LMFDB - Embedded newform 7605.2.a.bt.1.2

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:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
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			1.2
Root			$2.17009$ 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)

$f(q)$	$=$	$q-0.539189 q^{2} -1.70928 q^{4} -1.00000 q^{5} +4.80098 q^{7} +2.00000 q^{8} +O(q^{10})$	$q-0.539189 q^{2} -1.70928 q^{4} -1.00000 q^{5} +4.80098 q^{7} +2.00000 q^{8} +0.539189 q^{10} -3.41855 q^{11} -2.58864 q^{14} +2.34017 q^{16} -2.87936 q^{17} -4.97107 q^{19} +1.70928 q^{20} +1.84324 q^{22} -8.49693 q^{23} +1.00000 q^{25} -8.20620 q^{28} +3.51026 q^{29} -3.04945 q^{31} -5.26180 q^{32} +1.55252 q^{34} -4.80098 q^{35} -2.68035 q^{37} +2.68035 q^{38} -2.00000 q^{40} -3.75154 q^{41} +1.58864 q^{43} +5.84324 q^{44} +4.58145 q^{46} -0.539189 q^{47} +16.0494 q^{49} -0.539189 q^{50} +13.7587 q^{53} +3.41855 q^{55} +9.60197 q^{56} -1.89269 q^{58} +8.40522 q^{59} -3.04945 q^{61} +1.64423 q^{62} -1.84324 q^{64} +12.8504 q^{67} +4.92162 q^{68} +2.58864 q^{70} +2.09171 q^{71} +5.53919 q^{73} +1.44521 q^{74} +8.49693 q^{76} -16.4124 q^{77} +2.21235 q^{79} -2.34017 q^{80} +2.02279 q^{82} +4.34017 q^{83} +2.87936 q^{85} -0.856576 q^{86} -6.83710 q^{88} -7.01333 q^{89} +14.5236 q^{92} +0.290725 q^{94} +4.97107 q^{95} +14.2690 q^{97} -8.65368 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{4} - 3 q^{5} + 5 q^{7} + 6 q^{8}+O(q^{10}) $ 3 * q + 2 * q^4 - 3 * q^5 + 5 * q^7 + 6 * q^8	$ 3 q + 2 q^{4} - 3 q^{5} + 5 q^{7} + 6 q^{8} + 4 q^{11} + 12 q^{14} - 4 q^{16} + 4 q^{17} - 2 q^{20} + 12 q^{22} - 8 q^{23} + 3 q^{25} - 6 q^{29} + 9 q^{31} - 8 q^{32} + 4 q^{34} - 5 q^{35} + 14 q^{37} - 14 q^{38} - 6 q^{40} - 20 q^{41} - 15 q^{43} + 24 q^{44} + 28 q^{46} + 30 q^{49} + 16 q^{53} - 4 q^{55} + 10 q^{56} + 6 q^{58} + 10 q^{59} + 9 q^{61} + 2 q^{62} - 12 q^{64} + 11 q^{67} + 18 q^{68} - 12 q^{70} + 4 q^{71} + 15 q^{73} - 8 q^{74} + 8 q^{76} + 17 q^{79} + 4 q^{80} - 14 q^{82} + 2 q^{83} - 4 q^{85} - 10 q^{86} + 8 q^{88} - 22 q^{89} + 28 q^{92} + 8 q^{94} + q^{97} - 2 q^{98}+O(q^{100}) $ 3 * q + 2 * q^4 - 3 * q^5 + 5 * q^7 + 6 * q^8 + 4 * q^11 + 12 * q^14 - 4 * q^16 + 4 * q^17 - 2 * q^20 + 12 * q^22 - 8 * q^23 + 3 * q^25 - 6 * q^29 + 9 * q^31 - 8 * q^32 + 4 * q^34 - 5 * q^35 + 14 * q^37 - 14 * q^38 - 6 * q^40 - 20 * q^41 - 15 * q^43 + 24 * q^44 + 28 * q^46 + 30 * q^49 + 16 * q^53 - 4 * q^55 + 10 * q^56 + 6 * q^58 + 10 * q^59 + 9 * q^61 + 2 * q^62 - 12 * q^64 + 11 * q^67 + 18 * q^68 - 12 * q^70 + 4 * q^71 + 15 * q^73 - 8 * q^74 + 8 * q^76 + 17 * q^79 + 4 * q^80 - 14 * q^82 + 2 * q^83 - 4 * q^85 - 10 * q^86 + 8 * q^88 - 22 * q^89 + 28 * q^92 + 8 * q^94 + q^97 - 2 * 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 $

$2$	−0.539189	−0.381264	−0.190632	−	0.981662i	$-0.561054\pi$
$2$	−0.539189	−0.381264	−0.190632	+	0.981662i	$0.561054\pi$
$3$	0	0
$3$	0	0
$4$	−1.70928	−0.854638
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	4.80098	1.81460	0.907301	−	0.420482i	$-0.138139\pi$
$7$	4.80098	1.81460	0.907301	+	0.420482i	$0.138139\pi$
$8$	2.00000	0.707107
$9$	0	0
$10$	0.539189	0.170506
$11$	−3.41855	−1.03073	−0.515366	−	0.856970i	$-0.672344\pi$
$11$	−3.41855	−1.03073	−0.515366	+	0.856970i	$0.672344\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	−2.58864	−0.691842
$15$	0	0
$16$	2.34017	0.585043
$17$	−2.87936	−0.698348	−0.349174	−	0.937058i	$-0.613538\pi$
$17$	−2.87936	−0.698348	−0.349174	+	0.937058i	$0.613538\pi$
$18$	0	0
$19$	−4.97107	−1.14044	−0.570221	−	0.821491i	$-0.693142\pi$
$19$	−4.97107	−1.14044	−0.570221	+	0.821491i	$0.693142\pi$
$20$	1.70928	0.382206
$21$	0	0
$22$	1.84324	0.392981
$23$	−8.49693	−1.77173	−0.885866	−	0.463941i	$-0.846435\pi$
$23$	−8.49693	−1.77173	−0.885866	+	0.463941i	$0.846435\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	−8.20620	−1.55083
$29$	3.51026	0.651839	0.325919	−	0.945398i	$-0.394326\pi$
$29$	3.51026	0.651839	0.325919	+	0.945398i	$0.394326\pi$
$30$	0	0
$31$	−3.04945	−0.547697	−0.273849	−	0.961773i	$-0.588297\pi$
$31$	−3.04945	−0.547697	−0.273849	+	0.961773i	$0.588297\pi$
$32$	−5.26180	−0.930163
$33$	0	0
$34$	1.55252	0.266255
$35$	−4.80098	−0.811514
$36$	0	0
$37$	−2.68035	−0.440646	−0.220323	−	0.975427i	$-0.570711\pi$
$37$	−2.68035	−0.440646	−0.220323	+	0.975427i	$0.570711\pi$
$38$	2.68035	0.434810
$39$	0	0
$40$	−2.00000	−0.316228
$41$	−3.75154	−0.585891	−0.292946	−	0.956129i	$-0.594636\pi$
$41$	−3.75154	−0.585891	−0.292946	+	0.956129i	$0.594636\pi$
$42$	0	0
$43$	1.58864	0.242265	0.121132	−	0.992636i	$-0.461347\pi$
$43$	1.58864	0.242265	0.121132	+	0.992636i	$0.461347\pi$
$44$	5.84324	0.880902
$45$	0	0
$46$	4.58145	0.675498
$47$	−0.539189	−0.0786488	−0.0393244	−	0.999226i	$-0.512521\pi$
$47$	−0.539189	−0.0786488	−0.0393244	+	0.999226i	$0.512521\pi$
$48$	0	0
$49$	16.0494	2.29278
$50$	−0.539189	−0.0762528
$51$	0	0
$52$	0	0
$53$	13.7587	1.88991	0.944953	−	0.327206i	$-0.106107\pi$
$53$	13.7587	1.88991	0.944953	+	0.327206i	$0.106107\pi$
$54$	0	0
$55$	3.41855	0.460957
$56$	9.60197	1.28312
$57$	0	0
$58$	−1.89269	−0.248523
$59$	8.40522	1.09427	0.547133	−	0.837046i	$-0.315719\pi$
$59$	8.40522	1.09427	0.547133	+	0.837046i	$0.315719\pi$
$60$	0	0
$61$	−3.04945	−0.390442	−0.195221	−	0.980759i	$-0.562542\pi$
$61$	−3.04945	−0.390442	−0.195221	+	0.980759i	$0.562542\pi$
$62$	1.64423	0.208817
$63$	0	0
$64$	−1.84324	−0.230406
$65$	0	0
$66$	0	0
$67$	12.8504	1.56993	0.784965	−	0.619540i	$-0.212681\pi$
$67$	12.8504	1.56993	0.784965	+	0.619540i	$0.212681\pi$
$68$	4.92162	0.596834
$69$	0	0
$70$	2.58864	0.309401
$71$	2.09171	0.248240	0.124120	−	0.992267i	$-0.460389\pi$
$71$	2.09171	0.248240	0.124120	+	0.992267i	$0.460389\pi$
$72$	0	0
$73$	5.53919	0.648313	0.324157	−	0.946003i	$-0.394920\pi$
$73$	5.53919	0.648313	0.324157	+	0.946003i	$0.394920\pi$
$74$	1.44521	0.168003
$75$	0	0
$76$	8.49693	0.974665
$77$	−16.4124	−1.87037
$78$	0	0
$79$	2.21235	0.248908	0.124454	−	0.992225i	$-0.460282\pi$
$79$	2.21235	0.248908	0.124454	+	0.992225i	$0.460282\pi$
$80$	−2.34017	−0.261639
$81$	0	0
$82$	2.02279	0.223379
$83$	4.34017	0.476396	0.238198	−	0.971217i	$-0.423443\pi$
$83$	4.34017	0.476396	0.238198	+	0.971217i	$0.423443\pi$
$84$	0	0
$85$	2.87936	0.312311
$86$	−0.856576	−0.0923669
$87$	0	0
$88$	−6.83710	−0.728837
$89$	−7.01333	−0.743412	−0.371706	−	0.928351i	$-0.621227\pi$
$89$	−7.01333	−0.743412	−0.371706	+	0.928351i	$0.621227\pi$
$90$	0	0
$91$	0	0
$92$	14.5236	1.51419
$93$	0	0
$94$	0.290725	0.0299860
$95$	4.97107	0.510021
$96$	0	0
$97$	14.2690	1.44880	0.724398	−	0.689382i	$-0.242118\pi$
$97$	14.2690	1.44880	0.724398	+	0.689382i	$0.242118\pi$
$98$	−8.65368	−0.874154
$99$	0	0
$100$	−1.70928	−0.170928
$101$	−12.4969	−1.24349	−0.621745	−	0.783219i	$-0.713576\pi$
$101$	−12.4969	−1.24349	−0.621745	+	0.783219i	$0.713576\pi$
$102$	0	0
$103$	6.85043	0.674993	0.337497	−	0.941327i	$-0.390420\pi$
$103$	6.85043	0.674993	0.337497	+	0.941327i	$0.390420\pi$
$104$	0	0
$105$	0	0
$106$	−7.41855	−0.720553
$107$	−15.5597	−1.50421	−0.752107	−	0.659041i	$-0.770962\pi$
$107$	−15.5597	−1.50421	−0.752107	+	0.659041i	$0.770962\pi$
$108$	0	0
$109$	−15.8371	−1.51692	−0.758460	−	0.651720i	$-0.774048\pi$
$109$	−15.8371	−1.51692	−0.758460	+	0.651720i	$0.774048\pi$
$110$	−1.84324	−0.175746
$111$	0	0
$112$	11.2351	1.06162
$113$	3.07838	0.289589	0.144795	−	0.989462i	$-0.453748\pi$
$113$	3.07838	0.289589	0.144795	+	0.989462i	$0.453748\pi$
$114$	0	0
$115$	8.49693	0.792343
$116$	−6.00000	−0.557086
$117$	0	0
$118$	−4.53200	−0.417205
$119$	−13.8238	−1.26722
$120$	0	0
$121$	0.686489	0.0624081
$122$	1.64423	0.148861
$123$	0	0
$124$	5.21235	0.468083
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−7.45467	−0.661495	−0.330747	−	0.943719i	$-0.607301\pi$
$127$	−7.45467	−0.661495	−0.330747	+	0.943719i	$0.607301\pi$
$128$	11.5174	1.01801
$129$	0	0
$130$	0	0
$131$	−11.6937	−1.02168	−0.510841	−	0.859675i	$-0.670666\pi$
$131$	−11.6937	−1.02168	−0.510841	+	0.859675i	$0.670666\pi$
$132$	0	0
$133$	−23.8660	−2.06945
$134$	−6.92881	−0.598558
$135$	0	0
$136$	−5.75872	−0.493806
$137$	−9.27739	−0.792621	−0.396311	−	0.918116i	$-0.629710\pi$
$137$	−9.27739	−0.792621	−0.396311	+	0.918116i	$0.629710\pi$
$138$	0	0
$139$	12.9155	1.09548	0.547738	−	0.836650i	$-0.315489\pi$
$139$	12.9155	1.09548	0.547738	+	0.836650i	$0.315489\pi$
$140$	8.20620	0.693551
$141$	0	0
$142$	−1.12783	−0.0946451
$143$	0	0
$144$	0	0
$145$	−3.51026	−0.291511
$146$	−2.98667	−0.247178
$147$	0	0
$148$	4.58145	0.376593
$149$	−2.00000	−0.163846	−0.0819232	−	0.996639i	$-0.526106\pi$
$149$	−2.00000	−0.163846	−0.0819232	+	0.996639i	$0.526106\pi$
$150$	0	0
$151$	8.60424	0.700203	0.350101	−	0.936712i	$-0.386147\pi$
$151$	8.60424	0.700203	0.350101	+	0.936712i	$0.386147\pi$
$152$	−9.94214	−0.806414
$153$	0	0
$154$	8.84939	0.713104
$155$	3.04945	0.244938
$156$	0	0
$157$	0.908291	0.0724895	0.0362448	−	0.999343i	$-0.488460\pi$
$157$	0.908291	0.0724895	0.0362448	+	0.999343i	$0.488460\pi$
$158$	−1.19287	−0.0948999
$159$	0	0
$160$	5.26180	0.415981
$161$	−40.7936	−3.21499
$162$	0	0
$163$	14.6442	1.14702	0.573512	−	0.819197i	$-0.305580\pi$
$163$	14.6442	1.14702	0.573512	+	0.819197i	$0.305580\pi$
$164$	6.41241	0.500725
$165$	0	0
$166$	−2.34017	−0.181633
$167$	−4.89496	−0.378783	−0.189392	−	0.981902i	$-0.560652\pi$
$167$	−4.89496	−0.378783	−0.189392	+	0.981902i	$0.560652\pi$
$168$	0	0
$169$	0	0
$170$	−1.55252	−0.119073
$171$	0	0
$172$	−2.71542	−0.207049
$173$	17.5597	1.33504	0.667520	−	0.744592i	$-0.267356\pi$
$173$	17.5597	1.33504	0.667520	+	0.744592i	$0.267356\pi$
$174$	0	0
$175$	4.80098	0.362920
$176$	−8.00000	−0.603023
$177$	0	0
$178$	3.78151	0.283436
$179$	−1.38470	−0.103497	−0.0517487	−	0.998660i	$-0.516479\pi$
$179$	−1.38470	−0.103497	−0.0517487	+	0.998660i	$0.516479\pi$
$180$	0	0
$181$	−7.46800	−0.555092	−0.277546	−	0.960712i	$-0.589521\pi$
$181$	−7.46800	−0.555092	−0.277546	+	0.960712i	$0.589521\pi$
$182$	0	0
$183$	0	0
$184$	−16.9939	−1.25280
$185$	2.68035	0.197063
$186$	0	0
$187$	9.84324	0.719809
$188$	0.921622	0.0672162
$189$	0	0
$190$	−2.68035	−0.194453
$191$	2.40522	0.174036	0.0870178	−	0.996207i	$-0.472266\pi$
$191$	2.40522	0.174036	0.0870178	+	0.996207i	$0.472266\pi$
$192$	0	0
$193$	15.8215	1.13886	0.569428	−	0.822041i	$-0.307165\pi$
$193$	15.8215	1.13886	0.569428	+	0.822041i	$0.307165\pi$
$194$	−7.69368	−0.552374
$195$	0	0
$196$	−27.4329	−1.95949
$197$	18.4657	1.31563	0.657814	−	0.753180i	$-0.271481\pi$
$197$	18.4657	1.31563	0.657814	+	0.753180i	$0.271481\pi$
$198$	0	0
$199$	10.9421	0.775668	0.387834	−	0.921729i	$-0.373223\pi$
$199$	10.9421	0.775668	0.387834	+	0.921729i	$0.373223\pi$
$200$	2.00000	0.141421
$201$	0	0
$202$	6.73820	0.474098
$203$	16.8527	1.18283
$204$	0	0
$205$	3.75154	0.262019
$206$	−3.69368	−0.257351
$207$	0	0
$208$	0	0
$209$	16.9939	1.17549
$210$	0	0
$211$	−14.1773	−0.976004	−0.488002	−	0.872843i	$-0.662274\pi$
$211$	−14.1773	−0.976004	−0.488002	+	0.872843i	$0.662274\pi$
$212$	−23.5174	−1.61518
$213$	0	0
$214$	8.38962	0.573503
$215$	−1.58864	−0.108344
$216$	0	0
$217$	−14.6404	−0.993852
$218$	8.53919	0.578347
$219$	0	0
$220$	−5.84324	−0.393951
$221$	0	0
$222$	0	0
$223$	5.65983	0.379010	0.189505	−	0.981880i	$-0.439312\pi$
$223$	5.65983	0.379010	0.189505	+	0.981880i	$0.439312\pi$
$224$	−25.2618	−1.68787
$225$	0	0
$226$	−1.65983	−0.110410
$227$	20.8794	1.38581	0.692906	−	0.721028i	$-0.256330\pi$
$227$	20.8794	1.38581	0.692906	+	0.721028i	$0.256330\pi$
$228$	0	0
$229$	11.5525	0.763412	0.381706	−	0.924284i	$-0.375337\pi$
$229$	11.5525	0.763412	0.381706	+	0.924284i	$0.375337\pi$
$230$	−4.58145	−0.302092
$231$	0	0
$232$	7.02052	0.460920
$233$	3.73206	0.244495	0.122248	−	0.992500i	$-0.460990\pi$
$233$	3.73206	0.244495	0.122248	+	0.992500i	$0.460990\pi$
$234$	0	0
$235$	0.539189	0.0351728
$236$	−14.3668	−0.935201
$237$	0	0
$238$	7.45362	0.483147
$239$	27.0928	1.75248	0.876242	−	0.481871i	$-0.160043\pi$
$239$	27.0928	1.75248	0.876242	+	0.481871i	$0.160043\pi$
$240$	0	0
$241$	23.5981	1.52009	0.760043	−	0.649872i	$-0.225178\pi$
$241$	23.5981	1.52009	0.760043	+	0.649872i	$0.225178\pi$
$242$	−0.370147	−0.0237940
$243$	0	0
$244$	5.21235	0.333686
$245$	−16.0494	−1.02536
$246$	0	0
$247$	0	0
$248$	−6.09890	−0.387280
$249$	0	0
$250$	0.539189	0.0341013
$251$	26.8638	1.69563	0.847813	−	0.530296i	$-0.177919\pi$
$251$	26.8638	1.69563	0.847813	+	0.530296i	$0.177919\pi$
$252$	0	0
$253$	29.0472	1.82618
$254$	4.01947	0.252204
$255$	0	0
$256$	−2.52359	−0.157724
$257$	3.51867	0.219489	0.109744	−	0.993960i	$-0.464997\pi$
$257$	3.51867	0.219489	0.109744	+	0.993960i	$0.464997\pi$
$258$	0	0
$259$	−12.8683	−0.799597
$260$	0	0
$261$	0	0
$262$	6.30510	0.389530
$263$	8.85270	0.545881	0.272940	−	0.962031i	$-0.412004\pi$
$263$	8.85270	0.545881	0.272940	+	0.962031i	$0.412004\pi$
$264$	0	0
$265$	−13.7587	−0.845192
$266$	12.8683	0.789006
$267$	0	0
$268$	−21.9649	−1.34172
$269$	−27.2423	−1.66099	−0.830497	−	0.557023i	$-0.811943\pi$
$269$	−27.2423	−1.66099	−0.830497	+	0.557023i	$0.811943\pi$
$270$	0	0
$271$	−13.8638	−0.842164	−0.421082	−	0.907023i	$-0.638349\pi$
$271$	−13.8638	−0.842164	−0.421082	+	0.907023i	$0.638349\pi$
$272$	−6.73820	−0.408564
$273$	0	0
$274$	5.00227	0.302198
$275$	−3.41855	−0.206146
$276$	0	0
$277$	2.58145	0.155104	0.0775521	−	0.996988i	$-0.475290\pi$
$277$	2.58145	0.155104	0.0775521	+	0.996988i	$0.475290\pi$
$278$	−6.96388	−0.417666
$279$	0	0
$280$	−9.60197	−0.573827
$281$	−5.35350	−0.319363	−0.159682	−	0.987169i	$-0.551047\pi$
$281$	−5.35350	−0.319363	−0.159682	+	0.987169i	$0.551047\pi$
$282$	0	0
$283$	15.8937	0.944785	0.472392	−	0.881388i	$-0.343391\pi$
$283$	15.8937	0.944785	0.472392	+	0.881388i	$0.343391\pi$
$284$	−3.57531	−0.212155
$285$	0	0
$286$	0	0
$287$	−18.0111	−1.06316
$288$	0	0
$289$	−8.70928	−0.512310
$290$	1.89269	0.111143
$291$	0	0
$292$	−9.46800	−0.554073
$293$	−2.48133	−0.144961	−0.0724804	−	0.997370i	$-0.523091\pi$
$293$	−2.48133	−0.144961	−0.0724804	+	0.997370i	$0.523091\pi$
$294$	0	0
$295$	−8.40522	−0.489371
$296$	−5.36069	−0.311584
$297$	0	0
$298$	1.07838	0.0624687
$299$	0	0
$300$	0	0
$301$	7.62702	0.439614
$302$	−4.63931	−0.266962
$303$	0	0
$304$	−11.6332	−0.667208
$305$	3.04945	0.174611
$306$	0	0
$307$	22.2423	1.26944	0.634718	−	0.772744i	$-0.281116\pi$
$307$	22.2423	1.26944	0.634718	+	0.772744i	$0.281116\pi$
$308$	28.0533	1.59849
$309$	0	0
$310$	−1.64423	−0.0933859
$311$	0.405220	0.0229779	0.0114890	−	0.999934i	$-0.496343\pi$
$311$	0.405220	0.0229779	0.0114890	+	0.999934i	$0.496343\pi$
$312$	0	0
$313$	−0.353504	−0.0199812	−0.00999061	−	0.999950i	$-0.503180\pi$
$313$	−0.353504	−0.0199812	−0.00999061	+	0.999950i	$0.503180\pi$
$314$	−0.489741	−0.0276377
$315$	0	0
$316$	−3.78151	−0.212727
$317$	25.3028	1.42115	0.710574	−	0.703622i	$-0.248435\pi$
$317$	25.3028	1.42115	0.710574	+	0.703622i	$0.248435\pi$
$318$	0	0
$319$	−12.0000	−0.671871
$320$	1.84324	0.103041
$321$	0	0
$322$	21.9955	1.22576
$323$	14.3135	0.796425
$324$	0	0
$325$	0	0
$326$	−7.89601	−0.437319
$327$	0	0
$328$	−7.50307	−0.414288
$329$	−2.58864	−0.142716
$330$	0	0
$331$	−4.60197	−0.252947	−0.126474	−	0.991970i	$-0.540366\pi$
$331$	−4.60197	−0.252947	−0.126474	+	0.991970i	$0.540366\pi$
$332$	−7.41855	−0.407146
$333$	0	0
$334$	2.63931	0.144417
$335$	−12.8504	−0.702094
$336$	0	0
$337$	−31.0338	−1.69052	−0.845261	−	0.534354i	$-0.820555\pi$
$337$	−31.0338	−1.69052	−0.845261	+	0.534354i	$0.820555\pi$
$338$	0	0
$339$	0	0
$340$	−4.92162	−0.266912
$341$	10.4247	0.564529
$342$	0	0
$343$	43.4463	2.34588
$344$	3.17727	0.171307
$345$	0	0
$346$	−9.46800	−0.509003
$347$	17.9565	0.963956	0.481978	−	0.876183i	$-0.339918\pi$
$347$	17.9565	0.963956	0.481978	+	0.876183i	$0.339918\pi$
$348$	0	0
$349$	−0.101164	−0.00541519	−0.00270759	−	0.999996i	$-0.500862\pi$
$349$	−0.101164	−0.00541519	−0.00270759	+	0.999996i	$0.500862\pi$
$350$	−2.58864	−0.138368
$351$	0	0
$352$	17.9877	0.958748
$353$	−29.3607	−1.56271	−0.781356	−	0.624086i	$-0.785472\pi$
$353$	−29.3607	−1.56271	−0.781356	+	0.624086i	$0.785472\pi$
$354$	0	0
$355$	−2.09171	−0.111016
$356$	11.9877	0.635348
$357$	0	0
$358$	0.746615	0.0394598
$359$	−11.5369	−0.608895	−0.304448	−	0.952529i	$-0.598472\pi$
$359$	−11.5369	−0.608895	−0.304448	+	0.952529i	$0.598472\pi$
$360$	0	0
$361$	5.71154	0.300608
$362$	4.02666	0.211637
$363$	0	0
$364$	0	0
$365$	−5.53919	−0.289934
$366$	0	0
$367$	−35.6297	−1.85985	−0.929927	−	0.367744i	$-0.880130\pi$
$367$	−35.6297	−1.85985	−0.929927	+	0.367744i	$0.880130\pi$
$368$	−19.8843	−1.03654
$369$	0	0
$370$	−1.44521	−0.0751330
$371$	66.0554	3.42943
$372$	0	0
$373$	−9.61142	−0.497661	−0.248830	−	0.968547i	$-0.580046\pi$
$373$	−9.61142	−0.497661	−0.248830	+	0.968547i	$0.580046\pi$
$374$	−5.30737	−0.274437
$375$	0	0
$376$	−1.07838	−0.0556131
$377$	0	0
$378$	0	0
$379$	−7.02279	−0.360736	−0.180368	−	0.983599i	$-0.557729\pi$
$379$	−7.02279	−0.360736	−0.180368	+	0.983599i	$0.557729\pi$
$380$	−8.49693	−0.435883
$381$	0	0
$382$	−1.29687	−0.0663535
$383$	20.2134	1.03286	0.516428	−	0.856331i	$-0.327261\pi$
$383$	20.2134	1.03286	0.516428	+	0.856331i	$0.327261\pi$
$384$	0	0
$385$	16.4124	0.836454
$386$	−8.53078	−0.434205
$387$	0	0
$388$	−24.3896	−1.23820
$389$	15.7587	0.798999	0.399500	−	0.916733i	$-0.369184\pi$
$389$	15.7587	0.798999	0.399500	+	0.916733i	$0.369184\pi$
$390$	0	0
$391$	24.4657	1.23729
$392$	32.0989	1.62124
$393$	0	0
$394$	−9.95652	−0.501602
$395$	−2.21235	−0.111315
$396$	0	0
$397$	−11.3474	−0.569508	−0.284754	−	0.958601i	$-0.591912\pi$
$397$	−11.3474	−0.569508	−0.284754	+	0.958601i	$0.591912\pi$
$398$	−5.89988	−0.295734
$399$	0	0
$400$	2.34017	0.117009
$401$	26.6537	1.33102	0.665511	−	0.746388i	$-0.268214\pi$
$401$	26.6537	1.33102	0.665511	+	0.746388i	$0.268214\pi$
$402$	0	0
$403$	0	0
$404$	21.3607	1.06273
$405$	0	0
$406$	−9.08679	−0.450970
$407$	9.16290	0.454188
$408$	0	0
$409$	30.8927	1.52755	0.763773	−	0.645485i	$-0.223345\pi$
$409$	30.8927	1.52755	0.763773	+	0.645485i	$0.223345\pi$
$410$	−2.02279	−0.0998983
$411$	0	0
$412$	−11.7093	−0.576875
$413$	40.3533	1.98566
$414$	0	0
$415$	−4.34017	−0.213051
$416$	0	0
$417$	0	0
$418$	−9.16290	−0.448172
$419$	−34.3162	−1.67645	−0.838227	−	0.545321i	$-0.816408\pi$
$419$	−34.3162	−1.67645	−0.838227	+	0.545321i	$0.816408\pi$
$420$	0	0
$421$	13.5320	0.659509	0.329755	−	0.944067i	$-0.393034\pi$
$421$	13.5320	0.659509	0.329755	+	0.944067i	$0.393034\pi$
$422$	7.64423	0.372115
$423$	0	0
$424$	27.5174	1.33637
$425$	−2.87936	−0.139670
$426$	0	0
$427$	−14.6404	−0.708496
$428$	26.5958	1.28556
$429$	0	0
$430$	0.856576	0.0413077
$431$	15.2618	0.735135	0.367567	−	0.929997i	$-0.380191\pi$
$431$	15.2618	0.735135	0.367567	+	0.929997i	$0.380191\pi$
$432$	0	0
$433$	3.87936	0.186430	0.0932151	−	0.995646i	$-0.470286\pi$
$433$	3.87936	0.186430	0.0932151	+	0.995646i	$0.470286\pi$
$434$	7.89392	0.378920
$435$	0	0
$436$	27.0700	1.29642
$437$	42.2388	2.02056
$438$	0	0
$439$	−20.3112	−0.969403	−0.484701	−	0.874680i	$-0.661072\pi$
$439$	−20.3112	−0.969403	−0.484701	+	0.874680i	$0.661072\pi$
$440$	6.83710	0.325946
$441$	0	0
$442$	0	0
$443$	35.0772	1.66657	0.833283	−	0.552847i	$-0.186458\pi$
$443$	35.0772	1.66657	0.833283	+	0.552847i	$0.186458\pi$
$444$	0	0
$445$	7.01333	0.332464
$446$	−3.05172	−0.144503
$447$	0	0
$448$	−8.84939	−0.418094
$449$	7.97334	0.376285	0.188143	−	0.982142i	$-0.439753\pi$
$449$	7.97334	0.376285	0.188143	+	0.982142i	$0.439753\pi$
$450$	0	0
$451$	12.8248	0.603897
$452$	−5.26180	−0.247494
$453$	0	0
$454$	−11.2579	−0.528360
$455$	0	0
$456$	0	0
$457$	−2.60916	−0.122051	−0.0610256	−	0.998136i	$-0.519437\pi$
$457$	−2.60916	−0.122051	−0.0610256	+	0.998136i	$0.519437\pi$
$458$	−6.22899	−0.291062
$459$	0	0
$460$	−14.5236	−0.677166
$461$	−30.0794	−1.40094	−0.700469	−	0.713683i	$-0.747026\pi$
$461$	−30.0794	−1.40094	−0.700469	+	0.713683i	$0.747026\pi$
$462$	0	0
$463$	6.36788	0.295940	0.147970	−	0.988992i	$-0.452726\pi$
$463$	6.36788	0.295940	0.147970	+	0.988992i	$0.452726\pi$
$464$	8.21461	0.381354
$465$	0	0
$466$	−2.01229	−0.0932174
$467$	2.06892	0.0957383	0.0478692	−	0.998854i	$-0.484757\pi$
$467$	2.06892	0.0957383	0.0478692	+	0.998854i	$0.484757\pi$
$468$	0	0
$469$	61.6947	2.84880
$470$	−0.290725	−0.0134101
$471$	0	0
$472$	16.8104	0.773763
$473$	−5.43084	−0.249710
$474$	0	0
$475$	−4.97107	−0.228088
$476$	23.6286	1.08302
$477$	0	0
$478$	−14.6081	−0.668159
$479$	11.5103	0.525917	0.262959	−	0.964807i	$-0.415302\pi$
$479$	11.5103	0.525917	0.262959	+	0.964807i	$0.415302\pi$
$480$	0	0
$481$	0	0
$482$	−12.7238	−0.579555
$483$	0	0
$484$	−1.17340	−0.0533363
$485$	−14.2690	−0.647921
$486$	0	0
$487$	−20.1483	−0.913009	−0.456504	−	0.889721i	$-0.650899\pi$
$487$	−20.1483	−0.913009	−0.456504	+	0.889721i	$0.650899\pi$
$488$	−6.09890	−0.276084
$489$	0	0
$490$	8.65368	0.390934
$491$	21.8816	0.987504	0.493752	−	0.869603i	$-0.335625\pi$
$491$	21.8816	0.987504	0.493752	+	0.869603i	$0.335625\pi$
$492$	0	0
$493$	−10.1073	−0.455210
$494$	0	0
$495$	0	0
$496$	−7.13624	−0.320426
$497$	10.0423	0.450457
$498$	0	0
$499$	23.8225	1.06644	0.533222	−	0.845975i	$-0.320981\pi$
$499$	23.8225	1.06644	0.533222	+	0.845975i	$0.320981\pi$
$500$	1.70928	0.0764411
$501$	0	0
$502$	−14.4846	−0.646481
$503$	−4.30898	−0.192128	−0.0960639	−	0.995375i	$-0.530625\pi$
$503$	−4.30898	−0.192128	−0.0960639	+	0.995375i	$0.530625\pi$
$504$	0	0
$505$	12.4969	0.556106
$506$	−15.6619	−0.696257
$507$	0	0
$508$	12.7421	0.565338
$509$	0.993857	0.0440519	0.0220260	−	0.999757i	$-0.492988\pi$
$509$	0.993857	0.0440519	0.0220260	+	0.999757i	$0.492988\pi$
$510$	0	0
$511$	26.5936	1.17643
$512$	−21.6742	−0.957873
$513$	0	0
$514$	−1.89723	−0.0836831
$515$	−6.85043	−0.301866
$516$	0	0
$517$	1.84324	0.0810658
$518$	6.93844	0.304858
$519$	0	0
$520$	0	0
$521$	−9.75154	−0.427223	−0.213611	−	0.976919i	$-0.568523\pi$
$521$	−9.75154	−0.427223	−0.213611	+	0.976919i	$0.568523\pi$
$522$	0	0
$523$	−11.0205	−0.481894	−0.240947	−	0.970538i	$-0.577458\pi$
$523$	−11.0205	−0.481894	−0.240947	+	0.970538i	$0.577458\pi$
$524$	19.9877	0.873167
$525$	0	0
$526$	−4.77328	−0.208125
$527$	8.78047	0.382483
$528$	0	0
$529$	49.1978	2.13903
$530$	7.41855	0.322241
$531$	0	0
$532$	40.7936	1.76863
$533$	0	0
$534$	0	0
$535$	15.5597	0.672705
$536$	25.7009	1.11011
$537$	0	0
$538$	14.6888	0.633277
$539$	−54.8659	−2.36324
$540$	0	0
$541$	6.28846	0.270362	0.135181	−	0.990821i	$-0.456838\pi$
$541$	6.28846	0.270362	0.135181	+	0.990821i	$0.456838\pi$
$542$	7.47519	0.321087
$543$	0	0
$544$	15.1506	0.649577
$545$	15.8371	0.678387
$546$	0	0
$547$	27.6875	1.18383	0.591917	−	0.805999i	$-0.298371\pi$
$547$	27.6875	1.18383	0.591917	+	0.805999i	$0.298371\pi$
$548$	15.8576	0.677404
$549$	0	0
$550$	1.84324	0.0785962
$551$	−17.4497	−0.743384
$552$	0	0
$553$	10.6214	0.451670
$554$	−1.39189	−0.0591357
$555$	0	0
$556$	−22.0761	−0.936236
$557$	−21.4641	−0.909464	−0.454732	−	0.890628i	$-0.650265\pi$
$557$	−21.4641	−0.909464	−0.454732	+	0.890628i	$0.650265\pi$
$558$	0	0
$559$	0	0
$560$	−11.2351	−0.474771
$561$	0	0
$562$	2.88655	0.121762
$563$	15.6742	0.660589	0.330294	−	0.943878i	$-0.392852\pi$
$563$	15.6742	0.660589	0.330294	+	0.943878i	$0.392852\pi$
$564$	0	0
$565$	−3.07838	−0.129508
$566$	−8.56973	−0.360212
$567$	0	0
$568$	4.18342	0.175532
$569$	−16.4319	−0.688860	−0.344430	−	0.938812i	$-0.611928\pi$
$569$	−16.4319	−0.688860	−0.344430	+	0.938812i	$0.611928\pi$
$570$	0	0
$571$	14.6765	0.614191	0.307096	−	0.951679i	$-0.400643\pi$
$571$	14.6765	0.614191	0.307096	+	0.951679i	$0.400643\pi$
$572$	0	0
$573$	0	0
$574$	9.71137	0.405345
$575$	−8.49693	−0.354346
$576$	0	0
$577$	17.8622	0.743611	0.371806	−	0.928311i	$-0.378739\pi$
$577$	17.8622	0.743611	0.371806	+	0.928311i	$0.378739\pi$
$578$	4.69594	0.195326
$579$	0	0
$580$	6.00000	0.249136
$581$	20.8371	0.864469
$582$	0	0
$583$	−47.0349	−1.94799
$584$	11.0784	0.458427
$585$	0	0
$586$	1.33791	0.0552684
$587$	−4.42347	−0.182576	−0.0912881	−	0.995825i	$-0.529098\pi$
$587$	−4.42347	−0.182576	−0.0912881	+	0.995825i	$0.529098\pi$
$588$	0	0
$589$	15.1590	0.624617
$590$	4.53200	0.186580
$591$	0	0
$592$	−6.27247	−0.257797
$593$	12.5380	0.514873	0.257436	−	0.966295i	$-0.417122\pi$
$593$	12.5380	0.514873	0.257436	+	0.966295i	$0.417122\pi$
$594$	0	0
$595$	13.8238	0.566719
$596$	3.41855	0.140029
$597$	0	0
$598$	0	0
$599$	−23.5825	−0.963555	−0.481777	−	0.876294i	$-0.660009\pi$
$599$	−23.5825	−0.963555	−0.481777	+	0.876294i	$0.660009\pi$
$600$	0	0
$601$	3.28458	0.133981	0.0669904	−	0.997754i	$-0.478660\pi$
$601$	3.28458	0.133981	0.0669904	+	0.997754i	$0.478660\pi$
$602$	−4.11241	−0.167609
$603$	0	0
$604$	−14.7070	−0.598420
$605$	−0.686489	−0.0279097
$606$	0	0
$607$	−6.00841	−0.243874	−0.121937	−	0.992538i	$-0.538911\pi$
$607$	−6.00841	−0.243874	−0.121937	+	0.992538i	$0.538911\pi$
$608$	26.1568	1.06080
$609$	0	0
$610$	−1.64423	−0.0665729
$611$	0	0
$612$	0	0
$613$	5.53078	0.223386	0.111693	−	0.993743i	$-0.464373\pi$
$613$	5.53078	0.223386	0.111693	+	0.993743i	$0.464373\pi$
$614$	−11.9928	−0.483991
$615$	0	0
$616$	−32.8248	−1.32255
$617$	11.8843	0.478443	0.239222	−	0.970965i	$-0.423108\pi$
$617$	11.8843	0.478443	0.239222	+	0.970965i	$0.423108\pi$
$618$	0	0
$619$	−32.2183	−1.29496	−0.647482	−	0.762081i	$-0.724178\pi$
$619$	−32.2183	−1.29496	−0.647482	+	0.762081i	$0.724178\pi$
$620$	−5.21235	−0.209333
$621$	0	0
$622$	−0.218490	−0.00876065
$623$	−33.6709	−1.34900
$624$	0	0
$625$	1.00000	0.0400000
$626$	0.190605	0.00761812
$627$	0	0
$628$	−1.55252	−0.0619523
$629$	7.71769	0.307724
$630$	0	0
$631$	19.7442	0.786003	0.393002	−	0.919538i	$-0.371437\pi$
$631$	19.7442	0.786003	0.393002	+	0.919538i	$0.371437\pi$
$632$	4.42469	0.176005
$633$	0	0
$634$	−13.6430	−0.541833
$635$	7.45467	0.295829
$636$	0	0
$637$	0	0
$638$	6.47027	0.256160
$639$	0	0
$640$	−11.5174	−0.455267
$641$	−36.1133	−1.42639	−0.713194	−	0.700966i	$-0.752752\pi$
$641$	−36.1133	−1.42639	−0.713194	+	0.700966i	$0.752752\pi$
$642$	0	0
$643$	−39.1822	−1.54519	−0.772597	−	0.634896i	$-0.781043\pi$
$643$	−39.1822	−1.54519	−0.772597	+	0.634896i	$0.781043\pi$
$644$	69.7275	2.74765
$645$	0	0
$646$	−7.71769	−0.303648
$647$	42.1711	1.65792	0.828959	−	0.559309i	$-0.188934\pi$
$647$	42.1711	1.65792	0.828959	+	0.559309i	$0.188934\pi$
$648$	0	0
$649$	−28.7337	−1.12790
$650$	0	0
$651$	0	0
$652$	−25.0310	−0.980290
$653$	33.8466	1.32452	0.662259	−	0.749275i	$-0.269598\pi$
$653$	33.8466	1.32452	0.662259	+	0.749275i	$0.269598\pi$
$654$	0	0
$655$	11.6937	0.456910
$656$	−8.77924	−0.342772
$657$	0	0
$658$	1.39576	0.0544126
$659$	31.5318	1.22831	0.614153	−	0.789187i	$-0.289498\pi$
$659$	31.5318	1.22831	0.614153	+	0.789187i	$0.289498\pi$
$660$	0	0
$661$	6.37751	0.248057	0.124028	−	0.992279i	$-0.460419\pi$
$661$	6.37751	0.248057	0.124028	+	0.992279i	$0.460419\pi$
$662$	2.48133	0.0964396
$663$	0	0
$664$	8.68035	0.336863
$665$	23.8660	0.925485
$666$	0	0
$667$	−29.8264	−1.15488
$668$	8.36683	0.323723
$669$	0	0
$670$	6.92881	0.267683
$671$	10.4247	0.402441
$672$	0	0
$673$	9.72261	0.374779	0.187389	−	0.982286i	$-0.439997\pi$
$673$	9.72261	0.374779	0.187389	+	0.982286i	$0.439997\pi$
$674$	16.7331	0.644535
$675$	0	0
$676$	0	0
$677$	10.0845	0.387580	0.193790	−	0.981043i	$-0.437922\pi$
$677$	10.0845	0.387580	0.193790	+	0.981043i	$0.437922\pi$
$678$	0	0
$679$	68.5052	2.62899
$680$	5.75872	0.220837
$681$	0	0
$682$	−5.62088	−0.215235
$683$	−6.73698	−0.257783	−0.128892	−	0.991659i	$-0.541142\pi$
$683$	−6.73698	−0.257783	−0.128892	+	0.991659i	$0.541142\pi$
$684$	0	0
$685$	9.27739	0.354471
$686$	−23.4257	−0.894399
$687$	0	0
$688$	3.71769	0.141735
$689$	0	0
$690$	0	0
$691$	17.9444	0.682637	0.341319	−	0.939948i	$-0.389126\pi$
$691$	17.9444	0.682637	0.341319	+	0.939948i	$0.389126\pi$
$692$	−30.0144	−1.14098
$693$	0	0
$694$	−9.68195	−0.367522
$695$	−12.9155	−0.489912
$696$	0	0
$697$	10.8020	0.409156
$698$	0.0545466	0.00206462
$699$	0	0
$700$	−8.20620	−0.310165
$701$	−41.2762	−1.55898	−0.779490	−	0.626415i	$-0.784522\pi$
$701$	−41.2762	−1.55898	−0.779490	+	0.626415i	$0.784522\pi$
$702$	0	0
$703$	13.3242	0.502531
$704$	6.30122	0.237486
$705$	0	0
$706$	15.8310	0.595806
$707$	−59.9976	−2.25644
$708$	0	0
$709$	−1.46573	−0.0550467	−0.0275234	−	0.999621i	$-0.508762\pi$
$709$	−1.46573	−0.0550467	−0.0275234	+	0.999621i	$0.508762\pi$
$710$	1.12783	0.0423266
$711$	0	0
$712$	−14.0267	−0.525671
$713$	25.9109	0.970372
$714$	0	0
$715$	0	0
$716$	2.36683	0.0884528
$717$	0	0
$718$	6.22058	0.232150
$719$	9.61369	0.358530	0.179265	−	0.983801i	$-0.442628\pi$
$719$	9.61369	0.358530	0.179265	+	0.983801i	$0.442628\pi$
$720$	0	0
$721$	32.8888	1.22484
$722$	−3.07960	−0.114611
$723$	0	0
$724$	12.7649	0.474403
$725$	3.51026	0.130368
$726$	0	0
$727$	−42.6547	−1.58198	−0.790988	−	0.611831i	$-0.790433\pi$
$727$	−42.6547	−1.58198	−0.790988	+	0.611831i	$0.790433\pi$
$728$	0	0
$729$	0	0
$730$	2.98667	0.110542
$731$	−4.57426	−0.169185
$732$	0	0
$733$	6.48360	0.239477	0.119739	−	0.992805i	$-0.461794\pi$
$733$	6.48360	0.239477	0.119739	+	0.992805i	$0.461794\pi$
$734$	19.2111	0.709096
$735$	0	0
$736$	44.7091	1.64800
$737$	−43.9299	−1.61818
$738$	0	0
$739$	42.3689	1.55857	0.779283	−	0.626672i	$-0.215583\pi$
$739$	42.3689	1.55857	0.779283	+	0.626672i	$0.215583\pi$
$740$	−4.58145	−0.168417
$741$	0	0
$742$	−35.6163	−1.30752
$743$	−17.4296	−0.639431	−0.319715	−	0.947514i	$-0.603587\pi$
$743$	−17.4296	−0.639431	−0.319715	+	0.947514i	$0.603587\pi$
$744$	0	0
$745$	2.00000	0.0732743
$746$	5.18237	0.189740
$747$	0	0
$748$	−16.8248	−0.615176
$749$	−74.7019	−2.72955
$750$	0	0
$751$	13.3691	0.487845	0.243923	−	0.969795i	$-0.421566\pi$
$751$	13.3691	0.487845	0.243923	+	0.969795i	$0.421566\pi$
$752$	−1.26180	−0.0460129
$753$	0	0
$754$	0	0
$755$	−8.60424	−0.313140
$756$	0	0
$757$	27.6925	1.00650	0.503250	−	0.864141i	$-0.332138\pi$
$757$	27.6925	1.00650	0.503250	+	0.864141i	$0.332138\pi$
$758$	3.78661	0.137536
$759$	0	0
$760$	9.94214	0.360639
$761$	−0.523590	−0.0189801	−0.00949007	−	0.999955i	$-0.503021\pi$
$761$	−0.523590	−0.0189801	−0.00949007	+	0.999955i	$0.503021\pi$
$762$	0	0
$763$	−76.0337	−2.75260
$764$	−4.11118	−0.148737
$765$	0	0
$766$	−10.8988	−0.393791
$767$	0	0
$768$	0	0
$769$	14.1606	0.510645	0.255323	−	0.966856i	$-0.417818\pi$
$769$	14.1606	0.510645	0.255323	+	0.966856i	$0.417818\pi$
$770$	−8.84939	−0.318910
$771$	0	0
$772$	−27.0433	−0.973310
$773$	27.4186	0.986177	0.493088	−	0.869979i	$-0.335868\pi$
$773$	27.4186	0.986177	0.493088	+	0.869979i	$0.335868\pi$
$774$	0	0
$775$	−3.04945	−0.109539
$776$	28.5380	1.02445
$777$	0	0
$778$	−8.49693	−0.304630
$779$	18.6491	0.668175
$780$	0	0
$781$	−7.15061	−0.255869
$782$	−13.1917	−0.471732
$783$	0	0
$784$	37.5585	1.34137
$785$	−0.908291	−0.0324183
$786$	0	0
$787$	−21.3701	−0.761763	−0.380882	−	0.924624i	$-0.624380\pi$
$787$	−21.3701	−0.761763	−0.380882	+	0.924624i	$0.624380\pi$
$788$	−31.5630	−1.12439
$789$	0	0
$790$	1.19287	0.0424405
$791$	14.7792	0.525489
$792$	0	0
$793$	0	0
$794$	6.11837	0.217133
$795$	0	0
$796$	−18.7031	−0.662915
$797$	−3.88550	−0.137632	−0.0688158	−	0.997629i	$-0.521922\pi$
$797$	−3.88550	−0.137632	−0.0688158	+	0.997629i	$0.521922\pi$
$798$	0	0
$799$	1.55252	0.0549242
$800$	−5.26180	−0.186033
$801$	0	0
$802$	−14.3714	−0.507471
$803$	−18.9360	−0.668237
$804$	0	0
$805$	40.7936	1.43779
$806$	0	0
$807$	0	0
$808$	−24.9939	−0.879281
$809$	50.9048	1.78972	0.894859	−	0.446349i	$-0.147276\pi$
$809$	50.9048	1.78972	0.894859	+	0.446349i	$0.147276\pi$
$810$	0	0
$811$	42.5174	1.49299	0.746495	−	0.665391i	$-0.231735\pi$
$811$	42.5174	1.49299	0.746495	+	0.665391i	$0.231735\pi$
$812$	−28.8059	−1.01089
$813$	0	0
$814$	−4.94053	−0.173166
$815$	−14.6442	−0.512965
$816$	0	0
$817$	−7.89723	−0.276289
$818$	−16.6570	−0.582398
$819$	0	0
$820$	−6.41241	−0.223931
$821$	−3.99547	−0.139443	−0.0697213	−	0.997567i	$-0.522211\pi$
$821$	−3.99547	−0.139443	−0.0697213	+	0.997567i	$0.522211\pi$
$822$	0	0
$823$	−1.57531	−0.0549117	−0.0274559	−	0.999623i	$-0.508741\pi$
$823$	−1.57531	−0.0549117	−0.0274559	+	0.999623i	$0.508741\pi$
$824$	13.7009	0.477292
$825$	0	0
$826$	−21.7581	−0.757060
$827$	9.17850	0.319168	0.159584	−	0.987184i	$-0.448985\pi$
$827$	9.17850	0.319168	0.159584	+	0.987184i	$0.448985\pi$
$828$	0	0
$829$	49.4717	1.71822	0.859112	−	0.511788i	$-0.171017\pi$
$829$	49.4717	1.71822	0.859112	+	0.511788i	$0.171017\pi$
$830$	2.34017	0.0812286
$831$	0	0
$832$	0	0
$833$	−46.2122	−1.60116
$834$	0	0
$835$	4.89496	0.169397
$836$	−29.0472	−1.00462
$837$	0	0
$838$	18.5029	0.639172
$839$	9.29299	0.320830	0.160415	−	0.987050i	$-0.448717\pi$
$839$	9.29299	0.320830	0.160415	+	0.987050i	$0.448717\pi$
$840$	0	0
$841$	−16.6781	−0.575106
$842$	−7.29630	−0.251447
$843$	0	0
$844$	24.2329	0.834130
$845$	0	0
$846$	0	0
$847$	3.29582	0.113246
$848$	32.1978	1.10568
$849$	0	0
$850$	1.55252	0.0532510
$851$	22.7747	0.780707
$852$	0	0
$853$	11.0423	0.378080	0.189040	−	0.981969i	$-0.439462\pi$
$853$	11.0423	0.378080	0.189040	+	0.981969i	$0.439462\pi$
$854$	7.89392	0.270124
$855$	0	0
$856$	−31.1194	−1.06364
$857$	8.58476	0.293250	0.146625	−	0.989192i	$-0.453159\pi$
$857$	8.58476	0.293250	0.146625	+	0.989192i	$0.453159\pi$
$858$	0	0
$859$	−46.6202	−1.59066	−0.795331	−	0.606176i	$-0.792703\pi$
$859$	−46.6202	−1.59066	−0.795331	+	0.606176i	$0.792703\pi$
$860$	2.71542	0.0925950
$861$	0	0
$862$	−8.22899	−0.280281
$863$	10.6947	0.364053	0.182026	−	0.983294i	$-0.441734\pi$
$863$	10.6947	0.364053	0.182026	+	0.983294i	$0.441734\pi$
$864$	0	0
$865$	−17.5597	−0.597048
$866$	−2.09171	−0.0710792
$867$	0	0
$868$	25.0244	0.849383
$869$	−7.56302	−0.256558
$870$	0	0
$871$	0	0
$872$	−31.6742	−1.07262
$873$	0	0
$874$	−22.7747	−0.770366
$875$	−4.80098	−0.162303
$876$	0	0
$877$	10.5958	0.357796	0.178898	−	0.983868i	$-0.442747\pi$
$877$	10.5958	0.357796	0.178898	+	0.983868i	$0.442747\pi$
$878$	10.9516	0.369598
$879$	0	0
$880$	8.00000	0.269680
$881$	16.2290	0.546769	0.273384	−	0.961905i	$-0.411857\pi$
$881$	16.2290	0.546769	0.273384	+	0.961905i	$0.411857\pi$
$882$	0	0
$883$	12.6137	0.424484	0.212242	−	0.977217i	$-0.431923\pi$
$883$	12.6137	0.424484	0.212242	+	0.977217i	$0.431923\pi$
$884$	0	0
$885$	0	0
$886$	−18.9132	−0.635402
$887$	13.3907	0.449615	0.224807	−	0.974403i	$-0.427825\pi$
$887$	13.3907	0.449615	0.224807	+	0.974403i	$0.427825\pi$
$888$	0	0
$889$	−35.7897	−1.20035
$890$	−3.78151	−0.126757
$891$	0	0
$892$	−9.67420	−0.323916
$893$	2.68035	0.0896944
$894$	0	0
$895$	1.38470	0.0462854
$896$	55.2951	1.84728
$897$	0	0
$898$	−4.29914	−0.143464
$899$	−10.7044	−0.357010
$900$	0	0
$901$	−39.6163	−1.31981
$902$	−6.91500	−0.230244
$903$	0	0
$904$	6.15676	0.204771
$905$	7.46800	0.248245
$906$	0	0
$907$	−46.3051	−1.53754	−0.768768	−	0.639528i	$-0.779130\pi$
$907$	−46.3051	−1.53754	−0.768768	+	0.639528i	$0.779130\pi$
$908$	−35.6886	−1.18437
$909$	0	0
$910$	0	0
$911$	40.6947	1.34828	0.674138	−	0.738605i	$-0.264515\pi$
$911$	40.6947	1.34828	0.674138	+	0.738605i	$0.264515\pi$
$912$	0	0
$913$	−14.8371	−0.491036
$914$	1.40683	0.0465337
$915$	0	0
$916$	−19.7464	−0.652441
$917$	−56.1412	−1.85394
$918$	0	0
$919$	−44.5874	−1.47080	−0.735402	−	0.677632i	$-0.763006\pi$
$919$	−44.5874	−1.47080	−0.735402	+	0.677632i	$0.763006\pi$
$920$	16.9939	0.560271
$921$	0	0
$922$	16.2185	0.534128
$923$	0	0
$924$	0	0
$925$	−2.68035	−0.0881292
$926$	−3.43349	−0.112831
$927$	0	0
$928$	−18.4703	−0.606316
$929$	28.9770	0.950706	0.475353	−	0.879795i	$-0.342320\pi$
$929$	28.9770	0.950706	0.475353	+	0.879795i	$0.342320\pi$
$930$	0	0
$931$	−79.7829	−2.61478
$932$	−6.37912	−0.208955
$933$	0	0
$934$	−1.11554	−0.0365016
$935$	−9.84324	−0.321909
$936$	0	0
$937$	−53.1871	−1.73755	−0.868774	−	0.495209i	$-0.835091\pi$
$937$	−53.1871	−1.73755	−0.868774	+	0.495209i	$0.835091\pi$
$938$	−33.2651	−1.08614
$939$	0	0
$940$	−0.921622	−0.0300600
$941$	26.3617	0.859368	0.429684	−	0.902979i	$-0.358625\pi$
$941$	26.3617	0.859368	0.429684	+	0.902979i	$0.358625\pi$
$942$	0	0
$943$	31.8765	1.03804
$944$	19.6697	0.640193
$945$	0	0
$946$	2.92825	0.0952055
$947$	−50.7780	−1.65006	−0.825032	−	0.565086i	$-0.808843\pi$
$947$	−50.7780	−1.65006	−0.825032	+	0.565086i	$0.808843\pi$
$948$	0	0
$949$	0	0
$950$	2.68035	0.0869619
$951$	0	0
$952$	−27.6475	−0.896062
$953$	20.8326	0.674833	0.337417	−	0.941355i	$-0.390447\pi$
$953$	20.8326	0.674833	0.337417	+	0.941355i	$0.390447\pi$
$954$	0	0
$955$	−2.40522	−0.0778311
$956$	−46.3090	−1.49774
$957$	0	0
$958$	−6.20620	−0.200513
$959$	−44.5406	−1.43829
$960$	0	0
$961$	−21.7009	−0.700028
$962$	0	0
$963$	0	0
$964$	−40.3356	−1.29912
$965$	−15.8215	−0.509312
$966$	0	0
$967$	5.92777	0.190624	0.0953120	−	0.995447i	$-0.469615\pi$
$967$	5.92777	0.190624	0.0953120	+	0.995447i	$0.469615\pi$
$968$	1.37298	0.0441292
$969$	0	0
$970$	7.69368	0.247029
$971$	−10.4969	−0.336862	−0.168431	−	0.985713i	$-0.553870\pi$
$971$	−10.4969	−0.336862	−0.168431	+	0.985713i	$0.553870\pi$
$972$	0	0
$973$	62.0070	1.98785
$974$	10.8638	0.348097
$975$	0	0
$976$	−7.13624	−0.228425
$977$	53.3217	1.70591	0.852957	−	0.521981i	$-0.174807\pi$
$977$	53.3217	1.70591	0.852957	+	0.521981i	$0.174807\pi$
$978$	0	0
$979$	23.9754	0.766258
$980$	27.4329	0.876313
$981$	0	0
$982$	−11.7983	−0.376500
$983$	21.7275	0.693000	0.346500	−	0.938050i	$-0.387370\pi$
$983$	21.7275	0.693000	0.346500	+	0.938050i	$0.387370\pi$
$984$	0	0
$985$	−18.4657	−0.588367
$986$	5.44975	0.173555
$987$	0	0
$988$	0	0
$989$	−13.4985	−0.429228
$990$	0	0
$991$	−8.75646	−0.278158	−0.139079	−	0.990281i	$-0.544414\pi$
$991$	−8.75646	−0.278158	−0.139079	+	0.990281i	$0.544414\pi$
$992$	16.0456	0.509447
$993$	0	0
$994$	−5.41468	−0.171743
$995$	−10.9421	−0.346889
$996$	0	0
$997$	3.45920	0.109554	0.0547770	−	0.998499i	$-0.482555\pi$
$997$	3.45920	0.109554	0.0547770	+	0.998499i	$0.482555\pi$
$998$	−12.8449	−0.406597
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7605.2.a.bt.1.2		3
3.2	odd	2		2535.2.a.z.1.2		3
13.4	even	6		585.2.j.g.406.2		6
13.10	even	6		585.2.j.g.451.2		6
13.12	even	2		7605.2.a.bu.1.2		3
39.17	odd	6		195.2.i.e.16.2	✓	6
39.23	odd	6		195.2.i.e.61.2	yes	6
39.38	odd	2		2535.2.a.y.1.2		3
195.17	even	12		975.2.bb.j.874.3		12
195.23	even	12		975.2.bb.j.724.3		12
195.62	even	12		975.2.bb.j.724.4		12
195.134	odd	6		975.2.i.m.601.2		6
195.173	even	12		975.2.bb.j.874.4		12
195.179	odd	6		975.2.i.m.451.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
195.2.i.e.16.2	✓	6	39.17	odd	6
195.2.i.e.61.2	yes	6	39.23	odd	6
585.2.j.g.406.2		6	13.4	even	6
585.2.j.g.451.2		6	13.10	even	6
975.2.i.m.451.2		6	195.179	odd	6
975.2.i.m.601.2		6	195.134	odd	6
975.2.bb.j.724.3		12	195.23	even	12
975.2.bb.j.724.4		12	195.62	even	12
975.2.bb.j.874.3		12	195.17	even	12
975.2.bb.j.874.4		12	195.173	even	12
2535.2.a.y.1.2		3	39.38	odd	2
2535.2.a.z.1.2		3	3.2	odd	2
7605.2.a.bt.1.2		3	1.1	even	1	trivial
7605.2.a.bu.1.2		3	13.12	even	2

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)

\(f(q)\)	\(=\)	\(q-0.539189 q^{2} -1.70928 q^{4} -1.00000 q^{5} +4.80098 q^{7} +2.00000 q^{8} +O(q^{10})\)	\(q-0.539189 q^{2} -1.70928 q^{4} -1.00000 q^{5} +4.80098 q^{7} +2.00000 q^{8} +0.539189 q^{10} -3.41855 q^{11} -2.58864 q^{14} +2.34017 q^{16} -2.87936 q^{17} -4.97107 q^{19} +1.70928 q^{20} +1.84324 q^{22} -8.49693 q^{23} +1.00000 q^{25} -8.20620 q^{28} +3.51026 q^{29} -3.04945 q^{31} -5.26180 q^{32} +1.55252 q^{34} -4.80098 q^{35} -2.68035 q^{37} +2.68035 q^{38} -2.00000 q^{40} -3.75154 q^{41} +1.58864 q^{43} +5.84324 q^{44} +4.58145 q^{46} -0.539189 q^{47} +16.0494 q^{49} -0.539189 q^{50} +13.7587 q^{53} +3.41855 q^{55} +9.60197 q^{56} -1.89269 q^{58} +8.40522 q^{59} -3.04945 q^{61} +1.64423 q^{62} -1.84324 q^{64} +12.8504 q^{67} +4.92162 q^{68} +2.58864 q^{70} +2.09171 q^{71} +5.53919 q^{73} +1.44521 q^{74} +8.49693 q^{76} -16.4124 q^{77} +2.21235 q^{79} -2.34017 q^{80} +2.02279 q^{82} +4.34017 q^{83} +2.87936 q^{85} -0.856576 q^{86} -6.83710 q^{88} -7.01333 q^{89} +14.5236 q^{92} +0.290725 q^{94} +4.97107 q^{95} +14.2690 q^{97} -8.65368 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{4} - 3 q^{5} + 5 q^{7} + 6 q^{8}+O(q^{10}) \) 3 * q + 2 * q^4 - 3 * q^5 + 5 * q^7 + 6 * q^8	\( 3 q + 2 q^{4} - 3 q^{5} + 5 q^{7} + 6 q^{8} + 4 q^{11} + 12 q^{14} - 4 q^{16} + 4 q^{17} - 2 q^{20} + 12 q^{22} - 8 q^{23} + 3 q^{25} - 6 q^{29} + 9 q^{31} - 8 q^{32} + 4 q^{34} - 5 q^{35} + 14 q^{37} - 14 q^{38} - 6 q^{40} - 20 q^{41} - 15 q^{43} + 24 q^{44} + 28 q^{46} + 30 q^{49} + 16 q^{53} - 4 q^{55} + 10 q^{56} + 6 q^{58} + 10 q^{59} + 9 q^{61} + 2 q^{62} - 12 q^{64} + 11 q^{67} + 18 q^{68} - 12 q^{70} + 4 q^{71} + 15 q^{73} - 8 q^{74} + 8 q^{76} + 17 q^{79} + 4 q^{80} - 14 q^{82} + 2 q^{83} - 4 q^{85} - 10 q^{86} + 8 q^{88} - 22 q^{89} + 28 q^{92} + 8 q^{94} + q^{97} - 2 q^{98}+O(q^{100}) \) 3 * q + 2 * q^4 - 3 * q^5 + 5 * q^7 + 6 * q^8 + 4 * q^11 + 12 * q^14 - 4 * q^16 + 4 * q^17 - 2 * q^20 + 12 * q^22 - 8 * q^23 + 3 * q^25 - 6 * q^29 + 9 * q^31 - 8 * q^32 + 4 * q^34 - 5 * q^35 + 14 * q^37 - 14 * q^38 - 6 * q^40 - 20 * q^41 - 15 * q^43 + 24 * q^44 + 28 * q^46 + 30 * q^49 + 16 * q^53 - 4 * q^55 + 10 * q^56 + 6 * q^58 + 10 * q^59 + 9 * q^61 + 2 * q^62 - 12 * q^64 + 11 * q^67 + 18 * q^68 - 12 * q^70 + 4 * q^71 + 15 * q^73 - 8 * q^74 + 8 * q^76 + 17 * q^79 + 4 * q^80 - 14 * q^82 + 2 * q^83 - 4 * q^85 - 10 * q^86 + 8 * q^88 - 22 * q^89 + 28 * q^92 + 8 * q^94 + q^97 - 2 * q^98

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