$	16.5663	0.619110
$717$	0	0
$718$	18.4551	0.688737
$719$	12.9822	0.484153	0.242077	−	0.970257i	$-0.422171\pi$
$719$	12.9822	0.484153	0.242077	+	0.970257i	$0.422171\pi$
$720$	0	0
$721$	−17.6001	−0.655461
$722$	−16.2191	−0.603614
$723$	0	0
$724$	80.6071	2.99574
$725$	1.37755	0.0511610
$726$	0	0
$727$	17.3980	0.645255	0.322627	−	0.946526i	$-0.395434\pi$
$727$	17.3980	0.645255	0.322627	+	0.946526i	$0.395434\pi$
$728$	0	0
$729$	0	0
$730$	31.1799	1.15402
$731$	28.6563	1.05989
$732$	0	0
$733$	−45.8272	−1.69266	−0.846332	−	0.532655i	$-0.821194\pi$
$733$	−45.8272	−1.69266	−0.846332	+	0.532655i	$0.821194\pi$
$734$	15.4720	0.571081
$735$	0	0
$736$	−13.9056	−0.512567
$737$	−59.4069	−2.18828
$738$	0	0
$739$	−17.5708	−0.646353	−0.323176	−	0.946339i	$-0.604751\pi$
$739$	−17.5708	−0.646353	−0.323176	+	0.946339i	$0.604751\pi$
$740$	0.721292	0.0265152
$741$	0	0
$742$	12.9101	0.473946
$743$	−30.3562	−1.11366	−0.556831	−	0.830626i	$-0.687983\pi$
$743$	−30.3562	−1.11366	−0.556831	+	0.830626i	$0.687983\pi$
$744$	0	0
$745$	−3.24490	−0.118884
$746$	−31.2563	−1.14438
$747$	0	0
$748$	63.3305	2.31559
$749$	−15.7641	−0.576007
$750$	0	0
$751$	3.80323	0.138782	0.0693909	−	0.997590i	$-0.477894\pi$
$751$	3.80323	0.138782	0.0693909	+	0.997590i	$0.477894\pi$
$752$	5.64729	0.205935
$753$	0	0
$754$	0	0
$755$	−5.69996	−0.207443
$756$	0	0
$757$	−3.40786	−0.123861	−0.0619303	−	0.998080i	$-0.519726\pi$
$757$	−3.40786	−0.123861	−0.0619303	+	0.998080i	$0.519726\pi$
$758$	26.0462	0.946040
$759$	0	0
$760$	12.9101	0.468300
$761$	7.24490	0.262627	0.131314	−	0.991341i	$-0.458081\pi$
$761$	7.24490	0.262627	0.131314	+	0.991341i	$0.458081\pi$
$762$	0	0
$763$	2.87528	0.104092
$764$	−33.2022	−1.20121
$765$	0	0
$766$	−51.7496	−1.86979
$767$	0	0
$768$	0	0
$769$	−31.1764	−1.12425	−0.562124	−	0.827053i	$-0.690016\pi$
$769$	−31.1764	−1.12425	−0.562124	+	0.827053i	$0.690016\pi$
$770$	−12.9101	−0.465249
$771$	0	0
$772$	−24.8733	−0.895210
$773$	19.4605	0.699947	0.349973	−	0.936760i	$-0.386191\pi$
$773$	19.4605	0.699947	0.349973	+	0.936760i	$0.386191\pi$
$774$	0	0
$775$	−8.87085	−0.318650
$776$	26.9260	0.966587
$777$	0	0
$778$	−9.84498	−0.352959
$779$	−5.86292	−0.210061
$780$	0	0
$781$	−43.4675	−1.55539
$782$	22.6786	0.810986
$783$	0	0
$784$	2.83158	0.101128
$785$	−3.85395	−0.137553
$786$	0	0
$787$	−14.4034	−0.513427	−0.256713	−	0.966488i	$-0.582640\pi$
$787$	−14.4034	−0.513427	−0.256713	+	0.966488i	$0.582640\pi$
$788$	54.0250	1.92456
$789$	0	0
$790$	20.0641	0.713849
$791$	1.86292	0.0662378
$792$	0	0
$793$	0	0
$794$	17.2405	0.611841
$795$	0	0
$796$	−3.83704	−0.136000
$797$	23.6011	0.835994	0.417997	−	0.908448i	$-0.362732\pi$
$797$	23.6011	0.835994	0.417997	+	0.908448i	$0.362732\pi$
$798$	0	0
$799$	−48.4621	−1.71447
$800$	6.23150	0.220317
$801$	0	0
$802$	−47.1575	−1.66519
$803$	−62.3597	−2.20063
$804$	0	0
$805$	−2.81571	−0.0992407
$806$	0	0
$807$	0	0
$808$	−43.0204	−1.51345
$809$	−26.7551	−0.940659	−0.470330	−	0.882491i	$-0.655865\pi$
$809$	−26.7551	−0.940659	−0.470330	+	0.882491i	$0.655865\pi$
$810$	0	0
$811$	3.58421	0.125859	0.0629293	−	0.998018i	$-0.479956\pi$
$811$	3.58421	0.125859	0.0629293	+	0.998018i	$0.479956\pi$
$812$	5.41579	0.190057
$813$	0	0
$814$	−2.36858	−0.0830187
$815$	9.72480	0.340645
$816$	0	0
$817$	−32.6260	−1.14144
$818$	−38.3721	−1.34165
$819$	0	0
$820$	3.57081	0.124698
$821$	15.8361	0.552685	0.276342	−	0.961059i	$-0.410878\pi$
$821$	15.8361	0.552685	0.276342	+	0.961059i	$0.410878\pi$
$822$	0	0
$823$	7.41579	0.258498	0.129249	−	0.991612i	$-0.458743\pi$
$823$	7.41579	0.258498	0.129249	+	0.991612i	$0.458743\pi$
$824$	35.2002	1.22626
$825$	0	0
$826$	2.43712	0.0847983
$827$	−16.6036	−0.577363	−0.288682	−	0.957425i	$-0.593217\pi$
$827$	−16.6036	−0.577363	−0.288682	+	0.957425i	$0.593217\pi$
$828$	0	0
$829$	−29.2842	−1.01708	−0.508541	−	0.861038i	$-0.669815\pi$
$829$	−29.2842	−1.01708	−0.508541	+	0.861038i	$0.669815\pi$
$830$	18.6180	0.646241
$831$	0	0
$832$	0	0
$833$	−24.2991	−0.841915
$834$	0	0
$835$	−19.2787	−0.667167
$836$	−72.1035	−2.49375
$837$	0	0
$838$	4.74717	0.163988
$839$	−30.1620	−1.04131	−0.520655	−	0.853767i	$-0.674312\pi$
$839$	−30.1620	−1.04131	−0.520655	+	0.853767i	$0.674312\pi$
$840$	0	0
$841$	−27.1024	−0.934564
$842$	−89.7026	−3.09135
$843$	0	0
$844$	32.2147	1.10888
$845$	0	0
$846$	0	0
$847$	11.9404	0.410278
$848$	2.36858	0.0813374
$849$	0	0
$850$	−10.1630	−0.348587
$851$	−0.516589	−0.0177085
$852$	0	0
$853$	2.50670	0.0858277	0.0429139	−	0.999079i	$-0.486336\pi$
$853$	2.50670	0.0858277	0.0429139	+	0.999079i	$0.486336\pi$
$854$	−13.2405	−0.453080
$855$	0	0
$856$	31.5282	1.07761
$857$	24.4327	0.834605	0.417302	−	0.908768i	$-0.362976\pi$
$857$	24.4327	0.834605	0.417302	+	0.908768i	$0.362976\pi$
$858$	0	0
$859$	−7.10235	−0.242329	−0.121165	−	0.992632i	$-0.538663\pi$
$859$	−7.10235	−0.242329	−0.121165	+	0.992632i	$0.538663\pi$
$860$	19.8709	0.677590
$861$	0	0
$862$	50.0497	1.70470
$863$	−3.24490	−0.110458	−0.0552288	−	0.998474i	$-0.517589\pi$
$863$	−3.24490	−0.110458	−0.0552288	+	0.998474i	$0.517589\pi$
$864$	0	0
$865$	3.01691	0.102578
$866$	70.9216	2.41001
$867$	0	0
$868$	−34.8754	−1.18375
$869$	−40.1282	−1.36126
$870$	0	0
$871$	0	0
$872$	−5.75056	−0.194738
$873$	0	0
$874$	−25.8203	−0.873383
$875$	1.26180	0.0426567
$876$	0	0
$877$	−18.9866	−0.641132	−0.320566	−	0.947226i	$-0.603873\pi$
$877$	−18.9866	−0.641132	−0.320566	+	0.947226i	$0.603873\pi$
$878$	−9.75614	−0.329254
$879$	0	0
$880$	−2.36858	−0.0798448
$881$	19.4764	0.656176	0.328088	−	0.944647i	$-0.393596\pi$
$881$	19.4764	0.656176	0.328088	+	0.944647i	$0.393596\pi$
$882$	0	0
$883$	−50.5664	−1.70169	−0.850847	−	0.525413i	$-0.823911\pi$
$883$	−50.5664	−1.70169	−0.850847	+	0.525413i	$0.823911\pi$
$884$	0	0
$885$	0	0
$886$	1.11575	0.0374843
$887$	12.5271	0.420619	0.210310	−	0.977635i	$-0.432553\pi$
$887$	12.5271	0.420619	0.210310	+	0.977635i	$0.432553\pi$
$888$	0	0
$889$	−22.4417	−0.752669
$890$	−14.9787	−0.502086
$891$	0	0
$892$	−67.2092	−2.25033
$893$	55.1754	1.84638
$894$	0	0
$895$	5.31694	0.177726
$896$	21.5102	0.718606
$897$	0	0
$898$	52.7283	1.75957
$899$	−12.2201	−0.407562
$900$	0	0
$901$	−20.3259	−0.677154
$902$	−11.7258	−0.390428
$903$	0	0
$904$	−3.72584	−0.123920
$905$	25.8709	0.859976
$906$	0	0
$907$	28.0417	0.931111	0.465555	−	0.885019i	$-0.345855\pi$
$907$	28.0417	0.931111	0.465555	+	0.885019i	$0.345855\pi$
$908$	16.9687	0.563125
$909$	0	0
$910$	0	0
$911$	−16.1977	−0.536653	−0.268327	−	0.963328i	$-0.586471\pi$
$911$	−16.1977	−0.536653	−0.268327	+	0.963328i	$0.586471\pi$
$912$	0	0
$913$	−37.2360	−1.23233
$914$	15.0854	0.498982
$915$	0	0
$916$	−68.3329	−2.25778
$917$	−14.7248	−0.486256
$918$	0	0
$919$	−43.2778	−1.42760	−0.713801	−	0.700348i	$-0.753028\pi$
$919$	−43.2778	−1.42760	−0.713801	+	0.700348i	$0.753028\pi$
$920$	5.63142	0.185662
$921$	0	0
$922$	−65.8620	−2.16905
$923$	0	0
$924$	0	0
$925$	0.231499	0.00761163
$926$	−4.05618	−0.133294
$927$	0	0
$928$	8.58421	0.281791
$929$	13.7685	0.451730	0.225865	−	0.974159i	$-0.427479\pi$
$929$	13.7685	0.451730	0.225865	+	0.974159i	$0.427479\pi$
$930$	0	0
$931$	27.6652	0.906691
$932$	−78.8675	−2.58339
$933$	0	0
$934$	38.0417	1.24476
$935$	20.3259	0.664729
$936$	0	0
$937$	−38.1888	−1.24757	−0.623787	−	0.781594i	$-0.714407\pi$
$937$	−38.1888	−1.24757	−0.623787	+	0.781594i	$0.714407\pi$
$938$	37.4799	1.22376
$939$	0	0
$940$	−33.6046	−1.09606
$941$	−13.9618	−0.455140	−0.227570	−	0.973762i	$-0.573078\pi$
$941$	−13.9618	−0.455140	−0.227570	+	0.973762i	$0.573078\pi$
$942$	0	0
$943$	−2.55742	−0.0832809
$944$	0.447131	0.0145529
$945$	0	0
$946$	−65.2519	−2.12152
$947$	7.36962	0.239480	0.119740	−	0.992805i	$-0.461794\pi$
$947$	7.36962	0.239480	0.119740	+	0.992805i	$0.461794\pi$
$948$	0	0
$949$	0	0
$950$	11.5708	0.375407
$951$	0	0
$952$	−14.3080	−0.463724
$953$	−23.2181	−0.752108	−0.376054	−	0.926598i	$-0.622719\pi$
$953$	−23.2181	−0.752108	−0.376054	+	0.926598i	$0.622719\pi$
$954$	0	0
$955$	−10.6563	−0.344828
$956$	70.1779	2.26972
$957$	0	0
$958$	−79.5093	−2.56883
$959$	−25.3169	−0.817527
$960$	0	0
$961$	47.6920	1.53845
$962$	0	0
$963$	0	0
$964$	−34.6339	−1.11548
$965$	−7.98309	−0.256985
$966$	0	0
$967$	7.54402	0.242599	0.121300	−	0.992616i	$-0.461294\pi$
$967$	7.54402	0.242599	0.121300	+	0.992616i	$0.461294\pi$
$968$	−23.8809	−0.767560
$969$	0	0
$970$	24.1327	0.774853
$971$	30.5842	0.981494	0.490747	−	0.871302i	$-0.336724\pi$
$971$	30.5842	0.981494	0.490747	+	0.871302i	$0.336724\pi$
$972$	0	0
$973$	20.6170	0.660950
$974$	−19.5102	−0.625147
$975$	0	0
$976$	−2.42919	−0.0777564
$977$	−35.6811	−1.14154	−0.570770	−	0.821110i	$-0.693355\pi$
$977$	−35.6811	−1.14154	−0.570770	+	0.821110i	$0.693355\pi$
$978$	0	0
$979$	29.9573	0.957441
$980$	−16.8495	−0.538238
$981$	0	0
$982$	−52.5047	−1.67549
$983$	12.9866	0.414208	0.207104	−	0.978319i	$-0.433596\pi$
$983$	12.9866	0.414208	0.207104	+	0.978319i	$0.433596\pi$
$984$	0	0
$985$	17.3393	0.552477
$986$	−14.0000	−0.445851
$987$	0	0
$988$	0	0
$989$	−14.2315	−0.452535
$990$	0	0
$991$	−22.7472	−0.722588	−0.361294	−	0.932452i	$-0.617665\pi$
$991$	−22.7472	−0.722588	−0.361294	+	0.932452i	$0.617665\pi$
$992$	−55.2787	−1.75510
$993$	0	0
$994$	27.4237	0.869828
$995$	−1.23150	−0.0390411
$996$	0	0
$997$	−13.8460	−0.438508	−0.219254	−	0.975668i	$-0.570362\pi$
$997$	−13.8460	−0.438508	−0.219254	+	0.975668i	$0.570362\pi$
$998$	65.6811	2.07910
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7605.2.a.bw.1.1		3
3.2	odd	2		2535.2.a.ba.1.3		3
13.4	even	6		585.2.j.f.406.1		6
13.10	even	6		585.2.j.f.451.1		6
13.12	even	2		7605.2.a.bv.1.3		3
39.17	odd	6		195.2.i.d.16.3	✓	6
39.23	odd	6		195.2.i.d.61.3	yes	6
39.38	odd	2		2535.2.a.bb.1.1		3
195.17	even	12		975.2.bb.k.874.2		12
195.23	even	12		975.2.bb.k.724.2		12
195.62	even	12		975.2.bb.k.724.5		12
195.134	odd	6		975.2.i.l.601.1		6
195.173	even	12		975.2.bb.k.874.5		12
195.179	odd	6		975.2.i.l.451.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
195.2.i.d.16.3	✓	6	39.17	odd	6
195.2.i.d.61.3	yes	6	39.23	odd	6
585.2.j.f.406.1		6	13.4	even	6
585.2.j.f.451.1		6	13.10	even	6
975.2.i.l.451.1		6	195.179	odd	6
975.2.i.l.601.1		6	195.134	odd	6
975.2.bb.k.724.2		12	195.23	even	12
975.2.bb.k.724.5		12	195.62	even	12
975.2.bb.k.874.2		12	195.17	even	12
975.2.bb.k.874.5		12	195.173	even	12
2535.2.a.ba.1.3		3	3.2	odd	2
2535.2.a.bb.1.1		3	39.38	odd	2
7605.2.a.bv.1.3		3	13.12	even	2
7605.2.a.bw.1.1		3	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 7605 = 3^{2} \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7605.a (trivial)

\(f(q)\)	\(=\)	\(q-2.26180 q^{2} +3.11575 q^{4} +1.00000 q^{5} +1.26180 q^{7} -2.52360 q^{8} +O(q^{10})\)	\(q-2.26180 q^{2} +3.11575 q^{4} +1.00000 q^{5} +1.26180 q^{7} -2.52360 q^{8} -2.26180 q^{10} +4.52360 q^{11} -2.85395 q^{14} -0.523604 q^{16} +4.49330 q^{17} -5.11575 q^{19} +3.11575 q^{20} -10.2315 q^{22} -2.23150 q^{23} +1.00000 q^{25} +3.93146 q^{28} +1.37755 q^{29} -8.87085 q^{31} +6.23150 q^{32} -10.1630 q^{34} +1.26180 q^{35} +0.231499 q^{37} +11.5708 q^{38} -2.52360 q^{40} +1.14605 q^{41} +6.37755 q^{43} +14.0944 q^{44} +5.04721 q^{46} -10.7854 q^{47} -5.40786 q^{49} -2.26180 q^{50} -4.52360 q^{53} +4.52360 q^{55} -3.18429 q^{56} -3.11575 q^{58} -0.853947 q^{59} +4.63935 q^{61} +20.0641 q^{62} -13.0472 q^{64} -13.1327 q^{67} +14.0000 q^{68} -2.85395 q^{70} -9.60905 q^{71} -13.7854 q^{73} -0.523604 q^{74} -15.9394 q^{76} +5.70789 q^{77} -8.87085 q^{79} -0.523604 q^{80} -2.59214 q^{82} -8.23150 q^{83} +4.49330 q^{85} -14.4248 q^{86} -11.4158 q^{88} +6.62245 q^{89} -6.95279 q^{92} +24.3945 q^{94} -5.11575 q^{95} -10.6697 q^{97} +12.2315 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 6 q^{4} + 3 q^{5} - 3 q^{7} + 6 q^{8}+O(q^{10}) \) 3 * q + 6 * q^4 + 3 * q^5 - 3 * q^7 + 6 * q^8	\( 3 q + 6 q^{4} + 3 q^{5} - 3 q^{7} + 6 q^{8} - 12 q^{14} + 12 q^{16} - 12 q^{19} + 6 q^{20} - 24 q^{22} + 3 q^{25} - 12 q^{28} - 6 q^{29} - 3 q^{31} + 12 q^{32} - 3 q^{35} - 6 q^{37} - 6 q^{38} + 6 q^{40} + 9 q^{43} - 12 q^{44} - 12 q^{46} - 12 q^{47} - 6 q^{49} - 30 q^{56} - 6 q^{58} - 6 q^{59} - 3 q^{61} + 6 q^{62} - 12 q^{64} - 9 q^{67} + 42 q^{68} - 12 q^{70} - 12 q^{71} - 21 q^{73} + 12 q^{74} - 48 q^{76} + 24 q^{77} - 3 q^{79} + 12 q^{80} - 18 q^{82} - 18 q^{83} - 6 q^{86} - 48 q^{88} + 30 q^{89} - 48 q^{92} + 36 q^{94} - 12 q^{95} - 15 q^{97} + 30 q^{98}+O(q^{100}) \) 3 * q + 6 * q^4 + 3 * q^5 - 3 * q^7 + 6 * q^8 - 12 * q^14 + 12 * q^16 - 12 * q^19 + 6 * q^20 - 24 * q^22 + 3 * q^25 - 12 * q^28 - 6 * q^29 - 3 * q^31 + 12 * q^32 - 3 * q^35 - 6 * q^37 - 6 * q^38 + 6 * q^40 + 9 * q^43 - 12 * q^44 - 12 * q^46 - 12 * q^47 - 6 * q^49 - 30 * q^56 - 6 * q^58 - 6 * q^59 - 3 * q^61 + 6 * q^62 - 12 * q^64 - 9 * q^67 + 42 * q^68 - 12 * q^70 - 12 * q^71 - 21 * q^73 + 12 * q^74 - 48 * q^76 + 24 * q^77 - 3 * q^79 + 12 * q^80 - 18 * q^82 - 18 * q^83 - 6 * q^86 - 48 * q^88 + 30 * q^89 - 48 * q^92 + 36 * q^94 - 12 * q^95 - 15 * q^97 + 30 * q^98

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