LMFDB - Embedded newform 775.2.b.d.249.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [775,2,Mod(249,775)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(775, 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("775.249");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 775 = 5^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	775.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:	$6.18840615665$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 31)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			249.3
Root			$0.618034i$ of defining polynomial
Character	$\chi$	$=$	775.249
Dual form			775.2.b.d.249.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+0.618034i q^{2} +1.23607i q^{3} +1.61803 q^{4} -0.763932 q^{6} +4.23607i q^{7} +2.23607i q^{8} +1.47214 q^{9} +2.00000 q^{11} +2.00000i q^{12} +1.23607i q^{13} -2.61803 q^{14} +1.85410 q^{16} -5.23607i q^{17} +0.909830i q^{18} -2.23607 q^{19} -5.23607 q^{21} +1.23607i q^{22} -7.70820i q^{23} -2.76393 q^{24} -0.763932 q^{26} +5.52786i q^{27} +6.85410i q^{28} -7.23607 q^{29} +1.00000 q^{31} +5.61803i q^{32} +2.47214i q^{33} +3.23607 q^{34} +2.38197 q^{36} +2.00000i q^{37} -1.38197i q^{38} -1.52786 q^{39} +7.00000 q^{41} -3.23607i q^{42} -3.23607i q^{43} +3.23607 q^{44} +4.76393 q^{46} +6.47214i q^{47} +2.29180i q^{48} -10.9443 q^{49} +6.47214 q^{51} +2.00000i q^{52} -1.52786i q^{53} -3.41641 q^{54} -9.47214 q^{56} -2.76393i q^{57} -4.47214i q^{58} +2.23607 q^{59} -14.1803 q^{61} +0.618034i q^{62} +6.23607i q^{63} +0.236068 q^{64} -1.52786 q^{66} -8.00000i q^{67} -8.47214i q^{68} +9.52786 q^{69} +13.1803 q^{71} +3.29180i q^{72} -0.472136i q^{73} -1.23607 q^{74} -3.61803 q^{76} +8.47214i q^{77} -0.944272i q^{78} -1.70820 q^{79} -2.41641 q^{81} +4.32624i q^{82} +2.94427i q^{83} -8.47214 q^{84} +2.00000 q^{86} -8.94427i q^{87} +4.47214i q^{88} +1.70820 q^{89} -5.23607 q^{91} -12.4721i q^{92} +1.23607i q^{93} -4.00000 q^{94} -6.94427 q^{96} -1.94427i q^{97} -6.76393i q^{98} +2.94427 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{4} - 12 q^{6} - 12 q^{9} + 8 q^{11} - 6 q^{14} - 6 q^{16} - 12 q^{21} - 20 q^{24} - 12 q^{26} - 20 q^{29} + 4 q^{31} + 4 q^{34} + 14 q^{36} - 24 q^{39} + 28 q^{41} + 4 q^{44} + 28 q^{46} - 8 q^{49}+ \cdots - 24 q^{99}+O(q^{100}) $ 4 * q + 2 * q^4 - 12 * q^6 - 12 * q^9 + 8 * q^11 - 6 * q^14 - 6 * q^16 - 12 * q^21 - 20 * q^24 - 12 * q^26 - 20 * q^29 + 4 * q^31 + 4 * q^34 + 14 * q^36 - 24 * q^39 + 28 * q^41 + 4 * q^44 + 28 * q^46 - 8 * q^49 + 8 * q^51 + 40 * q^54 - 20 * q^56 - 12 * q^61 - 8 * q^64 - 24 * q^66 + 56 * q^69 + 8 * q^71 + 4 * q^74 - 10 * q^76 + 20 * q^79 + 44 * q^81 - 16 * q^84 + 8 * q^86 - 20 * q^89 - 12 * q^91 - 16 * q^94 + 8 * q^96 - 24 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/775\mathbb{Z}\right)^\times$.

$n$	$251$	$652$
$\chi(n)$	$1$	$-1$

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 $

$2$	0.618034i	0.437016i	0.975835	+	0.218508i	$0.0701190\pi$
$2$	0.618034i	0.437016i	−0.975835	+	0.218508i	$0.929881\pi$
$3$	1.23607i	0.713644i	0.934172	+	0.356822i	$0.116140\pi$
$3$	1.23607i	0.713644i	−0.934172	+	0.356822i	$0.883860\pi$
$4$	1.61803	0.809017
$5$	0	0
$5$	0	0
$6$	−0.763932	−0.311874
$7$	4.23607i	1.60108i	0.599277	+	0.800542i	$0.295455\pi$
$7$	4.23607i	1.60108i	−0.599277	+	0.800542i	$0.704545\pi$
$8$	2.23607i	0.790569i
$9$	1.47214	0.490712
$10$	0	0
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	2.00000i	0.577350i
$13$	1.23607i	0.342824i	0.985199	+	0.171412i	$0.0548329\pi$
$13$	1.23607i	0.342824i	−0.985199	+	0.171412i	$0.945167\pi$
$14$	−2.61803	−0.699699
$15$	0	0
$16$	1.85410	0.463525
$17$	− 5.23607i	− 1.26993i	−0.772540	−	0.634967i	$-0.781014\pi$
$17$	− 5.23607i	− 1.26993i	0.772540	−	0.634967i	$-0.218986\pi$
$18$	0.909830i	0.214449i
$19$	−2.23607	−0.512989	−0.256495	−	0.966546i	$-0.582568\pi$
$19$	−2.23607	−0.512989	−0.256495	+	0.966546i	$0.582568\pi$
$20$	0	0
$21$	−5.23607	−1.14260
$22$	1.23607i	0.263531i
$23$	− 7.70820i	− 1.60727i	−0.595121	−	0.803636i	$-0.702896\pi$
$23$	− 7.70820i	− 1.60727i	0.595121	−	0.803636i	$-0.297104\pi$
$24$	−2.76393	−0.564185
$25$	0	0
$26$	−0.763932	−0.149819
$27$	5.52786i	1.06384i
$28$	6.85410i	1.29530i
$29$	−7.23607	−1.34370	−0.671852	−	0.740685i	$-0.734501\pi$
$29$	−7.23607	−1.34370	−0.671852	+	0.740685i	$0.734501\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	5.61803i	0.993137i
$33$	2.47214i	0.430344i
$34$	3.23607	0.554981
$35$	0	0
$36$	2.38197	0.396994
$37$	2.00000i	0.328798i	0.986394	+	0.164399i	$0.0525685\pi$
$37$	2.00000i	0.328798i	−0.986394	+	0.164399i	$0.947432\pi$
$38$	− 1.38197i	− 0.224184i
$39$	−1.52786	−0.244654
$40$	0	0
$41$	7.00000	1.09322	0.546608	−	0.837389i	$-0.315919\pi$
$41$	7.00000	1.09322	0.546608	+	0.837389i	$0.315919\pi$
$42$	− 3.23607i	− 0.499336i
$43$	− 3.23607i	− 0.493496i	−0.969080	−	0.246748i	$-0.920638\pi$
$43$	− 3.23607i	− 0.493496i	0.969080	−	0.246748i	$-0.0793619\pi$
$44$	3.23607	0.487856
$45$	0	0
$46$	4.76393	0.702403
$47$	6.47214i	0.944058i	0.881583	+	0.472029i	$0.156478\pi$
$47$	6.47214i	0.944058i	−0.881583	+	0.472029i	$0.843522\pi$
$48$	2.29180i	0.330792i
$49$	−10.9443	−1.56347
$50$	0	0
$51$	6.47214	0.906280
$52$	2.00000i	0.277350i
$53$	− 1.52786i	− 0.209868i	−0.994479	−	0.104934i	$-0.966537\pi$
$53$	− 1.52786i	− 0.209868i	0.994479	−	0.104934i	$-0.0334632\pi$
$54$	−3.41641	−0.464914
$55$	0	0
$56$	−9.47214	−1.26577
$57$	− 2.76393i	− 0.366092i
$58$	− 4.47214i	− 0.587220i
$59$	2.23607	0.291111	0.145556	−	0.989350i	$-0.453503\pi$
$59$	2.23607	0.291111	0.145556	+	0.989350i	$0.453503\pi$
$60$	0	0
$61$	−14.1803	−1.81561	−0.907803	−	0.419396i	$-0.862242\pi$
$61$	−14.1803	−1.81561	−0.907803	+	0.419396i	$0.862242\pi$
$62$	0.618034i	0.0784904i
$63$	6.23607i	0.785671i
$64$	0.236068	0.0295085
$65$	0	0
$66$	−1.52786	−0.188067
$67$	− 8.00000i	− 0.977356i	−0.872464	−	0.488678i	$-0.837479\pi$
$67$	− 8.00000i	− 0.977356i	0.872464	−	0.488678i	$-0.162521\pi$
$68$	− 8.47214i	− 1.02740i
$69$	9.52786	1.14702
$70$	0	0
$71$	13.1803	1.56422	0.782109	−	0.623141i	$-0.214144\pi$
$71$	13.1803	1.56422	0.782109	+	0.623141i	$0.214144\pi$
$72$	3.29180i	0.387942i
$73$	− 0.472136i	− 0.0552593i	−0.999618	−	0.0276297i	$-0.991204\pi$
$73$	− 0.472136i	− 0.0552593i	0.999618	−	0.0276297i	$-0.00879592\pi$
$74$	−1.23607	−0.143690
$75$	0	0
$76$	−3.61803	−0.415017
$77$	8.47214i	0.965489i
$78$	− 0.944272i	− 0.106918i
$79$	−1.70820	−0.192188	−0.0960940	−	0.995372i	$-0.530635\pi$
$79$	−1.70820	−0.192188	−0.0960940	+	0.995372i	$0.530635\pi$
$80$	0	0
$81$	−2.41641	−0.268490
$82$	4.32624i	0.477753i
$83$	2.94427i	0.323176i	0.986858	+	0.161588i	$0.0516615\pi$
$83$	2.94427i	0.323176i	−0.986858	+	0.161588i	$0.948338\pi$
$84$	−8.47214	−0.924386
$85$	0	0
$86$	2.00000	0.215666
$87$	− 8.94427i	− 0.958927i
$88$	4.47214i	0.476731i
$89$	1.70820	0.181069	0.0905346	−	0.995893i	$-0.471142\pi$
$89$	1.70820	0.181069	0.0905346	+	0.995893i	$0.471142\pi$
$90$	0	0
$91$	−5.23607	−0.548889
$92$	− 12.4721i	− 1.30031i
$93$	1.23607i	0.128174i
$94$	−4.00000	−0.412568
$95$	0	0
$96$	−6.94427	−0.708747
$97$	− 1.94427i	− 0.197411i	−0.995117	−	0.0987055i	$-0.968530\pi$
$97$	− 1.94427i	− 0.197411i	0.995117	−	0.0987055i	$-0.0314702\pi$
$98$	− 6.76393i	− 0.683260i
$99$	2.94427	0.295910
$100$	0	0
$101$	−3.00000	−0.298511	−0.149256	−	0.988799i	$-0.547688\pi$
$101$	−3.00000	−0.298511	−0.149256	+	0.988799i	$0.547688\pi$
$102$	4.00000i	0.396059i
$103$	1.76393i	0.173805i	0.996217	+	0.0869027i	$0.0276969\pi$
$103$	1.76393i	0.173805i	−0.996217	+	0.0869027i	$0.972303\pi$
$104$	−2.76393	−0.271026
$105$	0	0
$106$	0.944272	0.0917158
$107$	− 10.2361i	− 0.989558i	−0.869019	−	0.494779i	$-0.835249\pi$
$107$	− 10.2361i	− 0.989558i	0.869019	−	0.494779i	$-0.164751\pi$
$108$	8.94427i	0.860663i
$109$	−3.94427	−0.377793	−0.188896	−	0.981997i	$-0.560491\pi$
$109$	−3.94427	−0.377793	−0.188896	+	0.981997i	$0.560491\pi$
$110$	0	0
$111$	−2.47214	−0.234645
$112$	7.85410i	0.742143i
$113$	− 5.47214i	− 0.514775i	−0.966308	−	0.257388i	$-0.917138\pi$
$113$	− 5.47214i	− 0.514775i	0.966308	−	0.257388i	$-0.0828617\pi$
$114$	1.70820	0.159988
$115$	0	0
$116$	−11.7082	−1.08708
$117$	1.81966i	0.168228i
$118$	1.38197i	0.127220i
$119$	22.1803	2.03327
$120$	0	0
$121$	−7.00000	−0.636364
$122$	− 8.76393i	− 0.793449i
$123$	8.65248i	0.780167i
$124$	1.61803	0.145304
$125$	0	0
$126$	−3.85410	−0.343351
$127$	− 3.52786i	− 0.313047i	−0.987674	−	0.156524i	$-0.949971\pi$
$127$	− 3.52786i	− 0.313047i	0.987674	−	0.156524i	$-0.0500287\pi$
$128$	11.3820i	1.00603i
$129$	4.00000	0.352180
$130$	0	0
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	4.00000i	0.348155i
$133$	− 9.47214i	− 0.821338i
$134$	4.94427	0.427120
$135$	0	0
$136$	11.7082	1.00397
$137$	− 19.7082i	− 1.68379i	−0.539645	−	0.841893i	$-0.681441\pi$
$137$	− 19.7082i	− 1.68379i	0.539645	−	0.841893i	$-0.318559\pi$
$138$	5.88854i	0.501266i
$139$	13.4164	1.13796	0.568982	−	0.822350i	$-0.307337\pi$
$139$	13.4164	1.13796	0.568982	+	0.822350i	$0.307337\pi$
$140$	0	0
$141$	−8.00000	−0.673722
$142$	8.14590i	0.683589i
$143$	2.47214i	0.206730i
$144$	2.72949	0.227458
$145$	0	0
$146$	0.291796	0.0241492
$147$	− 13.5279i	− 1.11576i
$148$	3.23607i	0.266003i
$149$	−10.0000	−0.819232	−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000	−0.819232	−0.409616	+	0.912258i	$0.634337\pi$
$150$	0	0
$151$	8.18034	0.665707	0.332853	−	0.942979i	$-0.391989\pi$
$151$	8.18034	0.665707	0.332853	+	0.942979i	$0.391989\pi$
$152$	− 5.00000i	− 0.405554i
$153$	− 7.70820i	− 0.623171i
$154$	−5.23607	−0.421934
$155$	0	0
$156$	−2.47214	−0.197929
$157$	14.8885i	1.18824i	0.804378	+	0.594118i	$0.202499\pi$
$157$	14.8885i	1.18824i	−0.804378	+	0.594118i	$0.797501\pi$
$158$	− 1.05573i	− 0.0839892i
$159$	1.88854	0.149771
$160$	0	0
$161$	32.6525	2.57338
$162$	− 1.49342i	− 0.117334i
$163$	− 2.70820i	− 0.212123i	−0.994360	−	0.106061i	$-0.966176\pi$
$163$	− 2.70820i	− 0.212123i	0.994360	−	0.106061i	$-0.0338240\pi$
$164$	11.3262	0.884431
$165$	0	0
$166$	−1.81966	−0.141233
$167$	− 2.47214i	− 0.191300i	−0.995415	−	0.0956498i	$-0.969507\pi$
$167$	− 2.47214i	− 0.191300i	0.995415	−	0.0956498i	$-0.0304929\pi$
$168$	− 11.7082i	− 0.903308i
$169$	11.4721	0.882472
$170$	0	0
$171$	−3.29180	−0.251730
$172$	− 5.23607i	− 0.399246i
$173$	− 14.9443i	− 1.13619i	−0.822962	−	0.568096i	$-0.807680\pi$
$173$	− 14.9443i	− 1.13619i	0.822962	−	0.568096i	$-0.192320\pi$
$174$	5.52786	0.419066
$175$	0	0
$176$	3.70820	0.279516
$177$	2.76393i	0.207750i
$178$	1.05573i	0.0791302i
$179$	11.7082	0.875112	0.437556	−	0.899191i	$-0.355844\pi$
$179$	11.7082	0.875112	0.437556	+	0.899191i	$0.355844\pi$
$180$	0	0
$181$	18.1803	1.35133	0.675667	−	0.737207i	$-0.263856\pi$
$181$	18.1803	1.35133	0.675667	+	0.737207i	$0.263856\pi$
$182$	− 3.23607i	− 0.239873i
$183$	− 17.5279i	− 1.29570i
$184$	17.2361	1.27066
$185$	0	0
$186$	−0.763932	−0.0560142
$187$	− 10.4721i	− 0.765798i
$188$	10.4721i	0.763759i
$189$	−23.4164	−1.70329
$190$	0	0
$191$	3.18034	0.230121	0.115061	−	0.993358i	$-0.463294\pi$
$191$	3.18034	0.230121	0.115061	+	0.993358i	$0.463294\pi$
$192$	0.291796i	0.0210586i
$193$	− 5.47214i	− 0.393893i	−0.980414	−	0.196946i	$-0.936897\pi$
$193$	− 5.47214i	− 0.393893i	0.980414	−	0.196946i	$-0.0631025\pi$
$194$	1.20163	0.0862717
$195$	0	0
$196$	−17.7082	−1.26487
$197$	15.4164i	1.09837i	0.835700	+	0.549187i	$0.185062\pi$
$197$	15.4164i	1.09837i	−0.835700	+	0.549187i	$0.814938\pi$
$198$	1.81966i	0.129318i
$199$	1.05573	0.0748386	0.0374193	−	0.999300i	$-0.488086\pi$
$199$	1.05573	0.0748386	0.0374193	+	0.999300i	$0.488086\pi$
$200$	0	0
$201$	9.88854	0.697484
$202$	− 1.85410i	− 0.130454i
$203$	− 30.6525i	− 2.15138i
$204$	10.4721	0.733196
$205$	0	0
$206$	−1.09017	−0.0759557
$207$	− 11.3475i	− 0.788707i
$208$	2.29180i	0.158907i
$209$	−4.47214	−0.309344
$210$	0	0
$211$	0.819660	0.0564277	0.0282139	−	0.999602i	$-0.491018\pi$
$211$	0.819660	0.0564277	0.0282139	+	0.999602i	$0.491018\pi$
$212$	− 2.47214i	− 0.169787i
$213$	16.2918i	1.11630i
$214$	6.32624	0.432453
$215$	0	0
$216$	−12.3607	−0.841038
$217$	4.23607i	0.287563i
$218$	− 2.43769i	− 0.165101i
$219$	0.583592	0.0394355
$220$	0	0
$221$	6.47214	0.435363
$222$	− 1.52786i	− 0.102544i
$223$	4.00000i	0.267860i	0.990991	+	0.133930i	$0.0427597\pi$
$223$	4.00000i	0.267860i	−0.990991	+	0.133930i	$0.957240\pi$
$224$	−23.7984	−1.59010
$225$	0	0
$226$	3.38197	0.224965
$227$	− 2.47214i	− 0.164081i	−0.996629	−	0.0820407i	$-0.973856\pi$
$227$	− 2.47214i	− 0.164081i	0.996629	−	0.0820407i	$-0.0261438\pi$
$228$	− 4.47214i	− 0.296174i
$229$	−13.4164	−0.886581	−0.443291	−	0.896378i	$-0.646189\pi$
$229$	−13.4164	−0.886581	−0.443291	+	0.896378i	$0.646189\pi$
$230$	0	0
$231$	−10.4721	−0.689016
$232$	− 16.1803i	− 1.06229i
$233$	0.0557281i	0.00365087i	0.999998	+	0.00182543i	$0.000581054\pi$
$233$	0.0557281i	0.00365087i	−0.999998	+	0.00182543i	$0.999419\pi$
$234$	−1.12461	−0.0735182
$235$	0	0
$236$	3.61803	0.235514
$237$	− 2.11146i	− 0.137154i
$238$	13.7082i	0.888571i
$239$	−1.70820	−0.110495	−0.0552473	−	0.998473i	$-0.517595\pi$
$239$	−1.70820	−0.110495	−0.0552473	+	0.998473i	$0.517595\pi$
$240$	0	0
$241$	−30.3607	−1.95570	−0.977852	−	0.209299i	$-0.932882\pi$
$241$	−30.3607	−1.95570	−0.977852	+	0.209299i	$0.932882\pi$
$242$	− 4.32624i	− 0.278101i
$243$	13.5967i	0.872232i
$244$	−22.9443	−1.46886
$245$	0	0
$246$	−5.34752	−0.340946
$247$	− 2.76393i	− 0.175865i
$248$	2.23607i	0.141990i
$249$	−3.63932	−0.230633
$250$	0	0
$251$	−24.1803	−1.52625	−0.763125	−	0.646251i	$-0.776336\pi$
$251$	−24.1803	−1.52625	−0.763125	+	0.646251i	$0.776336\pi$
$252$	10.0902i	0.635621i
$253$	− 15.4164i	− 0.969221i
$254$	2.18034	0.136807
$255$	0	0
$256$	−6.56231	−0.410144
$257$	15.9443i	0.994576i	0.867585	+	0.497288i	$0.165671\pi$
$257$	15.9443i	0.994576i	−0.867585	+	0.497288i	$0.834329\pi$
$258$	2.47214i	0.153908i
$259$	−8.47214	−0.526433
$260$	0	0
$261$	−10.6525	−0.659372
$262$	7.41641i	0.458187i
$263$	− 18.7639i	− 1.15703i	−0.815670	−	0.578517i	$-0.803632\pi$
$263$	− 18.7639i	− 1.15703i	0.815670	−	0.578517i	$-0.196368\pi$
$264$	−5.52786	−0.340217
$265$	0	0
$266$	5.85410	0.358938
$267$	2.11146i	0.129219i
$268$	− 12.9443i	− 0.790697i
$269$	28.9443	1.76476	0.882382	−	0.470534i	$-0.155939\pi$
$269$	28.9443	1.76476	0.882382	+	0.470534i	$0.155939\pi$
$270$	0	0
$271$	8.18034	0.496920	0.248460	−	0.968642i	$-0.420076\pi$
$271$	8.18034	0.496920	0.248460	+	0.968642i	$0.420076\pi$
$272$	− 9.70820i	− 0.588646i
$273$	− 6.47214i	− 0.391711i
$274$	12.1803	0.735841
$275$	0	0
$276$	15.4164	0.927959
$277$	− 18.6525i	− 1.12072i	−0.828250	−	0.560359i	$-0.810663\pi$
$277$	− 18.6525i	− 1.12072i	0.828250	−	0.560359i	$-0.189337\pi$
$278$	8.29180i	0.497309i
$279$	1.47214	0.0881345
$280$	0	0
$281$	17.0000	1.01413	0.507067	−	0.861906i	$-0.330729\pi$
$281$	17.0000	1.01413	0.507067	+	0.861906i	$0.330729\pi$
$282$	− 4.94427i	− 0.294427i
$283$	21.8885i	1.30114i	0.759447	+	0.650569i	$0.225470\pi$
$283$	21.8885i	1.30114i	−0.759447	+	0.650569i	$0.774530\pi$
$284$	21.3262	1.26548
$285$	0	0
$286$	−1.52786	−0.0903445
$287$	29.6525i	1.75033i
$288$	8.27051i	0.487344i
$289$	−10.4164	−0.612730
$290$	0	0
$291$	2.40325	0.140881
$292$	− 0.763932i	− 0.0447057i
$293$	8.47214i	0.494947i	0.968895	+	0.247474i	$0.0796004\pi$
$293$	8.47214i	0.494947i	−0.968895	+	0.247474i	$0.920400\pi$
$294$	8.36068	0.487605
$295$	0	0
$296$	−4.47214	−0.259938
$297$	11.0557i	0.641518i
$298$	− 6.18034i	− 0.358017i
$299$	9.52786	0.551011
$300$	0	0
$301$	13.7082	0.790128
$302$	5.05573i	0.290924i
$303$	− 3.70820i	− 0.213031i
$304$	−4.14590	−0.237784
$305$	0	0
$306$	4.76393	0.272336
$307$	15.2918i	0.872749i	0.899765	+	0.436374i	$0.143738\pi$
$307$	15.2918i	0.872749i	−0.899765	+	0.436374i	$0.856262\pi$
$308$	13.7082i	0.781097i
$309$	−2.18034	−0.124035
$310$	0	0
$311$	−6.81966	−0.386707	−0.193354	−	0.981129i	$-0.561936\pi$
$311$	−6.81966	−0.386707	−0.193354	+	0.981129i	$0.561936\pi$
$312$	− 3.41641i	− 0.193416i
$313$	21.2361i	1.20033i	0.799875	+	0.600167i	$0.204899\pi$
$313$	21.2361i	1.20033i	−0.799875	+	0.600167i	$0.795101\pi$
$314$	−9.20163	−0.519278
$315$	0	0
$316$	−2.76393	−0.155483
$317$	− 21.9443i	− 1.23251i	−0.787545	−	0.616257i	$-0.788648\pi$
$317$	− 21.9443i	− 1.23251i	0.787545	−	0.616257i	$-0.211352\pi$
$318$	1.16718i	0.0654524i
$319$	−14.4721	−0.810284
$320$	0	0
$321$	12.6525	0.706192
$322$	20.1803i	1.12461i
$323$	11.7082i	0.651462i
$324$	−3.90983	−0.217213
$325$	0	0
$326$	1.67376	0.0927011
$327$	− 4.87539i	− 0.269610i
$328$	15.6525i	0.864263i
$329$	−27.4164	−1.51152
$330$	0	0
$331$	2.00000	0.109930	0.0549650	−	0.998488i	$-0.482495\pi$
$331$	2.00000	0.109930	0.0549650	+	0.998488i	$0.482495\pi$
$332$	4.76393i	0.261455i
$333$	2.94427i	0.161345i
$334$	1.52786	0.0836010
$335$	0	0
$336$	−9.70820	−0.529626
$337$	19.2361i	1.04786i	0.851763	+	0.523928i	$0.175534\pi$
$337$	19.2361i	1.04786i	−0.851763	+	0.523928i	$0.824466\pi$
$338$	7.09017i	0.385654i
$339$	6.76393	0.367366
$340$	0	0
$341$	2.00000	0.108306
$342$	− 2.03444i	− 0.110010i
$343$	− 16.7082i	− 0.902158i
$344$	7.23607	0.390143
$345$	0	0
$346$	9.23607	0.496534
$347$	− 1.81966i	− 0.0976845i	−0.998807	−	0.0488422i	$-0.984447\pi$
$347$	− 1.81966i	− 0.0976845i	0.998807	−	0.0488422i	$-0.0155532\pi$
$348$	− 14.4721i	− 0.775788i
$349$	−27.8885	−1.49284	−0.746420	−	0.665475i	$-0.768229\pi$
$349$	−27.8885	−1.49284	−0.746420	+	0.665475i	$0.768229\pi$
$350$	0	0
$351$	−6.83282	−0.364709
$352$	11.2361i	0.598884i
$353$	− 19.4164i	− 1.03343i	−0.856157	−	0.516716i	$-0.827154\pi$
$353$	− 19.4164i	− 1.03343i	0.856157	−	0.516716i	$-0.172846\pi$
$354$	−1.70820	−0.0907900
$355$	0	0
$356$	2.76393	0.146488
$357$	27.4164i	1.45103i
$358$	7.23607i	0.382438i
$359$	−17.7639	−0.937544	−0.468772	−	0.883319i	$-0.655303\pi$
$359$	−17.7639	−0.937544	−0.468772	+	0.883319i	$0.655303\pi$
$360$	0	0
$361$	−14.0000	−0.736842
$362$	11.2361i	0.590555i
$363$	− 8.65248i	− 0.454137i
$364$	−8.47214	−0.444061
$365$	0	0
$366$	10.8328	0.566240
$367$	− 18.0000i	− 0.939592i	−0.882775	−	0.469796i	$-0.844327\pi$
$367$	− 18.0000i	− 0.939592i	0.882775	−	0.469796i	$-0.155673\pi$
$368$	− 14.2918i	− 0.745011i
$369$	10.3050	0.536454
$370$	0	0
$371$	6.47214	0.336017
$372$	2.00000i	0.103695i
$373$	19.0000i	0.983783i	0.870657	+	0.491891i	$0.163694\pi$
$373$	19.0000i	0.983783i	−0.870657	+	0.491891i	$0.836306\pi$
$374$	6.47214	0.334666
$375$	0	0
$376$	−14.4721	−0.746343
$377$	− 8.94427i	− 0.460653i
$378$	− 14.4721i	− 0.744366i
$379$	37.8885	1.94620	0.973102	−	0.230375i	$-0.0739953\pi$
$379$	37.8885	1.94620	0.973102	+	0.230375i	$0.0739953\pi$
$380$	0	0
$381$	4.36068	0.223404
$382$	1.96556i	0.100567i
$383$	11.8885i	0.607476i	0.952756	+	0.303738i	$0.0982348\pi$
$383$	11.8885i	0.607476i	−0.952756	+	0.303738i	$0.901765\pi$
$384$	−14.0689	−0.717950
$385$	0	0
$386$	3.38197	0.172138
$387$	− 4.76393i	− 0.242164i
$388$	− 3.14590i	− 0.159709i
$389$	−17.8885	−0.906985	−0.453493	−	0.891260i	$-0.649822\pi$
$389$	−17.8885	−0.906985	−0.453493	+	0.891260i	$0.649822\pi$
$390$	0	0
$391$	−40.3607	−2.04113
$392$	− 24.4721i	− 1.23603i
$393$	14.8328i	0.748217i
$394$	−9.52786	−0.480007
$395$	0	0
$396$	4.76393	0.239397
$397$	7.00000i	0.351320i	0.984451	+	0.175660i	$0.0562059\pi$
$397$	7.00000i	0.351320i	−0.984451	+	0.175660i	$0.943794\pi$
$398$	0.652476i	0.0327057i
$399$	11.7082	0.586143
$400$	0	0
$401$	15.8197	0.789996	0.394998	−	0.918682i	$-0.370745\pi$
$401$	15.8197	0.789996	0.394998	+	0.918682i	$0.370745\pi$
$402$	6.11146i	0.304812i
$403$	1.23607i	0.0615729i
$404$	−4.85410	−0.241501
$405$	0	0
$406$	18.9443	0.940188
$407$	4.00000i	0.198273i
$408$	14.4721i	0.716477i
$409$	26.1803	1.29453	0.647267	−	0.762263i	$-0.275912\pi$
$409$	26.1803	1.29453	0.647267	+	0.762263i	$0.275912\pi$
$410$	0	0
$411$	24.3607	1.20162
$412$	2.85410i	0.140612i
$413$	9.47214i	0.466093i
$414$	7.01316	0.344678
$415$	0	0
$416$	−6.94427	−0.340471
$417$	16.5836i	0.812102i
$418$	− 2.76393i	− 0.135188i
$419$	−30.1246	−1.47168	−0.735842	−	0.677153i	$-0.763213\pi$
$419$	−30.1246	−1.47168	−0.735842	+	0.677153i	$0.763213\pi$
$420$	0	0
$421$	−15.3607	−0.748634	−0.374317	−	0.927301i	$-0.622123\pi$
$421$	−15.3607	−0.748634	−0.374317	+	0.927301i	$0.622123\pi$
$422$	0.506578i	0.0246598i
$423$	9.52786i	0.463261i
$424$	3.41641	0.165915
$425$	0	0
$426$	−10.0689	−0.487839
$427$	− 60.0689i	− 2.90694i
$428$	− 16.5623i	− 0.800569i
$429$	−3.05573	−0.147532
$430$	0	0
$431$	12.0000	0.578020	0.289010	−	0.957326i	$-0.406674\pi$
$431$	12.0000	0.578020	0.289010	+	0.957326i	$0.406674\pi$
$432$	10.2492i	0.493116i
$433$	− 12.1803i	− 0.585350i	−0.956212	−	0.292675i	$-0.905455\pi$
$433$	− 12.1803i	− 0.585350i	0.956212	−	0.292675i	$-0.0945454\pi$
$434$	−2.61803	−0.125670
$435$	0	0
$436$	−6.38197	−0.305641
$437$	17.2361i	0.824513i
$438$	0.360680i	0.0172339i
$439$	−21.1803	−1.01088	−0.505441	−	0.862861i	$-0.668670\pi$
$439$	−21.1803	−1.01088	−0.505441	+	0.862861i	$0.668670\pi$
$440$	0	0
$441$	−16.1115	−0.767212
$442$	4.00000i	0.190261i
$443$	17.2918i	0.821558i	0.911735	+	0.410779i	$0.134743\pi$
$443$	17.2918i	0.821558i	−0.911735	+	0.410779i	$0.865257\pi$
$444$	−4.00000	−0.189832
$445$	0	0
$446$	−2.47214	−0.117059
$447$	− 12.3607i	− 0.584640i
$448$	1.00000i	0.0472456i
$449$	−31.3050	−1.47737	−0.738686	−	0.674050i	$-0.764553\pi$
$449$	−31.3050	−1.47737	−0.738686	+	0.674050i	$0.764553\pi$
$450$	0	0
$451$	14.0000	0.659234
$452$	− 8.85410i	− 0.416462i
$453$	10.1115i	0.475078i
$454$	1.52786	0.0717062
$455$	0	0
$456$	6.18034	0.289421
$457$	20.9443i	0.979732i	0.871798	+	0.489866i	$0.162954\pi$
$457$	20.9443i	0.979732i	−0.871798	+	0.489866i	$0.837046\pi$
$458$	− 8.29180i	− 0.387450i
$459$	28.9443	1.35100
$460$	0	0
$461$	−10.3607	−0.482545	−0.241272	−	0.970457i	$-0.577565\pi$
$461$	−10.3607	−0.482545	−0.241272	+	0.970457i	$0.577565\pi$
$462$	− 6.47214i	− 0.301111i
$463$	− 29.4164i	− 1.36710i	−0.729905	−	0.683548i	$-0.760436\pi$
$463$	− 29.4164i	− 1.36710i	0.729905	−	0.683548i	$-0.239564\pi$
$464$	−13.4164	−0.622841
$465$	0	0
$466$	−0.0344419	−0.00159549
$467$	8.70820i	0.402968i	0.979492	+	0.201484i	$0.0645763\pi$
$467$	8.70820i	0.402968i	−0.979492	+	0.201484i	$0.935424\pi$
$468$	2.94427i	0.136099i
$469$	33.8885	1.56483
$470$	0	0
$471$	−18.4033	−0.847977
$472$	5.00000i	0.230144i
$473$	− 6.47214i	− 0.297589i
$474$	1.30495	0.0599384
$475$	0	0
$476$	35.8885	1.64495
$477$	− 2.24922i	− 0.102985i
$478$	− 1.05573i	− 0.0482879i
$479$	−36.7082	−1.67724	−0.838620	−	0.544716i	$-0.816637\pi$
$479$	−36.7082	−1.67724	−0.838620	+	0.544716i	$0.816637\pi$
$480$	0	0
$481$	−2.47214	−0.112720
$482$	− 18.7639i	− 0.854674i
$483$	40.3607i	1.83647i
$484$	−11.3262	−0.514829
$485$	0	0
$486$	−8.40325	−0.381179
$487$	14.7639i	0.669018i	0.942393	+	0.334509i	$0.108570\pi$
$487$	14.7639i	0.669018i	−0.942393	+	0.334509i	$0.891430\pi$
$488$	− 31.7082i	− 1.43536i
$489$	3.34752	0.151380
$490$	0	0
$491$	−40.3607	−1.82145	−0.910726	−	0.413011i	$-0.864477\pi$
$491$	−40.3607	−1.82145	−0.910726	+	0.413011i	$0.864477\pi$
$492$	14.0000i	0.631169i
$493$	37.8885i	1.70641i
$494$	1.70820	0.0768557
$495$	0	0
$496$	1.85410	0.0832516
$497$	55.8328i	2.50444i
$498$	− 2.24922i	− 0.100790i
$499$	33.4164	1.49592	0.747962	−	0.663742i	$-0.231033\pi$
$499$	33.4164	1.49592	0.747962	+	0.663742i	$0.231033\pi$
$500$	0	0
$501$	3.05573	0.136520
$502$	− 14.9443i	− 0.666995i
$503$	− 1.65248i	− 0.0736803i	−0.999321	−	0.0368401i	$-0.988271\pi$
$503$	− 1.65248i	− 0.0736803i	0.999321	−	0.0368401i	$-0.0117292\pi$
$504$	−13.9443	−0.621127
$505$	0	0
$506$	9.52786	0.423565
$507$	14.1803i	0.629771i
$508$	− 5.70820i	− 0.253261i
$509$	19.5967	0.868611	0.434305	−	0.900766i	$-0.356994\pi$
$509$	19.5967	0.868611	0.434305	+	0.900766i	$0.356994\pi$
$510$	0	0
$511$	2.00000	0.0884748
$512$	18.7082i	0.826794i
$513$	− 12.3607i	− 0.545737i
$514$	−9.85410	−0.434646
$515$	0	0
$516$	6.47214	0.284920
$517$	12.9443i	0.569288i
$518$	− 5.23607i	− 0.230060i
$519$	18.4721	0.810837
$520$	0	0
$521$	2.00000	0.0876216	0.0438108	−	0.999040i	$-0.486050\pi$
$521$	2.00000	0.0876216	0.0438108	+	0.999040i	$0.486050\pi$
$522$	− 6.58359i	− 0.288156i
$523$	− 4.29180i	− 0.187667i	−0.995588	−	0.0938336i	$-0.970088\pi$
$523$	− 4.29180i	− 0.187667i	0.995588	−	0.0938336i	$-0.0299122\pi$
$524$	19.4164	0.848210
$525$	0	0
$526$	11.5967	0.505642
$527$	− 5.23607i	− 0.228087i
$528$	4.58359i	0.199475i
$529$	−36.4164	−1.58332
$530$	0	0
$531$	3.29180	0.142852
$532$	− 15.3262i	− 0.664477i
$533$	8.65248i	0.374780i
$534$	−1.30495	−0.0564708
$535$	0	0
$536$	17.8885	0.772667
$537$	14.4721i	0.624519i
$538$	17.8885i	0.771230i
$539$	−21.8885	−0.942806
$540$	0	0
$541$	19.3607	0.832381	0.416190	−	0.909278i	$-0.363365\pi$
$541$	19.3607	0.832381	0.416190	+	0.909278i	$0.363365\pi$
$542$	5.05573i	0.217162i
$543$	22.4721i	0.964372i
$544$	29.4164	1.26122
$545$	0	0
$546$	4.00000	0.171184
$547$	− 28.1246i	− 1.20252i	−0.799053	−	0.601261i	$-0.794665\pi$
$547$	− 28.1246i	− 1.20252i	0.799053	−	0.601261i	$-0.205335\pi$
$548$	− 31.8885i	− 1.36221i
$549$	−20.8754	−0.890940
$550$	0	0
$551$	16.1803	0.689306
$552$	21.3050i	0.906799i
$553$	− 7.23607i	− 0.307709i
$554$	11.5279	0.489772
$555$	0	0
$556$	21.7082	0.920633
$557$	12.0000i	0.508456i	0.967144	+	0.254228i	$0.0818214\pi$
$557$	12.0000i	0.508456i	−0.967144	+	0.254228i	$0.918179\pi$
$558$	0.909830i	0.0385162i
$559$	4.00000	0.169182
$560$	0	0
$561$	12.9443	0.546508
$562$	10.5066i	0.443193i
$563$	− 39.5410i	− 1.66646i	−0.552930	−	0.833228i	$-0.686490\pi$
$563$	− 39.5410i	− 1.66646i	0.552930	−	0.833228i	$-0.313510\pi$
$564$	−12.9443	−0.545052
$565$	0	0
$566$	−13.5279	−0.568619
$567$	− 10.2361i	− 0.429874i
$568$	29.4721i	1.23662i
$569$	−14.4721	−0.606704	−0.303352	−	0.952879i	$-0.598106\pi$
$569$	−14.4721	−0.606704	−0.303352	+	0.952879i	$0.598106\pi$
$570$	0	0
$571$	5.81966	0.243545	0.121773	−	0.992558i	$-0.461142\pi$
$571$	5.81966	0.243545	0.121773	+	0.992558i	$0.461142\pi$
$572$	4.00000i	0.167248i
$573$	3.93112i	0.164225i
$574$	−18.3262	−0.764922
$575$	0	0
$576$	0.347524	0.0144802
$577$	− 24.8328i	− 1.03380i	−0.856045	−	0.516902i	$-0.827085\pi$
$577$	− 24.8328i	− 1.03380i	0.856045	−	0.516902i	$-0.172915\pi$
$578$	− 6.43769i	− 0.267773i
$579$	6.76393	0.281099
$580$	0	0
$581$	−12.4721	−0.517431
$582$	1.48529i	0.0615673i
$583$	− 3.05573i	− 0.126555i
$584$	1.05573	0.0436863
$585$	0	0
$586$	−5.23607	−0.216300
$587$	− 2.47214i	− 0.102036i	−0.998698	−	0.0510180i	$-0.983753\pi$
$587$	− 2.47214i	− 0.102036i	0.998698	−	0.0510180i	$-0.0162466\pi$
$588$	− 21.8885i	− 0.902668i
$589$	−2.23607	−0.0921356
$590$	0	0
$591$	−19.0557	−0.783848
$592$	3.70820i	0.152406i
$593$	− 15.4721i	− 0.635364i	−0.948197	−	0.317682i	$-0.897095\pi$
$593$	− 15.4721i	− 0.635364i	0.948197	−	0.317682i	$-0.102905\pi$
$594$	−6.83282	−0.280354
$595$	0	0
$596$	−16.1803	−0.662773
$597$	1.30495i	0.0534081i
$598$	5.88854i	0.240800i
$599$	−34.5967	−1.41358	−0.706792	−	0.707421i	$-0.749859\pi$
$599$	−34.5967	−1.41358	−0.706792	+	0.707421i	$0.749859\pi$
$600$	0	0
$601$	−36.5410	−1.49054	−0.745270	−	0.666763i	$-0.767679\pi$
$601$	−36.5410	−1.49054	−0.745270	+	0.666763i	$0.767679\pi$
$602$	8.47214i	0.345298i
$603$	− 11.7771i	− 0.479600i
$604$	13.2361	0.538568
$605$	0	0
$606$	2.29180	0.0930979
$607$	− 13.5279i	− 0.549079i	−0.961576	−	0.274540i	$-0.911475\pi$
$607$	− 13.5279i	− 0.549079i	0.961576	−	0.274540i	$-0.0885254\pi$
$608$	− 12.5623i	− 0.509469i
$609$	37.8885	1.53532
$610$	0	0
$611$	−8.00000	−0.323645
$612$	− 12.4721i	− 0.504156i
$613$	− 8.11146i	− 0.327619i	−0.986492	−	0.163809i	$-0.947622\pi$
$613$	− 8.11146i	− 0.327619i	0.986492	−	0.163809i	$-0.0523782\pi$
$614$	−9.45085	−0.381405
$615$	0	0
$616$	−18.9443	−0.763286
$617$	− 23.5279i	− 0.947196i	−0.880741	−	0.473598i	$-0.842955\pi$
$617$	− 23.5279i	− 0.947196i	0.880741	−	0.473598i	$-0.157045\pi$
$618$	− 1.34752i	− 0.0542054i
$619$	16.1803	0.650343	0.325171	−	0.945655i	$-0.394578\pi$
$619$	16.1803	0.650343	0.325171	+	0.945655i	$0.394578\pi$
$620$	0	0
$621$	42.6099	1.70988
$622$	− 4.21478i	− 0.168997i
$623$	7.23607i	0.289907i
$624$	−2.83282	−0.113403
$625$	0	0
$626$	−13.1246	−0.524565
$627$	− 5.52786i	− 0.220762i
$628$	24.0902i	0.961302i
$629$	10.4721	0.417551
$630$	0	0
$631$	−10.3607	−0.412452	−0.206226	−	0.978504i	$-0.566118\pi$
$631$	−10.3607	−0.412452	−0.206226	+	0.978504i	$0.566118\pi$
$632$	− 3.81966i	− 0.151938i
$633$	1.01316i	0.0402693i
$634$	13.5623	0.538628
$635$	0	0
$636$	3.05573	0.121168
$637$	− 13.5279i	− 0.535993i
$638$	− 8.94427i	− 0.354107i
$639$	19.4033	0.767581
$640$	0	0
$641$	12.0000	0.473972	0.236986	−	0.971513i	$-0.423841\pi$
$641$	12.0000	0.473972	0.236986	+	0.971513i	$0.423841\pi$
$642$	7.81966i	0.308617i
$643$	28.4721i	1.12283i	0.827534	+	0.561416i	$0.189743\pi$
$643$	28.4721i	1.12283i	−0.827534	+	0.561416i	$0.810257\pi$
$644$	52.8328	2.08190
$645$	0	0
$646$	−7.23607	−0.284699
$647$	− 16.9443i	− 0.666148i	−0.942901	−	0.333074i	$-0.891914\pi$
$647$	− 16.9443i	− 0.666148i	0.942901	−	0.333074i	$-0.108086\pi$
$648$	− 5.40325i	− 0.212260i
$649$	4.47214	0.175547
$650$	0	0
$651$	−5.23607	−0.205218
$652$	− 4.38197i	− 0.171611i
$653$	15.3050i	0.598929i	0.954107	+	0.299465i	$0.0968080\pi$
$653$	15.3050i	0.598929i	−0.954107	+	0.299465i	$0.903192\pi$
$654$	3.01316	0.117824
$655$	0	0
$656$	12.9787	0.506734
$657$	− 0.695048i	− 0.0271164i
$658$	− 16.9443i	− 0.660556i
$659$	5.65248	0.220189	0.110095	−	0.993921i	$-0.464885\pi$
$659$	5.65248	0.220189	0.110095	+	0.993921i	$0.464885\pi$
$660$	0	0
$661$	−45.3607	−1.76433	−0.882163	−	0.470944i	$-0.843913\pi$
$661$	−45.3607	−1.76433	−0.882163	+	0.470944i	$0.843913\pi$
$662$	1.23607i	0.0480411i
$663$	8.00000i	0.310694i
$664$	−6.58359	−0.255493
$665$	0	0
$666$	−1.81966	−0.0705104
$667$	55.7771i	2.15970i
$668$	− 4.00000i	− 0.154765i
$669$	−4.94427	−0.191157
$670$	0	0
$671$	−28.3607	−1.09485
$672$	− 29.4164i	− 1.13476i
$673$	47.0132i	1.81222i	0.423038	+	0.906112i	$0.360964\pi$
$673$	47.0132i	1.81222i	−0.423038	+	0.906112i	$0.639036\pi$
$674$	−11.8885	−0.457930
$675$	0	0
$676$	18.5623	0.713935
$677$	− 42.7214i	− 1.64192i	−0.570989	−	0.820958i	$-0.693440\pi$
$677$	− 42.7214i	− 1.64192i	0.570989	−	0.820958i	$-0.306560\pi$
$678$	4.18034i	0.160545i
$679$	8.23607	0.316071
$680$	0	0
$681$	3.05573	0.117096
$682$	1.23607i	0.0473315i
$683$	− 17.1803i	− 0.657387i	−0.944437	−	0.328694i	$-0.893392\pi$
$683$	− 17.1803i	− 0.657387i	0.944437	−	0.328694i	$-0.106608\pi$
$684$	−5.32624	−0.203654
$685$	0	0
$686$	10.3262	0.394258
$687$	− 16.5836i	− 0.632704i
$688$	− 6.00000i	− 0.228748i
$689$	1.88854	0.0719478
$690$	0	0
$691$	−19.1803	−0.729655	−0.364827	−	0.931075i	$-0.618872\pi$
$691$	−19.1803	−0.729655	−0.364827	+	0.931075i	$0.618872\pi$
$692$	− 24.1803i	− 0.919199i
$693$	12.4721i	0.473777i
$694$	1.12461	0.0426897
$695$	0	0
$696$	20.0000	0.758098
$697$	− 36.6525i	− 1.38831i
$698$	− 17.2361i	− 0.652395i
$699$	−0.0688837	−0.00260542
$700$	0	0
$701$	7.00000	0.264386	0.132193	−	0.991224i	$-0.457798\pi$
$701$	7.00000	0.264386	0.132193	+	0.991224i	$0.457798\pi$
$702$	− 4.22291i	− 0.159384i
$703$	− 4.47214i	− 0.168670i
$704$	0.472136	0.0177943
$705$	0	0
$706$	12.0000	0.451626
$707$	− 12.7082i	− 0.477941i
$708$	4.47214i	0.168073i
$709$	−34.4721	−1.29463	−0.647314	−	0.762223i	$-0.724108\pi$
$709$	−34.4721	−1.29463	−0.647314	+	0.762223i	$0.724108\pi$
$710$	0	0
$711$	−2.51471	−0.0943089
$712$	3.81966i	0.143148i
$713$	− 7.70820i	− 0.288675i
$714$	−16.9443	−0.634123
$715$	0	0
$716$	18.9443	0.707981
$717$	− 2.11146i	− 0.0788538i
$718$	− 10.9787i	− 0.409722i
$719$	36.1803	1.34930	0.674649	−	0.738138i	$-0.264295\pi$
$719$	36.1803	1.34930	0.674649	+	0.738138i	$0.264295\pi$
$720$	0	0
$721$	−7.47214	−0.278277
$722$	− 8.65248i	− 0.322012i
$723$	− 37.5279i	− 1.39568i
$724$	29.4164	1.09325
$725$	0	0
$726$	5.34752	0.198465
$727$	39.7639i	1.47476i	0.675477	+	0.737381i	$0.263938\pi$
$727$	39.7639i	1.47476i	−0.675477	+	0.737381i	$0.736062\pi$
$728$	− 11.7082i	− 0.433935i
$729$	−24.0557	−0.890953
$730$	0	0
$731$	−16.9443	−0.626707
$732$	− 28.3607i	− 1.04824i
$733$	− 5.47214i	− 0.202118i	−0.994880	−	0.101059i	$-0.967777\pi$
$733$	− 5.47214i	− 0.202118i	0.994880	−	0.101059i	$-0.0322231\pi$
$734$	11.1246	0.410617
$735$	0	0
$736$	43.3050	1.59624
$737$	− 16.0000i	− 0.589368i
$738$	6.36881i	0.234439i
$739$	16.1803	0.595203	0.297602	−	0.954690i	$-0.403813\pi$
$739$	16.1803	0.595203	0.297602	+	0.954690i	$0.403813\pi$
$740$	0	0
$741$	3.41641	0.125505
$742$	4.00000i	0.146845i
$743$	27.8197i	1.02060i	0.859995	+	0.510302i	$0.170466\pi$
$743$	27.8197i	1.02060i	−0.859995	+	0.510302i	$0.829534\pi$
$744$	−2.76393	−0.101331
$745$	0	0
$746$	−11.7426	−0.429929
$747$	4.33437i	0.158586i
$748$	− 16.9443i	− 0.619544i
$749$	43.3607	1.58436
$750$	0	0
$751$	45.5410	1.66182	0.830908	−	0.556410i	$-0.187822\pi$
$751$	45.5410	1.66182	0.830908	+	0.556410i	$0.187822\pi$
$752$	12.0000i	0.437595i
$753$	− 29.8885i	− 1.08920i
$754$	5.52786	0.201313
$755$	0	0
$756$	−37.8885	−1.37799
$757$	22.6525i	0.823318i	0.911338	+	0.411659i	$0.135051\pi$
$757$	22.6525i	0.823318i	−0.911338	+	0.411659i	$0.864949\pi$
$758$	23.4164i	0.850522i
$759$	19.0557	0.691679
$760$	0	0
$761$	2.00000	0.0724999	0.0362500	−	0.999343i	$-0.488459\pi$
$761$	2.00000	0.0724999	0.0362500	+	0.999343i	$0.488459\pi$
$762$	2.69505i	0.0976313i
$763$	− 16.7082i	− 0.604878i
$764$	5.14590	0.186172
$765$	0	0
$766$	−7.34752	−0.265477
$767$	2.76393i	0.0997998i
$768$	− 8.11146i	− 0.292697i
$769$	2.63932	0.0951763	0.0475882	−	0.998867i	$-0.484846\pi$
$769$	2.63932	0.0951763	0.0475882	+	0.998867i	$0.484846\pi$
$770$	0	0
$771$	−19.7082	−0.709774
$772$	− 8.85410i	− 0.318666i
$773$	29.1246i	1.04754i	0.851860	+	0.523770i	$0.175475\pi$
$773$	29.1246i	1.04754i	−0.851860	+	0.523770i	$0.824525\pi$
$774$	2.94427	0.105830
$775$	0	0
$776$	4.34752	0.156067
$777$	− 10.4721i	− 0.375686i
$778$	− 11.0557i	− 0.396367i
$779$	−15.6525	−0.560808
$780$	0	0
$781$	26.3607	0.943259
$782$	− 24.9443i	− 0.892005i
$783$	− 40.0000i	− 1.42948i
$784$	−20.2918	−0.724707
$785$	0	0
$786$	−9.16718	−0.326983
$787$	− 38.6525i	− 1.37781i	−0.724851	−	0.688906i	$-0.758091\pi$
$787$	− 38.6525i	− 1.37781i	0.724851	−	0.688906i	$-0.241909\pi$
$788$	24.9443i	0.888603i
$789$	23.1935	0.825710
$790$	0	0
$791$	23.1803	0.824198
$792$	6.58359i	0.233938i
$793$	− 17.5279i	− 0.622433i
$794$	−4.32624	−0.153532
$795$	0	0
$796$	1.70820	0.0605457
$797$	28.5836i	1.01248i	0.862392	+	0.506241i	$0.168966\pi$
$797$	28.5836i	1.01248i	−0.862392	+	0.506241i	$0.831034\pi$
$798$	7.23607i	0.256154i
$799$	33.8885	1.19889
$800$	0	0
$801$	2.51471	0.0888529
$802$	9.77709i	0.345241i
$803$	− 0.944272i	− 0.0333226i
$804$	16.0000	0.564276
$805$	0	0
$806$	−0.763932	−0.0269084
$807$	35.7771i	1.25941i
$808$	− 6.70820i	− 0.235994i
$809$	3.41641	0.120115	0.0600573	−	0.998195i	$-0.480872\pi$
$809$	3.41641	0.120115	0.0600573	+	0.998195i	$0.480872\pi$
$810$	0	0
$811$	−28.0000	−0.983213	−0.491606	−	0.870817i	$-0.663590\pi$
$811$	−28.0000	−0.983213	−0.491606	+	0.870817i	$0.663590\pi$
$812$	− 49.5967i	− 1.74050i
$813$	10.1115i	0.354624i
$814$	−2.47214	−0.0866483
$815$	0	0
$816$	12.0000	0.420084
$817$	7.23607i	0.253158i
$818$	16.1803i	0.565732i
$819$	−7.70820	−0.269346
$820$	0	0
$821$	−36.5410	−1.27529	−0.637645	−	0.770330i	$-0.720091\pi$
$821$	−36.5410	−1.27529	−0.637645	+	0.770330i	$0.720091\pi$
$822$	15.0557i	0.525129i
$823$	− 27.7082i	− 0.965847i	−0.875662	−	0.482924i	$-0.839575\pi$
$823$	− 27.7082i	− 0.965847i	0.875662	−	0.482924i	$-0.160425\pi$
$824$	−3.94427	−0.137405
$825$	0	0
$826$	−5.85410	−0.203690
$827$	− 48.6525i	− 1.69181i	−0.533332	−	0.845906i	$-0.679060\pi$
$827$	− 48.6525i	− 1.69181i	0.533332	−	0.845906i	$-0.320940\pi$
$828$	− 18.3607i	− 0.638078i
$829$	36.8328	1.27926	0.639628	−	0.768684i	$-0.279088\pi$
$829$	36.8328	1.27926	0.639628	+	0.768684i	$0.279088\pi$
$830$	0	0
$831$	23.0557	0.799794
$832$	0.291796i	0.0101162i
$833$	57.3050i	1.98550i
$834$	−10.2492	−0.354902
$835$	0	0
$836$	−7.23607	−0.250265
$837$	5.52786i	0.191071i
$838$	− 18.6180i	− 0.643149i
$839$	−11.0557	−0.381686	−0.190843	−	0.981621i	$-0.561122\pi$
$839$	−11.0557	−0.381686	−0.190843	+	0.981621i	$0.561122\pi$
$840$	0	0
$841$	23.3607	0.805541
$842$	− 9.49342i	− 0.327165i
$843$	21.0132i	0.723732i
$844$	1.32624	0.0456510
$845$	0	0
$846$	−5.88854	−0.202452
$847$	− 29.6525i	− 1.01887i
$848$	− 2.83282i	− 0.0972793i
$849$	−27.0557	−0.928550
$850$	0	0
$851$	15.4164	0.528468
$852$	26.3607i	0.903102i
$853$	37.4164i	1.28111i	0.767911	+	0.640557i	$0.221296\pi$
$853$	37.4164i	1.28111i	−0.767911	+	0.640557i	$0.778704\pi$
$854$	37.1246	1.27038
$855$	0	0
$856$	22.8885	0.782314
$857$	− 51.6656i	− 1.76486i	−0.470440	−	0.882432i	$-0.655905\pi$
$857$	− 51.6656i	− 1.76486i	0.470440	−	0.882432i	$-0.344095\pi$
$858$	− 1.88854i	− 0.0644738i
$859$	−37.8885	−1.29274	−0.646370	−	0.763024i	$-0.723714\pi$
$859$	−37.8885	−1.29274	−0.646370	+	0.763024i	$0.723714\pi$
$860$	0	0
$861$	−36.6525	−1.24911
$862$	7.41641i	0.252604i
$863$	− 32.1803i	− 1.09543i	−0.836664	−	0.547716i	$-0.815498\pi$
$863$	− 32.1803i	− 1.09543i	0.836664	−	0.547716i	$-0.184502\pi$
$864$	−31.0557	−1.05654
$865$	0	0
$866$	7.52786	0.255807
$867$	− 12.8754i	− 0.437271i
$868$	6.85410i	0.232643i
$869$	−3.41641	−0.115894
$870$	0	0
$871$	9.88854	0.335061
$872$	− 8.81966i	− 0.298671i
$873$	− 2.86223i	− 0.0968719i
$874$	−10.6525	−0.360325
$875$	0	0
$876$	0.944272	0.0319040
$877$	35.9443i	1.21375i	0.794797	+	0.606876i	$0.207578\pi$
$877$	35.9443i	1.21375i	−0.794797	+	0.606876i	$0.792422\pi$
$878$	− 13.0902i	− 0.441772i
$879$	−10.4721	−0.353216
$880$	0	0
$881$	24.3607	0.820732	0.410366	−	0.911921i	$-0.365401\pi$
$881$	24.3607	0.820732	0.410366	+	0.911921i	$0.365401\pi$
$882$	− 9.95743i	− 0.335284i
$883$	39.7771i	1.33861i	0.742990	+	0.669303i	$0.233407\pi$
$883$	39.7771i	1.33861i	−0.742990	+	0.669303i	$0.766593\pi$
$884$	10.4721	0.352216
$885$	0	0
$886$	−10.6869	−0.359034
$887$	31.0689i	1.04319i	0.853193	+	0.521596i	$0.174663\pi$
$887$	31.0689i	1.04319i	−0.853193	+	0.521596i	$0.825337\pi$
$888$	− 5.52786i	− 0.185503i
$889$	14.9443	0.501215
$890$	0	0
$891$	−4.83282	−0.161905
$892$	6.47214i	0.216703i
$893$	− 14.4721i	− 0.484292i
$894$	7.63932	0.255497
$895$	0	0
$896$	−48.2148	−1.61074
$897$	11.7771i	0.393226i
$898$	− 19.3475i	− 0.645635i
$899$	−7.23607	−0.241336
$900$	0	0
$901$	−8.00000	−0.266519
$902$	8.65248i	0.288096i
$903$	16.9443i	0.563870i
$904$	12.2361	0.406966
$905$	0	0
$906$	−6.24922	−0.207617
$907$	19.7639i	0.656251i	0.944634	+	0.328125i	$0.106417\pi$
$907$	19.7639i	0.656251i	−0.944634	+	0.328125i	$0.893583\pi$
$908$	− 4.00000i	− 0.132745i
$909$	−4.41641	−0.146483
$910$	0	0
$911$	−4.18034	−0.138501	−0.0692504	−	0.997599i	$-0.522061\pi$
$911$	−4.18034	−0.138501	−0.0692504	+	0.997599i	$0.522061\pi$
$912$	− 5.12461i	− 0.169693i
$913$	5.88854i	0.194882i
$914$	−12.9443	−0.428158
$915$	0	0
$916$	−21.7082	−0.717259
$917$	50.8328i	1.67865i
$918$	17.8885i	0.590410i
$919$	5.52786	0.182347	0.0911737	−	0.995835i	$-0.470938\pi$
$919$	5.52786	0.182347	0.0911737	+	0.995835i	$0.470938\pi$
$920$	0	0
$921$	−18.9017	−0.622832
$922$	− 6.40325i	− 0.210880i
$923$	16.2918i	0.536251i
$924$	−16.9443	−0.557426
$925$	0	0
$926$	18.1803	0.597443
$927$	2.59675i	0.0852884i
$928$	− 40.6525i	− 1.33448i
$929$	20.0000	0.656179	0.328089	−	0.944647i	$-0.393595\pi$
$929$	20.0000	0.656179	0.328089	+	0.944647i	$0.393595\pi$
$930$	0	0
$931$	24.4721	0.802042
$932$	0.0901699i	0.00295361i
$933$	− 8.42956i	− 0.275972i
$934$	−5.38197	−0.176103
$935$	0	0
$936$	−4.06888	−0.132996
$937$	− 26.9443i	− 0.880231i	−0.897941	−	0.440115i	$-0.854937\pi$
$937$	− 26.9443i	− 0.880231i	0.897941	−	0.440115i	$-0.145063\pi$
$938$	20.9443i	0.683855i
$939$	−26.2492	−0.856611
$940$	0	0
$941$	−38.0000	−1.23876	−0.619382	−	0.785090i	$-0.712617\pi$
$941$	−38.0000	−1.23876	−0.619382	+	0.785090i	$0.712617\pi$
$942$	− 11.3738i	− 0.370580i
$943$	− 53.9574i	− 1.75710i
$944$	4.14590	0.134937
$945$	0	0
$946$	4.00000	0.130051
$947$	30.9443i	1.00555i	0.864416	+	0.502777i	$0.167688\pi$
$947$	30.9443i	1.00555i	−0.864416	+	0.502777i	$0.832312\pi$
$948$	− 3.41641i	− 0.110960i
$949$	0.583592	0.0189442
$950$	0	0
$951$	27.1246	0.879576
$952$	49.5967i	1.60744i
$953$	32.2918i	1.04603i	0.852322	+	0.523017i	$0.175194\pi$
$953$	32.2918i	1.04603i	−0.852322	+	0.523017i	$0.824806\pi$
$954$	1.39010	0.0450060
$955$	0	0
$956$	−2.76393	−0.0893920
$957$	− 17.8885i	− 0.578254i
$958$	− 22.6869i	− 0.732981i
$959$	83.4853	2.69588
$960$	0	0
$961$	1.00000	0.0322581
$962$	− 1.52786i	− 0.0492603i
$963$	− 15.0689i	− 0.485588i
$964$	−49.1246	−1.58220
$965$	0	0
$966$	−24.9443	−0.802569
$967$	− 15.6393i	− 0.502927i	−0.967867	−	0.251463i	$-0.919088\pi$
$967$	− 15.6393i	− 0.502927i	0.967867	−	0.251463i	$-0.0809118\pi$
$968$	− 15.6525i	− 0.503090i
$969$	−14.4721	−0.464912
$970$	0	0
$971$	−28.0000	−0.898563	−0.449281	−	0.893390i	$-0.648320\pi$
$971$	−28.0000	−0.898563	−0.449281	+	0.893390i	$0.648320\pi$
$972$	22.0000i	0.705650i
$973$	56.8328i	1.82198i
$974$	−9.12461	−0.292371
$975$	0	0
$976$	−26.2918	−0.841580
$977$	− 33.2492i	− 1.06374i	−0.846827	−	0.531868i	$-0.821490\pi$
$977$	− 33.2492i	− 1.06374i	0.846827	−	0.531868i	$-0.178510\pi$
$978$	2.06888i	0.0661556i
$979$	3.41641	0.109189
$980$	0	0
$981$	−5.80650	−0.185387
$982$	− 24.9443i	− 0.796004i
$983$	48.4721i	1.54602i	0.634393	+	0.773011i	$0.281250\pi$
$983$	48.4721i	1.54602i	−0.634393	+	0.773011i	$0.718750\pi$
$984$	−19.3475	−0.616777
$985$	0	0
$986$	−23.4164	−0.745730
$987$	− 33.8885i	− 1.07868i
$988$	− 4.47214i	− 0.142278i
$989$	−24.9443	−0.793182
$990$	0	0
$991$	50.5410	1.60549	0.802744	−	0.596324i	$-0.203372\pi$
$991$	50.5410	1.60549	0.802744	+	0.596324i	$0.203372\pi$
$992$	5.61803i	0.178373i
$993$	2.47214i	0.0784509i
$994$	−34.5066	−1.09448
$995$	0	0
$996$	−5.88854	−0.186586
$997$	− 15.3607i	− 0.486478i	−0.969966	−	0.243239i	$-0.921790\pi$
$997$	− 15.3607i	− 0.486478i	0.969966	−	0.243239i	$-0.0782099\pi$
$998$	20.6525i	0.653743i
$999$	−11.0557	−0.349788

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	775.2.b.d.249.3		4
5.2	odd	4		31.2.a.a.1.1	✓	2
5.3	odd	4		775.2.a.d.1.2		2
5.4	even	2	inner	775.2.b.d.249.2		4
15.2	even	4		279.2.a.a.1.2		2
15.8	even	4		6975.2.a.y.1.1		2
20.7	even	4		496.2.a.i.1.1		2
35.27	even	4		1519.2.a.a.1.1		2
40.27	even	4		1984.2.a.n.1.2		2
40.37	odd	4		1984.2.a.r.1.1		2
55.32	even	4		3751.2.a.b.1.2		2
60.47	odd	4		4464.2.a.bf.1.2		2
65.12	odd	4		5239.2.a.f.1.2		2
85.67	odd	4		8959.2.a.b.1.1		2
155.2	odd	20		961.2.d.c.531.1		4
155.7	odd	60		961.2.g.h.235.1		8
155.12	even	60		961.2.g.d.547.1		8
155.17	even	60		961.2.g.e.816.1		8
155.22	even	60		961.2.g.e.732.1		8
155.27	even	20		961.2.d.g.388.1		4
155.37	even	12		961.2.c.c.439.1		4
155.42	even	60		961.2.g.e.338.1		8
155.47	odd	20		961.2.d.c.628.1		4
155.52	even	60		961.2.g.d.844.1		8
155.57	even	12		961.2.c.c.521.1		4
155.67	odd	12		961.2.c.e.521.1		4
155.72	odd	60		961.2.g.a.844.1		8
155.77	even	20		961.2.d.a.628.1		4
155.82	odd	60		961.2.g.h.338.1		8
155.87	odd	12		961.2.c.e.439.1		4
155.92	even	4		961.2.a.f.1.1		2
155.97	odd	20		961.2.d.d.388.1		4
155.102	odd	60		961.2.g.h.732.1		8
155.107	odd	60		961.2.g.h.816.1		8
155.112	odd	60		961.2.g.a.547.1		8
155.117	even	60		961.2.g.e.235.1		8
155.122	even	20		961.2.d.a.531.1		4
155.127	even	60		961.2.g.d.846.1		8
155.132	odd	20		961.2.d.d.374.1		4
155.137	even	60		961.2.g.d.448.1		8
155.142	odd	60		961.2.g.a.448.1		8
155.147	even	20		961.2.d.g.374.1		4
155.152	odd	60		961.2.g.a.846.1		8
465.92	odd	4		8649.2.a.c.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
31.2.a.a.1.1	✓	2	5.2	odd	4
279.2.a.a.1.2		2	15.2	even	4
496.2.a.i.1.1		2	20.7	even	4
775.2.a.d.1.2		2	5.3	odd	4
775.2.b.d.249.2		4	5.4	even	2	inner
775.2.b.d.249.3		4	1.1	even	1	trivial
961.2.a.f.1.1		2	155.92	even	4
961.2.c.c.439.1		4	155.37	even	12
961.2.c.c.521.1		4	155.57	even	12
961.2.c.e.439.1		4	155.87	odd	12
961.2.c.e.521.1		4	155.67	odd	12
961.2.d.a.531.1		4	155.122	even	20
961.2.d.a.628.1		4	155.77	even	20
961.2.d.c.531.1		4	155.2	odd	20
961.2.d.c.628.1		4	155.47	odd	20
961.2.d.d.374.1		4	155.132	odd	20
961.2.d.d.388.1		4	155.97	odd	20
961.2.d.g.374.1		4	155.147	even	20
961.2.d.g.388.1		4	155.27	even	20
961.2.g.a.448.1		8	155.142	odd	60
961.2.g.a.547.1		8	155.112	odd	60
961.2.g.a.844.1		8	155.72	odd	60
961.2.g.a.846.1		8	155.152	odd	60
961.2.g.d.448.1		8	155.137	even	60
961.2.g.d.547.1		8	155.12	even	60
961.2.g.d.844.1		8	155.52	even	60
961.2.g.d.846.1		8	155.127	even	60
961.2.g.e.235.1		8	155.117	even	60
961.2.g.e.338.1		8	155.42	even	60
961.2.g.e.732.1		8	155.22	even	60
961.2.g.e.816.1		8	155.17	even	60
961.2.g.h.235.1		8	155.7	odd	60
961.2.g.h.338.1		8	155.82	odd	60
961.2.g.h.732.1		8	155.102	odd	60
961.2.g.h.816.1		8	155.107	odd	60
1519.2.a.a.1.1		2	35.27	even	4
1984.2.a.n.1.2		2	40.27	even	4
1984.2.a.r.1.1		2	40.37	odd	4
3751.2.a.b.1.2		2	55.32	even	4
4464.2.a.bf.1.2		2	60.47	odd	4
5239.2.a.f.1.2		2	65.12	odd	4
6975.2.a.y.1.1		2	15.8	even	4
8649.2.a.c.1.2		2	465.92	odd	4
8959.2.a.b.1.1		2	85.67	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:	\( N \)	\(=\)	\( 775 = 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	775.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.618034i q^{2} +1.23607i q^{3} +1.61803 q^{4} -0.763932 q^{6} +4.23607i q^{7} +2.23607i q^{8} +1.47214 q^{9} +2.00000 q^{11} +2.00000i q^{12} +1.23607i q^{13} -2.61803 q^{14} +1.85410 q^{16} -5.23607i q^{17} +0.909830i q^{18} -2.23607 q^{19} -5.23607 q^{21} +1.23607i q^{22} -7.70820i q^{23} -2.76393 q^{24} -0.763932 q^{26} +5.52786i q^{27} +6.85410i q^{28} -7.23607 q^{29} +1.00000 q^{31} +5.61803i q^{32} +2.47214i q^{33} +3.23607 q^{34} +2.38197 q^{36} +2.00000i q^{37} -1.38197i q^{38} -1.52786 q^{39} +7.00000 q^{41} -3.23607i q^{42} -3.23607i q^{43} +3.23607 q^{44} +4.76393 q^{46} +6.47214i q^{47} +2.29180i q^{48} -10.9443 q^{49} +6.47214 q^{51} +2.00000i q^{52} -1.52786i q^{53} -3.41641 q^{54} -9.47214 q^{56} -2.76393i q^{57} -4.47214i q^{58} +2.23607 q^{59} -14.1803 q^{61} +0.618034i q^{62} +6.23607i q^{63} +0.236068 q^{64} -1.52786 q^{66} -8.00000i q^{67} -8.47214i q^{68} +9.52786 q^{69} +13.1803 q^{71} +3.29180i q^{72} -0.472136i q^{73} -1.23607 q^{74} -3.61803 q^{76} +8.47214i q^{77} -0.944272i q^{78} -1.70820 q^{79} -2.41641 q^{81} +4.32624i q^{82} +2.94427i q^{83} -8.47214 q^{84} +2.00000 q^{86} -8.94427i q^{87} +4.47214i q^{88} +1.70820 q^{89} -5.23607 q^{91} -12.4721i q^{92} +1.23607i q^{93} -4.00000 q^{94} -6.94427 q^{96} -1.94427i q^{97} -6.76393i q^{98} +2.94427 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{4} - 12 q^{6} - 12 q^{9} + 8 q^{11} - 6 q^{14} - 6 q^{16} - 12 q^{21} - 20 q^{24} - 12 q^{26} - 20 q^{29} + 4 q^{31} + 4 q^{34} + 14 q^{36} - 24 q^{39} + 28 q^{41} + 4 q^{44} + 28 q^{46} - 8 q^{49}+ \cdots - 24 q^{99}+O(q^{100}) \) 4 * q + 2 * q^4 - 12 * q^6 - 12 * q^9 + 8 * q^11 - 6 * q^14 - 6 * q^16 - 12 * q^21 - 20 * q^24 - 12 * q^26 - 20 * q^29 + 4 * q^31 + 4 * q^34 + 14 * q^36 - 24 * q^39 + 28 * q^41 + 4 * q^44 + 28 * q^46 - 8 * q^49 + 8 * q^51 + 40 * q^54 - 20 * q^56 - 12 * q^61 - 8 * q^64 - 24 * q^66 + 56 * q^69 + 8 * q^71 + 4 * q^74 - 10 * q^76 + 20 * q^79 + 44 * q^81 - 16 * q^84 + 8 * q^86 - 20 * q^89 - 12 * q^91 - 16 * q^94 + 8 * q^96 - 24 * q^99

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