LMFDB - Embedded newform 1225.2.b.g.99.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1225,2,Mod(99,1225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1225, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

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

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

chi := DirichletCharacter("1225.99");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1225 = 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1225.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$9.78167424761$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 35)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			99.1
Root			$0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1225.99
Dual form			1225.2.b.g.99.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.41421i q^{2} -0.414214i q^{3} -3.82843 q^{4} -1.00000 q^{6} +4.41421i q^{8} +2.82843 q^{9} +O(q^{10})$	\(q-2.41421i q^{2} -0.414214i q^{3} -3.82843 q^{4} -1.00000 q^{6} +4.41421i q^{8} +2.82843 q^{9} -0.828427 q^{11} +1.58579i q^{12} -4.82843i q^{13} +3.00000 q^{16} -4.82843i q^{17} -6.82843i q^{18} -2.82843 q^{19} +2.00000i q^{22} +0.414214i q^{23} +1.82843 q^{24} -11.6569 q^{26} -2.41421i q^{27} +1.00000 q^{29} -6.00000 q^{31} +1.58579i q^{32} +0.343146i q^{33} -11.6569 q^{34} -10.8284 q^{36} +6.82843i q^{38} -2.00000 q^{39} -7.82843 q^{41} +3.58579i q^{43} +3.17157 q^{44} +1.00000 q^{46} -2.00000i q^{47} -1.24264i q^{48} -2.00000 q^{51} +18.4853i q^{52} -1.17157i q^{53} -5.82843 q^{54} +1.17157i q^{57} -2.41421i q^{58} -4.48528 q^{59} +5.48528 q^{61} +14.4853i q^{62} +9.82843 q^{64} +0.828427 q^{66} -9.58579i q^{67} +18.4853i q^{68} +0.171573 q^{69} +4.48528 q^{71} +12.4853i q^{72} -0.828427i q^{73} +10.8284 q^{76} +4.82843i q^{78} -14.8284 q^{79} +7.48528 q^{81} +18.8995i q^{82} +13.7279i q^{83} +8.65685 q^{86} -0.414214i q^{87} -3.65685i q^{88} +8.65685 q^{89} -1.58579i q^{92} +2.48528i q^{93} -4.82843 q^{94} +0.656854 q^{96} -11.6569i q^{97} -2.34315 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4} - 4 q^{6}+O(q^{10}) $ 4 * q - 4 * q^4 - 4 * q^6	$ 4 q - 4 q^{4} - 4 q^{6} + 8 q^{11} + 12 q^{16} - 4 q^{24} - 24 q^{26} + 4 q^{29} - 24 q^{31} - 24 q^{34} - 32 q^{36} - 8 q^{39} - 20 q^{41} + 24 q^{44} + 4 q^{46} - 8 q^{51} - 12 q^{54} + 16 q^{59} - 12 q^{61} + 28 q^{64} - 8 q^{66} + 12 q^{69} - 16 q^{71} + 32 q^{76} - 48 q^{79} - 4 q^{81} + 12 q^{86} + 12 q^{89} - 8 q^{94} - 20 q^{96} - 32 q^{99}+O(q^{100}) $ 4 * q - 4 * q^4 - 4 * q^6 + 8 * q^11 + 12 * q^16 - 4 * q^24 - 24 * q^26 + 4 * q^29 - 24 * q^31 - 24 * q^34 - 32 * q^36 - 8 * q^39 - 20 * q^41 + 24 * q^44 + 4 * q^46 - 8 * q^51 - 12 * q^54 + 16 * q^59 - 12 * q^61 + 28 * q^64 - 8 * q^66 + 12 * q^69 - 16 * q^71 + 32 * q^76 - 48 * q^79 - 4 * q^81 + 12 * q^86 + 12 * q^89 - 8 * q^94 - 20 * q^96 - 32 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1225\mathbb{Z}\right)^\times$.

$n$	$101$	$1177$
$\chi(n)$	$1$	$-1$

Coefficient data

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

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

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

$2$	− 2.41421i	− 1.70711i	−0.521005	−	0.853553i	$-0.674443\pi$
$2$	− 2.41421i	− 1.70711i	0.521005	−	0.853553i	$-0.325557\pi$
$3$	− 0.414214i	− 0.239146i	−0.992825	−	0.119573i	$-0.961847\pi$
$3$	− 0.414214i	− 0.239146i	0.992825	−	0.119573i	$-0.0381526\pi$
$4$	−3.82843	−1.91421
$5$	0	0
$5$	0	0
$6$	−1.00000	−0.408248
$7$	0	0
$7$	0	0
$8$	4.41421i	1.56066i
$9$	2.82843	0.942809
$10$	0	0
$11$	−0.828427	−0.249780	−0.124890	−	0.992171i	$-0.539858\pi$
$11$	−0.828427	−0.249780	−0.124890	+	0.992171i	$0.539858\pi$
$12$	1.58579i	0.457777i
$13$	− 4.82843i	− 1.33916i	−0.742738	−	0.669582i	$-0.766473\pi$
$13$	− 4.82843i	− 1.33916i	0.742738	−	0.669582i	$-0.233527\pi$
$14$	0	0
$15$	0	0
$16$	3.00000	0.750000
$17$	− 4.82843i	− 1.17107i	−0.810649	−	0.585533i	$-0.800885\pi$
$17$	− 4.82843i	− 1.17107i	0.810649	−	0.585533i	$-0.199115\pi$
$18$	− 6.82843i	− 1.60948i
$19$	−2.82843	−0.648886	−0.324443	−	0.945905i	$-0.605177\pi$
$19$	−2.82843	−0.648886	−0.324443	+	0.945905i	$0.605177\pi$
$20$	0	0
$21$	0	0
$22$	2.00000i	0.426401i
$23$	0.414214i	0.0863695i	0.999067	+	0.0431847i	$0.0137504\pi$
$23$	0.414214i	0.0863695i	−0.999067	+	0.0431847i	$0.986250\pi$
$24$	1.82843	0.373226
$25$	0	0
$26$	−11.6569	−2.28610
$27$	− 2.41421i	− 0.464616i
$28$	0	0
$29$	1.00000	0.185695	0.0928477	−	0.995680i	$-0.470403\pi$
$29$	1.00000	0.185695	0.0928477	+	0.995680i	$0.470403\pi$
$30$	0	0
$31$	−6.00000	−1.07763	−0.538816	−	0.842424i	$-0.681128\pi$
$31$	−6.00000	−1.07763	−0.538816	+	0.842424i	$0.681128\pi$
$32$	1.58579i	0.280330i
$33$	0.343146i	0.0597340i
$34$	−11.6569	−1.99913
$35$	0	0
$36$	−10.8284	−1.80474
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	6.82843i	1.10772i
$39$	−2.00000	−0.320256
$40$	0	0
$41$	−7.82843	−1.22259	−0.611297	−	0.791401i	$-0.709352\pi$
$41$	−7.82843	−1.22259	−0.611297	+	0.791401i	$0.709352\pi$
$42$	0	0
$43$	3.58579i	0.546827i	0.961897	+	0.273414i	$0.0881528\pi$
$43$	3.58579i	0.546827i	−0.961897	+	0.273414i	$0.911847\pi$
$44$	3.17157	0.478133
$45$	0	0
$46$	1.00000	0.147442
$47$	− 2.00000i	− 0.291730i	−0.989305	−	0.145865i	$-0.953403\pi$
$47$	− 2.00000i	− 0.291730i	0.989305	−	0.145865i	$-0.0465965\pi$
$48$	− 1.24264i	− 0.179360i
$49$	0	0
$50$	0	0
$51$	−2.00000	−0.280056
$52$	18.4853i	2.56345i
$53$	− 1.17157i	− 0.160928i	−0.996758	−	0.0804640i	$-0.974360\pi$
$53$	− 1.17157i	− 0.160928i	0.996758	−	0.0804640i	$-0.0256402\pi$
$54$	−5.82843	−0.793148
$55$	0	0
$56$	0	0
$57$	1.17157i	0.155179i
$58$	− 2.41421i	− 0.317002i
$59$	−4.48528	−0.583934	−0.291967	−	0.956428i	$-0.594310\pi$
$59$	−4.48528	−0.583934	−0.291967	+	0.956428i	$0.594310\pi$
$60$	0	0
$61$	5.48528	0.702318	0.351159	−	0.936316i	$-0.385788\pi$
$61$	5.48528	0.702318	0.351159	+	0.936316i	$0.385788\pi$
$62$	14.4853i	1.83963i
$63$	0	0
$64$	9.82843	1.22855
$65$	0	0
$66$	0.828427	0.101972
$67$	− 9.58579i	− 1.17109i	−0.810640	−	0.585545i	$-0.800881\pi$
$67$	− 9.58579i	− 1.17109i	0.810640	−	0.585545i	$-0.199119\pi$
$68$	18.4853i	2.24167i
$69$	0.171573	0.0206549
$70$	0	0
$71$	4.48528	0.532305	0.266152	−	0.963931i	$-0.414248\pi$
$71$	4.48528	0.532305	0.266152	+	0.963931i	$0.414248\pi$
$72$	12.4853i	1.47140i
$73$	− 0.828427i	− 0.0969601i	−0.998824	−	0.0484800i	$-0.984562\pi$
$73$	− 0.828427i	− 0.0969601i	0.998824	−	0.0484800i	$-0.0154377\pi$
$74$	0	0
$75$	0	0
$76$	10.8284	1.24211
$77$	0	0
$78$	4.82843i	0.546712i
$79$	−14.8284	−1.66833	−0.834164	−	0.551516i	$-0.814049\pi$
$79$	−14.8284	−1.66833	−0.834164	+	0.551516i	$0.814049\pi$
$80$	0	0
$81$	7.48528	0.831698
$82$	18.8995i	2.08710i
$83$	13.7279i	1.50684i	0.657542	+	0.753418i	$0.271596\pi$
$83$	13.7279i	1.50684i	−0.657542	+	0.753418i	$0.728404\pi$
$84$	0	0
$85$	0	0
$86$	8.65685	0.933493
$87$	− 0.414214i	− 0.0444084i
$88$	− 3.65685i	− 0.389822i
$89$	8.65685	0.917625	0.458812	−	0.888533i	$-0.348275\pi$
$89$	8.65685	0.917625	0.458812	+	0.888533i	$0.348275\pi$
$90$	0	0
$91$	0	0
$92$	− 1.58579i	− 0.165330i
$93$	2.48528i	0.257712i
$94$	−4.82843	−0.498014
$95$	0	0
$96$	0.656854	0.0670399
$97$	− 11.6569i	− 1.18357i	−0.806094	−	0.591787i	$-0.798423\pi$
$97$	− 11.6569i	− 1.18357i	0.806094	−	0.591787i	$-0.201577\pi$
$98$	0	0
$99$	−2.34315	−0.235495
$100$	0	0
$101$	−10.3137	−1.02625	−0.513126	−	0.858313i	$-0.671513\pi$
$101$	−10.3137	−1.02625	−0.513126	+	0.858313i	$0.671513\pi$
$102$	4.82843i	0.478086i
$103$	− 2.41421i	− 0.237880i	−0.992901	−	0.118940i	$-0.962050\pi$
$103$	− 2.41421i	− 0.237880i	0.992901	−	0.118940i	$-0.0379495\pi$
$104$	21.3137	2.08998
$105$	0	0
$106$	−2.82843	−0.274721
$107$	− 11.2426i	− 1.08687i	−0.839452	−	0.543434i	$-0.817124\pi$
$107$	− 11.2426i	− 1.08687i	0.839452	−	0.543434i	$-0.182876\pi$
$108$	9.24264i	0.889374i
$109$	13.4853	1.29166	0.645828	−	0.763483i	$-0.276512\pi$
$109$	13.4853	1.29166	0.645828	+	0.763483i	$0.276512\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 4.48528i	− 0.421940i	−0.977493	−	0.210970i	$-0.932338\pi$
$113$	− 4.48528i	− 0.421940i	0.977493	−	0.210970i	$-0.0676622\pi$
$114$	2.82843	0.264906
$115$	0	0
$116$	−3.82843	−0.355461
$117$	− 13.6569i	− 1.26258i
$118$	10.8284i	0.996838i
$119$	0	0
$120$	0	0
$121$	−10.3137	−0.937610
$122$	− 13.2426i	− 1.19893i
$123$	3.24264i	0.292379i
$124$	22.9706	2.06282
$125$	0	0
$126$	0	0
$127$	9.31371i	0.826458i	0.910627	+	0.413229i	$0.135599\pi$
$127$	9.31371i	0.826458i	−0.910627	+	0.413229i	$0.864401\pi$
$128$	− 20.5563i	− 1.81694i
$129$	1.48528	0.130772
$130$	0	0
$131$	−19.3137	−1.68745	−0.843723	−	0.536778i	$-0.819641\pi$
$131$	−19.3137	−1.68745	−0.843723	+	0.536778i	$0.819641\pi$
$132$	− 1.31371i	− 0.114344i
$133$	0	0
$134$	−23.1421	−1.99918
$135$	0	0
$136$	21.3137	1.82764
$137$	9.65685i	0.825041i	0.910948	+	0.412520i	$0.135351\pi$
$137$	9.65685i	0.825041i	−0.910948	+	0.412520i	$0.864649\pi$
$138$	− 0.414214i	− 0.0352602i
$139$	16.1421	1.36916	0.684579	−	0.728939i	$-0.259986\pi$
$139$	16.1421	1.36916	0.684579	+	0.728939i	$0.259986\pi$
$140$	0	0
$141$	−0.828427	−0.0697661
$142$	− 10.8284i	− 0.908701i
$143$	4.00000i	0.334497i
$144$	8.48528	0.707107
$145$	0	0
$146$	−2.00000	−0.165521
$147$	0	0
$148$	0	0
$149$	2.17157	0.177902	0.0889511	−	0.996036i	$-0.471649\pi$
$149$	2.17157	0.177902	0.0889511	+	0.996036i	$0.471649\pi$
$150$	0	0
$151$	11.6569	0.948621	0.474311	−	0.880358i	$-0.342697\pi$
$151$	11.6569	0.948621	0.474311	+	0.880358i	$0.342697\pi$
$152$	− 12.4853i	− 1.01269i
$153$	− 13.6569i	− 1.10409i
$154$	0	0
$155$	0	0
$156$	7.65685	0.613039
$157$	− 17.3137i	− 1.38178i	−0.722958	−	0.690892i	$-0.757218\pi$
$157$	− 17.3137i	− 1.38178i	0.722958	−	0.690892i	$-0.242782\pi$
$158$	35.7990i	2.84801i
$159$	−0.485281	−0.0384853
$160$	0	0
$161$	0	0
$162$	− 18.0711i	− 1.41980i
$163$	12.3431i	0.966790i	0.875402	+	0.483395i	$0.160596\pi$
$163$	12.3431i	0.966790i	−0.875402	+	0.483395i	$0.839404\pi$
$164$	29.9706	2.34031
$165$	0	0
$166$	33.1421	2.57233
$167$	− 22.4142i	− 1.73446i	−0.497904	−	0.867232i	$-0.665897\pi$
$167$	− 22.4142i	− 1.73446i	0.497904	−	0.867232i	$-0.334103\pi$
$168$	0	0
$169$	−10.3137	−0.793362
$170$	0	0
$171$	−8.00000	−0.611775
$172$	− 13.7279i	− 1.04674i
$173$	3.31371i	0.251937i	0.992034	+	0.125968i	$0.0402038\pi$
$173$	3.31371i	0.251937i	−0.992034	+	0.125968i	$0.959796\pi$
$174$	−1.00000	−0.0758098
$175$	0	0
$176$	−2.48528	−0.187335
$177$	1.85786i	0.139646i
$178$	− 20.8995i	− 1.56648i
$179$	10.0000	0.747435	0.373718	−	0.927543i	$-0.378083\pi$
$179$	10.0000	0.747435	0.373718	+	0.927543i	$0.378083\pi$
$180$	0	0
$181$	2.65685	0.197482	0.0987412	−	0.995113i	$-0.468518\pi$
$181$	2.65685	0.197482	0.0987412	+	0.995113i	$0.468518\pi$
$182$	0	0
$183$	− 2.27208i	− 0.167957i
$184$	−1.82843	−0.134793
$185$	0	0
$186$	6.00000	0.439941
$187$	4.00000i	0.292509i
$188$	7.65685i	0.558433i
$189$	0	0
$190$	0	0
$191$	−12.8284	−0.928232	−0.464116	−	0.885774i	$-0.653628\pi$
$191$	−12.8284	−0.928232	−0.464116	+	0.885774i	$0.653628\pi$
$192$	− 4.07107i	− 0.293804i
$193$	− 2.00000i	− 0.143963i	−0.997406	−	0.0719816i	$-0.977068\pi$
$193$	− 2.00000i	− 0.143963i	0.997406	−	0.0719816i	$-0.0229323\pi$
$194$	−28.1421	−2.02049
$195$	0	0
$196$	0	0
$197$	12.3431i	0.879413i	0.898142	+	0.439706i	$0.144917\pi$
$197$	12.3431i	0.879413i	−0.898142	+	0.439706i	$0.855083\pi$
$198$	5.65685i	0.402015i
$199$	9.65685	0.684556	0.342278	−	0.939599i	$-0.388801\pi$
$199$	9.65685	0.684556	0.342278	+	0.939599i	$0.388801\pi$
$200$	0	0
$201$	−3.97056	−0.280062
$202$	24.8995i	1.75192i
$203$	0	0
$204$	7.65685	0.536087
$205$	0	0
$206$	−5.82843	−0.406086
$207$	1.17157i	0.0814299i
$208$	− 14.4853i	− 1.00437i
$209$	2.34315	0.162079
$210$	0	0
$211$	20.4853	1.41026	0.705132	−	0.709076i	$-0.250888\pi$
$211$	20.4853	1.41026	0.705132	+	0.709076i	$0.250888\pi$
$212$	4.48528i	0.308050i
$213$	− 1.85786i	− 0.127299i
$214$	−27.1421	−1.85540
$215$	0	0
$216$	10.6569	0.725107
$217$	0	0
$218$	− 32.5563i	− 2.20499i
$219$	−0.343146	−0.0231876
$220$	0	0
$221$	−23.3137	−1.56825
$222$	0	0
$223$	0.343146i	0.0229787i	0.999934	+	0.0114894i	$0.00365726\pi$
$223$	0.343146i	0.0229787i	−0.999934	+	0.0114894i	$0.996343\pi$
$224$	0	0
$225$	0	0
$226$	−10.8284	−0.720296
$227$	6.97056i	0.462652i	0.972876	+	0.231326i	$0.0743065\pi$
$227$	6.97056i	0.462652i	−0.972876	+	0.231326i	$0.925694\pi$
$228$	− 4.48528i	− 0.297045i
$229$	11.6569	0.770307	0.385153	−	0.922853i	$-0.374149\pi$
$229$	11.6569	0.770307	0.385153	+	0.922853i	$0.374149\pi$
$230$	0	0
$231$	0	0
$232$	4.41421i	0.289807i
$233$	− 16.8284i	− 1.10247i	−0.834351	−	0.551233i	$-0.814157\pi$
$233$	− 16.8284i	− 1.10247i	0.834351	−	0.551233i	$-0.185843\pi$
$234$	−32.9706	−2.15535
$235$	0	0
$236$	17.1716	1.11777
$237$	6.14214i	0.398975i
$238$	0	0
$239$	21.3137	1.37867	0.689335	−	0.724443i	$-0.257903\pi$
$239$	21.3137	1.37867	0.689335	+	0.724443i	$0.257903\pi$
$240$	0	0
$241$	27.6569	1.78153	0.890767	−	0.454460i	$-0.150168\pi$
$241$	27.6569	1.78153	0.890767	+	0.454460i	$0.150168\pi$
$242$	24.8995i	1.60060i
$243$	− 10.3431i	− 0.663513i
$244$	−21.0000	−1.34439
$245$	0	0
$246$	7.82843	0.499122
$247$	13.6569i	0.868965i
$248$	− 26.4853i	− 1.68182i
$249$	5.68629	0.360354
$250$	0	0
$251$	−9.31371	−0.587876	−0.293938	−	0.955824i	$-0.594966\pi$
$251$	−9.31371	−0.587876	−0.293938	+	0.955824i	$0.594966\pi$
$252$	0	0
$253$	− 0.343146i	− 0.0215734i
$254$	22.4853	1.41085
$255$	0	0
$256$	−29.9706	−1.87316
$257$	− 6.34315i	− 0.395675i	−0.980235	−	0.197837i	$-0.936608\pi$
$257$	− 6.34315i	− 0.395675i	0.980235	−	0.197837i	$-0.0633918\pi$
$258$	− 3.58579i	− 0.223241i
$259$	0	0
$260$	0	0
$261$	2.82843	0.175075
$262$	46.6274i	2.88065i
$263$	− 29.0416i	− 1.79078i	−0.445279	−	0.895392i	$-0.646896\pi$
$263$	− 29.0416i	− 1.79078i	0.445279	−	0.895392i	$-0.353104\pi$
$264$	−1.51472	−0.0932245
$265$	0	0
$266$	0	0
$267$	− 3.58579i	− 0.219447i
$268$	36.6985i	2.24172i
$269$	−20.4558	−1.24721	−0.623607	−	0.781738i	$-0.714334\pi$
$269$	−20.4558	−1.24721	−0.623607	+	0.781738i	$0.714334\pi$
$270$	0	0
$271$	16.4853	1.00141	0.500705	−	0.865618i	$-0.333074\pi$
$271$	16.4853	1.00141	0.500705	+	0.865618i	$0.333074\pi$
$272$	− 14.4853i	− 0.878299i
$273$	0	0
$274$	23.3137	1.40843
$275$	0	0
$276$	−0.656854	−0.0395380
$277$	− 16.1421i	− 0.969887i	−0.874546	−	0.484943i	$-0.838840\pi$
$277$	− 16.1421i	− 0.969887i	0.874546	−	0.484943i	$-0.161160\pi$
$278$	− 38.9706i	− 2.33730i
$279$	−16.9706	−1.01600
$280$	0	0
$281$	−30.2843	−1.80661	−0.903304	−	0.429001i	$-0.858866\pi$
$281$	−30.2843	−1.80661	−0.903304	+	0.429001i	$0.858866\pi$
$282$	2.00000i	0.119098i
$283$	14.0000i	0.832214i	0.909316	+	0.416107i	$0.136606\pi$
$283$	14.0000i	0.832214i	−0.909316	+	0.416107i	$0.863394\pi$
$284$	−17.1716	−1.01895
$285$	0	0
$286$	9.65685	0.571022
$287$	0	0
$288$	4.48528i	0.264298i
$289$	−6.31371	−0.371395
$290$	0	0
$291$	−4.82843	−0.283047
$292$	3.17157i	0.185602i
$293$	− 16.0000i	− 0.934730i	−0.884064	−	0.467365i	$-0.845203\pi$
$293$	− 16.0000i	− 0.934730i	0.884064	−	0.467365i	$-0.154797\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	2.00000i	0.116052i
$298$	− 5.24264i	− 0.303698i
$299$	2.00000	0.115663
$300$	0	0
$301$	0	0
$302$	− 28.1421i	− 1.61940i
$303$	4.27208i	0.245424i
$304$	−8.48528	−0.486664
$305$	0	0
$306$	−32.9706	−1.88480
$307$	4.75736i	0.271517i	0.990742	+	0.135758i	$0.0433471\pi$
$307$	4.75736i	0.271517i	−0.990742	+	0.135758i	$0.956653\pi$
$308$	0	0
$309$	−1.00000	−0.0568880
$310$	0	0
$311$	−13.1716	−0.746891	−0.373446	−	0.927652i	$-0.621824\pi$
$311$	−13.1716	−0.746891	−0.373446	+	0.927652i	$0.621824\pi$
$312$	− 8.82843i	− 0.499811i
$313$	− 6.34315i	− 0.358536i	−0.983800	−	0.179268i	$-0.942627\pi$
$313$	− 6.34315i	− 0.358536i	0.983800	−	0.179268i	$-0.0573729\pi$
$314$	−41.7990	−2.35885
$315$	0	0
$316$	56.7696	3.19354
$317$	13.7990i	0.775028i	0.921864	+	0.387514i	$0.126666\pi$
$317$	13.7990i	0.775028i	−0.921864	+	0.387514i	$0.873334\pi$
$318$	1.17157i	0.0656985i
$319$	−0.828427	−0.0463830
$320$	0	0
$321$	−4.65685	−0.259920
$322$	0	0
$323$	13.6569i	0.759888i
$324$	−28.6569	−1.59205
$325$	0	0
$326$	29.7990	1.65041
$327$	− 5.58579i	− 0.308895i
$328$	− 34.5563i	− 1.90806i
$329$	0	0
$330$	0	0
$331$	22.9706	1.26258	0.631288	−	0.775548i	$-0.282527\pi$
$331$	22.9706	1.26258	0.631288	+	0.775548i	$0.282527\pi$
$332$	− 52.5563i	− 2.88440i
$333$	0	0
$334$	−54.1127	−2.96092
$335$	0	0
$336$	0	0
$337$	− 9.17157i	− 0.499607i	−0.968296	−	0.249804i	$-0.919634\pi$
$337$	− 9.17157i	− 0.499607i	0.968296	−	0.249804i	$-0.0803661\pi$
$338$	24.8995i	1.35435i
$339$	−1.85786	−0.100905
$340$	0	0
$341$	4.97056	0.269171
$342$	19.3137i	1.04437i
$343$	0	0
$344$	−15.8284	−0.853412
$345$	0	0
$346$	8.00000	0.430083
$347$	− 7.92893i	− 0.425647i	−0.977091	−	0.212824i	$-0.931734\pi$
$347$	− 7.92893i	− 0.425647i	0.977091	−	0.212824i	$-0.0682660\pi$
$348$	1.58579i	0.0850071i
$349$	15.3431	0.821300	0.410650	−	0.911793i	$-0.365302\pi$
$349$	15.3431	0.821300	0.410650	+	0.911793i	$0.365302\pi$
$350$	0	0
$351$	−11.6569	−0.622197
$352$	− 1.31371i	− 0.0700209i
$353$	− 26.8284i	− 1.42793i	−0.700180	−	0.713967i	$-0.746897\pi$
$353$	− 26.8284i	− 1.42793i	0.700180	−	0.713967i	$-0.253103\pi$
$354$	4.48528	0.238390
$355$	0	0
$356$	−33.1421	−1.75653
$357$	0	0
$358$	− 24.1421i	− 1.27595i
$359$	10.0000	0.527780	0.263890	−	0.964553i	$-0.414994\pi$
$359$	10.0000	0.527780	0.263890	+	0.964553i	$0.414994\pi$
$360$	0	0
$361$	−11.0000	−0.578947
$362$	− 6.41421i	− 0.337124i
$363$	4.27208i	0.224226i
$364$	0	0
$365$	0	0
$366$	−5.48528	−0.286720
$367$	2.75736i	0.143933i	0.997407	+	0.0719665i	$0.0229275\pi$
$367$	2.75736i	0.143933i	−0.997407	+	0.0719665i	$0.977073\pi$
$368$	1.24264i	0.0647771i
$369$	−22.1421	−1.15267
$370$	0	0
$371$	0	0
$372$	− 9.51472i	− 0.493315i
$373$	− 20.9706i	− 1.08581i	−0.839793	−	0.542907i	$-0.817323\pi$
$373$	− 20.9706i	− 1.08581i	0.839793	−	0.542907i	$-0.182677\pi$
$374$	9.65685	0.499344
$375$	0	0
$376$	8.82843	0.455291
$377$	− 4.82843i	− 0.248677i
$378$	0	0
$379$	−26.8284	−1.37808	−0.689042	−	0.724722i	$-0.741968\pi$
$379$	−26.8284	−1.37808	−0.689042	+	0.724722i	$0.741968\pi$
$380$	0	0
$381$	3.85786	0.197644
$382$	30.9706i	1.58459i
$383$	− 2.89949i	− 0.148157i	−0.997252	−	0.0740786i	$-0.976398\pi$
$383$	− 2.89949i	− 0.148157i	0.997252	−	0.0740786i	$-0.0236016\pi$
$384$	−8.51472	−0.434515
$385$	0	0
$386$	−4.82843	−0.245760
$387$	10.1421i	0.515554i
$388$	44.6274i	2.26561i
$389$	−23.6569	−1.19945	−0.599725	−	0.800206i	$-0.704723\pi$
$389$	−23.6569	−1.19945	−0.599725	+	0.800206i	$0.704723\pi$
$390$	0	0
$391$	2.00000	0.101144
$392$	0	0
$393$	8.00000i	0.403547i
$394$	29.7990	1.50125
$395$	0	0
$396$	8.97056	0.450788
$397$	16.6274i	0.834506i	0.908790	+	0.417253i	$0.137007\pi$
$397$	16.6274i	0.834506i	−0.908790	+	0.417253i	$0.862993\pi$
$398$	− 23.3137i	− 1.16861i
$399$	0	0
$400$	0	0
$401$	30.3137	1.51379	0.756897	−	0.653534i	$-0.226714\pi$
$401$	30.3137	1.51379	0.756897	+	0.653534i	$0.226714\pi$
$402$	9.58579i	0.478096i
$403$	28.9706i	1.44313i
$404$	39.4853	1.96447
$405$	0	0
$406$	0	0
$407$	0	0
$408$	− 8.82843i	− 0.437072i
$409$	14.7990	0.731763	0.365881	−	0.930661i	$-0.380768\pi$
$409$	14.7990	0.731763	0.365881	+	0.930661i	$0.380768\pi$
$410$	0	0
$411$	4.00000	0.197305
$412$	9.24264i	0.455352i
$413$	0	0
$414$	2.82843	0.139010
$415$	0	0
$416$	7.65685	0.375408
$417$	− 6.68629i	− 0.327429i
$418$	− 5.65685i	− 0.276686i
$419$	0.686292	0.0335275	0.0167638	−	0.999859i	$-0.494664\pi$
$419$	0.686292	0.0335275	0.0167638	+	0.999859i	$0.494664\pi$
$420$	0	0
$421$	13.4853	0.657232	0.328616	−	0.944464i	$-0.393418\pi$
$421$	13.4853	0.657232	0.328616	+	0.944464i	$0.393418\pi$
$422$	− 49.4558i	− 2.40747i
$423$	− 5.65685i	− 0.275046i
$424$	5.17157	0.251154
$425$	0	0
$426$	−4.48528	−0.217313
$427$	0	0
$428$	43.0416i	2.08050i
$429$	1.65685	0.0799937
$430$	0	0
$431$	17.7990	0.857347	0.428674	−	0.903459i	$-0.358981\pi$
$431$	17.7990	0.857347	0.428674	+	0.903459i	$0.358981\pi$
$432$	− 7.24264i	− 0.348462i
$433$	7.79899i	0.374796i	0.982284	+	0.187398i	$0.0600053\pi$
$433$	7.79899i	0.374796i	−0.982284	+	0.187398i	$0.939995\pi$
$434$	0	0
$435$	0	0
$436$	−51.6274	−2.47250
$437$	− 1.17157i	− 0.0560439i
$438$	0.828427i	0.0395838i
$439$	33.9411	1.61992	0.809961	−	0.586484i	$-0.199488\pi$
$439$	33.9411	1.61992	0.809961	+	0.586484i	$0.199488\pi$
$440$	0	0
$441$	0	0
$442$	56.2843i	2.67717i
$443$	30.2132i	1.43547i	0.696315	+	0.717736i	$0.254822\pi$
$443$	30.2132i	1.43547i	−0.696315	+	0.717736i	$0.745178\pi$
$444$	0	0
$445$	0	0
$446$	0.828427	0.0392272
$447$	− 0.899495i	− 0.0425447i
$448$	0	0
$449$	−3.82843	−0.180675	−0.0903373	−	0.995911i	$-0.528795\pi$
$449$	−3.82843	−0.180675	−0.0903373	+	0.995911i	$0.528795\pi$
$450$	0	0
$451$	6.48528	0.305380
$452$	17.1716i	0.807683i
$453$	− 4.82843i	− 0.226859i
$454$	16.8284	0.789797
$455$	0	0
$456$	−5.17157	−0.242181
$457$	− 24.2843i	− 1.13597i	−0.823039	−	0.567985i	$-0.807723\pi$
$457$	− 24.2843i	− 1.13597i	0.823039	−	0.567985i	$-0.192277\pi$
$458$	− 28.1421i	− 1.31500i
$459$	−11.6569	−0.544095
$460$	0	0
$461$	41.3137	1.92417	0.962086	−	0.272748i	$-0.0879324\pi$
$461$	41.3137	1.92417	0.962086	+	0.272748i	$0.0879324\pi$
$462$	0	0
$463$	− 37.0416i	− 1.72147i	−0.509053	−	0.860735i	$-0.670004\pi$
$463$	− 37.0416i	− 1.72147i	0.509053	−	0.860735i	$-0.329996\pi$
$464$	3.00000	0.139272
$465$	0	0
$466$	−40.6274	−1.88203
$467$	3.10051i	0.143474i	0.997424	+	0.0717371i	$0.0228543\pi$
$467$	3.10051i	0.143474i	−0.997424	+	0.0717371i	$0.977146\pi$
$468$	52.2843i	2.41684i
$469$	0	0
$470$	0	0
$471$	−7.17157	−0.330449
$472$	− 19.7990i	− 0.911322i
$473$	− 2.97056i	− 0.136587i
$474$	14.8284	0.681092
$475$	0	0
$476$	0	0
$477$	− 3.31371i	− 0.151724i
$478$	− 51.4558i	− 2.35354i
$479$	35.6569	1.62920	0.814602	−	0.580021i	$-0.196956\pi$
$479$	35.6569	1.62920	0.814602	+	0.580021i	$0.196956\pi$
$480$	0	0
$481$	0	0
$482$	− 66.7696i	− 3.04127i
$483$	0	0
$484$	39.4853	1.79479
$485$	0	0
$486$	−24.9706	−1.13269
$487$	− 4.34315i	− 0.196807i	−0.995147	−	0.0984034i	$-0.968626\pi$
$487$	− 4.34315i	− 0.196807i	0.995147	−	0.0984034i	$-0.0313735\pi$
$488$	24.2132i	1.09608i
$489$	5.11270	0.231204
$490$	0	0
$491$	9.31371	0.420322	0.210161	−	0.977667i	$-0.432601\pi$
$491$	9.31371	0.420322	0.210161	+	0.977667i	$0.432601\pi$
$492$	− 12.4142i	− 0.559676i
$493$	− 4.82843i	− 0.217461i
$494$	32.9706	1.48342
$495$	0	0
$496$	−18.0000	−0.808224
$497$	0	0
$498$	− 13.7279i	− 0.615163i
$499$	0.828427	0.0370855	0.0185427	−	0.999828i	$-0.494097\pi$
$499$	0.828427	0.0370855	0.0185427	+	0.999828i	$0.494097\pi$
$500$	0	0
$501$	−9.28427	−0.414791
$502$	22.4853i	1.00357i
$503$	− 15.8701i	− 0.707611i	−0.935319	−	0.353805i	$-0.884887\pi$
$503$	− 15.8701i	− 0.707611i	0.935319	−	0.353805i	$-0.115113\pi$
$504$	0	0
$505$	0	0
$506$	−0.828427	−0.0368281
$507$	4.27208i	0.189730i
$508$	− 35.6569i	− 1.58202i
$509$	−13.3431	−0.591425	−0.295712	−	0.955277i	$-0.595557\pi$
$509$	−13.3431	−0.591425	−0.295712	+	0.955277i	$0.595557\pi$
$510$	0	0
$511$	0	0
$512$	31.2426i	1.38074i
$513$	6.82843i	0.301482i
$514$	−15.3137	−0.675459
$515$	0	0
$516$	−5.68629	−0.250325
$517$	1.65685i	0.0728684i
$518$	0	0
$519$	1.37258	0.0602497
$520$	0	0
$521$	14.9706	0.655872	0.327936	−	0.944700i	$-0.393647\pi$
$521$	14.9706	0.655872	0.327936	+	0.944700i	$0.393647\pi$
$522$	− 6.82843i	− 0.298872i
$523$	35.6569i	1.55917i	0.626299	+	0.779583i	$0.284569\pi$
$523$	35.6569i	1.55917i	−0.626299	+	0.779583i	$0.715431\pi$
$524$	73.9411	3.23013
$525$	0	0
$526$	−70.1127	−3.05706
$527$	28.9706i	1.26198i
$528$	1.02944i	0.0448005i
$529$	22.8284	0.992540
$530$	0	0
$531$	−12.6863	−0.550538
$532$	0	0
$533$	37.7990i	1.63726i
$534$	−8.65685	−0.374619
$535$	0	0
$536$	42.3137	1.82767
$537$	− 4.14214i	− 0.178746i
$538$	49.3848i	2.12913i
$539$	0	0
$540$	0	0
$541$	7.34315	0.315706	0.157853	−	0.987463i	$-0.449543\pi$
$541$	7.34315	0.315706	0.157853	+	0.987463i	$0.449543\pi$
$542$	− 39.7990i	− 1.70951i
$543$	− 1.10051i	− 0.0472272i
$544$	7.65685	0.328285
$545$	0	0
$546$	0	0
$547$	− 24.8995i	− 1.06463i	−0.846548	−	0.532313i	$-0.821323\pi$
$547$	− 24.8995i	− 1.06463i	0.846548	−	0.532313i	$-0.178677\pi$
$548$	− 36.9706i	− 1.57930i
$549$	15.5147	0.662152
$550$	0	0
$551$	−2.82843	−0.120495
$552$	0.757359i	0.0322354i
$553$	0	0
$554$	−38.9706	−1.65570
$555$	0	0
$556$	−61.7990	−2.62086
$557$	− 22.2843i	− 0.944215i	−0.881541	−	0.472107i	$-0.843493\pi$
$557$	− 22.2843i	− 0.944215i	0.881541	−	0.472107i	$-0.156507\pi$
$558$	40.9706i	1.73442i
$559$	17.3137	0.732292
$560$	0	0
$561$	1.65685	0.0699524
$562$	73.1127i	3.08407i
$563$	41.7279i	1.75862i	0.476248	+	0.879311i	$0.341997\pi$
$563$	41.7279i	1.75862i	−0.476248	+	0.879311i	$0.658003\pi$
$564$	3.17157	0.133547
$565$	0	0
$566$	33.7990	1.42068
$567$	0	0
$568$	19.7990i	0.830747i
$569$	−7.65685	−0.320992	−0.160496	−	0.987036i	$-0.551309\pi$
$569$	−7.65685	−0.320992	−0.160496	+	0.987036i	$0.551309\pi$
$570$	0	0
$571$	9.17157	0.383818	0.191909	−	0.981413i	$-0.438532\pi$
$571$	9.17157	0.383818	0.191909	+	0.981413i	$0.438532\pi$
$572$	− 15.3137i	− 0.640298i
$573$	5.31371i	0.221983i
$574$	0	0
$575$	0	0
$576$	27.7990	1.15829
$577$	43.9411i	1.82929i	0.404255	+	0.914646i	$0.367531\pi$
$577$	43.9411i	1.82929i	−0.404255	+	0.914646i	$0.632469\pi$
$578$	15.2426i	0.634010i
$579$	−0.828427	−0.0344283
$580$	0	0
$581$	0	0
$582$	11.6569i	0.483192i
$583$	0.970563i	0.0401966i
$584$	3.65685	0.151322
$585$	0	0
$586$	−38.6274	−1.59568
$587$	34.2843i	1.41506i	0.706682	+	0.707532i	$0.250191\pi$
$587$	34.2843i	1.41506i	−0.706682	+	0.707532i	$0.749809\pi$
$588$	0	0
$589$	16.9706	0.699260
$590$	0	0
$591$	5.11270	0.210308
$592$	0	0
$593$	4.20101i	0.172515i	0.996273	+	0.0862574i	$0.0274907\pi$
$593$	4.20101i	0.172515i	−0.996273	+	0.0862574i	$0.972509\pi$
$594$	4.82843	0.198113
$595$	0	0
$596$	−8.31371	−0.340543
$597$	− 4.00000i	− 0.163709i
$598$	− 4.82843i	− 0.197449i
$599$	−6.34315	−0.259174	−0.129587	−	0.991568i	$-0.541365\pi$
$599$	−6.34315	−0.259174	−0.129587	+	0.991568i	$0.541365\pi$
$600$	0	0
$601$	19.6569	0.801820	0.400910	−	0.916117i	$-0.368694\pi$
$601$	19.6569	0.801820	0.400910	+	0.916117i	$0.368694\pi$
$602$	0	0
$603$	− 27.1127i	− 1.10411i
$604$	−44.6274	−1.81586
$605$	0	0
$606$	10.3137	0.418966
$607$	− 38.2132i	− 1.55103i	−0.631332	−	0.775513i	$-0.717491\pi$
$607$	− 38.2132i	− 1.55103i	0.631332	−	0.775513i	$-0.282509\pi$
$608$	− 4.48528i	− 0.181902i
$609$	0	0
$610$	0	0
$611$	−9.65685	−0.390675
$612$	52.2843i	2.11347i
$613$	35.4558i	1.43205i	0.698076	+	0.716024i	$0.254040\pi$
$613$	35.4558i	1.43205i	−0.698076	+	0.716024i	$0.745960\pi$
$614$	11.4853	0.463508
$615$	0	0
$616$	0	0
$617$	− 11.3137i	− 0.455473i	−0.973723	−	0.227736i	$-0.926868\pi$
$617$	− 11.3137i	− 0.455473i	0.973723	−	0.227736i	$-0.0731324\pi$
$618$	2.41421i	0.0971139i
$619$	−25.5147	−1.02552	−0.512762	−	0.858531i	$-0.671378\pi$
$619$	−25.5147	−1.02552	−0.512762	+	0.858531i	$0.671378\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	31.7990i	1.27502i
$623$	0	0
$624$	−6.00000	−0.240192
$625$	0	0
$626$	−15.3137	−0.612059
$627$	− 0.970563i	− 0.0387605i
$628$	66.2843i	2.64503i
$629$	0	0
$630$	0	0
$631$	−20.1421	−0.801846	−0.400923	−	0.916112i	$-0.631310\pi$
$631$	−20.1421	−0.801846	−0.400923	+	0.916112i	$0.631310\pi$
$632$	− 65.4558i	− 2.60369i
$633$	− 8.48528i	− 0.337260i
$634$	33.3137	1.32306
$635$	0	0
$636$	1.85786	0.0736691
$637$	0	0
$638$	2.00000i	0.0791808i
$639$	12.6863	0.501862
$640$	0	0
$641$	31.4853	1.24359	0.621797	−	0.783179i	$-0.286403\pi$
$641$	31.4853	1.24359	0.621797	+	0.783179i	$0.286403\pi$
$642$	11.2426i	0.443712i
$643$	26.2843i	1.03655i	0.855214	+	0.518275i	$0.173426\pi$
$643$	26.2843i	1.03655i	−0.855214	+	0.518275i	$0.826574\pi$
$644$	0	0
$645$	0	0
$646$	32.9706	1.29721
$647$	31.0416i	1.22037i	0.792258	+	0.610186i	$0.208905\pi$
$647$	31.0416i	1.22037i	−0.792258	+	0.610186i	$0.791095\pi$
$648$	33.0416i	1.29800i
$649$	3.71573	0.145855
$650$	0	0
$651$	0	0
$652$	− 47.2548i	− 1.85064i
$653$	− 19.1716i	− 0.750242i	−0.926976	−	0.375121i	$-0.877601\pi$
$653$	− 19.1716i	− 0.750242i	0.926976	−	0.375121i	$-0.122399\pi$
$654$	−13.4853	−0.527316
$655$	0	0
$656$	−23.4853	−0.916946
$657$	− 2.34315i	− 0.0914148i
$658$	0	0
$659$	−21.1716	−0.824727	−0.412364	−	0.911019i	$-0.635297\pi$
$659$	−21.1716	−0.824727	−0.412364	+	0.911019i	$0.635297\pi$
$660$	0	0
$661$	31.8284	1.23798	0.618991	−	0.785398i	$-0.287542\pi$
$661$	31.8284	1.23798	0.618991	+	0.785398i	$0.287542\pi$
$662$	− 55.4558i	− 2.15535i
$663$	9.65685i	0.375041i
$664$	−60.5980	−2.35166
$665$	0	0
$666$	0	0
$667$	0.414214i	0.0160384i
$668$	85.8112i	3.32013i
$669$	0.142136	0.00549528
$670$	0	0
$671$	−4.54416	−0.175425
$672$	0	0
$673$	29.6569i	1.14319i	0.820537	+	0.571594i	$0.193675\pi$
$673$	29.6569i	1.14319i	−0.820537	+	0.571594i	$0.806325\pi$
$674$	−22.1421	−0.852883
$675$	0	0
$676$	39.4853	1.51866
$677$	28.1421i	1.08159i	0.841154	+	0.540795i	$0.181877\pi$
$677$	28.1421i	1.08159i	−0.841154	+	0.540795i	$0.818123\pi$
$678$	4.48528i	0.172256i
$679$	0	0
$680$	0	0
$681$	2.88730	0.110642
$682$	− 12.0000i	− 0.459504i
$683$	− 34.7574i	− 1.32995i	−0.746864	−	0.664977i	$-0.768441\pi$
$683$	− 34.7574i	− 1.32995i	0.746864	−	0.664977i	$-0.231559\pi$
$684$	30.6274	1.17107
$685$	0	0
$686$	0	0
$687$	− 4.82843i	− 0.184216i
$688$	10.7574i	0.410120i
$689$	−5.65685	−0.215509
$690$	0	0
$691$	0.828427	0.0315149	0.0157574	−	0.999876i	$-0.494984\pi$
$691$	0.828427	0.0315149	0.0157574	+	0.999876i	$0.494984\pi$
$692$	− 12.6863i	− 0.482260i
$693$	0	0
$694$	−19.1421	−0.726626
$695$	0	0
$696$	1.82843	0.0693064
$697$	37.7990i	1.43174i
$698$	− 37.0416i	− 1.40205i
$699$	−6.97056	−0.263651
$700$	0	0
$701$	−3.20101	−0.120900	−0.0604502	−	0.998171i	$-0.519254\pi$
$701$	−3.20101	−0.120900	−0.0604502	+	0.998171i	$0.519254\pi$
$702$	28.1421i	1.06216i
$703$	0	0
$704$	−8.14214	−0.306868
$705$	0	0
$706$	−64.7696	−2.43763
$707$	0	0
$708$	− 7.11270i	− 0.267312i
$709$	−15.6863	−0.589111	−0.294556	−	0.955634i	$-0.595172\pi$
$709$	−15.6863	−0.589111	−0.294556	+	0.955634i	$0.595172\pi$
$710$	0	0
$711$	−41.9411	−1.57292
$712$	38.2132i	1.43210i
$713$	− 2.48528i	− 0.0930745i
$714$	0	0
$715$	0	0
$716$	−38.2843	−1.43075
$717$	− 8.82843i	− 0.329704i
$718$	− 24.1421i	− 0.900976i
$719$	−21.1127	−0.787371	−0.393685	−	0.919245i	$-0.628800\pi$
$719$	−21.1127	−0.787371	−0.393685	+	0.919245i	$0.628800\pi$
$720$	0	0
$721$	0	0
$722$	26.5563i	0.988325i
$723$	− 11.4558i	− 0.426047i
$724$	−10.1716	−0.378024
$725$	0	0
$726$	10.3137	0.382778
$727$	37.5858i	1.39398i	0.717081	+	0.696990i	$0.245478\pi$
$727$	37.5858i	1.39398i	−0.717081	+	0.696990i	$0.754522\pi$
$728$	0	0
$729$	18.1716	0.673021
$730$	0	0
$731$	17.3137	0.640371
$732$	8.69848i	0.321505i
$733$	− 22.0000i	− 0.812589i	−0.913742	−	0.406294i	$-0.866821\pi$
$733$	− 22.0000i	− 0.812589i	0.913742	−	0.406294i	$-0.133179\pi$
$734$	6.65685	0.245709
$735$	0	0
$736$	−0.656854	−0.0242120
$737$	7.94113i	0.292515i
$738$	53.4558i	1.96774i
$739$	21.1127	0.776643	0.388322	−	0.921524i	$-0.373055\pi$
$739$	21.1127	0.776643	0.388322	+	0.921524i	$0.373055\pi$
$740$	0	0
$741$	5.65685	0.207810
$742$	0	0
$743$	− 16.0711i	− 0.589590i	−0.955560	−	0.294795i	$-0.904749\pi$
$743$	− 16.0711i	− 0.589590i	0.955560	−	0.294795i	$-0.0952514\pi$
$744$	−10.9706	−0.402200
$745$	0	0
$746$	−50.6274	−1.85360
$747$	38.8284i	1.42066i
$748$	− 15.3137i	− 0.559925i
$749$	0	0
$750$	0	0
$751$	−30.3431	−1.10724	−0.553619	−	0.832770i	$-0.686753\pi$
$751$	−30.3431	−1.10724	−0.553619	+	0.832770i	$0.686753\pi$
$752$	− 6.00000i	− 0.218797i
$753$	3.85786i	0.140588i
$754$	−11.6569	−0.424518
$755$	0	0
$756$	0	0
$757$	31.4558i	1.14328i	0.820504	+	0.571641i	$0.193693\pi$
$757$	31.4558i	1.14328i	−0.820504	+	0.571641i	$0.806307\pi$
$758$	64.7696i	2.35254i
$759$	−0.142136	−0.00515920
$760$	0	0
$761$	9.31371	0.337622	0.168811	−	0.985648i	$-0.446007\pi$
$761$	9.31371	0.337622	0.168811	+	0.985648i	$0.446007\pi$
$762$	− 9.31371i	− 0.337400i
$763$	0	0
$764$	49.1127	1.77684
$765$	0	0
$766$	−7.00000	−0.252920
$767$	21.6569i	0.781984i
$768$	12.4142i	0.447959i
$769$	0.627417	0.0226252	0.0113126	−	0.999936i	$-0.496399\pi$
$769$	0.627417	0.0226252	0.0113126	+	0.999936i	$0.496399\pi$
$770$	0	0
$771$	−2.62742	−0.0946241
$772$	7.65685i	0.275576i
$773$	− 37.1127i	− 1.33485i	−0.744677	−	0.667425i	$-0.767396\pi$
$773$	− 37.1127i	− 1.33485i	0.744677	−	0.667425i	$-0.232604\pi$
$774$	24.4853	0.880105
$775$	0	0
$776$	51.4558	1.84716
$777$	0	0
$778$	57.1127i	2.04759i
$779$	22.1421	0.793324
$780$	0	0
$781$	−3.71573	−0.132959
$782$	− 4.82843i	− 0.172664i
$783$	− 2.41421i	− 0.0862770i
$784$	0	0
$785$	0	0
$786$	19.3137	0.688897
$787$	− 2.55635i	− 0.0911240i	−0.998962	−	0.0455620i	$-0.985492\pi$
$787$	− 2.55635i	− 0.0911240i	0.998962	−	0.0455620i	$-0.0145079\pi$
$788$	− 47.2548i	− 1.68338i
$789$	−12.0294	−0.428259
$790$	0	0
$791$	0	0
$792$	− 10.3431i	− 0.367528i
$793$	− 26.4853i	− 0.940520i
$794$	40.1421	1.42459
$795$	0	0
$796$	−36.9706	−1.31039
$797$	8.00000i	0.283375i	0.989911	+	0.141687i	$0.0452527\pi$
$797$	8.00000i	0.283375i	−0.989911	+	0.141687i	$0.954747\pi$
$798$	0	0
$799$	−9.65685	−0.341635
$800$	0	0
$801$	24.4853	0.865145
$802$	− 73.1838i	− 2.58421i
$803$	0.686292i	0.0242187i
$804$	15.2010	0.536098
$805$	0	0
$806$	69.9411	2.46357
$807$	8.47309i	0.298267i
$808$	− 45.5269i	− 1.60163i
$809$	−35.6274	−1.25259	−0.626297	−	0.779585i	$-0.715430\pi$
$809$	−35.6274	−1.25259	−0.626297	+	0.779585i	$0.715430\pi$
$810$	0	0
$811$	−20.6274	−0.724327	−0.362163	−	0.932115i	$-0.617962\pi$
$811$	−20.6274	−0.724327	−0.362163	+	0.932115i	$0.617962\pi$
$812$	0	0
$813$	− 6.82843i	− 0.239483i
$814$	0	0
$815$	0	0
$816$	−6.00000	−0.210042
$817$	− 10.1421i	− 0.354828i
$818$	− 35.7279i	− 1.24920i
$819$	0	0
$820$	0	0
$821$	47.9411	1.67316	0.836578	−	0.547847i	$-0.184552\pi$
$821$	47.9411	1.67316	0.836578	+	0.547847i	$0.184552\pi$
$822$	− 9.65685i	− 0.336821i
$823$	2.07107i	0.0721929i	0.999348	+	0.0360964i	$0.0114924\pi$
$823$	2.07107i	0.0721929i	−0.999348	+	0.0360964i	$0.988508\pi$
$824$	10.6569	0.371249
$825$	0	0
$826$	0	0
$827$	26.2132i	0.911522i	0.890102	+	0.455761i	$0.150633\pi$
$827$	26.2132i	0.911522i	−0.890102	+	0.455761i	$0.849367\pi$
$828$	− 4.48528i	− 0.155874i
$829$	−29.3137	−1.01811	−0.509054	−	0.860735i	$-0.670005\pi$
$829$	−29.3137	−1.01811	−0.509054	+	0.860735i	$0.670005\pi$
$830$	0	0
$831$	−6.68629	−0.231945
$832$	− 47.4558i	− 1.64524i
$833$	0	0
$834$	−16.1421	−0.558956
$835$	0	0
$836$	−8.97056	−0.310253
$837$	14.4853i	0.500685i
$838$	− 1.65685i	− 0.0572351i
$839$	−15.1716	−0.523781	−0.261890	−	0.965098i	$-0.584346\pi$
$839$	−15.1716	−0.523781	−0.261890	+	0.965098i	$0.584346\pi$
$840$	0	0
$841$	−28.0000	−0.965517
$842$	− 32.5563i	− 1.12197i
$843$	12.5442i	0.432044i
$844$	−78.4264	−2.69955
$845$	0	0
$846$	−13.6569	−0.469532
$847$	0	0
$848$	− 3.51472i	− 0.120696i
$849$	5.79899	0.199021
$850$	0	0
$851$	0	0
$852$	7.11270i	0.243677i
$853$	2.54416i	0.0871102i	0.999051	+	0.0435551i	$0.0138684\pi$
$853$	2.54416i	0.0871102i	−0.999051	+	0.0435551i	$0.986132\pi$
$854$	0	0
$855$	0	0
$856$	49.6274	1.69623
$857$	− 34.2843i	− 1.17113i	−0.810626	−	0.585564i	$-0.800873\pi$
$857$	− 34.2843i	− 1.17113i	0.810626	−	0.585564i	$-0.199127\pi$
$858$	− 4.00000i	− 0.136558i
$859$	−1.37258	−0.0468319	−0.0234160	−	0.999726i	$-0.507454\pi$
$859$	−1.37258	−0.0468319	−0.0234160	+	0.999726i	$0.507454\pi$
$860$	0	0
$861$	0	0
$862$	− 42.9706i	− 1.46358i
$863$	14.5563i	0.495504i	0.968823	+	0.247752i	$0.0796918\pi$
$863$	14.5563i	0.495504i	−0.968823	+	0.247752i	$0.920308\pi$
$864$	3.82843	0.130246
$865$	0	0
$866$	18.8284	0.639816
$867$	2.61522i	0.0888177i
$868$	0	0
$869$	12.2843	0.416715
$870$	0	0
$871$	−46.2843	−1.56828
$872$	59.5269i	2.01584i
$873$	− 32.9706i	− 1.11588i
$874$	−2.82843	−0.0956730
$875$	0	0
$876$	1.31371	0.0443861
$877$	− 25.1716i	− 0.849984i	−0.905197	−	0.424992i	$-0.860277\pi$
$877$	− 25.1716i	− 0.849984i	0.905197	−	0.424992i	$-0.139723\pi$
$878$	− 81.9411i	− 2.76538i
$879$	−6.62742	−0.223537
$880$	0	0
$881$	1.82843	0.0616013	0.0308006	−	0.999526i	$-0.490194\pi$
$881$	1.82843	0.0616013	0.0308006	+	0.999526i	$0.490194\pi$
$882$	0	0
$883$	18.2843i	0.615315i	0.951497	+	0.307657i	$0.0995450\pi$
$883$	18.2843i	0.615315i	−0.951497	+	0.307657i	$0.900455\pi$
$884$	89.2548	3.00196
$885$	0	0
$886$	72.9411	2.45051
$887$	− 29.9289i	− 1.00492i	−0.864602	−	0.502458i	$-0.832429\pi$
$887$	− 29.9289i	− 1.00492i	0.864602	−	0.502458i	$-0.167571\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−6.20101	−0.207742
$892$	− 1.31371i	− 0.0439862i
$893$	5.65685i	0.189299i
$894$	−2.17157	−0.0726283
$895$	0	0
$896$	0	0
$897$	− 0.828427i	− 0.0276604i
$898$	9.24264i	0.308431i
$899$	−6.00000	−0.200111
$900$	0	0
$901$	−5.65685	−0.188457
$902$	− 15.6569i	− 0.521316i
$903$	0	0
$904$	19.7990	0.658505
$905$	0	0
$906$	−11.6569	−0.387273
$907$	14.2132i	0.471942i	0.971760	+	0.235971i	$0.0758270\pi$
$907$	14.2132i	0.471942i	−0.971760	+	0.235971i	$0.924173\pi$
$908$	− 26.6863i	− 0.885616i
$909$	−29.1716	−0.967560
$910$	0	0
$911$	−10.2010	−0.337975	−0.168987	−	0.985618i	$-0.554050\pi$
$911$	−10.2010	−0.337975	−0.168987	+	0.985618i	$0.554050\pi$
$912$	3.51472i	0.116384i
$913$	− 11.3726i	− 0.376378i
$914$	−58.6274	−1.93922
$915$	0	0
$916$	−44.6274	−1.47453
$917$	0	0
$918$	28.1421i	0.928829i
$919$	43.1127	1.42216	0.711078	−	0.703113i	$-0.248207\pi$
$919$	43.1127	1.42216	0.711078	+	0.703113i	$0.248207\pi$
$920$	0	0
$921$	1.97056	0.0649323
$922$	− 99.7401i	− 3.28477i
$923$	− 21.6569i	− 0.712844i
$924$	0	0
$925$	0	0
$926$	−89.4264	−2.93873
$927$	− 6.82843i	− 0.224275i
$928$	1.58579i	0.0520560i
$929$	−5.48528	−0.179966	−0.0899831	−	0.995943i	$-0.528681\pi$
$929$	−5.48528	−0.179966	−0.0899831	+	0.995943i	$0.528681\pi$
$930$	0	0
$931$	0	0
$932$	64.4264i	2.11036i
$933$	5.45584i	0.178616i
$934$	7.48528	0.244926
$935$	0	0
$936$	60.2843	1.97045
$937$	− 34.6274i	− 1.13123i	−0.824670	−	0.565614i	$-0.808639\pi$
$937$	− 34.6274i	− 1.13123i	0.824670	−	0.565614i	$-0.191361\pi$
$938$	0	0
$939$	−2.62742	−0.0857425
$940$	0	0
$941$	−46.2843	−1.50882	−0.754412	−	0.656401i	$-0.772078\pi$
$941$	−46.2843	−1.50882	−0.754412	+	0.656401i	$0.772078\pi$
$942$	17.3137i	0.564111i
$943$	− 3.24264i	− 0.105595i
$944$	−13.4558	−0.437950
$945$	0	0
$946$	−7.17157	−0.233168
$947$	− 33.1838i	− 1.07833i	−0.842201	−	0.539164i	$-0.818740\pi$
$947$	− 33.1838i	− 1.07833i	0.842201	−	0.539164i	$-0.181260\pi$
$948$	− 23.5147i	− 0.763723i
$949$	−4.00000	−0.129845
$950$	0	0
$951$	5.71573	0.185345
$952$	0	0
$953$	− 13.6569i	− 0.442389i	−0.975230	−	0.221194i	$-0.929004\pi$
$953$	− 13.6569i	− 0.442389i	0.975230	−	0.221194i	$-0.0709955\pi$
$954$	−8.00000	−0.259010
$955$	0	0
$956$	−81.5980	−2.63907
$957$	0.343146i	0.0110923i
$958$	− 86.0833i	− 2.78122i
$959$	0	0
$960$	0	0
$961$	5.00000	0.161290
$962$	0	0
$963$	− 31.7990i	− 1.02471i
$964$	−105.882	−3.41024
$965$	0	0
$966$	0	0
$967$	− 37.5269i	− 1.20678i	−0.797445	−	0.603392i	$-0.793815\pi$
$967$	− 37.5269i	− 1.20678i	0.797445	−	0.603392i	$-0.206185\pi$
$968$	− 45.5269i	− 1.46329i
$969$	5.65685	0.181724
$970$	0	0
$971$	−24.0000	−0.770197	−0.385098	−	0.922876i	$-0.625832\pi$
$971$	−24.0000	−0.770197	−0.385098	+	0.922876i	$0.625832\pi$
$972$	39.5980i	1.27011i
$973$	0	0
$974$	−10.4853	−0.335970
$975$	0	0
$976$	16.4558	0.526739
$977$	1.31371i	0.0420293i	0.999779	+	0.0210146i	$0.00668966\pi$
$977$	1.31371i	0.0420293i	−0.999779	+	0.0210146i	$0.993310\pi$
$978$	− 12.3431i	− 0.394690i
$979$	−7.17157	−0.229204
$980$	0	0
$981$	38.1421	1.21778
$982$	− 22.4853i	− 0.717534i
$983$	28.2132i	0.899861i	0.893063	+	0.449931i	$0.148551\pi$
$983$	28.2132i	0.899861i	−0.893063	+	0.449931i	$0.851449\pi$
$984$	−14.3137	−0.456304
$985$	0	0
$986$	−11.6569	−0.371230
$987$	0	0
$988$	− 52.2843i	− 1.66338i
$989$	−1.48528	−0.0472292
$990$	0	0
$991$	−4.34315	−0.137965	−0.0689823	−	0.997618i	$-0.521975\pi$
$991$	−4.34315	−0.137965	−0.0689823	+	0.997618i	$0.521975\pi$
$992$	− 9.51472i	− 0.302093i
$993$	− 9.51472i	− 0.301940i
$994$	0	0
$995$	0	0
$996$	−21.7696	−0.689795
$997$	− 33.4558i	− 1.05956i	−0.848136	−	0.529779i	$-0.822275\pi$
$997$	− 33.4558i	− 1.05956i	0.848136	−	0.529779i	$-0.177725\pi$
$998$	− 2.00000i	− 0.0633089i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1225.2.b.g.99.1		4
5.2	odd	4		245.2.a.h.1.2		2
5.3	odd	4		1225.2.a.k.1.1		2
5.4	even	2	inner	1225.2.b.g.99.4		4
7.2	even	3		175.2.k.a.74.1		8
7.4	even	3		175.2.k.a.149.4		8
7.6	odd	2		1225.2.b.h.99.1		4
15.2	even	4		2205.2.a.n.1.1		2
20.7	even	4		3920.2.a.bq.1.2		2
35.2	odd	12		35.2.e.a.11.1	✓	4
35.4	even	6		175.2.k.a.149.1		8
35.9	even	6		175.2.k.a.74.4		8
35.12	even	12		245.2.e.e.116.1		4
35.13	even	4		1225.2.a.m.1.1		2
35.17	even	12		245.2.e.e.226.1		4
35.18	odd	12		175.2.e.c.51.2		4
35.23	odd	12		175.2.e.c.151.2		4
35.27	even	4		245.2.a.g.1.2		2
35.32	odd	12		35.2.e.a.16.1	yes	4
35.34	odd	2		1225.2.b.h.99.4		4
105.2	even	12		315.2.j.e.46.2		4
105.32	even	12		315.2.j.e.226.2		4
105.62	odd	4		2205.2.a.q.1.1		2
140.27	odd	4		3920.2.a.bv.1.1		2
140.67	even	12		560.2.q.k.401.1		4
140.107	even	12		560.2.q.k.81.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.2.e.a.11.1	✓	4	35.2	odd	12
35.2.e.a.16.1	yes	4	35.32	odd	12
175.2.e.c.51.2		4	35.18	odd	12
175.2.e.c.151.2		4	35.23	odd	12
175.2.k.a.74.1		8	7.2	even	3
175.2.k.a.74.4		8	35.9	even	6
175.2.k.a.149.1		8	35.4	even	6
175.2.k.a.149.4		8	7.4	even	3
245.2.a.g.1.2		2	35.27	even	4
245.2.a.h.1.2		2	5.2	odd	4
245.2.e.e.116.1		4	35.12	even	12
245.2.e.e.226.1		4	35.17	even	12
315.2.j.e.46.2		4	105.2	even	12
315.2.j.e.226.2		4	105.32	even	12
560.2.q.k.81.1		4	140.107	even	12
560.2.q.k.401.1		4	140.67	even	12
1225.2.a.k.1.1		2	5.3	odd	4
1225.2.a.m.1.1		2	35.13	even	4
1225.2.b.g.99.1		4	1.1	even	1	trivial
1225.2.b.g.99.4		4	5.4	even	2	inner
1225.2.b.h.99.1		4	7.6	odd	2
1225.2.b.h.99.4		4	35.34	odd	2
2205.2.a.n.1.1		2	15.2	even	4
2205.2.a.q.1.1		2	105.62	odd	4
3920.2.a.bq.1.2		2	20.7	even	4
3920.2.a.bv.1.1		2	140.27	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1225 = 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1225.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.41421i q^{2} -0.414214i q^{3} -3.82843 q^{4} -1.00000 q^{6} +4.41421i q^{8} +2.82843 q^{9} +O(q^{10})\)	\(q-2.41421i q^{2} -0.414214i q^{3} -3.82843 q^{4} -1.00000 q^{6} +4.41421i q^{8} +2.82843 q^{9} -0.828427 q^{11} +1.58579i q^{12} -4.82843i q^{13} +3.00000 q^{16} -4.82843i q^{17} -6.82843i q^{18} -2.82843 q^{19} +2.00000i q^{22} +0.414214i q^{23} +1.82843 q^{24} -11.6569 q^{26} -2.41421i q^{27} +1.00000 q^{29} -6.00000 q^{31} +1.58579i q^{32} +0.343146i q^{33} -11.6569 q^{34} -10.8284 q^{36} +6.82843i q^{38} -2.00000 q^{39} -7.82843 q^{41} +3.58579i q^{43} +3.17157 q^{44} +1.00000 q^{46} -2.00000i q^{47} -1.24264i q^{48} -2.00000 q^{51} +18.4853i q^{52} -1.17157i q^{53} -5.82843 q^{54} +1.17157i q^{57} -2.41421i q^{58} -4.48528 q^{59} +5.48528 q^{61} +14.4853i q^{62} +9.82843 q^{64} +0.828427 q^{66} -9.58579i q^{67} +18.4853i q^{68} +0.171573 q^{69} +4.48528 q^{71} +12.4853i q^{72} -0.828427i q^{73} +10.8284 q^{76} +4.82843i q^{78} -14.8284 q^{79} +7.48528 q^{81} +18.8995i q^{82} +13.7279i q^{83} +8.65685 q^{86} -0.414214i q^{87} -3.65685i q^{88} +8.65685 q^{89} -1.58579i q^{92} +2.48528i q^{93} -4.82843 q^{94} +0.656854 q^{96} -11.6569i q^{97} -2.34315 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4} - 4 q^{6}+O(q^{10}) \) 4 * q - 4 * q^4 - 4 * q^6	\( 4 q - 4 q^{4} - 4 q^{6} + 8 q^{11} + 12 q^{16} - 4 q^{24} - 24 q^{26} + 4 q^{29} - 24 q^{31} - 24 q^{34} - 32 q^{36} - 8 q^{39} - 20 q^{41} + 24 q^{44} + 4 q^{46} - 8 q^{51} - 12 q^{54} + 16 q^{59} - 12 q^{61} + 28 q^{64} - 8 q^{66} + 12 q^{69} - 16 q^{71} + 32 q^{76} - 48 q^{79} - 4 q^{81} + 12 q^{86} + 12 q^{89} - 8 q^{94} - 20 q^{96} - 32 q^{99}+O(q^{100}) \) 4 * q - 4 * q^4 - 4 * q^6 + 8 * q^11 + 12 * q^16 - 4 * q^24 - 24 * q^26 + 4 * q^29 - 24 * q^31 - 24 * q^34 - 32 * q^36 - 8 * q^39 - 20 * q^41 + 24 * q^44 + 4 * q^46 - 8 * q^51 - 12 * q^54 + 16 * q^59 - 12 * q^61 + 28 * q^64 - 8 * q^66 + 12 * q^69 - 16 * q^71 + 32 * q^76 - 48 * q^79 - 4 * q^81 + 12 * q^86 + 12 * q^89 - 8 * q^94 - 20 * q^96 - 32 * q^99

Introduction

L-functions

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