LMFDB - Embedded newform 650.6.b.h.599.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [650,6,Mod(599,650)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(650, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("650.599");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$N$	$=$	$650 = 2 \cdot 5^{2} \cdot 13$
Weight:	$k$	$=$	$6$
Character orbit:	$[\chi]$	$=$	650.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:	$104.249482878$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{849})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} + 425x^{2} + 44944$ x^4 + 425*x^2 + 44944
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 26)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			599.2
Root			$15.0688i$ of defining polynomial
Character	$\chi$	$=$	650.599
Dual form			650.6.b.h.599.3

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.00000i q^{2} +19.0688i q^{3} -16.0000 q^{4} +76.2752 q^{6} +53.6192i q^{7} +64.0000i q^{8} -120.619 q^{9} +239.651 q^{11} -305.101i q^{12} -169.000i q^{13} +214.477 q^{14} +256.000 q^{16} +1973.88i q^{17} +482.477i q^{18} +373.872 q^{19} -1022.45 q^{21} -958.605i q^{22} +51.4306i q^{23} -1220.40 q^{24} -676.000 q^{26} +2333.65i q^{27} -857.908i q^{28} -4796.16 q^{29} +6906.01 q^{31} -1024.00i q^{32} +4569.86i q^{33} +7895.50 q^{34} +1929.91 q^{36} -11481.2i q^{37} -1495.49i q^{38} +3222.63 q^{39} +12547.8 q^{41} +4089.82i q^{42} +1156.07i q^{43} -3834.42 q^{44} +205.723 q^{46} +18644.9i q^{47} +4881.61i q^{48} +13932.0 q^{49} -37639.4 q^{51} +2704.00i q^{52} +9318.90i q^{53} +9334.62 q^{54} -3431.63 q^{56} +7129.29i q^{57} +19184.7i q^{58} +5066.63 q^{59} +54271.7 q^{61} -27624.0i q^{62} -6467.51i q^{63} -4096.00 q^{64} +18279.4 q^{66} -40241.0i q^{67} -31582.0i q^{68} -980.721 q^{69} -65236.5 q^{71} -7719.63i q^{72} +68506.2i q^{73} -45924.7 q^{74} -5981.95 q^{76} +12849.9i q^{77} -12890.5i q^{78} -10627.3 q^{79} -73810.5 q^{81} -50191.1i q^{82} -2357.98i q^{83} +16359.3 q^{84} +4624.27 q^{86} -91457.1i q^{87} +15337.7i q^{88} +93620.0 q^{89} +9061.65 q^{91} -822.890i q^{92} +131689. i q^{93} +74579.5 q^{94} +19526.5 q^{96} +31195.8i q^{97} -55727.9i q^{98} -28906.5 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$4 q - 64 q^{4} + 72 q^{6} + 42 q^{9} - 440 q^{11} - 1240 q^{14} + 1024 q^{16} + 4992 q^{19} - 6246 q^{21} - 1152 q^{24} - 2704 q^{26} - 3800 q^{29} + 5596 q^{31} + 1512 q^{34} - 672 q^{36} + 3042 q^{39}+ \cdots - 188004 q^{99}+O(q^{100})$ 4 * q - 64 * q^4 + 72 * q^6 + 42 * q^9 - 440 * q^11 - 1240 * q^14 + 1024 * q^16 + 4992 * q^19 - 6246 * q^21 - 1152 * q^24 - 2704 * q^26 - 3800 * q^29 + 5596 * q^31 + 1512 * q^34 - 672 * q^36 + 3042 * q^39 + 23268 * q^41 + 7040 * q^44 - 24352 * q^46 - 25566 * q^49 - 111222 * q^51 - 12312 * q^54 + 19840 * q^56 + 46840 * q^59 + 193660 * q^61 - 16384 * q^64 + 73584 * q^66 - 64296 * q^69 - 73358 * q^71 - 142440 * q^74 - 79872 * q^76 - 104048 * q^79 - 156780 * q^81 + 99936 * q^84 - 32552 * q^86 + 210144 * q^89 - 52390 * q^91 + 203912 * q^94 + 18432 * q^96 - 188004 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/650\mathbb{Z}\right)^\times$ .

$n$	$27$	$301$
$\chi(n)$	$-1$	$1$

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

)

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

$2$	− 4.00000i	− 0.707107i
$2$	− 4.00000i	− 0.707107i
$3$	19.0688i	1.22326i	0.791142	+	0.611632i	$0.209487\pi$
$3$	19.0688i	1.22326i	−0.791142	+	0.611632i	$0.790513\pi$
$4$	−16.0000	−0.500000
$5$	0	0
$5$	0	0
$6$	76.2752	0.864978
$7$	53.6192i	0.413595i	0.978384	+	0.206798i	$0.0663041\pi$
$7$	53.6192i	0.413595i	−0.978384	+	0.206798i	$0.933696\pi$
$8$	64.0000i	0.353553i
$9$	−120.619	−0.496375
$10$	0	0
$11$	239.651	0.597170	0.298585	−	0.954383i	$-0.403485\pi$
$11$	239.651	0.597170	0.298585	+	0.954383i	$0.403485\pi$
$12$	− 305.101i	− 0.611632i
$13$	− 169.000i	− 0.277350i
$13$	− 169.000i	− 0.277350i
$14$	214.477	0.292456
$15$	0	0
$16$	256.000	0.250000
$17$	1973.88i	1.65652i	0.560342	+	0.828261i	$0.310670\pi$
$17$	1973.88i	1.65652i	−0.560342	+	0.828261i	$0.689330\pi$
$18$	482.477i	0.350990i
$19$	373.872	0.237596	0.118798	−	0.992918i	$-0.462096\pi$
$19$	373.872	0.237596	0.118798	+	0.992918i	$0.462096\pi$
$20$	0	0
$21$	−1022.45	−0.505936
$22$	− 958.605i	− 0.422263i
$23$	51.4306i	0.0202723i	0.999949	+	0.0101361i	$0.00322649\pi$
$23$	51.4306i	0.0202723i	−0.999949	+	0.0101361i	$0.996774\pi$
$24$	−1220.40	−0.432489
$25$	0	0
$26$	−676.000	−0.196116
$27$	2333.65i	0.616066i
$28$	− 857.908i	− 0.206798i
$29$	−4796.16	−1.05901	−0.529504	−	0.848308i	$-0.677622\pi$
$29$	−4796.16	−1.05901	−0.529504	+	0.848308i	$0.677622\pi$
$30$	0	0
$31$	6906.01	1.29069	0.645346	−	0.763890i	$-0.276713\pi$
$31$	6906.01	1.29069	0.645346	+	0.763890i	$0.276713\pi$
$32$	− 1024.00i	− 0.176777i
$33$	4569.86i	0.730497i
$34$	7895.50	1.17134
$35$	0	0
$36$	1929.91	0.248188
$37$	− 11481.2i	− 1.37874i	−0.724410	−	0.689370i	$-0.757888\pi$
$37$	− 11481.2i	− 1.37874i	0.724410	−	0.689370i	$-0.242112\pi$
$38$	− 1495.49i	− 0.168006i
$39$	3222.63	0.339272
$40$	0	0
$41$	12547.8	1.16576	0.582878	−	0.812560i	$-0.301927\pi$
$41$	12547.8	1.16576	0.582878	+	0.812560i	$0.301927\pi$
$42$	4089.82i	0.357751i
$43$	1156.07i	0.0953481i	0.998863	+	0.0476741i	$0.0151809\pi$
$43$	1156.07i	0.0953481i	−0.998863	+	0.0476741i	$0.984819\pi$
$44$	−3834.42	−0.298585
$45$	0	0
$46$	205.723	0.0143347
$47$	18644.9i	1.23116i	0.788074	+	0.615580i	$0.211078\pi$
$47$	18644.9i	1.23116i	−0.788074	+	0.615580i	$0.788922\pi$
$48$	4881.61i	0.305816i
$49$	13932.0	0.828939
$50$	0	0
$51$	−37639.4	−2.02637
$52$	2704.00i	0.138675i
$53$	9318.90i	0.455696i	0.973697	+	0.227848i	$0.0731689\pi$
$53$	9318.90i	0.455696i	−0.973697	+	0.227848i	$0.926831\pi$
$54$	9334.62	0.435624
$55$	0	0
$56$	−3431.63	−0.146228
$57$	7129.29i	0.290642i
$58$	19184.7i	0.748831i
$59$	5066.63	0.189491	0.0947456	−	0.995502i	$-0.469796\pi$
$59$	5066.63	0.189491	0.0947456	+	0.995502i	$0.469796\pi$
$60$	0	0
$61$	54271.7	1.86745	0.933724	−	0.357994i	$-0.116539\pi$
$61$	54271.7	1.86745	0.933724	+	0.357994i	$0.116539\pi$
$62$	− 27624.0i	− 0.912657i
$63$	− 6467.51i	− 0.205298i
$64$	−4096.00	−0.125000
$65$	0	0
$66$	18279.4	0.516539
$67$	− 40241.0i	− 1.09517i	−0.836750	−	0.547585i	$-0.815547\pi$
$67$	− 40241.0i	− 1.09517i	0.836750	−	0.547585i	$-0.184453\pi$
$68$	− 31582.0i	− 0.828261i
$69$	−980.721	−0.0247983
$70$	0	0
$71$	−65236.5	−1.53584	−0.767918	−	0.640549i	$-0.778707\pi$
$71$	−65236.5	−1.53584	−0.767918	+	0.640549i	$0.778707\pi$
$72$	− 7719.63i	− 0.175495i
$73$	68506.2i	1.50461i	0.658818	+	0.752303i	$0.271057\pi$
$73$	68506.2i	1.50461i	−0.658818	+	0.752303i	$0.728943\pi$
$74$	−45924.7	−0.974916
$75$	0	0
$76$	−5981.95	−0.118798
$77$	12849.9i	0.246987i
$78$	− 12890.5i	− 0.239902i
$79$	−10627.3	−0.191583	−0.0957915	−	0.995401i	$-0.530538\pi$
$79$	−10627.3	−0.191583	−0.0957915	+	0.995401i	$0.530538\pi$
$80$	0	0
$81$	−73810.5	−1.24999
$82$	− 50191.1i	− 0.824314i
$83$	− 2357.98i	− 0.0375703i	−0.999824	−	0.0187851i	$-0.994020\pi$
$83$	− 2357.98i	− 0.0375703i	0.999824	−	0.0187851i	$-0.00597985\pi$
$84$	16359.3	0.252968
$85$	0	0
$86$	4624.27	0.0674213
$87$	− 91457.1i	− 1.29545i
$88$	15337.7i	0.211131i
$89$	93620.0	1.25283	0.626417	−	0.779488i	$-0.284521\pi$
$89$	93620.0	1.25283	0.626417	+	0.779488i	$0.284521\pi$
$90$	0	0
$91$	9061.65	0.114711
$92$	− 822.890i	− 0.0101361i
$93$	131689.i	1.57886i
$94$	74579.5	0.870562
$95$	0	0
$96$	19526.5	0.216245
$97$	31195.8i	0.336641i	0.985732	+	0.168321i	$0.0538344\pi$
$97$	31195.8i	0.336641i	−0.985732	+	0.168321i	$0.946166\pi$
$98$	− 55727.9i	− 0.586148i
$99$	−28906.5	−0.296421
$100$	0	0
$101$	−86948.6	−0.848124	−0.424062	−	0.905633i	$-0.639396\pi$
$101$	−86948.6	−0.848124	−0.424062	+	0.905633i	$0.639396\pi$
$102$	150558.i	1.43286i
$103$	− 43111.1i	− 0.400402i	−0.979755	−	0.200201i	$-0.935841\pi$
$103$	− 43111.1i	− 0.400402i	0.979755	−	0.200201i	$-0.0641595\pi$
$104$	10816.0	0.0980581
$105$	0	0
$106$	37275.6	0.322226
$107$	175966.i	1.48583i	0.669388	+	0.742913i	$0.266557\pi$
$107$	175966.i	1.48583i	−0.669388	+	0.742913i	$0.733443\pi$
$108$	− 37338.5i	− 0.308033i
$109$	−119578.	−0.964020	−0.482010	−	0.876166i	$-0.660093\pi$
$109$	−119578.	−0.964020	−0.482010	+	0.876166i	$0.660093\pi$
$110$	0	0
$111$	218932.	1.68656
$112$	13726.5i	0.103399i
$113$	47558.0i	0.350370i	0.984536	+	0.175185i	$0.0560524\pi$
$113$	47558.0i	0.350370i	−0.984536	+	0.175185i	$0.943948\pi$
$114$	28517.2	0.205515
$115$	0	0
$116$	76738.6	0.529504
$117$	20384.6i	0.137670i
$118$	− 20266.5i	− 0.133990i
$119$	−105838.	−0.685130
$120$	0	0
$121$	−103618.	−0.643388
$122$	− 217087.i	− 1.32049i
$123$	239271.i	1.42603i
$124$	−110496.	−0.645346
$125$	0	0
$126$	−25870.0	−0.145168
$127$	108774.i	0.598433i	0.954185	+	0.299217i	$0.0967253\pi$
$127$	108774.i	0.598433i	−0.954185	+	0.299217i	$0.903275\pi$
$128$	16384.0i	0.0883883i
$129$	−22044.8	−0.116636
$130$	0	0
$131$	−340486.	−1.73349	−0.866744	−	0.498753i	$-0.833791\pi$
$131$	−340486.	−1.73349	−0.866744	+	0.498753i	$0.833791\pi$
$132$	− 73117.8i	− 0.365248i
$133$	20046.7i	0.0982685i
$134$	−160964.	−0.774402
$135$	0	0
$136$	−126328.	−0.585669
$137$	− 48143.4i	− 0.219147i	−0.993979	−	0.109574i	$-0.965051\pi$
$137$	− 48143.4i	− 0.219147i	0.993979	−	0.109574i	$-0.0349485\pi$
$138$	3922.88i	0.0175351i
$139$	−323803.	−1.42149	−0.710745	−	0.703450i	$-0.751642\pi$
$139$	−323803.	−1.42149	−0.710745	+	0.703450i	$0.751642\pi$
$140$	0	0
$141$	−355535.	−1.50603
$142$	260946.i	1.08600i
$143$	− 40501.1i	− 0.165625i
$144$	−30878.5	−0.124094
$145$	0	0
$146$	274025.	1.06392
$147$	265666.i	1.01401i
$148$	183699.i	0.689370i
$149$	−372945.	−1.37619	−0.688096	−	0.725620i	$-0.741553\pi$
$149$	−372945.	−1.37619	−0.688096	+	0.725620i	$0.741553\pi$
$150$	0	0
$151$	−159735.	−0.570110	−0.285055	−	0.958511i	$-0.592012\pi$
$151$	−159735.	−0.570110	−0.285055	+	0.958511i	$0.592012\pi$
$152$	23927.8i	0.0840028i
$153$	− 238087.i	− 0.822257i
$154$	51399.7	0.174646
$155$	0	0
$156$	−51562.0	−0.169636
$157$	484134.i	1.56753i	0.621057	+	0.783765i	$0.286704\pi$
$157$	484134.i	1.56753i	−0.621057	+	0.783765i	$0.713296\pi$
$158$	42509.4i	0.135470i
$159$	−177700.	−0.557437
$160$	0	0
$161$	−2757.67	−0.00838451
$162$	295242.i	0.883874i
$163$	− 140172.i	− 0.413232i	−0.978422	−	0.206616i	$-0.933755\pi$
$163$	− 140172.i	− 0.413232i	0.978422	−	0.206616i	$-0.0662450\pi$
$164$	−200765.	−0.582878
$165$	0	0
$166$	−9431.91	−0.0265662
$167$	− 218046.i	− 0.605002i	−0.953149	−	0.302501i	$-0.902178\pi$
$167$	− 218046.i	− 0.605002i	0.953149	−	0.302501i	$-0.0978216\pi$
$168$	− 65437.1i	− 0.178875i
$169$	−28561.0	−0.0769231
$170$	0	0
$171$	−45096.1	−0.117937
$172$	− 18497.1i	− 0.0476741i
$173$	− 384852.i	− 0.977638i	−0.872385	−	0.488819i	$-0.837428\pi$
$173$	− 384852.i	− 0.977638i	0.872385	−	0.488819i	$-0.162572\pi$
$174$	−365828.	−0.916019
$175$	0	0
$176$	61350.7	0.149293
$177$	96614.5i	0.231798i
$178$	− 374480.i	− 0.885888i
$179$	−498273.	−1.16234	−0.581172	−	0.813781i	$-0.697405\pi$
$179$	−498273.	−1.16234	−0.581172	+	0.813781i	$0.697405\pi$
$180$	0	0
$181$	152201.	0.345320	0.172660	−	0.984981i	$-0.444764\pi$
$181$	152201.	0.345320	0.172660	+	0.984981i	$0.444764\pi$
$182$	− 36246.6i	− 0.0811127i
$183$	1.03490e6i	2.28438i
$184$	−3291.56	−0.00716733
$185$	0	0
$186$	526757.	1.11642
$187$	473042.i	0.989226i
$188$	− 298318.i	− 0.615580i
$189$	−125129.	−0.254802
$190$	0	0
$191$	600284.	1.19062	0.595310	−	0.803496i	$-0.297029\pi$
$191$	600284.	1.19062	0.595310	+	0.803496i	$0.297029\pi$
$192$	− 78105.8i	− 0.152908i
$193$	350804.i	0.677909i	0.940803	+	0.338955i	$0.110073\pi$
$193$	350804.i	0.677909i	−0.940803	+	0.338955i	$0.889927\pi$
$194$	124783.	0.238041
$195$	0	0
$196$	−222912.	−0.414470
$197$	321145.i	0.589571i	0.955563	+	0.294785i	$0.0952482\pi$
$197$	321145.i	0.589571i	−0.955563	+	0.294785i	$0.904752\pi$
$198$	115626.i	0.209601i
$199$	−561416.	−1.00497	−0.502484	−	0.864586i	$-0.667580\pi$
$199$	−561416.	−1.00497	−0.502484	+	0.864586i	$0.667580\pi$
$200$	0	0
$201$	767347.	1.33968
$202$	347794.i	0.599714i
$203$	− 257167.i	− 0.438000i
$204$	602231.	1.01318
$205$	0	0
$206$	−172444.	−0.283127
$207$	− 6203.52i	− 0.0100627i
$208$	− 43264.0i	− 0.0693375i
$209$	89598.9	0.141885
$210$	0	0
$211$	−751002.	−1.16127	−0.580637	−	0.814162i	$-0.697196\pi$
$211$	−751002.	−1.16127	−0.580637	+	0.814162i	$0.697196\pi$
$212$	− 149102.i	− 0.227848i
$213$	− 1.24398e6i	− 1.87873i
$214$	703862.	1.05064
$215$	0	0
$216$	−149354.	−0.217812
$217$	370295.i	0.533824i
$218$	478313.i	0.681665i
$219$	−1.30633e6	−1.84053
$220$	0	0
$221$	333585.	0.459437
$222$	− 875729.i	− 1.19258i
$223$	− 598906.i	− 0.806485i	−0.915093	−	0.403243i	$-0.867883\pi$
$223$	− 598906.i	− 0.806485i	0.915093	−	0.403243i	$-0.132117\pi$
$224$	54906.1	0.0731140
$225$	0	0
$226$	190232.	0.247749
$227$	1.32303e6i	1.70414i	0.523426	+	0.852071i	$0.324654\pi$
$227$	1.32303e6i	1.70414i	−0.523426	+	0.852071i	$0.675346\pi$
$228$	− 114069.i	− 0.145321i
$229$	−115752.	−0.145861	−0.0729307	−	0.997337i	$-0.523235\pi$
$229$	−115752.	−0.145861	−0.0729307	+	0.997337i	$0.523235\pi$
$230$	0	0
$231$	−245032.	−0.302130
$232$	− 306954.i	− 0.374416i
$233$	1.21925e6i	1.47131i	0.677357	+	0.735655i	$0.263125\pi$
$233$	1.21925e6i	1.47131i	−0.677357	+	0.735655i	$0.736875\pi$
$234$	81538.6	0.0973472
$235$	0	0
$236$	−81066.0	−0.0947456
$237$	− 202651.i	− 0.234357i
$238$	423351.i	0.484460i
$239$	1.66823e6	1.88913	0.944565	−	0.328323i	$-0.106484\pi$
$239$	1.66823e6	1.88913	0.944565	+	0.328323i	$0.106484\pi$
$240$	0	0
$241$	847710.	0.940167	0.470083	−	0.882622i	$-0.344224\pi$
$241$	847710.	0.940167	0.470083	+	0.882622i	$0.344224\pi$
$242$	414473.i	0.454944i
$243$	− 840399.i	− 0.912998i
$244$	−868347.	−0.933724
$245$	0	0
$246$	957085.	1.00835
$247$	− 63184.3i	− 0.0658972i
$248$	441984.i	0.456329i
$249$	44963.8	0.0459584
$250$	0	0
$251$	−1.09148e6	−1.09353	−0.546765	−	0.837286i	$-0.684141\pi$
$251$	−1.09148e6	−1.09353	−0.546765	+	0.837286i	$0.684141\pi$
$252$	103480.i	0.102649i
$253$	12325.4i	0.0121060i
$254$	435096.	0.423156
$255$	0	0
$256$	65536.0	0.0625000
$257$	10510.6i	0.00992644i	0.999988	+	0.00496322i	$0.00157985\pi$
$257$	10510.6i	0.00992644i	−0.999988	+	0.00496322i	$0.998420\pi$
$258$	88179.3i	0.0824741i
$259$	615612.	0.570240
$260$	0	0
$261$	578510.	0.525665
$262$	1.36194e6i	1.22576i
$263$	1.66350e6i	1.48297i	0.670967	+	0.741487i	$0.265879\pi$
$263$	1.66350e6i	1.48297i	−0.670967	+	0.741487i	$0.734121\pi$
$264$	−292471.	−0.258270
$265$	0	0
$266$	80186.9	0.0694863
$267$	1.78522e6i	1.53255i
$268$	643856.i	0.547585i
$269$	123097.	0.103721	0.0518606	−	0.998654i	$-0.483485\pi$
$269$	123097.	0.103721	0.0518606	+	0.998654i	$0.483485\pi$
$270$	0	0
$271$	2.14956e6	1.77798	0.888991	−	0.457925i	$-0.151407\pi$
$271$	2.14956e6	1.77798	0.888991	+	0.457925i	$0.151407\pi$
$272$	505312.i	0.414131i
$273$	172795.i	0.140321i
$274$	−192574.	−0.154960
$275$	0	0
$276$	15691.5	0.0123992
$277$	− 818582.i	− 0.641007i	−0.947247	−	0.320503i	$-0.896148\pi$
$277$	− 818582.i	− 0.641007i	0.947247	−	0.320503i	$-0.103852\pi$
$278$	1.29521e6i	1.00514i
$279$	−832997.	−0.640668
$280$	0	0
$281$	1.38003e6	1.04261	0.521305	−	0.853370i	$-0.325445\pi$
$281$	1.38003e6	1.04261	0.521305	+	0.853370i	$0.325445\pi$
$282$	1.42214e6i	1.06493i
$283$	395823.i	0.293788i	0.989152	+	0.146894i	$0.0469277\pi$
$283$	395823.i	0.293788i	−0.989152	+	0.146894i	$0.953072\pi$
$284$	1.04378e6	0.767918
$285$	0	0
$286$	−162004.	−0.117115
$287$	672803.i	0.482151i
$288$	123514.i	0.0877476i
$289$	−2.47633e6	−1.74407
$290$	0	0
$291$	−594867.	−0.411801
$292$	− 1.09610e6i	− 0.752303i
$293$	382789.i	0.260490i	0.991482	+	0.130245i	$0.0415763\pi$
$293$	382789.i	0.260490i	−0.991482	+	0.130245i	$0.958424\pi$
$294$	1.06266e6	0.717014
$295$	0	0
$296$	734795.	0.487458
$297$	559263.i	0.367896i
$298$	1.49178e6i	0.973115i
$299$	8691.78	0.00562252
$300$	0	0
$301$	−61987.4	−0.0394355
$302$	638941.i	0.403129i
$303$	− 1.65801e6i	− 1.03748i
$304$	95711.2	0.0593990
$305$	0	0
$306$	−952349.	−0.581424
$307$	2.95976e6i	1.79230i	0.443751	+	0.896150i	$0.353648\pi$
$307$	2.95976e6i	1.79230i	−0.443751	+	0.896150i	$0.646352\pi$
$308$	− 205599.i	− 0.123493i
$309$	822077.	0.489797
$310$	0	0
$311$	−536900.	−0.314770	−0.157385	−	0.987537i	$-0.550306\pi$
$311$	−536900.	−0.314770	−0.157385	+	0.987537i	$0.550306\pi$
$312$	206248.i	0.119951i
$313$	− 3.33301e6i	− 1.92299i	−0.274829	−	0.961493i	$-0.588621\pi$
$313$	− 3.33301e6i	− 1.92299i	0.274829	−	0.961493i	$-0.411379\pi$
$314$	1.93653e6	1.10841
$315$	0	0
$316$	170038.	0.0957915
$317$	− 1.50100e6i	− 0.838943i	−0.907768	−	0.419472i	$-0.862215\pi$
$317$	− 1.50100e6i	− 0.838943i	0.907768	−	0.419472i	$-0.137785\pi$
$318$	710801.i	0.394167i
$319$	−1.14941e6	−0.632407
$320$	0	0
$321$	−3.35545e6	−1.81756
$322$	11030.7i	0.00592875i
$323$	737977.i	0.393583i
$324$	1.18097e6	0.624993
$325$	0	0
$326$	−560690.	−0.292199
$327$	− 2.28022e6i	− 1.17925i
$328$	803058.i	0.412157i
$329$	−999723.	−0.509202
$330$	0	0
$331$	2.17404e6	1.09068	0.545341	−	0.838214i	$-0.316400\pi$
$331$	2.17404e6	1.09068	0.545341	+	0.838214i	$0.316400\pi$
$332$	37727.7i	0.0187851i
$333$	1.38485e6i	0.684372i
$334$	−872184.	−0.427801
$335$	0	0
$336$	−261748.	−0.126484
$337$	− 821494.i	− 0.394030i	−0.980400	−	0.197015i	$-0.936875\pi$
$337$	− 821494.i	− 0.394030i	0.980400	−	0.197015i	$-0.0631248\pi$
$338$	114244.i	0.0543928i
$339$	−906874.	−0.428595
$340$	0	0
$341$	1.65503e6	0.770763
$342$	180385.i	0.0833939i
$343$	1.64820e6i	0.756440i
$344$	−73988.3	−0.0337107
$345$	0	0
$346$	−1.53941e6	−0.691295
$347$	− 1.75897e6i	− 0.784216i	−0.919919	−	0.392108i	$-0.871746\pi$
$347$	− 1.75897e6i	− 0.784216i	0.919919	−	0.392108i	$-0.128254\pi$
$348$	1.46331e6i	0.647723i
$349$	3.15359e6	1.38593	0.692965	−	0.720971i	$-0.256304\pi$
$349$	3.15359e6	1.38593	0.692965	+	0.720971i	$0.256304\pi$
$350$	0	0
$351$	394388.	0.170866
$352$	− 245403.i	− 0.105566i
$353$	− 1.51456e6i	− 0.646918i	−0.946242	−	0.323459i	$-0.895154\pi$
$353$	− 1.51456e6i	− 0.646918i	0.946242	−	0.323459i	$-0.104846\pi$
$354$	386458.	0.163906
$355$	0	0
$356$	−1.49792e6	−0.626417
$357$	− 2.01820e6i	− 0.838095i
$358$	1.99309e6i	0.821901i
$359$	−1.68935e6	−0.691803	−0.345902	−	0.938271i	$-0.612427\pi$
$359$	−1.68935e6	−0.691803	−0.345902	+	0.938271i	$0.612427\pi$
$360$	0	0
$361$	−2.33632e6	−0.943548
$362$	− 608805.i	− 0.244178i
$363$	− 1.97588e6i	− 0.787033i
$364$	−144986.	−0.0573553
$365$	0	0
$366$	4.13958e6	1.61530
$367$	1.87184e6i	0.725443i	0.931898	+	0.362722i	$0.118152\pi$
$367$	1.87184e6i	0.725443i	−0.931898	+	0.362722i	$0.881848\pi$
$368$	13166.2i	0.00506807i
$369$	−1.51350e6	−0.578652
$370$	0	0
$371$	−499672.	−0.188474
$372$	− 2.10703e6i	− 0.789429i
$373$	628003.i	0.233717i	0.993149	+	0.116858i	$0.0372824\pi$
$373$	628003.i	0.233717i	−0.993149	+	0.116858i	$0.962718\pi$
$374$	1.89217e6	0.699488
$375$	0	0
$376$	−1.19327e6	−0.435281
$377$	810552.i	0.293716i
$378$	500515.i	0.180172i
$379$	−478225.	−0.171015	−0.0855076	−	0.996338i	$-0.527251\pi$
$379$	−478225.	−0.171015	−0.0855076	+	0.996338i	$0.527251\pi$
$380$	0	0
$381$	−2.07419e6	−0.732042
$382$	− 2.40114e6i	− 0.841895i
$383$	− 4.58548e6i	− 1.59731i	−0.601791	−	0.798653i	$-0.705546\pi$
$383$	− 4.58548e6i	− 1.59731i	0.601791	−	0.798653i	$-0.294454\pi$
$384$	−312423.	−0.108122
$385$	0	0
$386$	1.40322e6	0.479354
$387$	− 139444.i	− 0.0473285i
$388$	− 499133.i	− 0.168321i
$389$	−2.62136e6	−0.878318	−0.439159	−	0.898409i	$-0.644724\pi$
$389$	−2.62136e6	−0.878318	−0.439159	+	0.898409i	$0.644724\pi$
$390$	0	0
$391$	−101518.	−0.0335815
$392$	891647.i	0.293074i
$393$	− 6.49266e6i	− 2.12051i
$394$	1.28458e6	0.416890
$395$	0	0
$396$	462505.	0.148210
$397$	− 961154.i	− 0.306067i	−0.988221	−	0.153034i	$-0.951096\pi$
$397$	− 961154.i	− 0.306067i	0.988221	−	0.153034i	$-0.0489042\pi$
$398$	2.24567e6i	0.710620i
$399$	−382267.	−0.120208
$400$	0	0
$401$	−4.00559e6	−1.24396	−0.621978	−	0.783034i	$-0.713671\pi$
$401$	−4.00559e6	−1.24396	−0.621978	+	0.783034i	$0.713671\pi$
$402$	− 3.06939e6i	− 0.947299i
$403$	− 1.16712e6i	− 0.357974i
$404$	1.39118e6	0.424062
$405$	0	0
$406$	−1.02867e6	−0.309713
$407$	− 2.75148e6i	− 0.823342i
$408$	− 2.40892e6i	− 0.716428i
$409$	3.12530e6	0.923813	0.461906	−	0.886929i	$-0.347166\pi$
$409$	3.12530e6	0.923813	0.461906	+	0.886929i	$0.347166\pi$
$410$	0	0
$411$	918038.	0.268075
$412$	689777.i	0.200201i
$413$	271669.i	0.0783726i
$414$	−24814.1	−0.00711537
$415$	0	0
$416$	−173056.	−0.0490290
$417$	− 6.17453e6i	− 1.73886i
$418$	− 358395.i	− 0.100328i
$419$	3.34877e6	0.931859	0.465929	−	0.884822i	$-0.345720\pi$
$419$	3.34877e6	0.931859	0.465929	+	0.884822i	$0.345720\pi$
$420$	0	0
$421$	−3.01332e6	−0.828591	−0.414295	−	0.910143i	$-0.635972\pi$
$421$	−3.01332e6	−0.828591	−0.414295	+	0.910143i	$0.635972\pi$
$422$	3.00401e6i	0.821145i
$423$	− 2.24893e6i	− 0.611118i
$424$	−596410.	−0.161113
$425$	0	0
$426$	−4.97593e6	−1.32846
$427$	2.91000e6i	0.772367i
$428$	− 2.81545e6i	− 0.742913i
$429$	772307.	0.202603
$430$	0	0
$431$	3.35224e6	0.869245	0.434623	−	0.900613i	$-0.356882\pi$
$431$	3.35224e6	0.869245	0.434623	+	0.900613i	$0.356882\pi$
$432$	597416.i	0.154016i
$433$	1.62188e6i	0.415719i	0.978159	+	0.207859i	$0.0666496\pi$
$433$	1.62188e6i	0.415719i	−0.978159	+	0.207859i	$0.933350\pi$
$434$	1.48118e6	0.377471
$435$	0	0
$436$	1.91325e6	0.482010
$437$	19228.5i	0.00481661i
$438$	5.22532e6i	1.30145i
$439$	−7.48948e6	−1.85477	−0.927386	−	0.374106i	$-0.877950\pi$
$439$	−7.48948e6	−1.85477	−0.927386	+	0.374106i	$0.877950\pi$
$440$	0	0
$441$	−1.68046e6	−0.411465
$442$	− 1.33434e6i	− 0.324871i
$443$	1.90156e6i	0.460364i	0.973148	+	0.230182i	$0.0739322\pi$
$443$	1.90156e6i	0.460364i	−0.973148	+	0.230182i	$0.926068\pi$
$444$	−3.50292e6	−0.843281
$445$	0	0
$446$	−2.39562e6	−0.570271
$447$	− 7.11161e6i	− 1.68345i
$448$	− 219624.i	− 0.0516994i
$449$	−1.62580e6	−0.380585	−0.190293	−	0.981727i	$-0.560944\pi$
$449$	−1.62580e6	−0.380585	−0.190293	+	0.981727i	$0.560944\pi$
$450$	0	0
$451$	3.00709e6	0.696154
$452$	− 760928.i	− 0.175185i
$453$	− 3.04596e6i	− 0.697395i
$454$	5.29213e6	1.20501
$455$	0	0
$456$	−456274.	−0.102758
$457$	− 1.50323e6i	− 0.336694i	−0.985728	−	0.168347i	$-0.946157\pi$
$457$	− 1.50323e6i	− 0.336694i	0.985728	−	0.168347i	$-0.0538430\pi$
$458$	463009.i	0.103140i
$459$	−4.60634e6	−1.02053
$460$	0	0
$461$	549546.	0.120435	0.0602173	−	0.998185i	$-0.480821\pi$
$461$	549546.	0.120435	0.0602173	+	0.998185i	$0.480821\pi$
$462$	980130.i	0.213638i
$463$	− 3.34547e6i	− 0.725279i	−0.931929	−	0.362640i	$-0.881876\pi$
$463$	− 3.34547e6i	− 0.725279i	0.931929	−	0.362640i	$-0.118124\pi$
$464$	−1.22782e6	−0.264752
$465$	0	0
$466$	4.87701e6	1.04037
$467$	− 2.20487e6i	− 0.467833i	−0.972257	−	0.233916i	$-0.924846\pi$
$467$	− 2.20487e6i	− 0.467833i	0.972257	−	0.233916i	$-0.0751542\pi$
$468$	− 326154.i	− 0.0688349i
$469$	2.15769e6	0.452957
$470$	0	0
$471$	−9.23185e6	−1.91750
$472$	324264.i	0.0669952i
$473$	277053.i	0.0569390i
$474$	−810603.	−0.165715
$475$	0	0
$476$	1.69340e6	0.342565
$477$	− 1.12404e6i	− 0.226196i
$478$	− 6.67293e6i	− 1.33582i
$479$	655516.	0.130540	0.0652702	−	0.997868i	$-0.479209\pi$
$479$	655516.	0.130540	0.0652702	+	0.997868i	$0.479209\pi$
$480$	0	0
$481$	−1.94032e6	−0.382393
$482$	− 3.39084e6i	− 0.664798i
$483$	− 52585.5i	− 0.0102565i
$484$	1.65789e6	0.321694
$485$	0	0
$486$	−3.36160e6	−0.645587
$487$	− 1.70140e6i	− 0.325076i	−0.986702	−	0.162538i	$-0.948032\pi$
$487$	− 1.70140e6i	− 0.325076i	0.986702	−	0.162538i	$-0.0519680\pi$
$488$	3.47339e6i	0.660243i
$489$	2.67292e6	0.505491
$490$	0	0
$491$	1.54564e6	0.289338	0.144669	−	0.989480i	$-0.453788\pi$
$491$	1.54564e6	0.289338	0.144669	+	0.989480i	$0.453788\pi$
$492$	− 3.82834e6i	− 0.713014i
$493$	− 9.46703e6i	− 1.75427i
$494$	−252737.	−0.0465964
$495$	0	0
$496$	1.76794e6	0.322673
$497$	− 3.49793e6i	− 0.635214i
$498$	− 179855.i	− 0.0324975i
$499$	−5.57437e6	−1.00218	−0.501089	−	0.865396i	$-0.667067\pi$
$499$	−5.57437e6	−1.00218	−0.501089	+	0.865396i	$0.667067\pi$
$500$	0	0
$501$	4.15788e6	0.740078
$502$	4.36591e6i	0.773243i
$503$	318350.i	0.0561028i	0.999606	+	0.0280514i	$0.00893021\pi$
$503$	318350.i	0.0561028i	−0.999606	+	0.0280514i	$0.991070\pi$
$504$	413921.	0.0725840
$505$	0	0
$506$	49301.7	0.00856023
$507$	− 544624.i	− 0.0940972i
$508$	− 1.74038e6i	− 0.299217i
$509$	−265194.	−0.0453701	−0.0226851	−	0.999743i	$-0.507221\pi$
$509$	−265194.	−0.0453701	−0.0226851	+	0.999743i	$0.507221\pi$
$510$	0	0
$511$	−3.67325e6	−0.622297
$512$	− 262144.i	− 0.0441942i
$513$	872488.i	0.146375i
$514$	42042.3	0.00701906
$515$	0	0
$516$	352717.	0.0583180
$517$	4.46827e6i	0.735212i
$518$	− 2.46245e6i	− 0.403220i
$519$	7.33866e6	1.19591
$520$	0	0
$521$	1.91132e6	0.308489	0.154245	−	0.988033i	$-0.450706\pi$
$521$	1.91132e6	0.308489	0.154245	+	0.988033i	$0.450706\pi$
$522$	− 2.31404e6i	− 0.371701i
$523$	4.75910e6i	0.760801i	0.924822	+	0.380400i	$0.124214\pi$
$523$	4.75910e6i	0.760801i	−0.924822	+	0.380400i	$0.875786\pi$
$524$	5.44777e6	0.866744
$525$	0	0
$526$	6.65400e6	1.04862
$527$	1.36316e7i	2.13806i
$528$	1.16988e6i	0.182624i
$529$	6.43370e6	0.999589
$530$	0	0
$531$	−611132.	−0.0940587
$532$	− 320747.i	− 0.0491342i
$533$	− 2.12058e6i	− 0.323322i
$534$	7.14089e6	1.08367
$535$	0	0
$536$	2.57542e6	0.387201
$537$	− 9.50146e6i	− 1.42185i
$538$	− 492389.i	− 0.0733420i
$539$	3.33882e6	0.495018
$540$	0	0
$541$	−1.03919e7	−1.52652	−0.763261	−	0.646090i	$-0.776403\pi$
$541$	−1.03919e7	−1.52652	−0.763261	+	0.646090i	$0.776403\pi$
$542$	− 8.59826e6i	− 1.25722i
$543$	2.90230e6i	0.422418i
$544$	2.02125e6	0.292835
$545$	0	0
$546$	691179.	0.0992222
$547$	9.00790e6i	1.28723i	0.765350	+	0.643614i	$0.222566\pi$
$547$	9.00790e6i	1.28723i	−0.765350	+	0.643614i	$0.777434\pi$
$548$	770295.i	0.109574i
$549$	−6.54621e6	−0.926955
$550$	0	0
$551$	−1.79315e6	−0.251616
$552$	− 62766.1i	− 0.00876754i
$553$	− 569830.i	− 0.0792378i
$554$	−3.27433e6	−0.453260
$555$	0	0
$556$	5.18084e6	0.710745
$557$	− 1.61153e6i	− 0.220090i	−0.993927	−	0.110045i	$-0.964901\pi$
$557$	− 1.61153e6i	− 0.220090i	0.993927	−	0.110045i	$-0.0350995\pi$
$558$	3.33199e6i	0.453021i
$559$	195375.	0.0264448
$560$	0	0
$561$	−9.02034e6	−1.21008
$562$	− 5.52011e6i	− 0.737237i
$563$	− 5.73380e6i	− 0.762380i	−0.924497	−	0.381190i	$-0.875514\pi$
$563$	− 5.73380e6i	− 0.762380i	0.924497	−	0.381190i	$-0.124486\pi$
$564$	5.68856e6	0.753017
$565$	0	0
$566$	1.58329e6	0.207740
$567$	− 3.95766e6i	− 0.516988i
$568$	− 4.17513e6i	− 0.543000i
$569$	−4.81958e6	−0.624063	−0.312032	−	0.950072i	$-0.601009\pi$
$569$	−4.81958e6	−0.624063	−0.312032	+	0.950072i	$0.601009\pi$
$570$	0	0
$571$	7.86132e6	1.00903	0.504517	−	0.863402i	$-0.331671\pi$
$571$	7.86132e6	1.00903	0.504517	+	0.863402i	$0.331671\pi$
$572$	648017.i	0.0828126i
$573$	1.14467e7i	1.45644i
$574$	2.69121e6	0.340932
$575$	0	0
$576$	494056.	0.0620469
$577$	− 1.55342e7i	− 1.94244i	−0.238179	−	0.971221i	$-0.576550\pi$
$577$	− 1.55342e7i	− 1.94244i	0.238179	−	0.971221i	$-0.423450\pi$
$578$	9.90531e6i	1.23324i
$579$	−6.68942e6	−0.829262
$580$	0	0
$581$	126433.	0.0155389
$582$	2.37947e6i	0.291187i
$583$	2.23329e6i	0.272128i
$584$	−4.38440e6	−0.531958
$585$	0	0
$586$	1.53116e6	0.184194
$587$	1.84016e6i	0.220425i	0.993908	+	0.110212i	$0.0351531\pi$
$587$	1.84016e6i	0.220425i	−0.993908	+	0.110212i	$0.964847\pi$
$588$	− 4.25066e6i	− 0.507006i
$589$	2.58196e6	0.306663
$590$	0	0
$591$	−6.12386e6	−0.721201
$592$	− 2.93918e6i	− 0.344685i
$593$	1.11019e7i	1.29646i	0.761445	+	0.648230i	$0.224490\pi$
$593$	1.11019e7i	1.29646i	−0.761445	+	0.648230i	$0.775510\pi$
$594$	2.23705e6	0.260142
$595$	0	0
$596$	5.96712e6	0.688096
$597$	− 1.07055e7i	− 1.22934i
$598$	− 34767.1i	− 0.00397572i
$599$	1.34553e7	1.53224	0.766122	−	0.642696i	$-0.222184\pi$
$599$	1.34553e7	1.53224	0.766122	+	0.642696i	$0.222184\pi$
$600$	0	0
$601$	1.49829e7	1.69204	0.846020	−	0.533151i	$-0.178992\pi$
$601$	1.49829e7	1.69204	0.846020	+	0.533151i	$0.178992\pi$
$602$	247950.i	0.0278851i
$603$	4.85384e6i	0.543616i
$604$	2.55577e6	0.285055
$605$	0	0
$606$	−6.63202e6	−0.733609
$607$	1.18436e7i	1.30471i	0.757915	+	0.652353i	$0.226218\pi$
$607$	1.18436e7i	1.30471i	−0.757915	+	0.652353i	$0.773782\pi$
$608$	− 382845.i	− 0.0420014i
$609$	4.90386e6	0.535790
$610$	0	0
$611$	3.15098e6	0.341462
$612$	3.80940e6i	0.411129i
$613$	6.44924e6i	0.693198i	0.938013	+	0.346599i	$0.112664\pi$
$613$	6.44924e6i	0.693198i	−0.938013	+	0.346599i	$0.887336\pi$
$614$	1.18390e7	1.26735
$615$	0	0
$616$	−822394.	−0.0873229
$617$	1.10487e7i	1.16842i	0.811602	+	0.584210i	$0.198596\pi$
$617$	1.10487e7i	1.16842i	−0.811602	+	0.584210i	$0.801404\pi$
$618$	− 3.28831e6i	− 0.346339i
$619$	−7.71392e6	−0.809186	−0.404593	−	0.914497i	$-0.632587\pi$
$619$	−7.71392e6	−0.809186	−0.404593	+	0.914497i	$0.632587\pi$
$620$	0	0
$621$	−120021.	−0.0124891
$622$	2.14760e6i	0.222576i
$623$	5.01983e6i	0.518166i
$624$	824993.	0.0848181
$625$	0	0
$626$	−1.33321e7	−1.35976
$627$	1.70854e6i	0.173563i
$628$	− 7.74614e6i	− 0.783765i
$629$	2.26624e7	2.28391
$630$	0	0
$631$	−9.79932e6	−0.979767	−0.489883	−	0.871788i	$-0.662961\pi$
$631$	−9.79932e6	−0.979767	−0.489883	+	0.871788i	$0.662961\pi$
$632$	− 680150.i	− 0.0677348i
$633$	− 1.43207e7i	− 1.42055i
$634$	−6.00400e6	−0.593222
$635$	0	0
$636$	2.84321e6	0.278718
$637$	− 2.35450e6i	− 0.229906i
$638$	4.59763e6i	0.447180i
$639$	7.86877e6	0.762351
$640$	0	0
$641$	1.53393e7	1.47455	0.737275	−	0.675593i	$-0.236112\pi$
$641$	1.53393e7	1.47455	0.737275	+	0.675593i	$0.236112\pi$
$642$	1.34218e7i	1.28521i
$643$	1.45341e7i	1.38631i	0.720789	+	0.693155i	$0.243780\pi$
$643$	1.45341e7i	1.38631i	−0.720789	+	0.693155i	$0.756220\pi$
$644$	44122.7	0.00419226
$645$	0	0
$646$	2.95191e6	0.278305
$647$	1.26902e7i	1.19181i	0.803055	+	0.595904i	$0.203206\pi$
$647$	1.26902e7i	1.19181i	−0.803055	+	0.595904i	$0.796794\pi$
$648$	− 4.72387e6i	− 0.441937i
$649$	1.21422e6	0.113158
$650$	0	0
$651$	−7.06108e6	−0.653008
$652$	2.24276e6i	0.206616i
$653$	− 4.48209e6i	− 0.411337i	−0.978622	−	0.205669i	$-0.934063\pi$
$653$	− 4.48209e6i	− 0.411337i	0.978622	−	0.205669i	$-0.0659369\pi$
$654$	−9.12086e6	−0.833857
$655$	0	0
$656$	3.21223e6	0.291439
$657$	− 8.26316e6i	− 0.746849i
$658$	3.99889e6i	0.360060i
$659$	1.76267e7	1.58109	0.790545	−	0.612403i	$-0.209797\pi$
$659$	1.76267e7	1.58109	0.790545	+	0.612403i	$0.209797\pi$
$660$	0	0
$661$	1.03733e7	0.923447	0.461724	−	0.887024i	$-0.347231\pi$
$661$	1.03733e7	0.923447	0.461724	+	0.887024i	$0.347231\pi$
$662$	− 8.69617e6i	− 0.771229i
$663$	6.36107e6i	0.562013i
$664$	150911.	0.0132831
$665$	0	0
$666$	5.53940e6	0.483924
$667$	− 246670.i	− 0.0214685i
$668$	3.48874e6i	0.302501i
$669$	1.14204e7	0.986545
$670$	0	0
$671$	1.30063e7	1.11518
$672$	1.04699e6i	0.0894377i
$673$	4.76273e6i	0.405339i	0.979247	+	0.202669i	$0.0649616\pi$
$673$	4.76273e6i	0.405339i	−0.979247	+	0.202669i	$0.935038\pi$
$674$	−3.28598e6	−0.278622
$675$	0	0
$676$	456976.	0.0384615
$677$	− 1.68783e7i	− 1.41533i	−0.706548	−	0.707665i	$-0.749749\pi$
$677$	− 1.68783e7i	− 1.41533i	0.706548	−	0.707665i	$-0.250251\pi$
$678$	3.62750e6i	0.303063i
$679$	−1.67270e6	−0.139233
$680$	0	0
$681$	−2.52286e7	−2.08462
$682$	− 6.62013e6i	− 0.545012i
$683$	− 1.87331e7i	− 1.53659i	−0.640097	−	0.768294i	$-0.721106\pi$
$683$	− 1.87331e7i	− 1.53659i	0.640097	−	0.768294i	$-0.278894\pi$
$684$	721538.	0.0589684
$685$	0	0
$686$	6.59280e6	0.534884
$687$	− 2.20726e6i	− 0.178427i
$688$	295953.i	0.0238370i
$689$	1.57489e6	0.126387
$690$	0	0
$691$	2.12121e7	1.69000	0.845002	−	0.534763i	$-0.179599\pi$
$691$	2.12121e7	1.69000	0.845002	+	0.534763i	$0.179599\pi$
$692$	6.15763e6i	0.488819i
$693$	− 1.54995e6i	− 0.122598i
$694$	−7.03590e6	−0.554525
$695$	0	0
$696$	5.85325e6	0.458009
$697$	2.47678e7i	1.93110i
$698$	− 1.26143e7i	− 0.980000i
$699$	−2.32497e7	−1.79980
$700$	0	0
$701$	2.61831e6	0.201246	0.100623	−	0.994925i	$-0.467916\pi$
$701$	2.61831e6	0.201246	0.100623	+	0.994925i	$0.467916\pi$
$702$	− 1.57755e6i	− 0.120820i
$703$	− 4.29249e6i	− 0.327583i
$704$	−981612.	−0.0746463
$705$	0	0
$706$	−6.05824e6	−0.457440
$707$	− 4.66212e6i	− 0.350780i
$708$	− 1.54583e6i	− 0.115899i
$709$	−1.41825e7	−1.05959	−0.529794	−	0.848127i	$-0.677731\pi$
$709$	−1.41825e7	−1.05959	−0.529794	+	0.848127i	$0.677731\pi$
$710$	0	0
$711$	1.28186e6	0.0950971
$712$	5.99168e6i	0.442944i
$713$	355180.i	0.0261653i
$714$	−8.07279e6	−0.592622
$715$	0	0
$716$	7.97236e6	0.581172
$717$	3.18112e7i	2.31091i
$718$	6.75739e6i	0.489179i
$719$	−1.39362e6	−0.100536	−0.0502681	−	0.998736i	$-0.516008\pi$
$719$	−1.39362e6	−0.100536	−0.0502681	+	0.998736i	$0.516008\pi$
$720$	0	0
$721$	2.31158e6	0.165604
$722$	9.34528e6i	0.667189i
$723$	1.61648e7i	1.15007i
$724$	−2.43522e6	−0.172660
$725$	0	0
$726$	−7.90351e6	−0.556517
$727$	− 5.67972e6i	− 0.398557i	−0.979943	−	0.199279i	$-0.936140\pi$
$727$	− 5.67972e6i	− 0.398557i	0.979943	−	0.199279i	$-0.0638599\pi$
$728$	579945.i	0.0405563i
$729$	−1.91054e6	−0.133149
$730$	0	0
$731$	−2.28193e6	−0.157946
$732$	− 1.65583e7i	− 1.14219i
$733$	− 1.55611e7i	− 1.06974i	−0.844933	−	0.534872i	$-0.820360\pi$
$733$	− 1.55611e7i	− 1.06974i	0.844933	−	0.534872i	$-0.179640\pi$
$734$	7.48736e6	0.512966
$735$	0	0
$736$	52665.0	0.00358367
$737$	− 9.64380e6i	− 0.654003i
$738$	6.05402e6i	0.409169i
$739$	−1.54351e7	−1.03968	−0.519839	−	0.854264i	$-0.674008\pi$
$739$	−1.54351e7	−1.03968	−0.519839	+	0.854264i	$0.674008\pi$
$740$	0	0
$741$	1.20485e6	0.0806097
$742$	1.99869e6i	0.133271i
$743$	− 8.72110e6i	− 0.579561i	−0.957093	−	0.289781i	$-0.906418\pi$
$743$	− 8.72110e6i	− 0.579561i	0.957093	−	0.289781i	$-0.0935824\pi$
$744$	−8.42811e6	−0.558211
$745$	0	0
$746$	2.51201e6	0.165263
$747$	284417.i	0.0186490i
$748$	− 7.56867e6i	− 0.494613i
$749$	−9.43514e6	−0.614531
$750$	0	0
$751$	1.07940e7	0.698365	0.349183	−	0.937055i	$-0.386459\pi$
$751$	1.07940e7	0.698365	0.349183	+	0.937055i	$0.386459\pi$
$752$	4.77309e6i	0.307790i
$753$	− 2.08132e7i	− 1.33768i
$754$	3.24221e6	0.207688
$755$	0	0
$756$	2.00206e6	0.127401
$757$	− 1.87309e7i	− 1.18801i	−0.804462	−	0.594005i	$-0.797546\pi$
$757$	− 1.87309e7i	− 1.18801i	0.804462	−	0.594005i	$-0.202454\pi$
$758$	1.91290e6i	0.120926i
$759$	−235031.	−0.0148088
$760$	0	0
$761$	−1.41785e7	−0.887498	−0.443749	−	0.896151i	$-0.646352\pi$
$761$	−1.41785e7	−0.887498	−0.443749	+	0.896151i	$0.646352\pi$
$762$	8.29676e6i	0.517632i
$763$	− 6.41170e6i	− 0.398714i
$764$	−9.60454e6	−0.595310
$765$	0	0
$766$	−1.83419e7	−1.12947
$767$	− 856260.i	− 0.0525554i
$768$	1.24969e6i	0.0764540i
$769$	1.90780e7	1.16337	0.581683	−	0.813416i	$-0.302395\pi$
$769$	1.90780e7	1.16337	0.581683	+	0.813416i	$0.302395\pi$
$770$	0	0
$771$	−200424.	−0.0121427
$772$	− 5.61287e6i	− 0.338955i
$773$	1.31787e7i	0.793276i	0.917975	+	0.396638i	$0.129823\pi$
$773$	1.31787e7i	0.793276i	−0.917975	+	0.396638i	$0.870177\pi$
$774$	−557776.	−0.0334663
$775$	0	0
$776$	−1.99653e6	−0.119021
$777$	1.17390e7i	0.697554i
$778$	1.04854e7i	0.621065i
$779$	4.69126e6	0.276979
$780$	0	0
$781$	−1.56340e7	−0.917155
$782$	406071.i	0.0237457i
$783$	− 1.11926e7i	− 0.652418i
$784$	3.56659e6	0.207235
$785$	0	0
$786$	−2.59706e7	−1.49943
$787$	2.47284e7i	1.42318i	0.702596	+	0.711589i	$0.252024\pi$
$787$	2.47284e7i	1.42318i	−0.702596	+	0.711589i	$0.747976\pi$
$788$	− 5.13832e6i	− 0.294785i
$789$	−3.17210e7	−1.81407
$790$	0	0
$791$	−2.55002e6	−0.144911
$792$	− 1.85002e6i	− 0.104800i
$793$	− 9.17191e6i	− 0.517937i
$794$	−3.84461e6	−0.216422
$795$	0	0
$796$	8.98266e6	0.502484
$797$	2.11191e7i	1.17769i	0.808248	+	0.588843i	$0.200416\pi$
$797$	2.11191e7i	1.17769i	−0.808248	+	0.588843i	$0.799584\pi$
$798$	1.52907e6i	0.0850001i
$799$	−3.68026e7	−2.03945
$800$	0	0
$801$	−1.12924e7	−0.621876
$802$	1.60223e7i	0.879610i
$803$	1.64176e7i	0.898505i
$804$	−1.22776e7	−0.669841
$805$	0	0
$806$	−4.66846e6	−0.253126
$807$	2.34732e6i	0.126878i
$808$	− 5.56471e6i	− 0.299857i
$809$	−1.62945e7	−0.875326	−0.437663	−	0.899139i	$-0.644194\pi$
$809$	−1.62945e7	−0.875326	−0.437663	+	0.899139i	$0.644194\pi$
$810$	0	0
$811$	2.00401e7	1.06991	0.534954	−	0.844881i	$-0.320329\pi$
$811$	2.00401e7	1.06991	0.534954	+	0.844881i	$0.320329\pi$
$812$	4.11467e6i	0.219000i
$813$	4.09896e7i	2.17494i
$814$	−1.10059e7	−0.582191
$815$	0	0
$816$	−9.63570e6	−0.506591
$817$	432221.i	0.0226543i
$818$	− 1.25012e7i	− 0.653234i
$819$	−1.09301e6	−0.0569395
$820$	0	0
$821$	3.51296e7	1.81893	0.909463	−	0.415784i	$-0.136493\pi$
$821$	3.51296e7	1.81893	0.909463	+	0.415784i	$0.136493\pi$
$822$	− 3.67215e6i	− 0.189558i
$823$	− 2.51324e7i	− 1.29341i	−0.762742	−	0.646703i	$-0.776147\pi$
$823$	− 2.51324e7i	− 1.29341i	0.762742	−	0.646703i	$-0.223853\pi$
$824$	2.75911e6	0.141563
$825$	0	0
$826$	1.08667e6	0.0554178
$827$	1.66576e7i	0.846934i	0.905911	+	0.423467i	$0.139187\pi$
$827$	1.66576e7i	0.846934i	−0.905911	+	0.423467i	$0.860813\pi$
$828$	99256.4i	0.00503133i
$829$	1.46404e7	0.739887	0.369944	−	0.929054i	$-0.379377\pi$
$829$	1.46404e7	0.739887	0.369944	+	0.929054i	$0.379377\pi$
$830$	0	0
$831$	1.56094e7	0.784121
$832$	692224.i	0.0346688i
$833$	2.75000e7i	1.37316i
$834$	−2.46981e7	−1.22956
$835$	0	0
$836$	−1.43358e6	−0.0709425
$837$	1.61162e7i	0.795152i
$838$	− 1.33951e7i	− 0.658924i
$839$	−2.79895e7	−1.37275	−0.686374	−	0.727249i	$-0.740799\pi$
$839$	−2.79895e7	−1.37275	−0.686374	+	0.727249i	$0.740799\pi$
$840$	0	0
$841$	2.49204e6	0.121497
$842$	1.20533e7i	0.585902i
$843$	2.63155e7i	1.27539i
$844$	1.20160e7	0.580637
$845$	0	0
$846$	−8.99572e6	−0.432125
$847$	− 5.55593e6i	− 0.266102i
$848$	2.38564e6i	0.113924i
$849$	−7.54787e6	−0.359381
$850$	0	0
$851$	590484.	0.0279502
$852$	1.99037e7i	0.939366i
$853$	− 2.22248e7i	− 1.04584i	−0.852382	−	0.522919i	$-0.824843\pi$
$853$	− 2.22248e7i	− 1.04584i	0.852382	−	0.522919i	$-0.175157\pi$
$854$	1.16400e7	0.546146
$855$	0	0
$856$	−1.12618e7	−0.525319
$857$	− 7.83995e6i	− 0.364637i	−0.983239	−	0.182319i	$-0.941640\pi$
$857$	− 7.83995e6i	− 0.364637i	0.983239	−	0.182319i	$-0.0583603\pi$
$858$	− 3.08923e6i	− 0.143262i
$859$	3.08215e7	1.42518	0.712592	−	0.701579i	$-0.247521\pi$
$859$	3.08215e7	1.42518	0.712592	+	0.701579i	$0.247521\pi$
$860$	0	0
$861$	−1.28295e7	−0.589798
$862$	− 1.34090e7i	− 0.614649i
$863$	− 4.53439e6i	− 0.207249i	−0.994617	−	0.103624i	$-0.966956\pi$
$863$	− 4.53439e6i	− 0.207249i	0.994617	−	0.103624i	$-0.0330439\pi$
$864$	2.38966e6	0.108906
$865$	0	0
$866$	6.48753e6	0.293957
$867$	− 4.72206e7i	− 2.13346i
$868$	− 5.92472e6i	− 0.266912i
$869$	−2.54686e6	−0.114408
$870$	0	0
$871$	−6.80073e6	−0.303746
$872$	− 7.65301e6i	− 0.340833i
$873$	− 3.76282e6i	− 0.167100i
$874$	76913.9	0.00340586
$875$	0	0
$876$	2.09013e7	0.920265
$877$	− 3.31312e7i	− 1.45458i	−0.686328	−	0.727292i	$-0.740779\pi$
$877$	− 3.31312e7i	− 1.45458i	0.686328	−	0.727292i	$-0.259221\pi$
$878$	2.99579e7i	1.31152i
$879$	−7.29933e6	−0.318648
$880$	0	0
$881$	6.36611e6	0.276334	0.138167	−	0.990409i	$-0.455879\pi$
$881$	6.36611e6	0.276334	0.138167	+	0.990409i	$0.455879\pi$
$882$	6.72186e6i	0.290950i
$883$	− 3.09630e7i	− 1.33641i	−0.743975	−	0.668207i	$-0.767062\pi$
$883$	− 3.09630e7i	− 1.33641i	0.743975	−	0.668207i	$-0.232938\pi$
$884$	−5.33736e6	−0.229718
$885$	0	0
$886$	7.60626e6	0.325527
$887$	4.17692e7i	1.78257i	0.453440	+	0.891287i	$0.350197\pi$
$887$	4.17692e7i	1.78257i	−0.453440	+	0.891287i	$0.649803\pi$
$888$	1.40117e7i	0.596290i
$889$	−5.83238e6	−0.247509
$890$	0	0
$891$	−1.76888e7	−0.746455
$892$	9.58250e6i	0.403243i
$893$	6.97079e6i	0.292519i
$894$	−2.84465e7	−1.19038
$895$	0	0
$896$	−878497.	−0.0365570
$897$	165742.i	0.00687782i
$898$	6.50321e6i	0.269114i
$899$	−3.31223e7	−1.36685
$900$	0	0
$901$	−1.83944e7	−0.754871
$902$	− 1.20284e7i	− 0.492255i
$903$	− 1.18203e6i	− 0.0482401i
$904$	−3.04371e6	−0.123875
$905$	0	0
$906$	−1.21838e7	−0.493133
$907$	4.51182e7i	1.82110i	0.413399	+	0.910550i	$0.364342\pi$
$907$	4.51182e7i	1.82110i	−0.413399	+	0.910550i	$0.635658\pi$
$908$	− 2.11685e7i	− 0.852071i
$909$	1.04877e7	0.420988
$910$	0	0
$911$	2.93224e6	0.117059	0.0585293	−	0.998286i	$-0.481359\pi$
$911$	2.93224e6	0.117059	0.0585293	+	0.998286i	$0.481359\pi$
$912$	1.82510e6i	0.0726606i
$913$	− 565092.i	− 0.0224359i
$914$	−6.01293e6	−0.238079
$915$	0	0
$916$	1.85203e6	0.0729307
$917$	− 1.82566e7i	− 0.716962i
$918$	1.84254e7i	0.721622i
$919$	−1.06534e7	−0.416101	−0.208051	−	0.978118i	$-0.566712\pi$
$919$	−1.06534e7	−0.416101	−0.208051	+	0.978118i	$0.566712\pi$
$920$	0	0
$921$	−5.64391e7	−2.19246
$922$	− 2.19818e6i	− 0.0851602i
$923$	1.10250e7i	0.425964i
$924$	3.92052e6	0.151065
$925$	0	0
$926$	−1.33819e7	−0.512850
$927$	5.20003e6i	0.198750i
$928$	4.91127e6i	0.187208i
$929$	1.93436e7	0.735355	0.367677	−	0.929953i	$-0.380153\pi$
$929$	1.93436e7	0.735355	0.367677	+	0.929953i	$0.380153\pi$
$930$	0	0
$931$	5.20878e6	0.196952
$932$	− 1.95081e7i	− 0.735655i
$933$	− 1.02380e7i	− 0.385046i
$934$	−8.81947e6	−0.330808
$935$	0	0
$936$	−1.30462e6	−0.0486736
$937$	5.62445e6i	0.209282i	0.994510	+	0.104641i	$0.0333693\pi$
$937$	5.62445e6i	0.209282i	−0.994510	+	0.104641i	$0.966631\pi$
$938$	− 8.63076e6i	− 0.320289i
$939$	6.35566e7	2.35232
$940$	0	0
$941$	−2.18961e7	−0.806106	−0.403053	−	0.915177i	$-0.632051\pi$
$941$	−2.18961e7	−0.806106	−0.403053	+	0.915177i	$0.632051\pi$
$942$	3.69274e7i	1.35588i
$943$	645341.i	0.0236325i
$944$	1.29706e6	0.0473728
$945$	0	0
$946$	1.10821e6	0.0402620
$947$	− 931486.i	− 0.0337522i	−0.999858	−	0.0168761i	$-0.994628\pi$
$947$	− 931486.i	− 0.0337522i	0.999858	−	0.0168761i	$-0.00537208\pi$
$948$	3.24241e6i	0.117178i
$949$	1.15775e7	0.417302
$950$	0	0
$951$	2.86223e7	1.02625
$952$	− 6.77361e6i	− 0.242230i
$953$	4.06914e7i	1.45135i	0.688040	+	0.725673i	$0.258471\pi$
$953$	4.06914e7i	1.45135i	−0.688040	+	0.725673i	$0.741529\pi$
$954$	−4.49616e6	−0.159945
$955$	0	0
$956$	−2.66917e7	−0.944565
$957$	− 2.19178e7i	− 0.773601i
$958$	− 2.62206e6i	− 0.0923059i
$959$	2.58141e6	0.0906382
$960$	0	0
$961$	1.90638e7	0.665887
$962$	7.76128e6i	0.270393i
$963$	− 2.12248e7i	− 0.737528i
$964$	−1.35634e7	−0.470083
$965$	0	0
$966$	−210342.	−0.00725242
$967$	− 1.43608e7i	− 0.493869i	−0.969032	−	0.246934i	$-0.920577\pi$
$967$	− 1.43608e7i	− 0.493869i	0.969032	−	0.246934i	$-0.0794232\pi$
$968$	− 6.63157e6i	− 0.227472i
$969$	−1.40723e7	−0.481456
$970$	0	0
$971$	4.38037e6	0.149095	0.0745475	−	0.997217i	$-0.476249\pi$
$971$	4.38037e6	0.149095	0.0745475	+	0.997217i	$0.476249\pi$
$972$	1.34464e7i	0.456499i
$973$	− 1.73621e7i	− 0.587921i
$974$	−6.80562e6	−0.229864
$975$	0	0
$976$	1.38935e7	0.466862
$977$	− 2.87093e7i	− 0.962247i	−0.876653	−	0.481123i	$-0.840229\pi$
$977$	− 2.87093e7i	− 0.962247i	0.876653	−	0.481123i	$-0.159771\pi$
$978$	− 1.06917e7i	− 0.357436i
$979$	2.24362e7	0.748155
$980$	0	0
$981$	1.44234e7	0.478516
$982$	− 6.18257e6i	− 0.204593i
$983$	5.36798e7i	1.77185i	0.463829	+	0.885925i	$0.346475\pi$
$983$	5.36798e7i	1.77185i	−0.463829	+	0.885925i	$0.653525\pi$
$984$	−1.53134e7	−0.504177
$985$	0	0
$986$	−3.78681e7	−1.24046
$987$	− 1.90635e7i	− 0.622888i
$988$	1.01095e6i	0.0329486i
$989$	−59457.3	−0.00193292
$990$	0	0
$991$	2.18434e7	0.706539	0.353269	−	0.935522i	$-0.385070\pi$
$991$	2.18434e7	0.706539	0.353269	+	0.935522i	$0.385070\pi$
$992$	− 7.07175e6i	− 0.228164i
$993$	4.14564e7i	1.33419i
$994$	−1.39917e7	−0.449164
$995$	0	0
$996$	−719421.	−0.0229792
$997$	− 4.14924e7i	− 1.32200i	−0.750387	−	0.660999i	$-0.770133\pi$
$997$	− 4.14924e7i	− 1.32200i	0.750387	−	0.660999i	$-0.229867\pi$
$998$	2.22975e7i	0.708646i
$999$	2.67931e7	0.849394

Display

a_p

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p

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a_n

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a_n

instead) (See

a_n

instead) Display

a_n

with

n

up to: 50 250 1000 (See only

a_p

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a_p

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a_p

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	650.6.b.h.599.2		4
5.2	odd	4		26.6.a.c.1.2	✓	2
5.3	odd	4		650.6.a.b.1.1		2
5.4	even	2	inner	650.6.b.h.599.3		4
15.2	even	4		234.6.a.h.1.2		2
20.7	even	4		208.6.a.g.1.1		2
40.27	even	4		832.6.a.m.1.2		2
40.37	odd	4		832.6.a.k.1.1		2
65.12	odd	4		338.6.a.f.1.2		2
65.47	even	4		338.6.b.b.337.4		4
65.57	even	4		338.6.b.b.337.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
26.6.a.c.1.2	✓	2	5.2	odd	4
208.6.a.g.1.1		2	20.7	even	4
234.6.a.h.1.2		2	15.2	even	4
338.6.a.f.1.2		2	65.12	odd	4
338.6.b.b.337.2		4	65.57	even	4
338.6.b.b.337.4		4	65.47	even	4
650.6.a.b.1.1		2	5.3	odd	4
650.6.b.h.599.2		4	1.1	even	1	trivial
650.6.b.h.599.3		4	5.4	even	2	inner
832.6.a.k.1.1		2	40.37	odd	4
832.6.a.m.1.2		2	40.27	even	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 $\F_{q}$
Belyi maps

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	650.6.b.h.599.2

Level	$650$
Weight	$6$
Character	650.599
Analytic conductor	$104.249$
Analytic rank	$0$
Dimension	$4$
Inner twists	$2$