LMFDB - Embedded newform 425.2.b.c.324.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [425,2,Mod(324,425)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(425, 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("425.324");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 425 = 5^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	425.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:	$3.39364208590$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 85)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			324.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	425.324
Dual form			425.2.b.c.324.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000i q^{2} -2.00000i q^{3} +1.00000 q^{4} +2.00000 q^{6} -2.00000i q^{7} +3.00000i q^{8} -1.00000 q^{9} +2.00000 q^{11} -2.00000i q^{12} -2.00000i q^{13} +2.00000 q^{14} -1.00000 q^{16} +1.00000i q^{17} -1.00000i q^{18} -4.00000 q^{21} +2.00000i q^{22} -6.00000i q^{23} +6.00000 q^{24} +2.00000 q^{26} -4.00000i q^{27} -2.00000i q^{28} +6.00000 q^{29} -10.0000 q^{31} +5.00000i q^{32} -4.00000i q^{33} -1.00000 q^{34} -1.00000 q^{36} +2.00000i q^{37} -4.00000 q^{39} +10.0000 q^{41} -4.00000i q^{42} -4.00000i q^{43} +2.00000 q^{44} +6.00000 q^{46} +12.0000i q^{47} +2.00000i q^{48} +3.00000 q^{49} +2.00000 q^{51} -2.00000i q^{52} +10.0000i q^{53} +4.00000 q^{54} +6.00000 q^{56} +6.00000i q^{58} -8.00000 q^{59} -14.0000 q^{61} -10.0000i q^{62} +2.00000i q^{63} -7.00000 q^{64} +4.00000 q^{66} +8.00000i q^{67} +1.00000i q^{68} -12.0000 q^{69} -2.00000 q^{71} -3.00000i q^{72} +14.0000i q^{73} -2.00000 q^{74} -4.00000i q^{77} -4.00000i q^{78} +14.0000 q^{79} -11.0000 q^{81} +10.0000i q^{82} -4.00000i q^{83} -4.00000 q^{84} +4.00000 q^{86} -12.0000i q^{87} +6.00000i q^{88} -6.00000 q^{89} -4.00000 q^{91} -6.00000i q^{92} +20.0000i q^{93} -12.0000 q^{94} +10.0000 q^{96} +2.00000i q^{97} +3.00000i q^{98} -2.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{4} + 4 q^{6} - 2 q^{9} + 4 q^{11} + 4 q^{14} - 2 q^{16} - 8 q^{21} + 12 q^{24} + 4 q^{26} + 12 q^{29} - 20 q^{31} - 2 q^{34} - 2 q^{36} - 8 q^{39} + 20 q^{41} + 4 q^{44} + 12 q^{46} + 6 q^{49}+ \cdots - 4 q^{99}+O(q^{100}) $ 2 * q + 2 * q^4 + 4 * q^6 - 2 * q^9 + 4 * q^11 + 4 * q^14 - 2 * q^16 - 8 * q^21 + 12 * q^24 + 4 * q^26 + 12 * q^29 - 20 * q^31 - 2 * q^34 - 2 * q^36 - 8 * q^39 + 20 * q^41 + 4 * q^44 + 12 * q^46 + 6 * q^49 + 4 * q^51 + 8 * q^54 + 12 * q^56 - 16 * q^59 - 28 * q^61 - 14 * q^64 + 8 * q^66 - 24 * q^69 - 4 * q^71 - 4 * q^74 + 28 * q^79 - 22 * q^81 - 8 * q^84 + 8 * q^86 - 12 * q^89 - 8 * q^91 - 24 * q^94 + 20 * q^96 - 4 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/425\mathbb{Z}\right)^\times$.

$n$	$52$	$326$
$\chi(n)$	$-1$	$1$

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 $

$2$	1.00000i	0.707107i	0.935414	+	0.353553i	$0.115027\pi$
$2$	1.00000i	0.707107i	−0.935414	+	0.353553i	$0.884973\pi$
$3$	− 2.00000i	− 1.15470i	−0.816497	−	0.577350i	$-0.804087\pi$
$3$	− 2.00000i	− 1.15470i	0.816497	−	0.577350i	$-0.195913\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	2.00000	0.816497
$7$	− 2.00000i	− 0.755929i	−0.925820	−	0.377964i	$-0.876624\pi$
$7$	− 2.00000i	− 0.755929i	0.925820	−	0.377964i	$-0.123376\pi$
$8$	3.00000i	1.06066i
$9$	−1.00000	−0.333333
$10$	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
$13$	− 2.00000i	− 0.554700i	−0.960769	−	0.277350i	$-0.910544\pi$
$13$	− 2.00000i	− 0.554700i	0.960769	−	0.277350i	$-0.0894562\pi$
$14$	2.00000	0.534522
$15$	0	0
$16$	−1.00000	−0.250000
$17$	1.00000i	0.242536i
$17$	1.00000i	0.242536i
$18$	− 1.00000i	− 0.235702i
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	−4.00000	−0.872872
$22$	2.00000i	0.426401i
$23$	− 6.00000i	− 1.25109i	−0.780189	−	0.625543i	$-0.784877\pi$
$23$	− 6.00000i	− 1.25109i	0.780189	−	0.625543i	$-0.215123\pi$
$24$	6.00000	1.22474
$25$	0	0
$26$	2.00000	0.392232
$27$	− 4.00000i	− 0.769800i
$28$	− 2.00000i	− 0.377964i
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	−10.0000	−1.79605	−0.898027	−	0.439941i	$-0.854999\pi$
$31$	−10.0000	−1.79605	−0.898027	+	0.439941i	$0.854999\pi$
$32$	5.00000i	0.883883i
$33$	− 4.00000i	− 0.696311i
$34$	−1.00000	−0.171499
$35$	0	0
$36$	−1.00000	−0.166667
$37$	2.00000i	0.328798i	0.986394	+	0.164399i	$0.0525685\pi$
$37$	2.00000i	0.328798i	−0.986394	+	0.164399i	$0.947432\pi$
$38$	0	0
$39$	−4.00000	−0.640513
$40$	0	0
$41$	10.0000	1.56174	0.780869	−	0.624695i	$-0.214777\pi$
$41$	10.0000	1.56174	0.780869	+	0.624695i	$0.214777\pi$
$42$	− 4.00000i	− 0.617213i
$43$	− 4.00000i	− 0.609994i	−0.952353	−	0.304997i	$-0.901344\pi$
$43$	− 4.00000i	− 0.609994i	0.952353	−	0.304997i	$-0.0986555\pi$
$44$	2.00000	0.301511
$45$	0	0
$46$	6.00000	0.884652
$47$	12.0000i	1.75038i	0.483779	+	0.875190i	$0.339264\pi$
$47$	12.0000i	1.75038i	−0.483779	+	0.875190i	$0.660736\pi$
$48$	2.00000i	0.288675i
$49$	3.00000	0.428571
$50$	0	0
$51$	2.00000	0.280056
$52$	− 2.00000i	− 0.277350i
$53$	10.0000i	1.37361i	0.726844	+	0.686803i	$0.240986\pi$
$53$	10.0000i	1.37361i	−0.726844	+	0.686803i	$0.759014\pi$
$54$	4.00000	0.544331
$55$	0	0
$56$	6.00000	0.801784
$57$	0	0
$58$	6.00000i	0.787839i
$59$	−8.00000	−1.04151	−0.520756	−	0.853706i	$-0.674350\pi$
$59$	−8.00000	−1.04151	−0.520756	+	0.853706i	$0.674350\pi$
$60$	0	0
$61$	−14.0000	−1.79252	−0.896258	−	0.443533i	$-0.853725\pi$
$61$	−14.0000	−1.79252	−0.896258	+	0.443533i	$0.853725\pi$
$62$	− 10.0000i	− 1.27000i
$63$	2.00000i	0.251976i
$64$	−7.00000	−0.875000
$65$	0	0
$66$	4.00000	0.492366
$67$	8.00000i	0.977356i	0.872464	+	0.488678i	$0.162521\pi$
$67$	8.00000i	0.977356i	−0.872464	+	0.488678i	$0.837479\pi$
$68$	1.00000i	0.121268i
$69$	−12.0000	−1.44463
$70$	0	0
$71$	−2.00000	−0.237356	−0.118678	−	0.992933i	$-0.537866\pi$
$71$	−2.00000	−0.237356	−0.118678	+	0.992933i	$0.537866\pi$
$72$	− 3.00000i	− 0.353553i
$73$	14.0000i	1.63858i	0.573382	+	0.819288i	$0.305631\pi$
$73$	14.0000i	1.63858i	−0.573382	+	0.819288i	$0.694369\pi$
$74$	−2.00000	−0.232495
$75$	0	0
$76$	0	0
$77$	− 4.00000i	− 0.455842i
$78$	− 4.00000i	− 0.452911i
$79$	14.0000	1.57512	0.787562	−	0.616236i	$-0.211343\pi$
$79$	14.0000	1.57512	0.787562	+	0.616236i	$0.211343\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	10.0000i	1.10432i
$83$	− 4.00000i	− 0.439057i	−0.975606	−	0.219529i	$-0.929548\pi$
$83$	− 4.00000i	− 0.439057i	0.975606	−	0.219529i	$-0.0704519\pi$
$84$	−4.00000	−0.436436
$85$	0	0
$86$	4.00000	0.431331
$87$	− 12.0000i	− 1.28654i
$88$	6.00000i	0.639602i
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	−4.00000	−0.419314
$92$	− 6.00000i	− 0.625543i
$93$	20.0000i	2.07390i
$94$	−12.0000	−1.23771
$95$	0	0
$96$	10.0000	1.02062
$97$	2.00000i	0.203069i	0.994832	+	0.101535i	$0.0323753\pi$
$97$	2.00000i	0.203069i	−0.994832	+	0.101535i	$0.967625\pi$
$98$	3.00000i	0.303046i
$99$	−2.00000	−0.201008
$100$	0	0
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	2.00000i	0.198030i
$103$	− 12.0000i	− 1.18240i	−0.806527	−	0.591198i	$-0.798655\pi$
$103$	− 12.0000i	− 1.18240i	0.806527	−	0.591198i	$-0.201345\pi$
$104$	6.00000	0.588348
$105$	0	0
$106$	−10.0000	−0.971286
$107$	2.00000i	0.193347i	0.995316	+	0.0966736i	$0.0308203\pi$
$107$	2.00000i	0.193347i	−0.995316	+	0.0966736i	$0.969180\pi$
$108$	− 4.00000i	− 0.384900i
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	0	0
$111$	4.00000	0.379663
$112$	2.00000i	0.188982i
$113$	6.00000i	0.564433i	0.959351	+	0.282216i	$0.0910696\pi$
$113$	6.00000i	0.564433i	−0.959351	+	0.282216i	$0.908930\pi$
$114$	0	0
$115$	0	0
$116$	6.00000	0.557086
$117$	2.00000i	0.184900i
$118$	− 8.00000i	− 0.736460i
$119$	2.00000	0.183340
$120$	0	0
$121$	−7.00000	−0.636364
$122$	− 14.0000i	− 1.26750i
$123$	− 20.0000i	− 1.80334i
$124$	−10.0000	−0.898027
$125$	0	0
$126$	−2.00000	−0.178174
$127$	8.00000i	0.709885i	0.934888	+	0.354943i	$0.115500\pi$
$127$	8.00000i	0.709885i	−0.934888	+	0.354943i	$0.884500\pi$
$128$	3.00000i	0.265165i
$129$	−8.00000	−0.704361
$130$	0	0
$131$	6.00000	0.524222	0.262111	−	0.965038i	$-0.415581\pi$
$131$	6.00000	0.524222	0.262111	+	0.965038i	$0.415581\pi$
$132$	− 4.00000i	− 0.348155i
$133$	0	0
$134$	−8.00000	−0.691095
$135$	0	0
$136$	−3.00000	−0.257248
$137$	− 2.00000i	− 0.170872i	−0.996344	−	0.0854358i	$-0.972772\pi$
$137$	− 2.00000i	− 0.170872i	0.996344	−	0.0854358i	$-0.0272282\pi$
$138$	− 12.0000i	− 1.02151i
$139$	−14.0000	−1.18746	−0.593732	−	0.804663i	$-0.702346\pi$
$139$	−14.0000	−1.18746	−0.593732	+	0.804663i	$0.702346\pi$
$140$	0	0
$141$	24.0000	2.02116
$142$	− 2.00000i	− 0.167836i
$143$	− 4.00000i	− 0.334497i
$144$	1.00000	0.0833333
$145$	0	0
$146$	−14.0000	−1.15865
$147$	− 6.00000i	− 0.494872i
$148$	2.00000i	0.164399i
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	0	0
$151$	−12.0000	−0.976546	−0.488273	−	0.872691i	$-0.662373\pi$
$151$	−12.0000	−0.976546	−0.488273	+	0.872691i	$0.662373\pi$
$152$	0	0
$153$	− 1.00000i	− 0.0808452i
$154$	4.00000	0.322329
$155$	0	0
$156$	−4.00000	−0.320256
$157$	6.00000i	0.478852i	0.970915	+	0.239426i	$0.0769593\pi$
$157$	6.00000i	0.478852i	−0.970915	+	0.239426i	$0.923041\pi$
$158$	14.0000i	1.11378i
$159$	20.0000	1.58610
$160$	0	0
$161$	−12.0000	−0.945732
$162$	− 11.0000i	− 0.864242i
$163$	2.00000i	0.156652i	0.996928	+	0.0783260i	$0.0249575\pi$
$163$	2.00000i	0.156652i	−0.996928	+	0.0783260i	$0.975042\pi$
$164$	10.0000	0.780869
$165$	0	0
$166$	4.00000	0.310460
$167$	2.00000i	0.154765i	0.997001	+	0.0773823i	$0.0246562\pi$
$167$	2.00000i	0.154765i	−0.997001	+	0.0773823i	$0.975344\pi$
$168$	− 12.0000i	− 0.925820i
$169$	9.00000	0.692308
$170$	0	0
$171$	0	0
$172$	− 4.00000i	− 0.304997i
$173$	6.00000i	0.456172i	0.973641	+	0.228086i	$0.0732467\pi$
$173$	6.00000i	0.456172i	−0.973641	+	0.228086i	$0.926753\pi$
$174$	12.0000	0.909718
$175$	0	0
$176$	−2.00000	−0.150756
$177$	16.0000i	1.20263i
$178$	− 6.00000i	− 0.449719i
$179$	24.0000	1.79384	0.896922	−	0.442189i	$-0.145798\pi$
$179$	24.0000	1.79384	0.896922	+	0.442189i	$0.145798\pi$
$180$	0	0
$181$	26.0000	1.93256	0.966282	−	0.257485i	$-0.0828937\pi$
$181$	26.0000	1.93256	0.966282	+	0.257485i	$0.0828937\pi$
$182$	− 4.00000i	− 0.296500i
$183$	28.0000i	2.06982i
$184$	18.0000	1.32698
$185$	0	0
$186$	−20.0000	−1.46647
$187$	2.00000i	0.146254i
$188$	12.0000i	0.875190i
$189$	−8.00000	−0.581914
$190$	0	0
$191$	8.00000	0.578860	0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000	0.578860	0.289430	+	0.957199i	$0.406534\pi$
$192$	14.0000i	1.01036i
$193$	− 18.0000i	− 1.29567i	−0.761781	−	0.647834i	$-0.775675\pi$
$193$	− 18.0000i	− 1.29567i	0.761781	−	0.647834i	$-0.224325\pi$
$194$	−2.00000	−0.143592
$195$	0	0
$196$	3.00000	0.214286
$197$	2.00000i	0.142494i	0.997459	+	0.0712470i	$0.0226979\pi$
$197$	2.00000i	0.142494i	−0.997459	+	0.0712470i	$0.977302\pi$
$198$	− 2.00000i	− 0.142134i
$199$	−6.00000	−0.425329	−0.212664	−	0.977125i	$-0.568214\pi$
$199$	−6.00000	−0.425329	−0.212664	+	0.977125i	$0.568214\pi$
$200$	0	0
$201$	16.0000	1.12855
$202$	− 6.00000i	− 0.422159i
$203$	− 12.0000i	− 0.842235i
$204$	2.00000	0.140028
$205$	0	0
$206$	12.0000	0.836080
$207$	6.00000i	0.417029i
$208$	2.00000i	0.138675i
$209$	0	0
$210$	0	0
$211$	2.00000	0.137686	0.0688428	−	0.997628i	$-0.478069\pi$
$211$	2.00000	0.137686	0.0688428	+	0.997628i	$0.478069\pi$
$212$	10.0000i	0.686803i
$213$	4.00000i	0.274075i
$214$	−2.00000	−0.136717
$215$	0	0
$216$	12.0000	0.816497
$217$	20.0000i	1.35769i
$218$	− 2.00000i	− 0.135457i
$219$	28.0000	1.89206
$220$	0	0
$221$	2.00000	0.134535
$222$	4.00000i	0.268462i
$223$	− 16.0000i	− 1.07144i	−0.844396	−	0.535720i	$-0.820040\pi$
$223$	− 16.0000i	− 1.07144i	0.844396	−	0.535720i	$-0.179960\pi$
$224$	10.0000	0.668153
$225$	0	0
$226$	−6.00000	−0.399114
$227$	− 2.00000i	− 0.132745i	−0.997795	−	0.0663723i	$-0.978857\pi$
$227$	− 2.00000i	− 0.132745i	0.997795	−	0.0663723i	$-0.0211425\pi$
$228$	0	0
$229$	−10.0000	−0.660819	−0.330409	−	0.943838i	$-0.607187\pi$
$229$	−10.0000	−0.660819	−0.330409	+	0.943838i	$0.607187\pi$
$230$	0	0
$231$	−8.00000	−0.526361
$232$	18.0000i	1.18176i
$233$	14.0000i	0.917170i	0.888650	+	0.458585i	$0.151644\pi$
$233$	14.0000i	0.917170i	−0.888650	+	0.458585i	$0.848356\pi$
$234$	−2.00000	−0.130744
$235$	0	0
$236$	−8.00000	−0.520756
$237$	− 28.0000i	− 1.81880i
$238$	2.00000i	0.129641i
$239$	8.00000	0.517477	0.258738	−	0.965947i	$-0.416693\pi$
$239$	8.00000	0.517477	0.258738	+	0.965947i	$0.416693\pi$
$240$	0	0
$241$	−6.00000	−0.386494	−0.193247	−	0.981150i	$-0.561902\pi$
$241$	−6.00000	−0.386494	−0.193247	+	0.981150i	$0.561902\pi$
$242$	− 7.00000i	− 0.449977i
$243$	10.0000i	0.641500i
$244$	−14.0000	−0.896258
$245$	0	0
$246$	20.0000	1.27515
$247$	0	0
$248$	− 30.0000i	− 1.90500i
$249$	−8.00000	−0.506979
$250$	0	0
$251$	28.0000	1.76734	0.883672	−	0.468106i	$-0.155064\pi$
$251$	28.0000	1.76734	0.883672	+	0.468106i	$0.155064\pi$
$252$	2.00000i	0.125988i
$253$	− 12.0000i	− 0.754434i
$254$	−8.00000	−0.501965
$255$	0	0
$256$	−17.0000	−1.06250
$257$	− 2.00000i	− 0.124757i	−0.998053	−	0.0623783i	$-0.980131\pi$
$257$	− 2.00000i	− 0.124757i	0.998053	−	0.0623783i	$-0.0198685\pi$
$258$	− 8.00000i	− 0.498058i
$259$	4.00000	0.248548
$260$	0	0
$261$	−6.00000	−0.371391
$262$	6.00000i	0.370681i
$263$	8.00000i	0.493301i	0.969104	+	0.246651i	$0.0793300\pi$
$263$	8.00000i	0.493301i	−0.969104	+	0.246651i	$0.920670\pi$
$264$	12.0000	0.738549
$265$	0	0
$266$	0	0
$267$	12.0000i	0.734388i
$268$	8.00000i	0.488678i
$269$	−10.0000	−0.609711	−0.304855	−	0.952399i	$-0.598608\pi$
$269$	−10.0000	−0.609711	−0.304855	+	0.952399i	$0.598608\pi$
$270$	0	0
$271$	−16.0000	−0.971931	−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000	−0.971931	−0.485965	+	0.873978i	$0.661532\pi$
$272$	− 1.00000i	− 0.0606339i
$273$	8.00000i	0.484182i
$274$	2.00000	0.120824
$275$	0	0
$276$	−12.0000	−0.722315
$277$	− 30.0000i	− 1.80253i	−0.433273	−	0.901263i	$-0.642641\pi$
$277$	− 30.0000i	− 1.80253i	0.433273	−	0.901263i	$-0.357359\pi$
$278$	− 14.0000i	− 0.839664i
$279$	10.0000	0.598684
$280$	0	0
$281$	−6.00000	−0.357930	−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000	−0.357930	−0.178965	+	0.983855i	$0.557275\pi$
$282$	24.0000i	1.42918i
$283$	− 6.00000i	− 0.356663i	−0.983970	−	0.178331i	$-0.942930\pi$
$283$	− 6.00000i	− 0.356663i	0.983970	−	0.178331i	$-0.0570699\pi$
$284$	−2.00000	−0.118678
$285$	0	0
$286$	4.00000	0.236525
$287$	− 20.0000i	− 1.18056i
$288$	− 5.00000i	− 0.294628i
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	4.00000	0.234484
$292$	14.0000i	0.819288i
$293$	− 30.0000i	− 1.75262i	−0.481749	−	0.876309i	$-0.659998\pi$
$293$	− 30.0000i	− 1.75262i	0.481749	−	0.876309i	$-0.340002\pi$
$294$	6.00000	0.349927
$295$	0	0
$296$	−6.00000	−0.348743
$297$	− 8.00000i	− 0.464207i
$298$	− 6.00000i	− 0.347571i
$299$	−12.0000	−0.693978
$300$	0	0
$301$	−8.00000	−0.461112
$302$	− 12.0000i	− 0.690522i
$303$	12.0000i	0.689382i
$304$	0	0
$305$	0	0
$306$	1.00000	0.0571662
$307$	− 16.0000i	− 0.913168i	−0.889680	−	0.456584i	$-0.849073\pi$
$307$	− 16.0000i	− 0.913168i	0.889680	−	0.456584i	$-0.150927\pi$
$308$	− 4.00000i	− 0.227921i
$309$	−24.0000	−1.36531
$310$	0	0
$311$	6.00000	0.340229	0.170114	−	0.985424i	$-0.445586\pi$
$311$	6.00000	0.340229	0.170114	+	0.985424i	$0.445586\pi$
$312$	− 12.0000i	− 0.679366i
$313$	− 2.00000i	− 0.113047i	−0.998401	−	0.0565233i	$-0.981998\pi$
$313$	− 2.00000i	− 0.113047i	0.998401	−	0.0565233i	$-0.0180015\pi$
$314$	−6.00000	−0.338600
$315$	0	0
$316$	14.0000	0.787562
$317$	10.0000i	0.561656i	0.959758	+	0.280828i	$0.0906090\pi$
$317$	10.0000i	0.561656i	−0.959758	+	0.280828i	$0.909391\pi$
$318$	20.0000i	1.12154i
$319$	12.0000	0.671871
$320$	0	0
$321$	4.00000	0.223258
$322$	− 12.0000i	− 0.668734i
$323$	0	0
$324$	−11.0000	−0.611111
$325$	0	0
$326$	−2.00000	−0.110770
$327$	4.00000i	0.221201i
$328$	30.0000i	1.65647i
$329$	24.0000	1.32316
$330$	0	0
$331$	16.0000	0.879440	0.439720	−	0.898135i	$-0.355078\pi$
$331$	16.0000	0.879440	0.439720	+	0.898135i	$0.355078\pi$
$332$	− 4.00000i	− 0.219529i
$333$	− 2.00000i	− 0.109599i
$334$	−2.00000	−0.109435
$335$	0	0
$336$	4.00000	0.218218
$337$	− 6.00000i	− 0.326841i	−0.986557	−	0.163420i	$-0.947747\pi$
$337$	− 6.00000i	− 0.326841i	0.986557	−	0.163420i	$-0.0522527\pi$
$338$	9.00000i	0.489535i
$339$	12.0000	0.651751
$340$	0	0
$341$	−20.0000	−1.08306
$342$	0	0
$343$	− 20.0000i	− 1.07990i
$344$	12.0000	0.646997
$345$	0	0
$346$	−6.00000	−0.322562
$347$	− 26.0000i	− 1.39575i	−0.716218	−	0.697877i	$-0.754128\pi$
$347$	− 26.0000i	− 1.39575i	0.716218	−	0.697877i	$-0.245872\pi$
$348$	− 12.0000i	− 0.643268i
$349$	−22.0000	−1.17763	−0.588817	−	0.808267i	$-0.700406\pi$
$349$	−22.0000	−1.17763	−0.588817	+	0.808267i	$0.700406\pi$
$350$	0	0
$351$	−8.00000	−0.427008
$352$	10.0000i	0.533002i
$353$	14.0000i	0.745145i	0.928003	+	0.372572i	$0.121524\pi$
$353$	14.0000i	0.745145i	−0.928003	+	0.372572i	$0.878476\pi$
$354$	−16.0000	−0.850390
$355$	0	0
$356$	−6.00000	−0.317999
$357$	− 4.00000i	− 0.211702i
$358$	24.0000i	1.26844i
$359$	−20.0000	−1.05556	−0.527780	−	0.849381i	$-0.676975\pi$
$359$	−20.0000	−1.05556	−0.527780	+	0.849381i	$0.676975\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	26.0000i	1.36653i
$363$	14.0000i	0.734809i
$364$	−4.00000	−0.209657
$365$	0	0
$366$	−28.0000	−1.46358
$367$	− 34.0000i	− 1.77479i	−0.461014	−	0.887393i	$-0.652514\pi$
$367$	− 34.0000i	− 1.77479i	0.461014	−	0.887393i	$-0.347486\pi$
$368$	6.00000i	0.312772i
$369$	−10.0000	−0.520579
$370$	0	0
$371$	20.0000	1.03835
$372$	20.0000i	1.03695i
$373$	− 10.0000i	− 0.517780i	−0.965907	−	0.258890i	$-0.916643\pi$
$373$	− 10.0000i	− 0.517780i	0.965907	−	0.258890i	$-0.0833568\pi$
$374$	−2.00000	−0.103418
$375$	0	0
$376$	−36.0000	−1.85656
$377$	− 12.0000i	− 0.618031i
$378$	− 8.00000i	− 0.411476i
$379$	2.00000	0.102733	0.0513665	−	0.998680i	$-0.483642\pi$
$379$	2.00000	0.102733	0.0513665	+	0.998680i	$0.483642\pi$
$380$	0	0
$381$	16.0000	0.819705
$382$	8.00000i	0.409316i
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	6.00000	0.306186
$385$	0	0
$386$	18.0000	0.916176
$387$	4.00000i	0.203331i
$388$	2.00000i	0.101535i
$389$	−26.0000	−1.31825	−0.659126	−	0.752032i	$-0.729074\pi$
$389$	−26.0000	−1.31825	−0.659126	+	0.752032i	$0.729074\pi$
$390$	0	0
$391$	6.00000	0.303433
$392$	9.00000i	0.454569i
$393$	− 12.0000i	− 0.605320i
$394$	−2.00000	−0.100759
$395$	0	0
$396$	−2.00000	−0.100504
$397$	− 14.0000i	− 0.702640i	−0.936255	−	0.351320i	$-0.885733\pi$
$397$	− 14.0000i	− 0.702640i	0.936255	−	0.351320i	$-0.114267\pi$
$398$	− 6.00000i	− 0.300753i
$399$	0	0
$400$	0	0
$401$	18.0000	0.898877	0.449439	−	0.893311i	$-0.351624\pi$
$401$	18.0000	0.898877	0.449439	+	0.893311i	$0.351624\pi$
$402$	16.0000i	0.798007i
$403$	20.0000i	0.996271i
$404$	−6.00000	−0.298511
$405$	0	0
$406$	12.0000	0.595550
$407$	4.00000i	0.198273i
$408$	6.00000i	0.297044i
$409$	38.0000	1.87898	0.939490	−	0.342578i	$-0.111300\pi$
$409$	38.0000	1.87898	0.939490	+	0.342578i	$0.111300\pi$
$410$	0	0
$411$	−4.00000	−0.197305
$412$	− 12.0000i	− 0.591198i
$413$	16.0000i	0.787309i
$414$	−6.00000	−0.294884
$415$	0	0
$416$	10.0000	0.490290
$417$	28.0000i	1.37117i
$418$	0	0
$419$	−18.0000	−0.879358	−0.439679	−	0.898155i	$-0.644908\pi$
$419$	−18.0000	−0.879358	−0.439679	+	0.898155i	$0.644908\pi$
$420$	0	0
$421$	26.0000	1.26716	0.633581	−	0.773676i	$-0.281584\pi$
$421$	26.0000	1.26716	0.633581	+	0.773676i	$0.281584\pi$
$422$	2.00000i	0.0973585i
$423$	− 12.0000i	− 0.583460i
$424$	−30.0000	−1.45693
$425$	0	0
$426$	−4.00000	−0.193801
$427$	28.0000i	1.35501i
$428$	2.00000i	0.0966736i
$429$	−8.00000	−0.386244
$430$	0	0
$431$	−38.0000	−1.83040	−0.915198	−	0.403005i	$-0.867966\pi$
$431$	−38.0000	−1.83040	−0.915198	+	0.403005i	$0.867966\pi$
$432$	4.00000i	0.192450i
$433$	34.0000i	1.63394i	0.576683	+	0.816968i	$0.304347\pi$
$433$	34.0000i	1.63394i	−0.576683	+	0.816968i	$0.695653\pi$
$434$	−20.0000	−0.960031
$435$	0	0
$436$	−2.00000	−0.0957826
$437$	0	0
$438$	28.0000i	1.33789i
$439$	−6.00000	−0.286364	−0.143182	−	0.989696i	$-0.545733\pi$
$439$	−6.00000	−0.286364	−0.143182	+	0.989696i	$0.545733\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	2.00000i	0.0951303i
$443$	− 24.0000i	− 1.14027i	−0.821549	−	0.570137i	$-0.806890\pi$
$443$	− 24.0000i	− 1.14027i	0.821549	−	0.570137i	$-0.193110\pi$
$444$	4.00000	0.189832
$445$	0	0
$446$	16.0000	0.757622
$447$	12.0000i	0.567581i
$448$	14.0000i	0.661438i
$449$	−18.0000	−0.849473	−0.424736	−	0.905317i	$-0.639633\pi$
$449$	−18.0000	−0.849473	−0.424736	+	0.905317i	$0.639633\pi$
$450$	0	0
$451$	20.0000	0.941763
$452$	6.00000i	0.282216i
$453$	24.0000i	1.12762i
$454$	2.00000	0.0938647
$455$	0	0
$456$	0	0
$457$	6.00000i	0.280668i	0.990104	+	0.140334i	$0.0448177\pi$
$457$	6.00000i	0.280668i	−0.990104	+	0.140334i	$0.955182\pi$
$458$	− 10.0000i	− 0.467269i
$459$	4.00000	0.186704
$460$	0	0
$461$	−26.0000	−1.21094	−0.605470	−	0.795868i	$-0.707015\pi$
$461$	−26.0000	−1.21094	−0.605470	+	0.795868i	$0.707015\pi$
$462$	− 8.00000i	− 0.372194i
$463$	36.0000i	1.67306i	0.547920	+	0.836531i	$0.315420\pi$
$463$	36.0000i	1.67306i	−0.547920	+	0.836531i	$0.684580\pi$
$464$	−6.00000	−0.278543
$465$	0	0
$466$	−14.0000	−0.648537
$467$	28.0000i	1.29569i	0.761774	+	0.647843i	$0.224329\pi$
$467$	28.0000i	1.29569i	−0.761774	+	0.647843i	$0.775671\pi$
$468$	2.00000i	0.0924500i
$469$	16.0000	0.738811
$470$	0	0
$471$	12.0000	0.552931
$472$	− 24.0000i	− 1.10469i
$473$	− 8.00000i	− 0.367840i
$474$	28.0000	1.28608
$475$	0	0
$476$	2.00000	0.0916698
$477$	− 10.0000i	− 0.457869i
$478$	8.00000i	0.365911i
$479$	18.0000	0.822441	0.411220	−	0.911536i	$-0.365103\pi$
$479$	18.0000	0.822441	0.411220	+	0.911536i	$0.365103\pi$
$480$	0	0
$481$	4.00000	0.182384
$482$	− 6.00000i	− 0.273293i
$483$	24.0000i	1.09204i
$484$	−7.00000	−0.318182
$485$	0	0
$486$	−10.0000	−0.453609
$487$	22.0000i	0.996915i	0.866914	+	0.498458i	$0.166100\pi$
$487$	22.0000i	0.996915i	−0.866914	+	0.498458i	$0.833900\pi$
$488$	− 42.0000i	− 1.90125i
$489$	4.00000	0.180886
$490$	0	0
$491$	−16.0000	−0.722070	−0.361035	−	0.932552i	$-0.617576\pi$
$491$	−16.0000	−0.722070	−0.361035	+	0.932552i	$0.617576\pi$
$492$	− 20.0000i	− 0.901670i
$493$	6.00000i	0.270226i
$494$	0	0
$495$	0	0
$496$	10.0000	0.449013
$497$	4.00000i	0.179425i
$498$	− 8.00000i	− 0.358489i
$499$	14.0000	0.626726	0.313363	−	0.949633i	$-0.398544\pi$
$499$	14.0000	0.626726	0.313363	+	0.949633i	$0.398544\pi$
$500$	0	0
$501$	4.00000	0.178707
$502$	28.0000i	1.24970i
$503$	− 6.00000i	− 0.267527i	−0.991013	−	0.133763i	$-0.957294\pi$
$503$	− 6.00000i	− 0.267527i	0.991013	−	0.133763i	$-0.0427062\pi$
$504$	−6.00000	−0.267261
$505$	0	0
$506$	12.0000	0.533465
$507$	− 18.0000i	− 0.799408i
$508$	8.00000i	0.354943i
$509$	34.0000	1.50702	0.753512	−	0.657434i	$-0.228358\pi$
$509$	34.0000	1.50702	0.753512	+	0.657434i	$0.228358\pi$
$510$	0	0
$511$	28.0000	1.23865
$512$	− 11.0000i	− 0.486136i
$513$	0	0
$514$	2.00000	0.0882162
$515$	0	0
$516$	−8.00000	−0.352180
$517$	24.0000i	1.05552i
$518$	4.00000i	0.175750i
$519$	12.0000	0.526742
$520$	0	0
$521$	−30.0000	−1.31432	−0.657162	−	0.753749i	$-0.728243\pi$
$521$	−30.0000	−1.31432	−0.657162	+	0.753749i	$0.728243\pi$
$522$	− 6.00000i	− 0.262613i
$523$	− 32.0000i	− 1.39926i	−0.714504	−	0.699631i	$-0.753348\pi$
$523$	− 32.0000i	− 1.39926i	0.714504	−	0.699631i	$-0.246652\pi$
$524$	6.00000	0.262111
$525$	0	0
$526$	−8.00000	−0.348817
$527$	− 10.0000i	− 0.435607i
$528$	4.00000i	0.174078i
$529$	−13.0000	−0.565217
$530$	0	0
$531$	8.00000	0.347170
$532$	0	0
$533$	− 20.0000i	− 0.866296i
$534$	−12.0000	−0.519291
$535$	0	0
$536$	−24.0000	−1.03664
$537$	− 48.0000i	− 2.07135i
$538$	− 10.0000i	− 0.431131i
$539$	6.00000	0.258438
$540$	0	0
$541$	−14.0000	−0.601907	−0.300954	−	0.953639i	$-0.597305\pi$
$541$	−14.0000	−0.601907	−0.300954	+	0.953639i	$0.597305\pi$
$542$	− 16.0000i	− 0.687259i
$543$	− 52.0000i	− 2.23153i
$544$	−5.00000	−0.214373
$545$	0	0
$546$	−8.00000	−0.342368
$547$	− 22.0000i	− 0.940652i	−0.882493	−	0.470326i	$-0.844136\pi$
$547$	− 22.0000i	− 0.940652i	0.882493	−	0.470326i	$-0.155864\pi$
$548$	− 2.00000i	− 0.0854358i
$549$	14.0000	0.597505
$550$	0	0
$551$	0	0
$552$	− 36.0000i	− 1.53226i
$553$	− 28.0000i	− 1.19068i
$554$	30.0000	1.27458
$555$	0	0
$556$	−14.0000	−0.593732
$557$	2.00000i	0.0847427i	0.999102	+	0.0423714i	$0.0134913\pi$
$557$	2.00000i	0.0847427i	−0.999102	+	0.0423714i	$0.986509\pi$
$558$	10.0000i	0.423334i
$559$	−8.00000	−0.338364
$560$	0	0
$561$	4.00000	0.168880
$562$	− 6.00000i	− 0.253095i
$563$	− 4.00000i	− 0.168580i	−0.996441	−	0.0842900i	$-0.973138\pi$
$563$	− 4.00000i	− 0.168580i	0.996441	−	0.0842900i	$-0.0268622\pi$
$564$	24.0000	1.01058
$565$	0	0
$566$	6.00000	0.252199
$567$	22.0000i	0.923913i
$568$	− 6.00000i	− 0.251754i
$569$	38.0000	1.59304	0.796521	−	0.604610i	$-0.206671\pi$
$569$	38.0000	1.59304	0.796521	+	0.604610i	$0.206671\pi$
$570$	0	0
$571$	18.0000	0.753277	0.376638	−	0.926360i	$-0.377080\pi$
$571$	18.0000	0.753277	0.376638	+	0.926360i	$0.377080\pi$
$572$	− 4.00000i	− 0.167248i
$573$	− 16.0000i	− 0.668410i
$574$	20.0000	0.834784
$575$	0	0
$576$	7.00000	0.291667
$577$	6.00000i	0.249783i	0.992170	+	0.124892i	$0.0398583\pi$
$577$	6.00000i	0.249783i	−0.992170	+	0.124892i	$0.960142\pi$
$578$	− 1.00000i	− 0.0415945i
$579$	−36.0000	−1.49611
$580$	0	0
$581$	−8.00000	−0.331896
$582$	4.00000i	0.165805i
$583$	20.0000i	0.828315i
$584$	−42.0000	−1.73797
$585$	0	0
$586$	30.0000	1.23929
$587$	− 12.0000i	− 0.495293i	−0.968850	−	0.247647i	$-0.920343\pi$
$587$	− 12.0000i	− 0.495293i	0.968850	−	0.247647i	$-0.0796572\pi$
$588$	− 6.00000i	− 0.247436i
$589$	0	0
$590$	0	0
$591$	4.00000	0.164538
$592$	− 2.00000i	− 0.0821995i
$593$	30.0000i	1.23195i	0.787765	+	0.615976i	$0.211238\pi$
$593$	30.0000i	1.23195i	−0.787765	+	0.615976i	$0.788762\pi$
$594$	8.00000	0.328244
$595$	0	0
$596$	−6.00000	−0.245770
$597$	12.0000i	0.491127i
$598$	− 12.0000i	− 0.490716i
$599$	−16.0000	−0.653742	−0.326871	−	0.945069i	$-0.605994\pi$
$599$	−16.0000	−0.653742	−0.326871	+	0.945069i	$0.605994\pi$
$600$	0	0
$601$	10.0000	0.407909	0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000	0.407909	0.203954	+	0.978980i	$0.434621\pi$
$602$	− 8.00000i	− 0.326056i
$603$	− 8.00000i	− 0.325785i
$604$	−12.0000	−0.488273
$605$	0	0
$606$	−12.0000	−0.487467
$607$	18.0000i	0.730597i	0.930890	+	0.365299i	$0.119033\pi$
$607$	18.0000i	0.730597i	−0.930890	+	0.365299i	$0.880967\pi$
$608$	0	0
$609$	−24.0000	−0.972529
$610$	0	0
$611$	24.0000	0.970936
$612$	− 1.00000i	− 0.0404226i
$613$	− 46.0000i	− 1.85792i	−0.370177	−	0.928961i	$-0.620703\pi$
$613$	− 46.0000i	− 1.85792i	0.370177	−	0.928961i	$-0.379297\pi$
$614$	16.0000	0.645707
$615$	0	0
$616$	12.0000	0.483494
$617$	42.0000i	1.69086i	0.534089	+	0.845428i	$0.320655\pi$
$617$	42.0000i	1.69086i	−0.534089	+	0.845428i	$0.679345\pi$
$618$	− 24.0000i	− 0.965422i
$619$	14.0000	0.562708	0.281354	−	0.959604i	$-0.409217\pi$
$619$	14.0000	0.562708	0.281354	+	0.959604i	$0.409217\pi$
$620$	0	0
$621$	−24.0000	−0.963087
$622$	6.00000i	0.240578i
$623$	12.0000i	0.480770i
$624$	4.00000	0.160128
$625$	0	0
$626$	2.00000	0.0799361
$627$	0	0
$628$	6.00000i	0.239426i
$629$	−2.00000	−0.0797452
$630$	0	0
$631$	−4.00000	−0.159237	−0.0796187	−	0.996825i	$-0.525370\pi$
$631$	−4.00000	−0.159237	−0.0796187	+	0.996825i	$0.525370\pi$
$632$	42.0000i	1.67067i
$633$	− 4.00000i	− 0.158986i
$634$	−10.0000	−0.397151
$635$	0	0
$636$	20.0000	0.793052
$637$	− 6.00000i	− 0.237729i
$638$	12.0000i	0.475085i
$639$	2.00000	0.0791188
$640$	0	0
$641$	−30.0000	−1.18493	−0.592464	−	0.805597i	$-0.701845\pi$
$641$	−30.0000	−1.18493	−0.592464	+	0.805597i	$0.701845\pi$
$642$	4.00000i	0.157867i
$643$	14.0000i	0.552106i	0.961142	+	0.276053i	$0.0890266\pi$
$643$	14.0000i	0.552106i	−0.961142	+	0.276053i	$0.910973\pi$
$644$	−12.0000	−0.472866
$645$	0	0
$646$	0	0
$647$	28.0000i	1.10079i	0.834903	+	0.550397i	$0.185524\pi$
$647$	28.0000i	1.10079i	−0.834903	+	0.550397i	$0.814476\pi$
$648$	− 33.0000i	− 1.29636i
$649$	−16.0000	−0.628055
$650$	0	0
$651$	40.0000	1.56772
$652$	2.00000i	0.0783260i
$653$	14.0000i	0.547862i	0.961749	+	0.273931i	$0.0883240\pi$
$653$	14.0000i	0.547862i	−0.961749	+	0.273931i	$0.911676\pi$
$654$	−4.00000	−0.156412
$655$	0	0
$656$	−10.0000	−0.390434
$657$	− 14.0000i	− 0.546192i
$658$	24.0000i	0.935617i
$659$	44.0000	1.71400	0.856998	−	0.515319i	$-0.172327\pi$
$659$	44.0000	1.71400	0.856998	+	0.515319i	$0.172327\pi$
$660$	0	0
$661$	46.0000	1.78919	0.894596	−	0.446875i	$-0.147463\pi$
$661$	46.0000	1.78919	0.894596	+	0.446875i	$0.147463\pi$
$662$	16.0000i	0.621858i
$663$	− 4.00000i	− 0.155347i
$664$	12.0000	0.465690
$665$	0	0
$666$	2.00000	0.0774984
$667$	− 36.0000i	− 1.39393i
$668$	2.00000i	0.0773823i
$669$	−32.0000	−1.23719
$670$	0	0
$671$	−28.0000	−1.08093
$672$	− 20.0000i	− 0.771517i
$673$	− 10.0000i	− 0.385472i	−0.981251	−	0.192736i	$-0.938264\pi$
$673$	− 10.0000i	− 0.385472i	0.981251	−	0.192736i	$-0.0617360\pi$
$674$	6.00000	0.231111
$675$	0	0
$676$	9.00000	0.346154
$677$	− 30.0000i	− 1.15299i	−0.817099	−	0.576497i	$-0.804419\pi$
$677$	− 30.0000i	− 1.15299i	0.817099	−	0.576497i	$-0.195581\pi$
$678$	12.0000i	0.460857i
$679$	4.00000	0.153506
$680$	0	0
$681$	−4.00000	−0.153280
$682$	− 20.0000i	− 0.765840i
$683$	− 18.0000i	− 0.688751i	−0.938832	−	0.344375i	$-0.888091\pi$
$683$	− 18.0000i	− 0.688751i	0.938832	−	0.344375i	$-0.111909\pi$
$684$	0	0
$685$	0	0
$686$	20.0000	0.763604
$687$	20.0000i	0.763048i
$688$	4.00000i	0.152499i
$689$	20.0000	0.761939
$690$	0	0
$691$	2.00000	0.0760836	0.0380418	−	0.999276i	$-0.487888\pi$
$691$	2.00000	0.0760836	0.0380418	+	0.999276i	$0.487888\pi$
$692$	6.00000i	0.228086i
$693$	4.00000i	0.151947i
$694$	26.0000	0.986947
$695$	0	0
$696$	36.0000	1.36458
$697$	10.0000i	0.378777i
$698$	− 22.0000i	− 0.832712i
$699$	28.0000	1.05906
$700$	0	0
$701$	−30.0000	−1.13308	−0.566542	−	0.824033i	$-0.691719\pi$
$701$	−30.0000	−1.13308	−0.566542	+	0.824033i	$0.691719\pi$
$702$	− 8.00000i	− 0.301941i
$703$	0	0
$704$	−14.0000	−0.527645
$705$	0	0
$706$	−14.0000	−0.526897
$707$	12.0000i	0.451306i
$708$	16.0000i	0.601317i
$709$	6.00000	0.225335	0.112667	−	0.993633i	$-0.464061\pi$
$709$	6.00000	0.225335	0.112667	+	0.993633i	$0.464061\pi$
$710$	0	0
$711$	−14.0000	−0.525041
$712$	− 18.0000i	− 0.674579i
$713$	60.0000i	2.24702i
$714$	4.00000	0.149696
$715$	0	0
$716$	24.0000	0.896922
$717$	− 16.0000i	− 0.597531i
$718$	− 20.0000i	− 0.746393i
$719$	−14.0000	−0.522112	−0.261056	−	0.965324i	$-0.584071\pi$
$719$	−14.0000	−0.522112	−0.261056	+	0.965324i	$0.584071\pi$
$720$	0	0
$721$	−24.0000	−0.893807
$722$	− 19.0000i	− 0.707107i
$723$	12.0000i	0.446285i
$724$	26.0000	0.966282
$725$	0	0
$726$	−14.0000	−0.519589
$727$	− 12.0000i	− 0.445055i	−0.974926	−	0.222528i	$-0.928569\pi$
$727$	− 12.0000i	− 0.445055i	0.974926	−	0.222528i	$-0.0714308\pi$
$728$	− 12.0000i	− 0.444750i
$729$	−13.0000	−0.481481
$730$	0	0
$731$	4.00000	0.147945
$732$	28.0000i	1.03491i
$733$	2.00000i	0.0738717i	0.999318	+	0.0369358i	$0.0117597\pi$
$733$	2.00000i	0.0738717i	−0.999318	+	0.0369358i	$0.988240\pi$
$734$	34.0000	1.25496
$735$	0	0
$736$	30.0000	1.10581
$737$	16.0000i	0.589368i
$738$	− 10.0000i	− 0.368105i
$739$	8.00000	0.294285	0.147142	−	0.989115i	$-0.452992\pi$
$739$	8.00000	0.294285	0.147142	+	0.989115i	$0.452992\pi$
$740$	0	0
$741$	0	0
$742$	20.0000i	0.734223i
$743$	6.00000i	0.220119i	0.993925	+	0.110059i	$0.0351041\pi$
$743$	6.00000i	0.220119i	−0.993925	+	0.110059i	$0.964896\pi$
$744$	−60.0000	−2.19971
$745$	0	0
$746$	10.0000	0.366126
$747$	4.00000i	0.146352i
$748$	2.00000i	0.0731272i
$749$	4.00000	0.146157
$750$	0	0
$751$	−2.00000	−0.0729810	−0.0364905	−	0.999334i	$-0.511618\pi$
$751$	−2.00000	−0.0729810	−0.0364905	+	0.999334i	$0.511618\pi$
$752$	− 12.0000i	− 0.437595i
$753$	− 56.0000i	− 2.04075i
$754$	12.0000	0.437014
$755$	0	0
$756$	−8.00000	−0.290957
$757$	− 22.0000i	− 0.799604i	−0.916602	−	0.399802i	$-0.869079\pi$
$757$	− 22.0000i	− 0.799604i	0.916602	−	0.399802i	$-0.130921\pi$
$758$	2.00000i	0.0726433i
$759$	−24.0000	−0.871145
$760$	0	0
$761$	6.00000	0.217500	0.108750	−	0.994069i	$-0.465315\pi$
$761$	6.00000	0.217500	0.108750	+	0.994069i	$0.465315\pi$
$762$	16.0000i	0.579619i
$763$	4.00000i	0.144810i
$764$	8.00000	0.289430
$765$	0	0
$766$	0	0
$767$	16.0000i	0.577727i
$768$	34.0000i	1.22687i
$769$	18.0000	0.649097	0.324548	−	0.945869i	$-0.394788\pi$
$769$	18.0000	0.649097	0.324548	+	0.945869i	$0.394788\pi$
$770$	0	0
$771$	−4.00000	−0.144056
$772$	− 18.0000i	− 0.647834i
$773$	6.00000i	0.215805i	0.994161	+	0.107903i	$0.0344134\pi$
$773$	6.00000i	0.215805i	−0.994161	+	0.107903i	$0.965587\pi$
$774$	−4.00000	−0.143777
$775$	0	0
$776$	−6.00000	−0.215387
$777$	− 8.00000i	− 0.286998i
$778$	− 26.0000i	− 0.932145i
$779$	0	0
$780$	0	0
$781$	−4.00000	−0.143131
$782$	6.00000i	0.214560i
$783$	− 24.0000i	− 0.857690i
$784$	−3.00000	−0.107143
$785$	0	0
$786$	12.0000	0.428026
$787$	− 6.00000i	− 0.213877i	−0.994266	−	0.106938i	$-0.965895\pi$
$787$	− 6.00000i	− 0.213877i	0.994266	−	0.106938i	$-0.0341048\pi$
$788$	2.00000i	0.0712470i
$789$	16.0000	0.569615
$790$	0	0
$791$	12.0000	0.426671
$792$	− 6.00000i	− 0.213201i
$793$	28.0000i	0.994309i
$794$	14.0000	0.496841
$795$	0	0
$796$	−6.00000	−0.212664
$797$	− 2.00000i	− 0.0708436i	−0.999372	−	0.0354218i	$-0.988723\pi$
$797$	− 2.00000i	− 0.0708436i	0.999372	−	0.0354218i	$-0.0112775\pi$
$798$	0	0
$799$	−12.0000	−0.424529
$800$	0	0
$801$	6.00000	0.212000
$802$	18.0000i	0.635602i
$803$	28.0000i	0.988099i
$804$	16.0000	0.564276
$805$	0	0
$806$	−20.0000	−0.704470
$807$	20.0000i	0.704033i
$808$	− 18.0000i	− 0.633238i
$809$	−26.0000	−0.914111	−0.457056	−	0.889438i	$-0.651096\pi$
$809$	−26.0000	−0.914111	−0.457056	+	0.889438i	$0.651096\pi$
$810$	0	0
$811$	−2.00000	−0.0702295	−0.0351147	−	0.999383i	$-0.511180\pi$
$811$	−2.00000	−0.0702295	−0.0351147	+	0.999383i	$0.511180\pi$
$812$	− 12.0000i	− 0.421117i
$813$	32.0000i	1.12229i
$814$	−4.00000	−0.140200
$815$	0	0
$816$	−2.00000	−0.0700140
$817$	0	0
$818$	38.0000i	1.32864i
$819$	4.00000	0.139771
$820$	0	0
$821$	2.00000	0.0698005	0.0349002	−	0.999391i	$-0.488889\pi$
$821$	2.00000	0.0698005	0.0349002	+	0.999391i	$0.488889\pi$
$822$	− 4.00000i	− 0.139516i
$823$	34.0000i	1.18517i	0.805510	+	0.592583i	$0.201892\pi$
$823$	34.0000i	1.18517i	−0.805510	+	0.592583i	$0.798108\pi$
$824$	36.0000	1.25412
$825$	0	0
$826$	−16.0000	−0.556711
$827$	− 42.0000i	− 1.46048i	−0.683189	−	0.730242i	$-0.739408\pi$
$827$	− 42.0000i	− 1.46048i	0.683189	−	0.730242i	$-0.260592\pi$
$828$	6.00000i	0.208514i
$829$	2.00000	0.0694629	0.0347314	−	0.999397i	$-0.488942\pi$
$829$	2.00000	0.0694629	0.0347314	+	0.999397i	$0.488942\pi$
$830$	0	0
$831$	−60.0000	−2.08138
$832$	14.0000i	0.485363i
$833$	3.00000i	0.103944i
$834$	−28.0000	−0.969561
$835$	0	0
$836$	0	0
$837$	40.0000i	1.38260i
$838$	− 18.0000i	− 0.621800i
$839$	−26.0000	−0.897620	−0.448810	−	0.893627i	$-0.648152\pi$
$839$	−26.0000	−0.897620	−0.448810	+	0.893627i	$0.648152\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	26.0000i	0.896019i
$843$	12.0000i	0.413302i
$844$	2.00000	0.0688428
$845$	0	0
$846$	12.0000	0.412568
$847$	14.0000i	0.481046i
$848$	− 10.0000i	− 0.343401i
$849$	−12.0000	−0.411839
$850$	0	0
$851$	12.0000	0.411355
$852$	4.00000i	0.137038i
$853$	− 34.0000i	− 1.16414i	−0.813139	−	0.582069i	$-0.802243\pi$
$853$	− 34.0000i	− 1.16414i	0.813139	−	0.582069i	$-0.197757\pi$
$854$	−28.0000	−0.958140
$855$	0	0
$856$	−6.00000	−0.205076
$857$	34.0000i	1.16142i	0.814111	+	0.580709i	$0.197225\pi$
$857$	34.0000i	1.16142i	−0.814111	+	0.580709i	$0.802775\pi$
$858$	− 8.00000i	− 0.273115i
$859$	24.0000	0.818869	0.409435	−	0.912339i	$-0.365726\pi$
$859$	24.0000	0.818869	0.409435	+	0.912339i	$0.365726\pi$
$860$	0	0
$861$	−40.0000	−1.36320
$862$	− 38.0000i	− 1.29429i
$863$	4.00000i	0.136162i	0.997680	+	0.0680808i	$0.0216876\pi$
$863$	4.00000i	0.136162i	−0.997680	+	0.0680808i	$0.978312\pi$
$864$	20.0000	0.680414
$865$	0	0
$866$	−34.0000	−1.15537
$867$	2.00000i	0.0679236i
$868$	20.0000i	0.678844i
$869$	28.0000	0.949835
$870$	0	0
$871$	16.0000	0.542139
$872$	− 6.00000i	− 0.203186i
$873$	− 2.00000i	− 0.0676897i
$874$	0	0
$875$	0	0
$876$	28.0000	0.946032
$877$	− 14.0000i	− 0.472746i	−0.971662	−	0.236373i	$-0.924041\pi$
$877$	− 14.0000i	− 0.472746i	0.971662	−	0.236373i	$-0.0759588\pi$
$878$	− 6.00000i	− 0.202490i
$879$	−60.0000	−2.02375
$880$	0	0
$881$	42.0000	1.41502	0.707508	−	0.706705i	$-0.249819\pi$
$881$	42.0000	1.41502	0.707508	+	0.706705i	$0.249819\pi$
$882$	− 3.00000i	− 0.101015i
$883$	− 16.0000i	− 0.538443i	−0.963078	−	0.269221i	$-0.913234\pi$
$883$	− 16.0000i	− 0.538443i	0.963078	−	0.269221i	$-0.0867663\pi$
$884$	2.00000	0.0672673
$885$	0	0
$886$	24.0000	0.806296
$887$	− 22.0000i	− 0.738688i	−0.929293	−	0.369344i	$-0.879582\pi$
$887$	− 22.0000i	− 0.738688i	0.929293	−	0.369344i	$-0.120418\pi$
$888$	12.0000i	0.402694i
$889$	16.0000	0.536623
$890$	0	0
$891$	−22.0000	−0.737028
$892$	− 16.0000i	− 0.535720i
$893$	0	0
$894$	−12.0000	−0.401340
$895$	0	0
$896$	6.00000	0.200446
$897$	24.0000i	0.801337i
$898$	− 18.0000i	− 0.600668i
$899$	−60.0000	−2.00111
$900$	0	0
$901$	−10.0000	−0.333148
$902$	20.0000i	0.665927i
$903$	16.0000i	0.532447i
$904$	−18.0000	−0.598671
$905$	0	0
$906$	−24.0000	−0.797347
$907$	30.0000i	0.996134i	0.867139	+	0.498067i	$0.165957\pi$
$907$	30.0000i	0.996134i	−0.867139	+	0.498067i	$0.834043\pi$
$908$	− 2.00000i	− 0.0663723i
$909$	6.00000	0.199007
$910$	0	0
$911$	−38.0000	−1.25900	−0.629498	−	0.777002i	$-0.716739\pi$
$911$	−38.0000	−1.25900	−0.629498	+	0.777002i	$0.716739\pi$
$912$	0	0
$913$	− 8.00000i	− 0.264761i
$914$	−6.00000	−0.198462
$915$	0	0
$916$	−10.0000	−0.330409
$917$	− 12.0000i	− 0.396275i
$918$	4.00000i	0.132020i
$919$	−16.0000	−0.527791	−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000	−0.527791	−0.263896	+	0.964551i	$0.585007\pi$
$920$	0	0
$921$	−32.0000	−1.05444
$922$	− 26.0000i	− 0.856264i
$923$	4.00000i	0.131662i
$924$	−8.00000	−0.263181
$925$	0	0
$926$	−36.0000	−1.18303
$927$	12.0000i	0.394132i
$928$	30.0000i	0.984798i
$929$	−58.0000	−1.90292	−0.951459	−	0.307775i	$-0.900416\pi$
$929$	−58.0000	−1.90292	−0.951459	+	0.307775i	$0.900416\pi$
$930$	0	0
$931$	0	0
$932$	14.0000i	0.458585i
$933$	− 12.0000i	− 0.392862i
$934$	−28.0000	−0.916188
$935$	0	0
$936$	−6.00000	−0.196116
$937$	− 38.0000i	− 1.24141i	−0.784046	−	0.620703i	$-0.786847\pi$
$937$	− 38.0000i	− 1.24141i	0.784046	−	0.620703i	$-0.213153\pi$
$938$	16.0000i	0.522419i
$939$	−4.00000	−0.130535
$940$	0	0
$941$	34.0000	1.10837	0.554184	−	0.832394i	$-0.313030\pi$
$941$	34.0000	1.10837	0.554184	+	0.832394i	$0.313030\pi$
$942$	12.0000i	0.390981i
$943$	− 60.0000i	− 1.95387i
$944$	8.00000	0.260378
$945$	0	0
$946$	8.00000	0.260102
$947$	26.0000i	0.844886i	0.906389	+	0.422443i	$0.138827\pi$
$947$	26.0000i	0.844886i	−0.906389	+	0.422443i	$0.861173\pi$
$948$	− 28.0000i	− 0.909398i
$949$	28.0000	0.908918
$950$	0	0
$951$	20.0000	0.648544
$952$	6.00000i	0.194461i
$953$	18.0000i	0.583077i	0.956559	+	0.291539i	$0.0941672\pi$
$953$	18.0000i	0.583077i	−0.956559	+	0.291539i	$0.905833\pi$
$954$	10.0000	0.323762
$955$	0	0
$956$	8.00000	0.258738
$957$	− 24.0000i	− 0.775810i
$958$	18.0000i	0.581554i
$959$	−4.00000	−0.129167
$960$	0	0
$961$	69.0000	2.22581
$962$	4.00000i	0.128965i
$963$	− 2.00000i	− 0.0644491i
$964$	−6.00000	−0.193247
$965$	0	0
$966$	−24.0000	−0.772187
$967$	32.0000i	1.02905i	0.857475	+	0.514525i	$0.172032\pi$
$967$	32.0000i	1.02905i	−0.857475	+	0.514525i	$0.827968\pi$
$968$	− 21.0000i	− 0.674966i
$969$	0	0
$970$	0	0
$971$	−8.00000	−0.256732	−0.128366	−	0.991727i	$-0.540973\pi$
$971$	−8.00000	−0.256732	−0.128366	+	0.991727i	$0.540973\pi$
$972$	10.0000i	0.320750i
$973$	28.0000i	0.897639i
$974$	−22.0000	−0.704925
$975$	0	0
$976$	14.0000	0.448129
$977$	34.0000i	1.08776i	0.839164	+	0.543878i	$0.183045\pi$
$977$	34.0000i	1.08776i	−0.839164	+	0.543878i	$0.816955\pi$
$978$	4.00000i	0.127906i
$979$	−12.0000	−0.383522
$980$	0	0
$981$	2.00000	0.0638551
$982$	− 16.0000i	− 0.510581i
$983$	6.00000i	0.191370i	0.995412	+	0.0956851i	$0.0305042\pi$
$983$	6.00000i	0.191370i	−0.995412	+	0.0956851i	$0.969496\pi$
$984$	60.0000	1.91273
$985$	0	0
$986$	−6.00000	−0.191079
$987$	− 48.0000i	− 1.52786i
$988$	0	0
$989$	−24.0000	−0.763156
$990$	0	0
$991$	−58.0000	−1.84243	−0.921215	−	0.389053i	$-0.872802\pi$
$991$	−58.0000	−1.84243	−0.921215	+	0.389053i	$0.872802\pi$
$992$	− 50.0000i	− 1.58750i
$993$	− 32.0000i	− 1.01549i
$994$	−4.00000	−0.126872
$995$	0	0
$996$	−8.00000	−0.253490
$997$	− 62.0000i	− 1.96356i	−0.190022	−	0.981780i	$-0.560856\pi$
$997$	− 62.0000i	− 1.96356i	0.190022	−	0.981780i	$-0.439144\pi$
$998$	14.0000i	0.443162i
$999$	8.00000	0.253109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	425.2.b.c.324.2		2
5.2	odd	4		425.2.a.a.1.1		1
5.3	odd	4		85.2.a.a.1.1	✓	1
5.4	even	2	inner	425.2.b.c.324.1		2
15.2	even	4		3825.2.a.l.1.1		1
15.8	even	4		765.2.a.a.1.1		1
20.3	even	4		1360.2.a.b.1.1		1
20.7	even	4		6800.2.a.v.1.1		1
35.13	even	4		4165.2.a.l.1.1		1
40.3	even	4		5440.2.a.x.1.1		1
40.13	odd	4		5440.2.a.e.1.1		1
85.13	odd	4		1445.2.d.a.866.2		2
85.33	odd	4		1445.2.a.c.1.1		1
85.38	odd	4		1445.2.d.a.866.1		2
85.67	odd	4		7225.2.a.d.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
85.2.a.a.1.1	✓	1	5.3	odd	4
425.2.a.a.1.1		1	5.2	odd	4
425.2.b.c.324.1		2	5.4	even	2	inner
425.2.b.c.324.2		2	1.1	even	1	trivial
765.2.a.a.1.1		1	15.8	even	4
1360.2.a.b.1.1		1	20.3	even	4
1445.2.a.c.1.1		1	85.33	odd	4
1445.2.d.a.866.1		2	85.38	odd	4
1445.2.d.a.866.2		2	85.13	odd	4
3825.2.a.l.1.1		1	15.2	even	4
4165.2.a.l.1.1		1	35.13	even	4
5440.2.a.e.1.1		1	40.13	odd	4
5440.2.a.x.1.1		1	40.3	even	4
6800.2.a.v.1.1		1	20.7	even	4
7225.2.a.d.1.1		1	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:	\( N \)	\(=\)	\( 425 = 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	425.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -2.00000i q^{3} +1.00000 q^{4} +2.00000 q^{6} -2.00000i q^{7} +3.00000i q^{8} -1.00000 q^{9} +2.00000 q^{11} -2.00000i q^{12} -2.00000i q^{13} +2.00000 q^{14} -1.00000 q^{16} +1.00000i q^{17} -1.00000i q^{18} -4.00000 q^{21} +2.00000i q^{22} -6.00000i q^{23} +6.00000 q^{24} +2.00000 q^{26} -4.00000i q^{27} -2.00000i q^{28} +6.00000 q^{29} -10.0000 q^{31} +5.00000i q^{32} -4.00000i q^{33} -1.00000 q^{34} -1.00000 q^{36} +2.00000i q^{37} -4.00000 q^{39} +10.0000 q^{41} -4.00000i q^{42} -4.00000i q^{43} +2.00000 q^{44} +6.00000 q^{46} +12.0000i q^{47} +2.00000i q^{48} +3.00000 q^{49} +2.00000 q^{51} -2.00000i q^{52} +10.0000i q^{53} +4.00000 q^{54} +6.00000 q^{56} +6.00000i q^{58} -8.00000 q^{59} -14.0000 q^{61} -10.0000i q^{62} +2.00000i q^{63} -7.00000 q^{64} +4.00000 q^{66} +8.00000i q^{67} +1.00000i q^{68} -12.0000 q^{69} -2.00000 q^{71} -3.00000i q^{72} +14.0000i q^{73} -2.00000 q^{74} -4.00000i q^{77} -4.00000i q^{78} +14.0000 q^{79} -11.0000 q^{81} +10.0000i q^{82} -4.00000i q^{83} -4.00000 q^{84} +4.00000 q^{86} -12.0000i q^{87} +6.00000i q^{88} -6.00000 q^{89} -4.00000 q^{91} -6.00000i q^{92} +20.0000i q^{93} -12.0000 q^{94} +10.0000 q^{96} +2.00000i q^{97} +3.00000i q^{98} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{4} + 4 q^{6} - 2 q^{9} + 4 q^{11} + 4 q^{14} - 2 q^{16} - 8 q^{21} + 12 q^{24} + 4 q^{26} + 12 q^{29} - 20 q^{31} - 2 q^{34} - 2 q^{36} - 8 q^{39} + 20 q^{41} + 4 q^{44} + 12 q^{46} + 6 q^{49}+ \cdots - 4 q^{99}+O(q^{100}) \) 2 * q + 2 * q^4 + 4 * q^6 - 2 * q^9 + 4 * q^11 + 4 * q^14 - 2 * q^16 - 8 * q^21 + 12 * q^24 + 4 * q^26 + 12 * q^29 - 20 * q^31 - 2 * q^34 - 2 * q^36 - 8 * q^39 + 20 * q^41 + 4 * q^44 + 12 * q^46 + 6 * q^49 + 4 * q^51 + 8 * q^54 + 12 * q^56 - 16 * q^59 - 28 * q^61 - 14 * q^64 + 8 * q^66 - 24 * q^69 - 4 * q^71 - 4 * q^74 + 28 * q^79 - 22 * q^81 - 8 * q^84 + 8 * q^86 - 12 * q^89 - 8 * q^91 - 24 * q^94 + 20 * q^96 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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