LMFDB - Embedded newform 7605.2.a.ci.1.3

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:	$0$
Dimension:	$4$
Coefficient field:	4.4.13824.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 6x^{2} + 6 $ x^4 - 6*x^2 + 6
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			1.3
Root			$1.12603$ 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)

$f(q)$	$=$	$q+1.12603 q^{2} -0.732051 q^{4} +1.00000 q^{5} -0.606018 q^{7} -3.07638 q^{8} +O(q^{10})$	$q+1.12603 q^{2} -0.732051 q^{4} +1.00000 q^{5} -0.606018 q^{7} -3.07638 q^{8} +1.12603 q^{10} -3.07638 q^{11} -0.682396 q^{14} -2.00000 q^{16} -1.95035 q^{17} -4.88481 q^{19} -0.732051 q^{20} -3.46410 q^{22} -1.86434 q^{23} +1.00000 q^{25} +0.443636 q^{28} +2.78171 q^{29} +9.15276 q^{31} +3.90069 q^{32} -2.19615 q^{34} -0.606018 q^{35} +3.74793 q^{37} -5.50045 q^{38} -3.07638 q^{40} +1.07012 q^{41} -5.01084 q^{43} +2.25207 q^{44} -2.09931 q^{46} -10.2024 q^{47} -6.63274 q^{49} +1.12603 q^{50} +10.6569 q^{53} -3.07638 q^{55} +1.86434 q^{56} +3.13229 q^{58} +11.5742 q^{59} -6.26795 q^{61} +10.3063 q^{62} +8.39230 q^{64} +1.97786 q^{67} +1.42775 q^{68} -0.682396 q^{70} +5.90695 q^{71} +4.35395 q^{73} +4.22030 q^{74} +3.57593 q^{76} +1.86434 q^{77} -3.29546 q^{79} -2.00000 q^{80} +1.20499 q^{82} +6.97707 q^{83} -1.95035 q^{85} -5.64237 q^{86} +9.46410 q^{88} -16.2993 q^{89} +1.36479 q^{92} -11.4882 q^{94} -4.88481 q^{95} +13.2024 q^{97} -7.46868 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{4} + 4 q^{5}+O(q^{10}) $ 4 * q + 4 * q^4 + 4 * q^5	$ 4 q + 4 q^{4} + 4 q^{5} + 12 q^{14} - 8 q^{16} + 12 q^{19} + 4 q^{20} + 4 q^{25} + 12 q^{28} + 12 q^{29} + 12 q^{31} + 12 q^{34} + 24 q^{37} - 12 q^{41} + 16 q^{43} - 24 q^{46} - 24 q^{47} - 4 q^{49} + 12 q^{58} + 12 q^{59} - 32 q^{61} - 8 q^{64} - 12 q^{67} + 12 q^{70} + 12 q^{71} + 24 q^{73} - 24 q^{74} + 24 q^{76} - 8 q^{79} - 8 q^{80} - 12 q^{82} - 12 q^{86} + 24 q^{88} - 12 q^{89} - 24 q^{92} - 12 q^{94} + 12 q^{95} + 36 q^{97} + 24 q^{98}+O(q^{100}) $ 4 * q + 4 * q^4 + 4 * q^5 + 12 * q^14 - 8 * q^16 + 12 * q^19 + 4 * q^20 + 4 * q^25 + 12 * q^28 + 12 * q^29 + 12 * q^31 + 12 * q^34 + 24 * q^37 - 12 * q^41 + 16 * q^43 - 24 * q^46 - 24 * q^47 - 4 * q^49 + 12 * q^58 + 12 * q^59 - 32 * q^61 - 8 * q^64 - 12 * q^67 + 12 * q^70 + 12 * q^71 + 24 * q^73 - 24 * q^74 + 24 * q^76 - 8 * q^79 - 8 * q^80 - 12 * q^82 - 12 * q^86 + 24 * q^88 - 12 * q^89 - 24 * q^92 - 12 * q^94 + 12 * q^95 + 36 * q^97 + 24 * 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 $

$2$	1.12603	0.796225	0.398113	−	0.917337i	$-0.369665\pi$
$2$	1.12603	0.796225	0.398113	+	0.917337i	$0.369665\pi$
$3$	0	0
$3$	0	0
$4$	−0.732051	−0.366025
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−0.606018	−0.229053	−0.114527	−	0.993420i	$-0.536535\pi$
$7$	−0.606018	−0.229053	−0.114527	+	0.993420i	$0.536535\pi$
$8$	−3.07638	−1.08766
$9$	0	0
$10$	1.12603	0.356083
$11$	−3.07638	−0.927563	−0.463781	−	0.885950i	$-0.653508\pi$
$11$	−3.07638	−0.927563	−0.463781	+	0.885950i	$0.653508\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	−0.682396	−0.182378
$15$	0	0
$16$	−2.00000	−0.500000
$17$	−1.95035	−0.473028	−0.236514	−	0.971628i	$-0.576005\pi$
$17$	−1.95035	−0.473028	−0.236514	+	0.971628i	$0.576005\pi$
$18$	0	0
$19$	−4.88481	−1.12065	−0.560326	−	0.828272i	$-0.689324\pi$
$19$	−4.88481	−1.12065	−0.560326	+	0.828272i	$0.689324\pi$
$20$	−0.732051	−0.163692
$21$	0	0
$22$	−3.46410	−0.738549
$23$	−1.86434	−0.388742	−0.194371	−	0.980928i	$-0.562267\pi$
$23$	−1.86434	−0.388742	−0.194371	+	0.980928i	$0.562267\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0.443636	0.0838394
$29$	2.78171	0.516550	0.258275	−	0.966071i	$-0.416846\pi$
$29$	2.78171	0.516550	0.258275	+	0.966071i	$0.416846\pi$
$30$	0	0
$31$	9.15276	1.64388	0.821942	−	0.569572i	$-0.192891\pi$
$31$	9.15276	1.64388	0.821942	+	0.569572i	$0.192891\pi$
$32$	3.90069	0.689551
$33$	0	0
$34$	−2.19615	−0.376637
$35$	−0.606018	−0.102436
$36$	0	0
$37$	3.74793	0.616157	0.308078	−	0.951361i	$-0.400314\pi$
$37$	3.74793	0.616157	0.308078	+	0.951361i	$0.400314\pi$
$38$	−5.50045	−0.892291
$39$	0	0
$40$	−3.07638	−0.486418
$41$	1.07012	0.167125	0.0835623	−	0.996503i	$-0.473370\pi$
$41$	1.07012	0.167125	0.0835623	+	0.996503i	$0.473370\pi$
$42$	0	0
$43$	−5.01084	−0.764146	−0.382073	−	0.924132i	$-0.624790\pi$
$43$	−5.01084	−0.764146	−0.382073	+	0.924132i	$0.624790\pi$
$44$	2.25207	0.339512
$45$	0	0
$46$	−2.09931	−0.309526
$47$	−10.2024	−1.48817	−0.744087	−	0.668082i	$-0.767115\pi$
$47$	−10.2024	−1.48817	−0.744087	+	0.668082i	$0.767115\pi$
$48$	0	0
$49$	−6.63274	−0.947535
$50$	1.12603	0.159245
$51$	0	0
$52$	0	0
$53$	10.6569	1.46384	0.731918	−	0.681393i	$-0.238625\pi$
$53$	10.6569	1.46384	0.731918	+	0.681393i	$0.238625\pi$
$54$	0	0
$55$	−3.07638	−0.414819
$56$	1.86434	0.249133
$57$	0	0
$58$	3.13229	0.411290
$59$	11.5742	1.50684	0.753419	−	0.657540i	$-0.228403\pi$
$59$	11.5742	1.50684	0.753419	+	0.657540i	$0.228403\pi$
$60$	0	0
$61$	−6.26795	−0.802529	−0.401264	−	0.915962i	$-0.631429\pi$
$61$	−6.26795	−0.802529	−0.401264	+	0.915962i	$0.631429\pi$
$62$	10.3063	1.30890
$63$	0	0
$64$	8.39230	1.04904
$65$	0	0
$66$	0	0
$67$	1.97786	0.241634	0.120817	−	0.992675i	$-0.461449\pi$
$67$	1.97786	0.241634	0.120817	+	0.992675i	$0.461449\pi$
$68$	1.42775	0.173140
$69$	0	0
$70$	−0.682396	−0.0815620
$71$	5.90695	0.701026	0.350513	−	0.936558i	$-0.386007\pi$
$71$	5.90695	0.701026	0.350513	+	0.936558i	$0.386007\pi$
$72$	0	0
$73$	4.35395	0.509592	0.254796	−	0.966995i	$-0.417992\pi$
$73$	4.35395	0.509592	0.254796	+	0.966995i	$0.417992\pi$
$74$	4.22030	0.490600
$75$	0	0
$76$	3.57593	0.410187
$77$	1.86434	0.212461
$78$	0	0
$79$	−3.29546	−0.370768	−0.185384	−	0.982666i	$-0.559353\pi$
$79$	−3.29546	−0.370768	−0.185384	+	0.982666i	$0.559353\pi$
$80$	−2.00000	−0.223607
$81$	0	0
$82$	1.20499	0.133069
$83$	6.97707	0.765833	0.382916	−	0.923783i	$-0.374920\pi$
$83$	6.97707	0.765833	0.382916	+	0.923783i	$0.374920\pi$
$84$	0	0
$85$	−1.95035	−0.211545
$86$	−5.64237	−0.608432
$87$	0	0
$88$	9.46410	1.00888
$89$	−16.2993	−1.72772	−0.863859	−	0.503734i	$-0.831959\pi$
$89$	−16.2993	−1.72772	−0.863859	+	0.503734i	$0.831959\pi$
$90$	0	0
$91$	0	0
$92$	1.36479	0.142289
$93$	0	0
$94$	−11.4882	−1.18492
$95$	−4.88481	−0.501171
$96$	0	0
$97$	13.2024	1.34050	0.670251	−	0.742135i	$-0.266187\pi$
$97$	13.2024	1.34050	0.670251	+	0.742135i	$0.266187\pi$
$98$	−7.46868	−0.754451
$99$	0	0
$100$	−0.732051	−0.0732051
$101$	7.86434	0.782531	0.391266	−	0.920278i	$-0.372037\pi$
$101$	7.86434	0.782531	0.391266	+	0.920278i	$0.372037\pi$
$102$	0	0
$103$	15.1101	1.48885	0.744424	−	0.667708i	$-0.232724\pi$
$103$	15.1101	1.48885	0.744424	+	0.667708i	$0.232724\pi$
$104$	0	0
$105$	0	0
$106$	12.0000	1.16554
$107$	8.20699	0.793400	0.396700	−	0.917948i	$-0.370155\pi$
$107$	8.20699	0.793400	0.396700	+	0.917948i	$0.370155\pi$
$108$	0	0
$109$	10.5927	1.01460	0.507299	−	0.861770i	$-0.330644\pi$
$109$	10.5927	1.01460	0.507299	+	0.861770i	$0.330644\pi$
$110$	−3.46410	−0.330289
$111$	0	0
$112$	1.21204	0.114527
$113$	−3.46410	−0.325875	−0.162938	−	0.986636i	$-0.552097\pi$
$113$	−3.46410	−0.325875	−0.162938	+	0.986636i	$0.552097\pi$
$114$	0	0
$115$	−1.86434	−0.173851
$116$	−2.03635	−0.189070
$117$	0	0
$118$	13.0330	1.19978
$119$	1.18195	0.108349
$120$	0	0
$121$	−1.53590	−0.139627
$122$	−7.05791	−0.638994
$123$	0	0
$124$	−6.70028	−0.601703
$125$	1.00000	0.0894427
$126$	0	0
$127$	18.0438	1.60113	0.800565	−	0.599246i	$-0.204533\pi$
$127$	18.0438	1.60113	0.800565	+	0.599246i	$0.204533\pi$
$128$	1.64863	0.145719
$129$	0	0
$130$	0	0
$131$	−1.11899	−0.0977662	−0.0488831	−	0.998805i	$-0.515566\pi$
$131$	−1.11899	−0.0977662	−0.0488831	+	0.998805i	$0.515566\pi$
$132$	0	0
$133$	2.96028	0.256689
$134$	2.22713	0.192395
$135$	0	0
$136$	6.00000	0.514496
$137$	−8.70654	−0.743850	−0.371925	−	0.928263i	$-0.621302\pi$
$137$	−8.70654	−0.743850	−0.371925	+	0.928263i	$0.621302\pi$
$138$	0	0
$139$	9.65689	0.819086	0.409543	−	0.912291i	$-0.365688\pi$
$139$	9.65689	0.819086	0.409543	+	0.912291i	$0.365688\pi$
$140$	0.443636	0.0374941
$141$	0	0
$142$	6.65142	0.558174
$143$	0	0
$144$	0	0
$145$	2.78171	0.231008
$146$	4.90269	0.405750
$147$	0	0
$148$	−2.74368	−0.225529
$149$	−5.17569	−0.424009	−0.212004	−	0.977269i	$-0.567999\pi$
$149$	−5.17569	−0.424009	−0.212004	+	0.977269i	$0.567999\pi$
$150$	0	0
$151$	22.1451	1.80215	0.901073	−	0.433668i	$-0.142781\pi$
$151$	22.1451	1.80215	0.901073	+	0.433668i	$0.142781\pi$
$152$	15.0275	1.21889
$153$	0	0
$154$	2.09931	0.169167
$155$	9.15276	0.735167
$156$	0	0
$157$	−14.3756	−1.14730	−0.573650	−	0.819100i	$-0.694473\pi$
$157$	−14.3756	−1.14730	−0.573650	+	0.819100i	$0.694473\pi$
$158$	−3.71080	−0.295215
$159$	0	0
$160$	3.90069	0.308377
$161$	1.12983	0.0890427
$162$	0	0
$163$	10.0701	0.788753	0.394376	−	0.918949i	$-0.370961\pi$
$163$	10.0701	0.788753	0.394376	+	0.918949i	$0.370961\pi$
$164$	−0.783382	−0.0611719
$165$	0	0
$166$	7.85641	0.609775
$167$	10.8430	0.839056	0.419528	−	0.907742i	$-0.362196\pi$
$167$	10.8430	0.839056	0.419528	+	0.907742i	$0.362196\pi$
$168$	0	0
$169$	0	0
$170$	−2.19615	−0.168437
$171$	0	0
$172$	3.66819	0.279697
$173$	−24.6623	−1.87504	−0.937518	−	0.347936i	$-0.886883\pi$
$173$	−24.6623	−1.87504	−0.937518	+	0.347936i	$0.886883\pi$
$174$	0	0
$175$	−0.606018	−0.0458107
$176$	6.15276	0.463781
$177$	0	0
$178$	−18.3535	−1.37565
$179$	22.8397	1.70712	0.853561	−	0.520993i	$-0.174438\pi$
$179$	22.8397	1.70712	0.853561	+	0.520993i	$0.174438\pi$
$180$	0	0
$181$	17.4616	1.29791	0.648957	−	0.760825i	$-0.275206\pi$
$181$	17.4616	1.29791	0.648957	+	0.760825i	$0.275206\pi$
$182$	0	0
$183$	0	0
$184$	5.73542	0.422821
$185$	3.74793	0.275554
$186$	0	0
$187$	6.00000	0.438763
$188$	7.46868	0.544710
$189$	0	0
$190$	−5.50045	−0.399045
$191$	−7.67356	−0.555239	−0.277620	−	0.960691i	$-0.589545\pi$
$191$	−7.67356	−0.555239	−0.277620	+	0.960691i	$0.589545\pi$
$192$	0	0
$193$	21.0108	1.51239	0.756197	−	0.654344i	$-0.227055\pi$
$193$	21.0108	1.51239	0.756197	+	0.654344i	$0.227055\pi$
$194$	14.8663	1.06734
$195$	0	0
$196$	4.85550	0.346822
$197$	−20.3776	−1.45185	−0.725923	−	0.687776i	$-0.758587\pi$
$197$	−20.3776	−1.45185	−0.725923	+	0.687776i	$0.758587\pi$
$198$	0	0
$199$	−14.8014	−1.04924	−0.524621	−	0.851336i	$-0.675793\pi$
$199$	−14.8014	−1.04924	−0.524621	+	0.851336i	$0.675793\pi$
$200$	−3.07638	−0.217533
$201$	0	0
$202$	8.85550	0.623071
$203$	−1.68576	−0.118317
$204$	0	0
$205$	1.07012	0.0747404
$206$	17.0145	1.18546
$207$	0	0
$208$	0	0
$209$	15.0275	1.03947
$210$	0	0
$211$	5.22030	0.359380	0.179690	−	0.983723i	$-0.442491\pi$
$211$	5.22030	0.359380	0.179690	+	0.983723i	$0.442491\pi$
$212$	−7.80138	−0.535801
$213$	0	0
$214$	9.24134	0.631725
$215$	−5.01084	−0.341736
$216$	0	0
$217$	−5.54674	−0.376537
$218$	11.9277	0.807848
$219$	0	0
$220$	2.25207	0.151834
$221$	0	0
$222$	0	0
$223$	−6.16685	−0.412963	−0.206481	−	0.978451i	$-0.566201\pi$
$223$	−6.16685	−0.412963	−0.206481	+	0.978451i	$0.566201\pi$
$224$	−2.36389	−0.157944
$225$	0	0
$226$	−3.90069	−0.259470
$227$	−17.1114	−1.13572	−0.567860	−	0.823125i	$-0.692229\pi$
$227$	−17.1114	−1.13572	−0.567860	+	0.823125i	$0.692229\pi$
$228$	0	0
$229$	−28.9206	−1.91113	−0.955563	−	0.294788i	$-0.904751\pi$
$229$	−28.9206	−1.91113	−0.955563	+	0.294788i	$0.904751\pi$
$230$	−2.09931	−0.138424
$231$	0	0
$232$	−8.55758	−0.561832
$233$	−11.3284	−0.742151	−0.371075	−	0.928603i	$-0.621011\pi$
$233$	−11.3284	−0.742151	−0.371075	+	0.928603i	$0.621011\pi$
$234$	0	0
$235$	−10.2024	−0.665532
$236$	−8.47294	−0.551541
$237$	0	0
$238$	1.33091	0.0862700
$239$	19.1298	1.23741	0.618703	−	0.785625i	$-0.287659\pi$
$239$	19.1298	1.23741	0.618703	+	0.785625i	$0.287659\pi$
$240$	0	0
$241$	−5.47415	−0.352621	−0.176311	−	0.984335i	$-0.556416\pi$
$241$	−5.47415	−0.352621	−0.176311	+	0.984335i	$0.556416\pi$
$242$	−1.72947	−0.111175
$243$	0	0
$244$	4.58846	0.293746
$245$	−6.63274	−0.423750
$246$	0	0
$247$	0	0
$248$	−28.1573	−1.78799
$249$	0	0
$250$	1.12603	0.0712165
$251$	−4.71985	−0.297914	−0.148957	−	0.988844i	$-0.547592\pi$
$251$	−4.71985	−0.297914	−0.148957	+	0.988844i	$0.547592\pi$
$252$	0	0
$253$	5.73542	0.360583
$254$	20.3179	1.27486
$255$	0	0
$256$	−14.9282	−0.933013
$257$	−7.27879	−0.454038	−0.227019	−	0.973890i	$-0.572898\pi$
$257$	−7.27879	−0.454038	−0.227019	+	0.973890i	$0.572898\pi$
$258$	0	0
$259$	−2.27132	−0.141133
$260$	0	0
$261$	0	0
$262$	−1.26001	−0.0778439
$263$	1.28672	0.0793428	0.0396714	−	0.999213i	$-0.487369\pi$
$263$	1.28672	0.0793428	0.0396714	+	0.999213i	$0.487369\pi$
$264$	0	0
$265$	10.6569	0.654647
$266$	3.33337	0.204382
$267$	0	0
$268$	−1.44789	−0.0884441
$269$	−18.9390	−1.15473	−0.577367	−	0.816485i	$-0.695920\pi$
$269$	−18.9390	−1.15473	−0.577367	+	0.816485i	$0.695920\pi$
$270$	0	0
$271$	11.0241	0.669669	0.334835	−	0.942277i	$-0.391320\pi$
$271$	11.0241	0.669669	0.334835	+	0.942277i	$0.391320\pi$
$272$	3.90069	0.236514
$273$	0	0
$274$	−9.80385	−0.592272
$275$	−3.07638	−0.185513
$276$	0	0
$277$	20.9215	1.25705	0.628525	−	0.777790i	$-0.283659\pi$
$277$	20.9215	1.25705	0.628525	+	0.777790i	$0.283659\pi$
$278$	10.8740	0.652177
$279$	0	0
$280$	1.86434	0.111416
$281$	−10.6653	−0.636238	−0.318119	−	0.948051i	$-0.603051\pi$
$281$	−10.6653	−0.636238	−0.318119	+	0.948051i	$0.603051\pi$
$282$	0	0
$283$	2.78507	0.165555	0.0827777	−	0.996568i	$-0.473621\pi$
$283$	2.78507	0.165555	0.0827777	+	0.996568i	$0.473621\pi$
$284$	−4.32419	−0.256593
$285$	0	0
$286$	0	0
$287$	−0.648512	−0.0382805
$288$	0	0
$289$	−13.1962	−0.776244
$290$	3.13229	0.183934
$291$	0	0
$292$	−3.18732	−0.186524
$293$	4.95861	0.289685	0.144842	−	0.989455i	$-0.453732\pi$
$293$	4.95861	0.289685	0.144842	+	0.989455i	$0.453732\pi$
$294$	0	0
$295$	11.5742	0.673879
$296$	−11.5301	−0.670171
$297$	0	0
$298$	−5.82799	−0.337606
$299$	0	0
$300$	0	0
$301$	3.03666	0.175030
$302$	24.9361	1.43491
$303$	0	0
$304$	9.76961	0.560326
$305$	−6.26795	−0.358902
$306$	0	0
$307$	−9.60723	−0.548314	−0.274157	−	0.961685i	$-0.588399\pi$
$307$	−9.60723	−0.548314	−0.274157	+	0.961685i	$0.588399\pi$
$308$	−1.36479	−0.0777663
$309$	0	0
$310$	10.3063	0.585359
$311$	26.0393	1.47655	0.738275	−	0.674499i	$-0.235641\pi$
$311$	26.0393	1.47655	0.738275	+	0.674499i	$0.235641\pi$
$312$	0	0
$313$	29.2311	1.65224	0.826121	−	0.563493i	$-0.190543\pi$
$313$	29.2311	1.65224	0.826121	+	0.563493i	$0.190543\pi$
$314$	−16.1874	−0.913509
$315$	0	0
$316$	2.41245	0.135711
$317$	8.62570	0.484467	0.242234	−	0.970218i	$-0.422120\pi$
$317$	8.62570	0.484467	0.242234	+	0.970218i	$0.422120\pi$
$318$	0	0
$319$	−8.55758	−0.479132
$320$	8.39230	0.469144
$321$	0	0
$322$	1.27222	0.0708980
$323$	9.52706	0.530100
$324$	0	0
$325$	0	0
$326$	11.3393	0.628025
$327$	0	0
$328$	−3.29209	−0.181775
$329$	6.18285	0.340871
$330$	0	0
$331$	20.3021	1.11591	0.557953	−	0.829872i	$-0.311587\pi$
$331$	20.3021	1.11591	0.557953	+	0.829872i	$0.311587\pi$
$332$	−5.10757	−0.280314
$333$	0	0
$334$	12.2096	0.668077
$335$	1.97786	0.108062
$336$	0	0
$337$	−17.7847	−0.968795	−0.484397	−	0.874848i	$-0.660961\pi$
$337$	−17.7847	−0.968795	−0.484397	+	0.874848i	$0.660961\pi$
$338$	0	0
$339$	0	0
$340$	1.42775	0.0774307
$341$	−28.1573	−1.52481
$342$	0	0
$343$	8.26169	0.446089
$344$	15.4152	0.831134
$345$	0	0
$346$	−27.7705	−1.49295
$347$	−8.59092	−0.461185	−0.230592	−	0.973050i	$-0.574066\pi$
$347$	−8.59092	−0.461185	−0.230592	+	0.973050i	$0.574066\pi$
$348$	0	0
$349$	30.1803	1.61551	0.807756	−	0.589516i	$-0.200682\pi$
$349$	30.1803	1.61551	0.807756	+	0.589516i	$0.200682\pi$
$350$	−0.682396	−0.0364756
$351$	0	0
$352$	−12.0000	−0.639602
$353$	28.6297	1.52381	0.761903	−	0.647692i	$-0.224265\pi$
$353$	28.6297	1.52381	0.761903	+	0.647692i	$0.224265\pi$
$354$	0	0
$355$	5.90695	0.313508
$356$	11.9319	0.632388
$357$	0	0
$358$	25.7183	1.35925
$359$	17.4624	0.921632	0.460816	−	0.887496i	$-0.347557\pi$
$359$	17.4624	0.921632	0.460816	+	0.887496i	$0.347557\pi$
$360$	0	0
$361$	4.86134	0.255860
$362$	19.6624	1.03343
$363$	0	0
$364$	0	0
$365$	4.35395	0.227896
$366$	0	0
$367$	32.4966	1.69631	0.848155	−	0.529748i	$-0.177714\pi$
$367$	32.4966	1.69631	0.848155	+	0.529748i	$0.177714\pi$
$368$	3.72868	0.194371
$369$	0	0
$370$	4.22030	0.219403
$371$	−6.45827	−0.335297
$372$	0	0
$373$	10.7517	0.556703	0.278352	−	0.960479i	$-0.410212\pi$
$373$	10.7517	0.556703	0.278352	+	0.960479i	$0.410212\pi$
$374$	6.75620	0.349355
$375$	0	0
$376$	31.3865	1.61863
$377$	0	0
$378$	0	0
$379$	−5.84141	−0.300053	−0.150027	−	0.988682i	$-0.547936\pi$
$379$	−5.84141	−0.300053	−0.150027	+	0.988682i	$0.547936\pi$
$380$	3.57593	0.183441
$381$	0	0
$382$	−8.64068	−0.442095
$383$	23.6439	1.20815	0.604074	−	0.796929i	$-0.293543\pi$
$383$	23.6439	1.20815	0.604074	+	0.796929i	$0.293543\pi$
$384$	0	0
$385$	1.86434	0.0950156
$386$	23.6589	1.20421
$387$	0	0
$388$	−9.66484	−0.490658
$389$	−16.5939	−0.841345	−0.420673	−	0.907212i	$-0.638206\pi$
$389$	−16.5939	−0.841345	−0.420673	+	0.907212i	$0.638206\pi$
$390$	0	0
$391$	3.63611	0.183886
$392$	20.4048	1.03060
$393$	0	0
$394$	−22.9459	−1.15600
$395$	−3.29546	−0.165813
$396$	0	0
$397$	23.4035	1.17459	0.587293	−	0.809375i	$-0.300194\pi$
$397$	23.4035	1.17459	0.587293	+	0.809375i	$0.300194\pi$
$398$	−16.6668	−0.835433
$399$	0	0
$400$	−2.00000	−0.100000
$401$	−0.111825	−0.00558428	−0.00279214	−	0.999996i	$-0.500889\pi$
$401$	−0.111825	−0.00558428	−0.00279214	+	0.999996i	$0.500889\pi$
$402$	0	0
$403$	0	0
$404$	−5.75710	−0.286426
$405$	0	0
$406$	−1.89823	−0.0942073
$407$	−11.5301	−0.571524
$408$	0	0
$409$	19.2805	0.953358	0.476679	−	0.879077i	$-0.341840\pi$
$409$	19.2805	0.953358	0.476679	+	0.879077i	$0.341840\pi$
$410$	1.20499	0.0595102
$411$	0	0
$412$	−11.0614	−0.544956
$413$	−7.01421	−0.345147
$414$	0	0
$415$	6.97707	0.342491
$416$	0	0
$417$	0	0
$418$	16.9215	0.827656
$419$	−22.1386	−1.08154	−0.540770	−	0.841171i	$-0.681867\pi$
$419$	−22.1386	−1.08154	−0.540770	+	0.841171i	$0.681867\pi$
$420$	0	0
$421$	0.914785	0.0445839	0.0222919	−	0.999752i	$-0.492904\pi$
$421$	0.914785	0.0445839	0.0222919	+	0.999752i	$0.492904\pi$
$422$	5.87822	0.286147
$423$	0	0
$424$	−32.7846	−1.59216
$425$	−1.95035	−0.0946057
$426$	0	0
$427$	3.79849	0.183822
$428$	−6.00793	−0.290404
$429$	0	0
$430$	−5.64237	−0.272099
$431$	36.9771	1.78112	0.890561	−	0.454863i	$-0.150312\pi$
$431$	36.9771	1.78112	0.890561	+	0.454863i	$0.150312\pi$
$432$	0	0
$433$	−28.6898	−1.37874	−0.689371	−	0.724408i	$-0.742113\pi$
$433$	−28.6898	−1.37874	−0.689371	+	0.724408i	$0.742113\pi$
$434$	−6.24581	−0.299808
$435$	0	0
$436$	−7.75440	−0.371369
$437$	9.10695	0.435644
$438$	0	0
$439$	5.26212	0.251147	0.125574	−	0.992084i	$-0.459923\pi$
$439$	5.26212	0.251147	0.125574	+	0.992084i	$0.459923\pi$
$440$	9.46410	0.451183
$441$	0	0
$442$	0	0
$443$	−28.8275	−1.36964	−0.684819	−	0.728714i	$-0.740119\pi$
$443$	−28.8275	−1.36964	−0.684819	+	0.728714i	$0.740119\pi$
$444$	0	0
$445$	−16.2993	−0.772659
$446$	−6.94407	−0.328811
$447$	0	0
$448$	−5.08589	−0.240286
$449$	3.00826	0.141969	0.0709843	−	0.997477i	$-0.477386\pi$
$449$	3.00826	0.141969	0.0709843	+	0.997477i	$0.477386\pi$
$450$	0	0
$451$	−3.29209	−0.155019
$452$	2.53590	0.119279
$453$	0	0
$454$	−19.2679	−0.904290
$455$	0	0
$456$	0	0
$457$	−8.88681	−0.415707	−0.207854	−	0.978160i	$-0.566648\pi$
$457$	−8.88681	−0.415707	−0.207854	+	0.978160i	$0.566648\pi$
$458$	−32.5655	−1.52169
$459$	0	0
$460$	1.36479	0.0636338
$461$	21.1089	0.983139	0.491570	−	0.870838i	$-0.336423\pi$
$461$	21.1089	0.983139	0.491570	+	0.870838i	$0.336423\pi$
$462$	0	0
$463$	24.4679	1.13712	0.568560	−	0.822642i	$-0.307501\pi$
$463$	24.4679	1.13712	0.568560	+	0.822642i	$0.307501\pi$
$464$	−5.56341	−0.258275
$465$	0	0
$466$	−12.7562	−0.590919
$467$	19.5058	0.902622	0.451311	−	0.892367i	$-0.350957\pi$
$467$	19.5058	0.902622	0.451311	+	0.892367i	$0.350957\pi$
$468$	0	0
$469$	−1.19862	−0.0553470
$470$	−11.4882	−0.529913
$471$	0	0
$472$	−35.6068	−1.63893
$473$	15.4152	0.708793
$474$	0	0
$475$	−4.88481	−0.224130
$476$	−0.865244	−0.0396584
$477$	0	0
$478$	21.5408	0.985253
$479$	−27.0304	−1.23505	−0.617526	−	0.786551i	$-0.711865\pi$
$479$	−27.0304	−1.23505	−0.617526	+	0.786551i	$0.711865\pi$
$480$	0	0
$481$	0	0
$482$	−6.16407	−0.280766
$483$	0	0
$484$	1.12436	0.0511071
$485$	13.2024	0.599491
$486$	0	0
$487$	−5.21751	−0.236428	−0.118214	−	0.992988i	$-0.537717\pi$
$487$	−5.21751	−0.236428	−0.118214	+	0.992988i	$0.537717\pi$
$488$	19.2826	0.872881
$489$	0	0
$490$	−7.46868	−0.337401
$491$	−14.7817	−0.667089	−0.333545	−	0.942734i	$-0.608245\pi$
$491$	−14.7817	−0.667089	−0.333545	+	0.942734i	$0.608245\pi$
$492$	0	0
$493$	−5.42529	−0.244343
$494$	0	0
$495$	0	0
$496$	−18.3055	−0.821942
$497$	−3.57972	−0.160572
$498$	0	0
$499$	−10.3171	−0.461859	−0.230929	−	0.972971i	$-0.574177\pi$
$499$	−10.3171	−0.461859	−0.230929	+	0.972971i	$0.574177\pi$
$500$	−0.732051	−0.0327383
$501$	0	0
$502$	−5.31470	−0.237207
$503$	−26.6343	−1.18756	−0.593782	−	0.804626i	$-0.702366\pi$
$503$	−26.6343	−1.18756	−0.593782	+	0.804626i	$0.702366\pi$
$504$	0	0
$505$	7.86434	0.349959
$506$	6.45827	0.287105
$507$	0	0
$508$	−13.2090	−0.586054
$509$	−1.42775	−0.0632840	−0.0316420	−	0.999499i	$-0.510074\pi$
$509$	−1.42775	−0.0632840	−0.0316420	+	0.999499i	$0.510074\pi$
$510$	0	0
$511$	−2.63858	−0.116724
$512$	−20.1069	−0.888608
$513$	0	0
$514$	−8.19615	−0.361517
$515$	15.1101	0.665833
$516$	0	0
$517$	31.3865	1.38038
$518$	−2.55758	−0.112373
$519$	0	0
$520$	0	0
$521$	−12.8623	−0.563509	−0.281755	−	0.959486i	$-0.590916\pi$
$521$	−12.8623	−0.563509	−0.281755	+	0.959486i	$0.590916\pi$
$522$	0	0
$523$	15.1002	0.660286	0.330143	−	0.943931i	$-0.392903\pi$
$523$	15.1002	0.660286	0.330143	+	0.943931i	$0.392903\pi$
$524$	0.819154	0.0357849
$525$	0	0
$526$	1.44889	0.0631747
$527$	−17.8510	−0.777603
$528$	0	0
$529$	−19.5242	−0.848880
$530$	12.0000	0.521247
$531$	0	0
$532$	−2.16708	−0.0939547
$533$	0	0
$534$	0	0
$535$	8.20699	0.354819
$536$	−6.08464	−0.262816
$537$	0	0
$538$	−21.3260	−0.919428
$539$	20.4048	0.878898
$540$	0	0
$541$	41.1084	1.76739	0.883695	−	0.468064i	$-0.155048\pi$
$541$	41.1084	1.76739	0.883695	+	0.468064i	$0.155048\pi$
$542$	12.4135	0.533207
$543$	0	0
$544$	−7.60770	−0.326177
$545$	10.5927	0.453742
$546$	0	0
$547$	30.1327	1.28838	0.644191	−	0.764864i	$-0.277194\pi$
$547$	30.1327	1.28838	0.644191	+	0.764864i	$0.277194\pi$
$548$	6.37363	0.272268
$549$	0	0
$550$	−3.46410	−0.147710
$551$	−13.5881	−0.578872
$552$	0	0
$553$	1.99711	0.0849258
$554$	23.5583	1.00089
$555$	0	0
$556$	−7.06933	−0.299806
$557$	25.2041	1.06793	0.533966	−	0.845506i	$-0.320701\pi$
$557$	25.2041	1.06793	0.533966	+	0.845506i	$0.320701\pi$
$558$	0	0
$559$	0	0
$560$	1.21204	0.0512179
$561$	0	0
$562$	−12.0095	−0.506589
$563$	7.19278	0.303140	0.151570	−	0.988447i	$-0.451567\pi$
$563$	7.19278	0.303140	0.151570	+	0.988447i	$0.451567\pi$
$564$	0	0
$565$	−3.46410	−0.145736
$566$	3.13608	0.131819
$567$	0	0
$568$	−18.1720	−0.762481
$569$	−18.3521	−0.769361	−0.384681	−	0.923050i	$-0.625688\pi$
$569$	−18.3521	−0.769361	−0.384681	+	0.923050i	$0.625688\pi$
$570$	0	0
$571$	−27.5433	−1.15265	−0.576325	−	0.817221i	$-0.695514\pi$
$571$	−27.5433	−1.15265	−0.576325	+	0.817221i	$0.695514\pi$
$572$	0	0
$573$	0	0
$574$	−0.730246	−0.0304799
$575$	−1.86434	−0.0777484
$576$	0	0
$577$	19.1378	0.796715	0.398358	−	0.917230i	$-0.369580\pi$
$577$	19.1378	0.796715	0.398358	+	0.917230i	$0.369580\pi$
$578$	−14.8593	−0.618065
$579$	0	0
$580$	−2.03635	−0.0845548
$581$	−4.22823	−0.175417
$582$	0	0
$583$	−32.7846	−1.35780
$584$	−13.3944	−0.554264
$585$	0	0
$586$	5.58355	0.230654
$587$	−17.5620	−0.724863	−0.362432	−	0.932010i	$-0.618053\pi$
$587$	−17.5620	−0.724863	−0.362432	+	0.932010i	$0.618053\pi$
$588$	0	0
$589$	−44.7094	−1.84222
$590$	13.0330	0.536559
$591$	0	0
$592$	−7.49587	−0.308078
$593$	−28.9248	−1.18780	−0.593901	−	0.804538i	$-0.702413\pi$
$593$	−28.9248	−1.18780	−0.593901	+	0.804538i	$0.702413\pi$
$594$	0	0
$595$	1.18195	0.0484550
$596$	3.78887	0.155198
$597$	0	0
$598$	0	0
$599$	46.1052	1.88381	0.941904	−	0.335882i	$-0.109034\pi$
$599$	46.1052	1.88381	0.941904	+	0.335882i	$0.109034\pi$
$600$	0	0
$601$	−19.5099	−0.795826	−0.397913	−	0.917423i	$-0.630265\pi$
$601$	−19.5099	−0.795826	−0.397913	+	0.917423i	$0.630265\pi$
$602$	3.41938	0.139363
$603$	0	0
$604$	−16.2114	−0.659631
$605$	−1.53590	−0.0624431
$606$	0	0
$607$	25.4253	1.03198	0.515990	−	0.856594i	$-0.327424\pi$
$607$	25.4253	1.03198	0.515990	+	0.856594i	$0.327424\pi$
$608$	−19.0541	−0.772747
$609$	0	0
$610$	−7.05791	−0.285767
$611$	0	0
$612$	0	0
$613$	−33.4569	−1.35131	−0.675656	−	0.737217i	$-0.736139\pi$
$613$	−33.4569	−1.35131	−0.675656	+	0.737217i	$0.736139\pi$
$614$	−10.8181	−0.436581
$615$	0	0
$616$	−5.73542	−0.231087
$617$	−13.9603	−0.562020	−0.281010	−	0.959705i	$-0.590669\pi$
$617$	−13.9603	−0.562020	−0.281010	+	0.959705i	$0.590669\pi$
$618$	0	0
$619$	−42.1677	−1.69486	−0.847432	−	0.530904i	$-0.821852\pi$
$619$	−42.1677	−1.69486	−0.847432	+	0.530904i	$0.821852\pi$
$620$	−6.70028	−0.269090
$621$	0	0
$622$	29.3210	1.17567
$623$	9.87765	0.395740
$624$	0	0
$625$	1.00000	0.0400000
$626$	32.9152	1.31556
$627$	0	0
$628$	10.5237	0.419941
$629$	−7.30977	−0.291460
$630$	0	0
$631$	39.9691	1.59115	0.795573	−	0.605858i	$-0.207170\pi$
$631$	39.9691	1.59115	0.795573	+	0.605858i	$0.207170\pi$
$632$	10.1381	0.403271
$633$	0	0
$634$	9.71281	0.385745
$635$	18.0438	0.716047
$636$	0	0
$637$	0	0
$638$	−9.63611	−0.381497
$639$	0	0
$640$	1.64863	0.0651677
$641$	26.6422	1.05230	0.526152	−	0.850390i	$-0.323634\pi$
$641$	26.6422	1.05230	0.526152	+	0.850390i	$0.323634\pi$
$642$	0	0
$643$	−16.6185	−0.655371	−0.327686	−	0.944787i	$-0.606269\pi$
$643$	−16.6185	−0.655371	−0.327686	+	0.944787i	$0.606269\pi$
$644$	−0.827089	−0.0325919
$645$	0	0
$646$	10.7278	0.422079
$647$	20.4729	0.804874	0.402437	−	0.915448i	$-0.368163\pi$
$647$	20.4729	0.804874	0.402437	+	0.915448i	$0.368163\pi$
$648$	0	0
$649$	−35.6068	−1.39769
$650$	0	0
$651$	0	0
$652$	−7.37184	−0.288704
$653$	−5.76740	−0.225696	−0.112848	−	0.993612i	$-0.535997\pi$
$653$	−5.76740	−0.225696	−0.112848	+	0.993612i	$0.535997\pi$
$654$	0	0
$655$	−1.11899	−0.0437224
$656$	−2.14024	−0.0835623
$657$	0	0
$658$	6.96209	0.271410
$659$	−30.9499	−1.20564	−0.602818	−	0.797879i	$-0.705956\pi$
$659$	−30.9499	−1.20564	−0.602818	+	0.797879i	$0.705956\pi$
$660$	0	0
$661$	30.8806	1.20111	0.600557	−	0.799582i	$-0.294946\pi$
$661$	30.8806	1.20111	0.600557	+	0.799582i	$0.294946\pi$
$662$	22.8609	0.888513
$663$	0	0
$664$	−21.4641	−0.832969
$665$	2.96028	0.114795
$666$	0	0
$667$	−5.18605	−0.200805
$668$	−7.93762	−0.307116
$669$	0	0
$670$	2.22713	0.0860416
$671$	19.2826	0.744396
$672$	0	0
$673$	23.5079	0.906164	0.453082	−	0.891469i	$-0.350325\pi$
$673$	23.5079	0.906164	0.453082	+	0.891469i	$0.350325\pi$
$674$	−20.0262	−0.771379
$675$	0	0
$676$	0	0
$677$	5.52213	0.212233	0.106116	−	0.994354i	$-0.466158\pi$
$677$	5.52213	0.212233	0.106116	+	0.994354i	$0.466158\pi$
$678$	0	0
$679$	−8.00090	−0.307046
$680$	6.00000	0.230089
$681$	0	0
$682$	−31.7061	−1.21409
$683$	−13.4725	−0.515510	−0.257755	−	0.966210i	$-0.582983\pi$
$683$	−13.4725	−0.515510	−0.257755	+	0.966210i	$0.582983\pi$
$684$	0	0
$685$	−8.70654	−0.332660
$686$	9.30293	0.355188
$687$	0	0
$688$	10.0217	0.382073
$689$	0	0
$690$	0	0
$691$	42.5023	1.61686	0.808432	−	0.588590i	$-0.200317\pi$
$691$	42.5023	1.61686	0.808432	+	0.588590i	$0.200317\pi$
$692$	18.0540	0.686311
$693$	0	0
$694$	−9.67366	−0.367207
$695$	9.65689	0.366307
$696$	0	0
$697$	−2.08710	−0.0790547
$698$	33.9840	1.28631
$699$	0	0
$700$	0.443636	0.0167679
$701$	0.553573	0.0209082	0.0104541	−	0.999945i	$-0.496672\pi$
$701$	0.553573	0.0209082	0.0104541	+	0.999945i	$0.496672\pi$
$702$	0	0
$703$	−18.3079	−0.690497
$704$	−25.8179	−0.973049
$705$	0	0
$706$	32.2380	1.21329
$707$	−4.76593	−0.179241
$708$	0	0
$709$	12.9783	0.487410	0.243705	−	0.969849i	$-0.421637\pi$
$709$	12.9783	0.487410	0.243705	+	0.969849i	$0.421637\pi$
$710$	6.65142	0.249623
$711$	0	0
$712$	50.1427	1.87918
$713$	−17.0639	−0.639047
$714$	0	0
$715$	0	0
$716$	−16.7198	−0.624850
$717$	0	0
$718$	19.6633	0.733826
$719$	22.4328	0.836601	0.418300	−	0.908309i	$-0.362626\pi$
$719$	22.4328	0.836601	0.418300	+	0.908309i	$0.362626\pi$
$720$	0	0
$721$	−9.15703	−0.341025
$722$	5.47402	0.203722
$723$	0	0
$724$	−12.7828	−0.475069
$725$	2.78171	0.103310
$726$	0	0
$727$	−48.3530	−1.79331	−0.896657	−	0.442725i	$-0.854012\pi$
$727$	−48.3530	−1.79331	−0.896657	+	0.442725i	$0.854012\pi$
$728$	0	0
$729$	0	0
$730$	4.90269	0.181457
$731$	9.77287	0.361463
$732$	0	0
$733$	5.72579	0.211487	0.105743	−	0.994393i	$-0.466278\pi$
$733$	5.72579	0.211487	0.105743	+	0.994393i	$0.466278\pi$
$734$	36.5922	1.35064
$735$	0	0
$736$	−7.27222	−0.268058
$737$	−6.08464	−0.224131
$738$	0	0
$739$	−20.8964	−0.768688	−0.384344	−	0.923190i	$-0.625572\pi$
$739$	−20.8964	−0.768688	−0.384344	+	0.923190i	$0.625572\pi$
$740$	−2.74368	−0.100860
$741$	0	0
$742$	−7.27222	−0.266972
$743$	29.5593	1.08442	0.542212	−	0.840242i	$-0.317587\pi$
$743$	29.5593	1.08442	0.542212	+	0.840242i	$0.317587\pi$
$744$	0	0
$745$	−5.17569	−0.189622
$746$	12.1068	0.443261
$747$	0	0
$748$	−4.39230	−0.160599
$749$	−4.97359	−0.181731
$750$	0	0
$751$	−21.0467	−0.768006	−0.384003	−	0.923332i	$-0.625455\pi$
$751$	−21.0467	−0.768006	−0.384003	+	0.923332i	$0.625455\pi$
$752$	20.4048	0.744087
$753$	0	0
$754$	0	0
$755$	22.1451	0.805944
$756$	0	0
$757$	14.0849	0.511925	0.255962	−	0.966687i	$-0.417608\pi$
$757$	14.0849	0.511925	0.255962	+	0.966687i	$0.417608\pi$
$758$	−6.57762	−0.238910
$759$	0	0
$760$	15.0275	0.545105
$761$	−45.3819	−1.64509	−0.822546	−	0.568698i	$-0.807447\pi$
$761$	−45.3819	−1.64509	−0.822546	+	0.568698i	$0.807447\pi$
$762$	0	0
$763$	−6.41938	−0.232397
$764$	5.61744	0.203232
$765$	0	0
$766$	26.6238	0.961957
$767$	0	0
$768$	0	0
$769$	44.1769	1.59306	0.796530	−	0.604599i	$-0.206667\pi$
$769$	44.1769	1.59306	0.796530	+	0.604599i	$0.206667\pi$
$770$	2.09931	0.0756538
$771$	0	0
$772$	−15.3810	−0.553574
$773$	−35.7458	−1.28569	−0.642843	−	0.765998i	$-0.722245\pi$
$773$	−35.7458	−1.28569	−0.642843	+	0.765998i	$0.722245\pi$
$774$	0	0
$775$	9.15276	0.328777
$776$	−40.6156	−1.45802
$777$	0	0
$778$	−18.6853	−0.669900
$779$	−5.22733	−0.187289
$780$	0	0
$781$	−18.1720	−0.650246
$782$	4.09438	0.146415
$783$	0	0
$784$	13.2655	0.473767
$785$	−14.3756	−0.513088
$786$	0	0
$787$	10.1819	0.362947	0.181474	−	0.983396i	$-0.441913\pi$
$787$	10.1819	0.362947	0.181474	+	0.983396i	$0.441913\pi$
$788$	14.9175	0.531413
$789$	0	0
$790$	−3.71080	−0.132024
$791$	2.09931	0.0746428
$792$	0	0
$793$	0	0
$794$	26.3531	0.935235
$795$	0	0
$796$	10.8354	0.384049
$797$	36.8856	1.30655	0.653277	−	0.757119i	$-0.273394\pi$
$797$	36.8856	1.30655	0.653277	+	0.757119i	$0.273394\pi$
$798$	0	0
$799$	19.8982	0.703949
$800$	3.90069	0.137910
$801$	0	0
$802$	−0.125919	−0.00444635
$803$	−13.3944	−0.472678
$804$	0	0
$805$	1.12983	0.0398211
$806$	0	0
$807$	0	0
$808$	−24.1937	−0.851131
$809$	12.1693	0.427849	0.213924	−	0.976850i	$-0.431375\pi$
$809$	12.1693	0.427849	0.213924	+	0.976850i	$0.431375\pi$
$810$	0	0
$811$	26.2312	0.921104	0.460552	−	0.887633i	$-0.347652\pi$
$811$	26.2312	0.921104	0.460552	+	0.887633i	$0.347652\pi$
$812$	1.23407	0.0433072
$813$	0	0
$814$	−12.9832	−0.455062
$815$	10.0701	0.352741
$816$	0	0
$817$	24.4770	0.856341
$818$	21.7104	0.759088
$819$	0	0
$820$	−0.783382	−0.0273569
$821$	16.5451	0.577427	0.288713	−	0.957416i	$-0.406773\pi$
$821$	16.5451	0.577427	0.288713	+	0.957416i	$0.406773\pi$
$822$	0	0
$823$	27.9291	0.973547	0.486774	−	0.873528i	$-0.338174\pi$
$823$	27.9291	0.973547	0.486774	+	0.873528i	$0.338174\pi$
$824$	−46.4845	−1.61937
$825$	0	0
$826$	−7.89823	−0.274814
$827$	−1.83852	−0.0639316	−0.0319658	−	0.999489i	$-0.510177\pi$
$827$	−1.83852	−0.0639316	−0.0319658	+	0.999489i	$0.510177\pi$
$828$	0	0
$829$	−34.9731	−1.21467	−0.607333	−	0.794447i	$-0.707761\pi$
$829$	−34.9731	−1.21467	−0.607333	+	0.794447i	$0.707761\pi$
$830$	7.85641	0.272700
$831$	0	0
$832$	0	0
$833$	12.9361	0.448211
$834$	0	0
$835$	10.8430	0.375237
$836$	−11.0009	−0.380474
$837$	0	0
$838$	−24.9287	−0.861149
$839$	12.7125	0.438884	0.219442	−	0.975626i	$-0.429576\pi$
$839$	12.7125	0.438884	0.219442	+	0.975626i	$0.429576\pi$
$840$	0	0
$841$	−21.2621	−0.733176
$842$	1.03008	0.0354988
$843$	0	0
$844$	−3.82152	−0.131542
$845$	0	0
$846$	0	0
$847$	0.930783	0.0319821
$848$	−21.3138	−0.731918
$849$	0	0
$850$	−2.19615	−0.0753274
$851$	−6.98743	−0.239526
$852$	0	0
$853$	−20.2430	−0.693106	−0.346553	−	0.938030i	$-0.612648\pi$
$853$	−20.2430	−0.693106	−0.346553	+	0.938030i	$0.612648\pi$
$854$	4.27723	0.146364
$855$	0	0
$856$	−25.2478	−0.862952
$857$	−53.5208	−1.82823	−0.914117	−	0.405450i	$-0.867115\pi$
$857$	−53.5208	−1.82823	−0.914117	+	0.405450i	$0.867115\pi$
$858$	0	0
$859$	−0.969120	−0.0330659	−0.0165330	−	0.999863i	$-0.505263\pi$
$859$	−0.969120	−0.0330659	−0.0165330	+	0.999863i	$0.505263\pi$
$860$	3.66819	0.125084
$861$	0	0
$862$	41.6374	1.41817
$863$	−5.67766	−0.193270	−0.0966349	−	0.995320i	$-0.530808\pi$
$863$	−5.67766	−0.193270	−0.0966349	+	0.995320i	$0.530808\pi$
$864$	0	0
$865$	−24.6623	−0.838542
$866$	−32.3056	−1.09779
$867$	0	0
$868$	4.06049	0.137822
$869$	10.1381	0.343911
$870$	0	0
$871$	0	0
$872$	−32.5872	−1.10354
$873$	0	0
$874$	10.2547	0.346871
$875$	−0.606018	−0.0204872
$876$	0	0
$877$	−32.2169	−1.08789	−0.543944	−	0.839121i	$-0.683070\pi$
$877$	−32.2169	−1.08789	−0.543944	+	0.839121i	$0.683070\pi$
$878$	5.92531	0.199970
$879$	0	0
$880$	6.15276	0.207409
$881$	−41.1152	−1.38520	−0.692602	−	0.721320i	$-0.743536\pi$
$881$	−41.1152	−1.38520	−0.692602	+	0.721320i	$0.743536\pi$
$882$	0	0
$883$	−14.6027	−0.491419	−0.245709	−	0.969344i	$-0.579021\pi$
$883$	−14.6027	−0.491419	−0.245709	+	0.969344i	$0.579021\pi$
$884$	0	0
$885$	0	0
$886$	−32.4607	−1.09054
$887$	−9.10558	−0.305736	−0.152868	−	0.988247i	$-0.548851\pi$
$887$	−9.10558	−0.305736	−0.152868	+	0.988247i	$0.548851\pi$
$888$	0	0
$889$	−10.9349	−0.366744
$890$	−18.3535	−0.615210
$891$	0	0
$892$	4.51445	0.151155
$893$	49.8368	1.66773
$894$	0	0
$895$	22.8397	0.763448
$896$	−0.999098	−0.0333775
$897$	0	0
$898$	3.38740	0.113039
$899$	25.4603	0.849148
$900$	0	0
$901$	−20.7846	−0.692436
$902$	−3.70700	−0.123430
$903$	0	0
$904$	10.6569	0.354443
$905$	17.4616	0.580444
$906$	0	0
$907$	−50.5160	−1.67736	−0.838679	−	0.544627i	$-0.816671\pi$
$907$	−50.5160	−1.67736	−0.838679	+	0.544627i	$0.816671\pi$
$908$	12.5264	0.415703
$909$	0	0
$910$	0	0
$911$	−45.3571	−1.50275	−0.751374	−	0.659876i	$-0.770609\pi$
$911$	−45.3571	−1.50275	−0.751374	+	0.659876i	$0.770609\pi$
$912$	0	0
$913$	−21.4641	−0.710358
$914$	−10.0068	−0.330997
$915$	0	0
$916$	21.1713	0.699521
$917$	0.678126	0.0223937
$918$	0	0
$919$	4.30220	0.141916	0.0709582	−	0.997479i	$-0.477394\pi$
$919$	4.30220	0.141916	0.0709582	+	0.997479i	$0.477394\pi$
$920$	5.73542	0.189091
$921$	0	0
$922$	23.7693	0.782800
$923$	0	0
$924$	0	0
$925$	3.74793	0.123231
$926$	27.5516	0.905403
$927$	0	0
$928$	10.8506	0.356188
$929$	−48.3874	−1.58754	−0.793769	−	0.608219i	$-0.791884\pi$
$929$	−48.3874	−1.58754	−0.793769	+	0.608219i	$0.791884\pi$
$930$	0	0
$931$	32.3997	1.06186
$932$	8.29300	0.271646
$933$	0	0
$934$	21.9642	0.718690
$935$	6.00000	0.196221
$936$	0	0
$937$	33.9291	1.10842	0.554208	−	0.832378i	$-0.313021\pi$
$937$	33.9291	1.10842	0.554208	+	0.832378i	$0.313021\pi$
$938$	−1.34968	−0.0440687
$939$	0	0
$940$	7.46868	0.243602
$941$	8.20272	0.267401	0.133701	−	0.991022i	$-0.457314\pi$
$941$	8.20272	0.267401	0.133701	+	0.991022i	$0.457314\pi$
$942$	0	0
$943$	−1.99507	−0.0649684
$944$	−23.1485	−0.753419
$945$	0	0
$946$	17.3581	0.564359
$947$	45.0093	1.46260	0.731302	−	0.682053i	$-0.238913\pi$
$947$	45.0093	1.46260	0.731302	+	0.682053i	$0.238913\pi$
$948$	0	0
$949$	0	0
$950$	−5.50045	−0.178458
$951$	0	0
$952$	−3.63611	−0.117847
$953$	−53.0188	−1.71745	−0.858724	−	0.512438i	$-0.828743\pi$
$953$	−53.0188	−1.71745	−0.858724	+	0.512438i	$0.828743\pi$
$954$	0	0
$955$	−7.67356	−0.248311
$956$	−14.0040	−0.452922
$957$	0	0
$958$	−30.4371	−0.983379
$959$	5.27632	0.170381
$960$	0	0
$961$	52.7729	1.70235
$962$	0	0
$963$	0	0
$964$	4.00736	0.129068
$965$	21.0108	0.676363
$966$	0	0
$967$	−31.7061	−1.01960	−0.509799	−	0.860293i	$-0.670280\pi$
$967$	−31.7061	−1.01960	−0.509799	+	0.860293i	$0.670280\pi$
$968$	4.72500	0.151867
$969$	0	0
$970$	14.8663	0.477330
$971$	29.6147	0.950381	0.475191	−	0.879883i	$-0.342379\pi$
$971$	29.6147	0.950381	0.475191	+	0.879883i	$0.342379\pi$
$972$	0	0
$973$	−5.85225	−0.187615
$974$	−5.87508	−0.188250
$975$	0	0
$976$	12.5359	0.401264
$977$	−1.23129	−0.0393924	−0.0196962	−	0.999806i	$-0.506270\pi$
$977$	−1.23129	−0.0393924	−0.0196962	+	0.999806i	$0.506270\pi$
$978$	0	0
$979$	50.1427	1.60257
$980$	4.85550	0.155103
$981$	0	0
$982$	−16.6447	−0.531153
$983$	−25.1632	−0.802581	−0.401290	−	0.915951i	$-0.631438\pi$
$983$	−25.1632	−0.802581	−0.401290	+	0.915951i	$0.631438\pi$
$984$	0	0
$985$	−20.3776	−0.649285
$986$	−6.10905	−0.194552
$987$	0	0
$988$	0	0
$989$	9.34192	0.297056
$990$	0	0
$991$	−59.4589	−1.88877	−0.944387	−	0.328836i	$-0.893344\pi$
$991$	−59.4589	−1.88877	−0.944387	+	0.328836i	$0.893344\pi$
$992$	35.7021	1.13354
$993$	0	0
$994$	−4.03088	−0.127852
$995$	−14.8014	−0.469235
$996$	0	0
$997$	37.9219	1.20100	0.600499	−	0.799625i	$-0.294968\pi$
$997$	37.9219	1.20100	0.600499	+	0.799625i	$0.294968\pi$
$998$	−11.6174	−0.367743
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7605.2.a.ci.1.3		4
3.2	odd	2		2535.2.a.bj.1.2		4
13.6	odd	12		585.2.bu.d.361.2		8
13.11	odd	12		585.2.bu.d.316.2		8
13.12	even	2		7605.2.a.ch.1.2		4
39.11	even	12		195.2.bb.b.121.3	✓	8
39.32	even	12		195.2.bb.b.166.3	yes	8
39.38	odd	2		2535.2.a.bk.1.3		4
195.32	odd	12		975.2.w.h.49.3		8
195.89	even	12		975.2.bc.j.901.2		8
195.128	odd	12		975.2.w.h.199.3		8
195.149	even	12		975.2.bc.j.751.2		8
195.167	odd	12		975.2.w.i.199.2		8
195.188	odd	12		975.2.w.i.49.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
195.2.bb.b.121.3	✓	8	39.11	even	12
195.2.bb.b.166.3	yes	8	39.32	even	12
585.2.bu.d.316.2		8	13.11	odd	12
585.2.bu.d.361.2		8	13.6	odd	12
975.2.w.h.49.3		8	195.32	odd	12
975.2.w.h.199.3		8	195.128	odd	12
975.2.w.i.49.2		8	195.188	odd	12
975.2.w.i.199.2		8	195.167	odd	12
975.2.bc.j.751.2		8	195.149	even	12
975.2.bc.j.901.2		8	195.89	even	12
2535.2.a.bj.1.2		4	3.2	odd	2
2535.2.a.bk.1.3		4	39.38	odd	2
7605.2.a.ch.1.2		4	13.12	even	2
7605.2.a.ci.1.3		4	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)

\(f(q)\)	\(=\)	\(q+1.12603 q^{2} -0.732051 q^{4} +1.00000 q^{5} -0.606018 q^{7} -3.07638 q^{8} +O(q^{10})\)	\(q+1.12603 q^{2} -0.732051 q^{4} +1.00000 q^{5} -0.606018 q^{7} -3.07638 q^{8} +1.12603 q^{10} -3.07638 q^{11} -0.682396 q^{14} -2.00000 q^{16} -1.95035 q^{17} -4.88481 q^{19} -0.732051 q^{20} -3.46410 q^{22} -1.86434 q^{23} +1.00000 q^{25} +0.443636 q^{28} +2.78171 q^{29} +9.15276 q^{31} +3.90069 q^{32} -2.19615 q^{34} -0.606018 q^{35} +3.74793 q^{37} -5.50045 q^{38} -3.07638 q^{40} +1.07012 q^{41} -5.01084 q^{43} +2.25207 q^{44} -2.09931 q^{46} -10.2024 q^{47} -6.63274 q^{49} +1.12603 q^{50} +10.6569 q^{53} -3.07638 q^{55} +1.86434 q^{56} +3.13229 q^{58} +11.5742 q^{59} -6.26795 q^{61} +10.3063 q^{62} +8.39230 q^{64} +1.97786 q^{67} +1.42775 q^{68} -0.682396 q^{70} +5.90695 q^{71} +4.35395 q^{73} +4.22030 q^{74} +3.57593 q^{76} +1.86434 q^{77} -3.29546 q^{79} -2.00000 q^{80} +1.20499 q^{82} +6.97707 q^{83} -1.95035 q^{85} -5.64237 q^{86} +9.46410 q^{88} -16.2993 q^{89} +1.36479 q^{92} -11.4882 q^{94} -4.88481 q^{95} +13.2024 q^{97} -7.46868 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{4} + 4 q^{5}+O(q^{10}) \) 4 * q + 4 * q^4 + 4 * q^5	\( 4 q + 4 q^{4} + 4 q^{5} + 12 q^{14} - 8 q^{16} + 12 q^{19} + 4 q^{20} + 4 q^{25} + 12 q^{28} + 12 q^{29} + 12 q^{31} + 12 q^{34} + 24 q^{37} - 12 q^{41} + 16 q^{43} - 24 q^{46} - 24 q^{47} - 4 q^{49} + 12 q^{58} + 12 q^{59} - 32 q^{61} - 8 q^{64} - 12 q^{67} + 12 q^{70} + 12 q^{71} + 24 q^{73} - 24 q^{74} + 24 q^{76} - 8 q^{79} - 8 q^{80} - 12 q^{82} - 12 q^{86} + 24 q^{88} - 12 q^{89} - 24 q^{92} - 12 q^{94} + 12 q^{95} + 36 q^{97} + 24 q^{98}+O(q^{100}) \) 4 * q + 4 * q^4 + 4 * q^5 + 12 * q^14 - 8 * q^16 + 12 * q^19 + 4 * q^20 + 4 * q^25 + 12 * q^28 + 12 * q^29 + 12 * q^31 + 12 * q^34 + 24 * q^37 - 12 * q^41 + 16 * q^43 - 24 * q^46 - 24 * q^47 - 4 * q^49 + 12 * q^58 + 12 * q^59 - 32 * q^61 - 8 * q^64 - 12 * q^67 + 12 * q^70 + 12 * q^71 + 24 * q^73 - 24 * q^74 + 24 * q^76 - 8 * q^79 - 8 * q^80 - 12 * q^82 - 12 * q^86 + 24 * q^88 - 12 * q^89 - 24 * q^92 - 12 * q^94 + 12 * q^95 + 36 * q^97 + 24 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more