LMFDB - Embedded newform 810.5.d.c.161.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [810,5,Mod(161,810)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(810, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("810.161");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 810 = 2 \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	810.d (of order $2$, degree $1$, 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:	$83.7296700979$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	no (minimal twist has level 90)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.15
Character	$\chi$	$=$	810.161
Dual form			810.5.d.c.161.16

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.82843i q^{2} -8.00000 q^{4} -11.1803i q^{5} -61.8752 q^{7} +22.6274i q^{8} +O(q^{10})$	\(q-2.82843i q^{2} -8.00000 q^{4} -11.1803i q^{5} -61.8752 q^{7} +22.6274i q^{8} -31.6228 q^{10} +9.95454i q^{11} +196.963 q^{13} +175.009i q^{14} +64.0000 q^{16} -68.7137i q^{17} -452.899 q^{19} +89.4427i q^{20} +28.1557 q^{22} +764.630i q^{23} -125.000 q^{25} -557.095i q^{26} +495.001 q^{28} +104.060i q^{29} +1483.60 q^{31} -181.019i q^{32} -194.352 q^{34} +691.785i q^{35} +1074.51 q^{37} +1280.99i q^{38} +252.982 q^{40} +2717.64i q^{41} -2657.70 q^{43} -79.6363i q^{44} +2162.70 q^{46} -878.444i q^{47} +1427.54 q^{49} +353.553i q^{50} -1575.70 q^{52} -494.940i q^{53} +111.295 q^{55} -1400.08i q^{56} +294.325 q^{58} -4422.41i q^{59} +4951.17 q^{61} -4196.27i q^{62} -512.000 q^{64} -2202.11i q^{65} +4574.29 q^{67} +549.710i q^{68} +1956.66 q^{70} -6469.29i q^{71} -4425.57 q^{73} -3039.18i q^{74} +3623.19 q^{76} -615.939i q^{77} -6667.61 q^{79} -715.542i q^{80} +7686.65 q^{82} -7535.17i q^{83} -768.243 q^{85} +7517.12i q^{86} -225.246 q^{88} +6482.42i q^{89} -12187.1 q^{91} -6117.04i q^{92} -2484.62 q^{94} +5063.56i q^{95} +9284.54 q^{97} -4037.68i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - 256 q^{4} - 104 q^{7}+O(q^{10}) $ 32 * q - 256 * q^4 - 104 * q^7	$ 32 q - 256 q^{4} - 104 q^{7} + 40 q^{13} + 2048 q^{16} - 200 q^{19} + 1344 q^{22} - 4000 q^{25} + 832 q^{28} - 1472 q^{31} - 384 q^{34} - 4136 q^{37} - 272 q^{43} - 2112 q^{46} + 22296 q^{49} - 320 q^{52} - 12344 q^{61} - 16384 q^{64} + 40936 q^{67} + 4800 q^{70} - 41432 q^{73} + 1600 q^{76} - 14048 q^{79} - 17664 q^{82} - 17400 q^{85} - 10752 q^{88} + 69392 q^{91} + 1344 q^{94} + 36664 q^{97}+O(q^{100}) $ 32 * q - 256 * q^4 - 104 * q^7 + 40 * q^13 + 2048 * q^16 - 200 * q^19 + 1344 * q^22 - 4000 * q^25 + 832 * q^28 - 1472 * q^31 - 384 * q^34 - 4136 * q^37 - 272 * q^43 - 2112 * q^46 + 22296 * q^49 - 320 * q^52 - 12344 * q^61 - 16384 * q^64 + 40936 * q^67 + 4800 * q^70 - 41432 * q^73 + 1600 * q^76 - 14048 * q^79 - 17664 * q^82 - 17400 * q^85 - 10752 * q^88 + 69392 * q^91 + 1344 * q^94 + 36664 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/810\mathbb{Z}\right)^\times$.

$n$	$487$	$731$
$\chi(n)$	$1$	$-1$

Coefficient data

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

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

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

$2$	− 2.82843i	− 0.707107i
$2$	− 2.82843i	− 0.707107i
$3$	0	0
$3$	0	0
$4$	−8.00000	−0.500000
$5$	− 11.1803i	− 0.447214i
$5$	− 11.1803i	− 0.447214i
$6$	0	0
$7$	−61.8752	−1.26276	−0.631379	−	0.775474i	$-0.717511\pi$
$7$	−61.8752	−1.26276	−0.631379	+	0.775474i	$0.717511\pi$
$8$	22.6274i	0.353553i
$9$	0	0
$10$	−31.6228	−0.316228
$11$	9.95454i	0.0822689i	0.999154	+	0.0411345i	$0.0130972\pi$
$11$	9.95454i	0.0822689i	−0.999154	+	0.0411345i	$0.986903\pi$
$12$	0	0
$13$	196.963	1.16546	0.582731	−	0.812665i	$-0.301984\pi$
$13$	196.963	1.16546	0.582731	+	0.812665i	$0.301984\pi$
$14$	175.009i	0.892905i
$15$	0	0
$16$	64.0000	0.250000
$17$	− 68.7137i	− 0.237764i	−0.992908	−	0.118882i	$-0.962069\pi$
$17$	− 68.7137i	− 0.237764i	0.992908	−	0.118882i	$-0.0379310\pi$
$18$	0	0
$19$	−452.899	−1.25457	−0.627284	−	0.778791i	$-0.715833\pi$
$19$	−452.899	−1.25457	−0.627284	+	0.778791i	$0.715833\pi$
$20$	89.4427i	0.223607i
$21$	0	0
$22$	28.1557	0.0581729
$23$	764.630i	1.44543i	0.691148	+	0.722713i	$0.257105\pi$
$23$	764.630i	1.44543i	−0.691148	+	0.722713i	$0.742895\pi$
$24$	0	0
$25$	−125.000	−0.200000
$26$	− 557.095i	− 0.824106i
$27$	0	0
$28$	495.001	0.631379
$29$	104.060i	0.123733i	0.998084	+	0.0618666i	$0.0197053\pi$
$29$	104.060i	0.123733i	−0.998084	+	0.0618666i	$0.980295\pi$
$30$	0	0
$31$	1483.60	1.54381	0.771906	−	0.635736i	$-0.219303\pi$
$31$	1483.60	1.54381	0.771906	+	0.635736i	$0.219303\pi$
$32$	− 181.019i	− 0.176777i
$33$	0	0
$34$	−194.352	−0.168124
$35$	691.785i	0.564723i
$36$	0	0
$37$	1074.51	0.784888	0.392444	−	0.919776i	$-0.371630\pi$
$37$	1074.51	0.784888	0.392444	+	0.919776i	$0.371630\pi$
$38$	1280.99i	0.887113i
$39$	0	0
$40$	252.982	0.158114
$41$	2717.64i	1.61668i	0.588715	+	0.808341i	$0.299634\pi$
$41$	2717.64i	1.61668i	−0.588715	+	0.808341i	$0.700366\pi$
$42$	0	0
$43$	−2657.70	−1.43737	−0.718687	−	0.695334i	$-0.755256\pi$
$43$	−2657.70	−1.43737	−0.718687	+	0.695334i	$0.755256\pi$
$44$	− 79.6363i	− 0.0411345i
$45$	0	0
$46$	2162.70	1.02207
$47$	− 878.444i	− 0.397666i	−0.980033	−	0.198833i	$-0.936285\pi$
$47$	− 878.444i	− 0.397666i	0.980033	−	0.198833i	$-0.0637152\pi$
$48$	0	0
$49$	1427.54	0.594559
$50$	353.553i	0.141421i
$51$	0	0
$52$	−1575.70	−0.582731
$53$	− 494.940i	− 0.176198i	−0.996112	−	0.0880990i	$-0.971921\pi$
$53$	− 494.940i	− 0.176198i	0.996112	−	0.0880990i	$-0.0280792\pi$
$54$	0	0
$55$	111.295	0.0367918
$56$	− 1400.08i	− 0.446453i
$57$	0	0
$58$	294.325	0.0874925
$59$	− 4422.41i	− 1.27044i	−0.772330	−	0.635221i	$-0.780909\pi$
$59$	− 4422.41i	− 1.27044i	0.772330	−	0.635221i	$-0.219091\pi$
$60$	0	0
$61$	4951.17	1.33060	0.665301	−	0.746575i	$-0.268303\pi$
$61$	4951.17	1.33060	0.665301	+	0.746575i	$0.268303\pi$
$62$	− 4196.27i	− 1.09164i
$63$	0	0
$64$	−512.000	−0.125000
$65$	− 2202.11i	− 0.521210i
$66$	0	0
$67$	4574.29	1.01900	0.509500	−	0.860470i	$-0.329830\pi$
$67$	4574.29	1.01900	0.509500	+	0.860470i	$0.329830\pi$
$68$	549.710i	0.118882i
$69$	0	0
$70$	1956.66	0.399319
$71$	− 6469.29i	− 1.28333i	−0.766983	−	0.641667i	$-0.778243\pi$
$71$	− 6469.29i	− 1.28333i	0.766983	−	0.641667i	$-0.221757\pi$
$72$	0	0
$73$	−4425.57	−0.830469	−0.415235	−	0.909714i	$-0.636301\pi$
$73$	−4425.57	−0.830469	−0.415235	+	0.909714i	$0.636301\pi$
$74$	− 3039.18i	− 0.555000i
$75$	0	0
$76$	3623.19	0.627284
$77$	− 615.939i	− 0.103886i
$78$	0	0
$79$	−6667.61	−1.06836	−0.534178	−	0.845372i	$-0.679379\pi$
$79$	−6667.61	−1.06836	−0.534178	+	0.845372i	$0.679379\pi$
$80$	− 715.542i	− 0.111803i
$81$	0	0
$82$	7686.65	1.14317
$83$	− 7535.17i	− 1.09380i	−0.837199	−	0.546899i	$-0.815808\pi$
$83$	− 7535.17i	− 1.09380i	0.837199	−	0.546899i	$-0.184192\pi$
$84$	0	0
$85$	−768.243	−0.106331
$86$	7517.12i	1.01638i
$87$	0	0
$88$	−225.246	−0.0290865
$89$	6482.42i	0.818384i	0.912448	+	0.409192i	$0.134189\pi$
$89$	6482.42i	0.818384i	−0.912448	+	0.409192i	$0.865811\pi$
$90$	0	0
$91$	−12187.1	−1.47170
$92$	− 6117.04i	− 0.722713i
$93$	0	0
$94$	−2484.62	−0.281192
$95$	5063.56i	0.561060i
$96$	0	0
$97$	9284.54	0.986772	0.493386	−	0.869810i	$-0.335759\pi$
$97$	9284.54	0.986772	0.493386	+	0.869810i	$0.335759\pi$
$98$	− 4037.68i	− 0.420417i
$99$	0	0
$100$	1000.00	0.100000
$101$	− 2962.26i	− 0.290389i	−0.989403	−	0.145194i	$-0.953619\pi$
$101$	− 2962.26i	− 0.290389i	0.989403	−	0.145194i	$-0.0463808\pi$
$102$	0	0
$103$	4584.60	0.432142	0.216071	−	0.976378i	$-0.430676\pi$
$103$	4584.60	0.432142	0.216071	+	0.976378i	$0.430676\pi$
$104$	4456.76i	0.412053i
$105$	0	0
$106$	−1399.90	−0.124591
$107$	2425.13i	0.211820i	0.994376	+	0.105910i	$0.0337756\pi$
$107$	2425.13i	0.211820i	−0.994376	+	0.105910i	$0.966224\pi$
$108$	0	0
$109$	16156.6	1.35987	0.679933	−	0.733274i	$-0.262009\pi$
$109$	16156.6	1.35987	0.679933	+	0.733274i	$0.262009\pi$
$110$	− 314.790i	− 0.0260157i
$111$	0	0
$112$	−3960.01	−0.315690
$113$	− 12846.9i	− 1.00610i	−0.864257	−	0.503050i	$-0.832211\pi$
$113$	− 12846.9i	− 1.00610i	0.864257	−	0.503050i	$-0.167789\pi$
$114$	0	0
$115$	8548.83	0.646414
$116$	− 832.476i	− 0.0618666i
$117$	0	0
$118$	−12508.5	−0.898339
$119$	4251.67i	0.300238i
$120$	0	0
$121$	14541.9	0.993232
$122$	− 14004.0i	− 0.940878i
$123$	0	0
$124$	−11868.8	−0.771906
$125$	1397.54i	0.0894427i
$126$	0	0
$127$	13760.5	0.853150	0.426575	−	0.904452i	$-0.359720\pi$
$127$	13760.5	0.853150	0.426575	+	0.904452i	$0.359720\pi$
$128$	1448.15i	0.0883883i
$129$	0	0
$130$	−6228.52	−0.368551
$131$	− 25549.6i	− 1.48882i	−0.667725	−	0.744408i	$-0.732732\pi$
$131$	− 25549.6i	− 1.48882i	0.667725	−	0.744408i	$-0.267268\pi$
$132$	0	0
$133$	28023.2	1.58422
$134$	− 12938.1i	− 0.720542i
$135$	0	0
$136$	1554.81	0.0840622
$137$	9419.31i	0.501855i	0.968006	+	0.250927i	$0.0807355\pi$
$137$	9419.31i	0.501855i	−0.968006	+	0.250927i	$0.919264\pi$
$138$	0	0
$139$	−34238.3	−1.77207	−0.886037	−	0.463614i	$-0.846552\pi$
$139$	−34238.3	−1.77207	−0.886037	+	0.463614i	$0.846552\pi$
$140$	− 5534.28i	− 0.282361i
$141$	0	0
$142$	−18297.9	−0.907455
$143$	1960.68i	0.0958813i
$144$	0	0
$145$	1163.42	0.0553351
$146$	12517.4i	0.587230i
$147$	0	0
$148$	−8596.10	−0.392444
$149$	− 31541.8i	− 1.42074i	−0.703830	−	0.710368i	$-0.748528\pi$
$149$	− 31541.8i	− 1.42074i	0.703830	−	0.710368i	$-0.251472\pi$
$150$	0	0
$151$	−2848.16	−0.124914	−0.0624570	−	0.998048i	$-0.519894\pi$
$151$	−2848.16	−0.124914	−0.0624570	+	0.998048i	$0.519894\pi$
$152$	− 10247.9i	− 0.443557i
$153$	0	0
$154$	−1742.14	−0.0734583
$155$	− 16587.2i	− 0.690414i
$156$	0	0
$157$	33061.8	1.34130	0.670651	−	0.741773i	$-0.266015\pi$
$157$	33061.8	1.34130	0.670651	+	0.741773i	$0.266015\pi$
$158$	18858.9i	0.755442i
$159$	0	0
$160$	−2023.86	−0.0790569
$161$	− 47311.6i	− 1.82522i
$162$	0	0
$163$	−15195.4	−0.571923	−0.285962	−	0.958241i	$-0.592313\pi$
$163$	−15195.4	−0.571923	−0.285962	+	0.958241i	$0.592313\pi$
$164$	− 21741.1i	− 0.808341i
$165$	0	0
$166$	−21312.7	−0.773432
$167$	− 44329.4i	− 1.58949i	−0.606941	−	0.794747i	$-0.707604\pi$
$167$	− 44329.4i	− 1.58949i	0.606941	−	0.794747i	$-0.292396\pi$
$168$	0	0
$169$	10233.4	0.358301
$170$	2172.92i	0.0751875i
$171$	0	0
$172$	21261.6	0.718687
$173$	− 3912.49i	− 0.130726i	−0.997862	−	0.0653628i	$-0.979180\pi$
$173$	− 3912.49i	− 0.130726i	0.997862	−	0.0653628i	$-0.0208205\pi$
$174$	0	0
$175$	7734.40	0.252552
$176$	637.091i	0.0205672i
$177$	0	0
$178$	18335.0	0.578685
$179$	32346.2i	1.00953i	0.863258	+	0.504763i	$0.168420\pi$
$179$	32346.2i	1.00953i	−0.863258	+	0.504763i	$0.831580\pi$
$180$	0	0
$181$	38573.5	1.17742	0.588710	−	0.808344i	$-0.299636\pi$
$181$	38573.5	1.17742	0.588710	+	0.808344i	$0.299636\pi$
$182$	34470.4i	1.04065i
$183$	0	0
$184$	−17301.6	−0.511035
$185$	− 12013.4i	− 0.351013i
$186$	0	0
$187$	684.014	0.0195606
$188$	7027.56i	0.198833i
$189$	0	0
$190$	14321.9	0.396729
$191$	− 19091.1i	− 0.523317i	−0.965161	−	0.261658i	$-0.915731\pi$
$191$	− 19091.1i	− 0.523317i	0.965161	−	0.261658i	$-0.0842694\pi$
$192$	0	0
$193$	−31302.1	−0.840347	−0.420174	−	0.907444i	$-0.638031\pi$
$193$	−31302.1	−0.840347	−0.420174	+	0.907444i	$0.638031\pi$
$194$	− 26260.6i	− 0.697753i
$195$	0	0
$196$	−11420.3	−0.297279
$197$	− 24444.9i	− 0.629876i	−0.949112	−	0.314938i	$-0.898016\pi$
$197$	− 24444.9i	− 0.629876i	0.949112	−	0.314938i	$-0.101984\pi$
$198$	0	0
$199$	−28458.8	−0.718640	−0.359320	−	0.933214i	$-0.616991\pi$
$199$	−28458.8	−0.718640	−0.359320	+	0.933214i	$0.616991\pi$
$200$	− 2828.43i	− 0.0707107i
$201$	0	0
$202$	−8378.52	−0.205336
$203$	− 6438.70i	− 0.156245i
$204$	0	0
$205$	30384.2	0.723002
$206$	− 12967.2i	− 0.305571i
$207$	0	0
$208$	12605.6	0.291365
$209$	− 4508.40i	− 0.103212i
$210$	0	0
$211$	42409.5	0.952574	0.476287	−	0.879290i	$-0.341982\pi$
$211$	42409.5	0.952574	0.476287	+	0.879290i	$0.341982\pi$
$212$	3959.52i	0.0880990i
$213$	0	0
$214$	6859.30	0.149780
$215$	29714.0i	0.642813i
$216$	0	0
$217$	−91798.2	−1.94946
$218$	− 45697.7i	− 0.961571i
$219$	0	0
$220$	−890.361	−0.0183959
$221$	− 13534.1i	− 0.277105i
$222$	0	0
$223$	54280.7	1.09153	0.545765	−	0.837938i	$-0.316239\pi$
$223$	54280.7	1.09153	0.545765	+	0.837938i	$0.316239\pi$
$224$	11200.6i	0.223226i
$225$	0	0
$226$	−36336.5	−0.711420
$227$	− 88101.0i	− 1.70974i	−0.518846	−	0.854868i	$-0.673638\pi$
$227$	− 88101.0i	− 1.70974i	0.518846	−	0.854868i	$-0.326362\pi$
$228$	0	0
$229$	−13598.9	−0.259319	−0.129659	−	0.991559i	$-0.541388\pi$
$229$	−13598.9	−0.259319	−0.129659	+	0.991559i	$0.541388\pi$
$230$	− 24179.7i	− 0.457084i
$231$	0	0
$232$	−2354.60	−0.0437463
$233$	9418.55i	0.173489i	0.996231	+	0.0867446i	$0.0276464\pi$
$233$	9418.55i	0.173489i	−0.996231	+	0.0867446i	$0.972354\pi$
$234$	0	0
$235$	−9821.31	−0.177842
$236$	35379.3i	0.635221i
$237$	0	0
$238$	12025.5	0.212300
$239$	69873.0i	1.22325i	0.791149	+	0.611623i	$0.209483\pi$
$239$	69873.0i	1.22325i	−0.791149	+	0.611623i	$0.790517\pi$
$240$	0	0
$241$	−96784.3	−1.66637	−0.833184	−	0.552996i	$-0.813484\pi$
$241$	−96784.3	−1.66637	−0.833184	+	0.552996i	$0.813484\pi$
$242$	− 41130.7i	− 0.702321i
$243$	0	0
$244$	−39609.4	−0.665301
$245$	− 15960.3i	− 0.265895i
$246$	0	0
$247$	−89204.3	−1.46215
$248$	33570.1i	0.545820i
$249$	0	0
$250$	3952.85	0.0632456
$251$	− 52831.5i	− 0.838582i	−0.907852	−	0.419291i	$-0.862279\pi$
$251$	− 52831.5i	− 0.838582i	0.907852	−	0.419291i	$-0.137721\pi$
$252$	0	0
$253$	−7611.54	−0.118914
$254$	− 38920.5i	− 0.603268i
$255$	0	0
$256$	4096.00	0.0625000
$257$	74915.9i	1.13425i	0.823632	+	0.567124i	$0.191944\pi$
$257$	74915.9i	1.13425i	−0.823632	+	0.567124i	$0.808056\pi$
$258$	0	0
$259$	−66485.6	−0.991124
$260$	17616.9i	0.260605i
$261$	0	0
$262$	−72265.1	−1.05275
$263$	− 115945.i	− 1.67626i	−0.545469	−	0.838131i	$-0.683648\pi$
$263$	− 115945.i	− 1.67626i	0.545469	−	0.838131i	$-0.316352\pi$
$264$	0	0
$265$	−5533.60	−0.0787982
$266$	− 79261.6i	− 1.12021i
$267$	0	0
$268$	−36594.3	−0.509500
$269$	− 1709.54i	− 0.0236252i	−0.999930	−	0.0118126i	$-0.996240\pi$
$269$	− 1709.54i	− 0.0236252i	0.999930	−	0.0118126i	$-0.00376015\pi$
$270$	0	0
$271$	142160.	1.93570	0.967849	−	0.251532i	$-0.0809343\pi$
$271$	142160.	1.93570	0.967849	+	0.251532i	$0.0809343\pi$
$272$	− 4397.68i	− 0.0594409i
$273$	0	0
$274$	26641.8	0.354865
$275$	− 1244.32i	− 0.0164538i
$276$	0	0
$277$	−65063.0	−0.847958	−0.423979	−	0.905672i	$-0.639367\pi$
$277$	−65063.0	−0.847958	−0.423979	+	0.905672i	$0.639367\pi$
$278$	96840.4i	1.25305i
$279$	0	0
$280$	−15653.3	−0.199660
$281$	153564.i	1.94481i	0.233297	+	0.972406i	$0.425049\pi$
$281$	153564.i	1.94481i	−0.233297	+	0.972406i	$0.574951\pi$
$282$	0	0
$283$	11472.2	0.143243	0.0716216	−	0.997432i	$-0.477183\pi$
$283$	11472.2	0.143243	0.0716216	+	0.997432i	$0.477183\pi$
$284$	51754.3i	0.641667i
$285$	0	0
$286$	5545.63	0.0677983
$287$	− 168155.i	− 2.04148i
$288$	0	0
$289$	78799.4	0.943468
$290$	− 3290.65i	− 0.0391278i
$291$	0	0
$292$	35404.6	0.415235
$293$	− 67199.4i	− 0.782762i	−0.920229	−	0.391381i	$-0.871997\pi$
$293$	− 67199.4i	− 0.782762i	0.920229	−	0.391381i	$-0.128003\pi$
$294$	0	0
$295$	−49444.1	−0.568159
$296$	24313.4i	0.277500i
$297$	0	0
$298$	−89213.6	−1.00461
$299$	150604.i	1.68459i
$300$	0	0
$301$	164446.	1.81506
$302$	8055.82i	0.0883275i
$303$	0	0
$304$	−28985.5	−0.313642
$305$	− 55355.8i	− 0.595064i
$306$	0	0
$307$	79669.3	0.845306	0.422653	−	0.906292i	$-0.361099\pi$
$307$	79669.3	0.845306	0.422653	+	0.906292i	$0.361099\pi$
$308$	4927.51i	0.0519429i
$309$	0	0
$310$	−46915.7	−0.488196
$311$	151714.i	1.56858i	0.620397	+	0.784288i	$0.286971\pi$
$311$	151714.i	1.56858i	−0.620397	+	0.784288i	$0.713029\pi$
$312$	0	0
$313$	123299.	1.25855	0.629275	−	0.777183i	$-0.283352\pi$
$313$	123299.	1.25855	0.629275	+	0.777183i	$0.283352\pi$
$314$	− 93512.8i	− 0.948444i
$315$	0	0
$316$	53340.9	0.534178
$317$	33957.9i	0.337927i	0.985622	+	0.168963i	$0.0540420\pi$
$317$	33957.9i	0.337927i	−0.985622	+	0.168963i	$0.945958\pi$
$318$	0	0
$319$	−1035.86	−0.0101794
$320$	5724.33i	0.0559017i
$321$	0	0
$322$	−133817.	−1.29063
$323$	31120.4i	0.298291i
$324$	0	0
$325$	−24620.4	−0.233092
$326$	42979.2i	0.404411i
$327$	0	0
$328$	−61493.2	−0.571583
$329$	54353.9i	0.502156i
$330$	0	0
$331$	−116313.	−1.06163	−0.530815	−	0.847488i	$-0.678114\pi$
$331$	−116313.	−1.06163	−0.530815	+	0.847488i	$0.678114\pi$
$332$	60281.4i	0.546899i
$333$	0	0
$334$	−125382.	−1.12394
$335$	− 51142.2i	− 0.455711i
$336$	0	0
$337$	157806.	1.38952	0.694760	−	0.719241i	$-0.255510\pi$
$337$	157806.	1.38952	0.694760	+	0.719241i	$0.255510\pi$
$338$	− 28944.5i	− 0.253357i
$339$	0	0
$340$	6145.94	0.0531656
$341$	14768.6i	0.127008i
$342$	0	0
$343$	60233.2	0.511974
$344$	− 60137.0i	− 0.508188i
$345$	0	0
$346$	−11066.2	−0.0924370
$347$	− 108925.i	− 0.904621i	−0.891860	−	0.452311i	$-0.850600\pi$
$347$	− 108925.i	− 0.904621i	0.891860	−	0.452311i	$-0.149400\pi$
$348$	0	0
$349$	−3922.53	−0.0322044	−0.0161022	−	0.999870i	$-0.505126\pi$
$349$	−3922.53	−0.0322044	−0.0161022	+	0.999870i	$0.505126\pi$
$350$	− 21876.2i	− 0.178581i
$351$	0	0
$352$	1801.96	0.0145432
$353$	− 148394.i	− 1.19088i	−0.803401	−	0.595439i	$-0.796978\pi$
$353$	− 148394.i	− 1.19088i	0.803401	−	0.595439i	$-0.203022\pi$
$354$	0	0
$355$	−72328.9	−0.573925
$356$	− 51859.3i	− 0.409192i
$357$	0	0
$358$	91489.0	0.713843
$359$	− 115124.i	− 0.893257i	−0.894719	−	0.446629i	$-0.852625\pi$
$359$	− 115124.i	− 0.893257i	0.894719	−	0.446629i	$-0.147375\pi$
$360$	0	0
$361$	74796.5	0.573940
$362$	− 109102.i	− 0.832562i
$363$	0	0
$364$	97496.9	0.735848
$365$	49479.4i	0.371397i
$366$	0	0
$367$	101198.	0.751345	0.375672	−	0.926753i	$-0.377412\pi$
$367$	101198.	0.751345	0.375672	+	0.926753i	$0.377412\pi$
$368$	48936.3i	0.361356i
$369$	0	0
$370$	−33979.1	−0.248203
$371$	30624.5i	0.222496i
$372$	0	0
$373$	−43994.3	−0.316212	−0.158106	−	0.987422i	$-0.550539\pi$
$373$	−43994.3	−0.316212	−0.158106	+	0.987422i	$0.550539\pi$
$374$	− 1934.68i	− 0.0138314i
$375$	0	0
$376$	19876.9	0.140596
$377$	20495.9i	0.144206i
$378$	0	0
$379$	23998.6	0.167074	0.0835369	−	0.996505i	$-0.473378\pi$
$379$	23998.6	0.167074	0.0835369	+	0.996505i	$0.473378\pi$
$380$	− 40508.5i	− 0.280530i
$381$	0	0
$382$	−53997.8	−0.370041
$383$	182996.i	1.24751i	0.781620	+	0.623755i	$0.214394\pi$
$383$	182996.i	1.24751i	−0.781620	+	0.623755i	$0.785606\pi$
$384$	0	0
$385$	−6886.40	−0.0464591
$386$	88535.7i	0.594215i
$387$	0	0
$388$	−74276.3	−0.493386
$389$	101696.i	0.672055i	0.941852	+	0.336028i	$0.109084\pi$
$389$	101696.i	0.672055i	−0.941852	+	0.336028i	$0.890916\pi$
$390$	0	0
$391$	52540.6	0.343670
$392$	32301.5i	0.210208i
$393$	0	0
$394$	−69140.5	−0.445390
$395$	74546.2i	0.477784i
$396$	0	0
$397$	97013.2	0.615531	0.307765	−	0.951462i	$-0.400419\pi$
$397$	97013.2	0.615531	0.307765	+	0.951462i	$0.400419\pi$
$398$	80493.8i	0.508155i
$399$	0	0
$400$	−8000.00	−0.0500000
$401$	− 19221.2i	− 0.119534i	−0.998212	−	0.0597672i	$-0.980964\pi$
$401$	− 19221.2i	− 0.119534i	0.998212	−	0.0597672i	$-0.0190358\pi$
$402$	0	0
$403$	292215.	1.79925
$404$	23698.0i	0.145194i
$405$	0	0
$406$	−18211.4	−0.110482
$407$	10696.3i	0.0645719i
$408$	0	0
$409$	247051.	1.47686	0.738431	−	0.674329i	$-0.235567\pi$
$409$	247051.	1.47686	0.738431	+	0.674329i	$0.235567\pi$
$410$	− 85939.4i	− 0.511240i
$411$	0	0
$412$	−36676.8	−0.216071
$413$	273637.i	1.60426i
$414$	0	0
$415$	−84245.8	−0.489161
$416$	− 35654.1i	− 0.206026i
$417$	0	0
$418$	−12751.7	−0.0729819
$419$	82428.4i	0.469514i	0.972054	+	0.234757i	$0.0754295\pi$
$419$	82428.4i	0.469514i	−0.972054	+	0.234757i	$0.924571\pi$
$420$	0	0
$421$	17970.2	0.101388	0.0506942	−	0.998714i	$-0.483857\pi$
$421$	17970.2	0.101388	0.0506942	+	0.998714i	$0.483857\pi$
$422$	− 119952.i	− 0.673571i
$423$	0	0
$424$	11199.2	0.0622954
$425$	8589.22i	0.0475528i
$426$	0	0
$427$	−306355.	−1.68023
$428$	− 19401.0i	− 0.105910i
$429$	0	0
$430$	84044.0	0.454537
$431$	− 191503.i	− 1.03091i	−0.856916	−	0.515456i	$-0.827623\pi$
$431$	− 191503.i	− 1.03091i	0.856916	−	0.515456i	$-0.172377\pi$
$432$	0	0
$433$	212894.	1.13550	0.567751	−	0.823200i	$-0.307814\pi$
$433$	212894.	1.13550	0.567751	+	0.823200i	$0.307814\pi$
$434$	259645.i	1.37848i
$435$	0	0
$436$	−129253.	−0.679933
$437$	− 346300.i	− 1.81338i
$438$	0	0
$439$	329767.	1.71111	0.855554	−	0.517713i	$-0.173217\pi$
$439$	329767.	1.71111	0.855554	+	0.517713i	$0.173217\pi$
$440$	2518.32i	0.0130079i
$441$	0	0
$442$	−38280.1	−0.195942
$443$	− 44340.5i	− 0.225940i	−0.993598	−	0.112970i	$-0.963964\pi$
$443$	− 44340.5i	− 0.225940i	0.993598	−	0.112970i	$-0.0360364\pi$
$444$	0	0
$445$	72475.6	0.365992
$446$	− 153529.i	− 0.771829i
$447$	0	0
$448$	31680.1	0.157845
$449$	370160.i	1.83610i	0.396465	+	0.918050i	$0.370237\pi$
$449$	370160.i	1.83610i	−0.396465	+	0.918050i	$0.629763\pi$
$450$	0	0
$451$	−27052.9	−0.133003
$452$	102775.i	0.503050i
$453$	0	0
$454$	−249187.	−1.20897
$455$	136256.i	0.658163i
$456$	0	0
$457$	80899.8	0.387360	0.193680	−	0.981065i	$-0.437958\pi$
$457$	80899.8	0.387360	0.193680	+	0.981065i	$0.437958\pi$
$458$	38463.6i	0.183366i
$459$	0	0
$460$	−68390.6	−0.323207
$461$	13947.5i	0.0656289i	0.999461	+	0.0328145i	$0.0104470\pi$
$461$	13947.5i	0.0656289i	−0.999461	+	0.0328145i	$0.989553\pi$
$462$	0	0
$463$	−44144.1	−0.205926	−0.102963	−	0.994685i	$-0.532832\pi$
$463$	−44144.1	−0.205926	−0.102963	+	0.994685i	$0.532832\pi$
$464$	6659.81i	0.0309333i
$465$	0	0
$466$	26639.7	0.122675
$467$	234450.i	1.07502i	0.843257	+	0.537511i	$0.180635\pi$
$467$	234450.i	1.07502i	−0.843257	+	0.537511i	$0.819365\pi$
$468$	0	0
$469$	−283035.	−1.28675
$470$	27778.9i	0.125753i
$471$	0	0
$472$	100068.	0.449169
$473$	− 26456.2i	− 0.118251i
$474$	0	0
$475$	56612.4	0.250914
$476$	− 34013.4i	− 0.150119i
$477$	0	0
$478$	197631.	0.864965
$479$	99102.2i	0.431929i	0.976401	+	0.215964i	$0.0692895\pi$
$479$	99102.2i	0.431929i	−0.976401	+	0.215964i	$0.930710\pi$
$480$	0	0
$481$	211639.	0.914757
$482$	273747.i	1.17830i
$483$	0	0
$484$	−116335.	−0.496616
$485$	− 103804.i	− 0.441298i
$486$	0	0
$487$	319957.	1.34907	0.674534	−	0.738243i	$-0.264344\pi$
$487$	319957.	1.34907	0.674534	+	0.738243i	$0.264344\pi$
$488$	112032.i	0.470439i
$489$	0	0
$490$	−45142.7	−0.188016
$491$	332287.i	1.37832i	0.724609	+	0.689161i	$0.242021\pi$
$491$	332287.i	1.37832i	−0.724609	+	0.689161i	$0.757979\pi$
$492$	0	0
$493$	7150.32	0.0294192
$494$	252308.i	1.03390i
$495$	0	0
$496$	94950.6	0.385953
$497$	400288.i	1.62054i
$498$	0	0
$499$	−186584.	−0.749330	−0.374665	−	0.927160i	$-0.622242\pi$
$499$	−186584.	−0.749330	−0.374665	+	0.927160i	$0.622242\pi$
$500$	− 11180.3i	− 0.0447214i
$501$	0	0
$502$	−149430.	−0.592967
$503$	477437.i	1.88704i	0.331320	+	0.943518i	$0.392506\pi$
$503$	477437.i	1.88704i	−0.331320	+	0.943518i	$0.607494\pi$
$504$	0	0
$505$	−33119.0	−0.129866
$506$	21528.7i	0.0840846i
$507$	0	0
$508$	−110084.	−0.426575
$509$	− 185062.i	− 0.714303i	−0.934046	−	0.357152i	$-0.883748\pi$
$509$	− 185062.i	− 0.714303i	0.934046	−	0.357152i	$-0.116252\pi$
$510$	0	0
$511$	273833.	1.04868
$512$	− 11585.2i	− 0.0441942i
$513$	0	0
$514$	211894.	0.802034
$515$	− 51257.4i	− 0.193260i
$516$	0	0
$517$	8744.51	0.0327156
$518$	188050.i	0.700831i
$519$	0	0
$520$	49828.1	0.184276
$521$	230292.i	0.848406i	0.905567	+	0.424203i	$0.139446\pi$
$521$	230292.i	0.848406i	−0.905567	+	0.424203i	$0.860554\pi$
$522$	0	0
$523$	317014.	1.15898	0.579488	−	0.814981i	$-0.303252\pi$
$523$	317014.	1.15898	0.579488	+	0.814981i	$0.303252\pi$
$524$	204397.i	0.744408i
$525$	0	0
$526$	−327943.	−1.18530
$527$	− 101944.i	− 0.367063i
$528$	0	0
$529$	−304818.	−1.08926
$530$	15651.4i	0.0557187i
$531$	0	0
$532$	−224186.	−0.792108
$533$	535275.i	1.88418i
$534$	0	0
$535$	27113.8	0.0947289
$536$	103504.i	0.360271i
$537$	0	0
$538$	−4835.31	−0.0167055
$539$	14210.5i	0.0489137i
$540$	0	0
$541$	−241663.	−0.825688	−0.412844	−	0.910802i	$-0.635464\pi$
$541$	−241663.	−0.825688	−0.412844	+	0.910802i	$0.635464\pi$
$542$	− 402088.i	− 1.36875i
$543$	0	0
$544$	−12438.5	−0.0420311
$545$	− 180636.i	− 0.608151i
$546$	0	0
$547$	53489.8	0.178771	0.0893853	−	0.995997i	$-0.471510\pi$
$547$	53489.8	0.178771	0.0893853	+	0.995997i	$0.471510\pi$
$548$	− 75354.5i	− 0.250927i
$549$	0	0
$550$	−3519.46	−0.0116346
$551$	− 47128.5i	− 0.155232i
$552$	0	0
$553$	412560.	1.34908
$554$	184026.i	0.599597i
$555$	0	0
$556$	273906.	0.886037
$557$	127327.i	0.410401i	0.978720	+	0.205201i	$0.0657847\pi$
$557$	127327.i	0.410401i	−0.978720	+	0.205201i	$0.934215\pi$
$558$	0	0
$559$	−523469.	−1.67520
$560$	44274.3i	0.141181i
$561$	0	0
$562$	434345.	1.37519
$563$	− 186908.i	− 0.589674i	−0.955548	−	0.294837i	$-0.904735\pi$
$563$	− 186908.i	− 0.589674i	0.955548	−	0.294837i	$-0.0952653\pi$
$564$	0	0
$565$	−143633.	−0.449942
$566$	− 32448.3i	− 0.101288i
$567$	0	0
$568$	146383.	0.453727
$569$	76274.8i	0.235590i	0.993038	+	0.117795i	$0.0375826\pi$
$569$	76274.8i	0.235590i	−0.993038	+	0.117795i	$0.962417\pi$
$570$	0	0
$571$	−88913.4	−0.272706	−0.136353	−	0.990660i	$-0.543538\pi$
$571$	−88913.4	−0.272706	−0.136353	+	0.990660i	$0.543538\pi$
$572$	− 15685.4i	− 0.0479406i
$573$	0	0
$574$	−475613.	−1.44354
$575$	− 95578.8i	− 0.289085i
$576$	0	0
$577$	18718.7	0.0562242	0.0281121	−	0.999605i	$-0.491050\pi$
$577$	18718.7	0.0562242	0.0281121	+	0.999605i	$0.491050\pi$
$578$	− 222878.i	− 0.667133i
$579$	0	0
$580$	−9307.37	−0.0276676
$581$	466240.i	1.38120i
$582$	0	0
$583$	4926.90	0.0144956
$584$	− 100139.i	− 0.293615i
$585$	0	0
$586$	−190069.	−0.553497
$587$	149761.i	0.434634i	0.976101	+	0.217317i	$0.0697305\pi$
$587$	149761.i	0.434634i	−0.976101	+	0.217317i	$0.930270\pi$
$588$	0	0
$589$	−671923.	−1.93682
$590$	139849.i	0.401749i
$591$	0	0
$592$	68768.8	0.196222
$593$	− 636352.i	− 1.80962i	−0.425814	−	0.904811i	$-0.640012\pi$
$593$	− 636352.i	− 1.80962i	0.425814	−	0.904811i	$-0.359988\pi$
$594$	0	0
$595$	47535.2	0.134271
$596$	252334.i	0.710368i
$597$	0	0
$598$	425972.	1.19118
$599$	478872.i	1.33464i	0.744769	+	0.667322i	$0.232559\pi$
$599$	478872.i	1.33464i	−0.744769	+	0.667322i	$0.767441\pi$
$600$	0	0
$601$	−233769.	−0.647198	−0.323599	−	0.946194i	$-0.604893\pi$
$601$	−233769.	−0.647198	−0.323599	+	0.946194i	$0.604893\pi$
$602$	− 465123.i	− 1.28344i
$603$	0	0
$604$	22785.3	0.0624570
$605$	− 162583.i	− 0.444187i
$606$	0	0
$607$	155285.	0.421455	0.210728	−	0.977545i	$-0.432417\pi$
$607$	155285.	0.421455	0.210728	+	0.977545i	$0.432417\pi$
$608$	81983.5i	0.221778i
$609$	0	0
$610$	−156570.	−0.420774
$611$	− 173021.i	− 0.463465i
$612$	0	0
$613$	−351436.	−0.935246	−0.467623	−	0.883928i	$-0.654890\pi$
$613$	−351436.	−0.935246	−0.467623	+	0.883928i	$0.654890\pi$
$614$	− 225339.i	− 0.597722i
$615$	0	0
$616$	13937.1	0.0367292
$617$	− 524669.i	− 1.37821i	−0.724662	−	0.689105i	$-0.758004\pi$
$617$	− 524669.i	− 1.37821i	0.724662	−	0.689105i	$-0.241996\pi$
$618$	0	0
$619$	−477718.	−1.24678	−0.623391	−	0.781910i	$-0.714246\pi$
$619$	−477718.	−1.24678	−0.623391	+	0.781910i	$0.714246\pi$
$620$	132698.i	0.345207i
$621$	0	0
$622$	429112.	1.10915
$623$	− 401101.i	− 1.03342i
$624$	0	0
$625$	15625.0	0.0400000
$626$	− 348742.i	− 0.889929i
$627$	0	0
$628$	−264494.	−0.670651
$629$	− 73833.7i	− 0.186618i
$630$	0	0
$631$	−43917.1	−0.110300	−0.0551499	−	0.998478i	$-0.517564\pi$
$631$	−43917.1	−0.110300	−0.0551499	+	0.998478i	$0.517564\pi$
$632$	− 150871.i	− 0.377721i
$633$	0	0
$634$	96047.5	0.238950
$635$	− 153847.i	− 0.381540i
$636$	0	0
$637$	281172.	0.692936
$638$	2929.87i	0.00719791i
$639$	0	0
$640$	16190.9	0.0395285
$641$	− 557772.i	− 1.35750i	−0.734368	−	0.678751i	$-0.762521\pi$
$641$	− 557772.i	− 1.35750i	0.734368	−	0.678751i	$-0.237479\pi$
$642$	0	0
$643$	519270.	1.25595	0.627974	−	0.778235i	$-0.283885\pi$
$643$	519270.	1.25595	0.627974	+	0.778235i	$0.283885\pi$
$644$	378493.i	0.912612i
$645$	0	0
$646$	88021.7	0.210923
$647$	76924.3i	0.183762i	0.995770	+	0.0918809i	$0.0292879\pi$
$647$	76924.3i	0.183762i	−0.995770	+	0.0918809i	$0.970712\pi$
$648$	0	0
$649$	44023.1	0.104518
$650$	69636.9i	0.164821i
$651$	0	0
$652$	121563.	0.285962
$653$	273840.i	0.642201i	0.947045	+	0.321101i	$0.104053\pi$
$653$	273840.i	0.642201i	−0.947045	+	0.321101i	$0.895947\pi$
$654$	0	0
$655$	−285653.	−0.665819
$656$	173929.i	0.404170i
$657$	0	0
$658$	153736.	0.355078
$659$	− 591257.i	− 1.36146i	−0.732534	−	0.680730i	$-0.761662\pi$
$659$	− 591257.i	− 1.36146i	0.732534	−	0.680730i	$-0.238338\pi$
$660$	0	0
$661$	727188.	1.66435	0.832173	−	0.554516i	$-0.187096\pi$
$661$	727188.	1.66435	0.832173	+	0.554516i	$0.187096\pi$
$662$	328983.i	0.750686i
$663$	0	0
$664$	170502.	0.386716
$665$	− 313309.i	− 0.708483i
$666$	0	0
$667$	−79567.1	−0.178847
$668$	354635.i	0.794747i
$669$	0	0
$670$	−144652.	−0.322236
$671$	49286.6i	0.109467i
$672$	0	0
$673$	319706.	0.705864	0.352932	−	0.935649i	$-0.385185\pi$
$673$	319706.	0.705864	0.352932	+	0.935649i	$0.385185\pi$
$674$	− 446344.i	− 0.982539i
$675$	0	0
$676$	−81867.4	−0.179150
$677$	− 387691.i	− 0.845880i	−0.906158	−	0.422940i	$-0.860998\pi$
$677$	− 387691.i	− 0.845880i	0.906158	−	0.422940i	$-0.139002\pi$
$678$	0	0
$679$	−574482.	−1.24605
$680$	− 17383.4i	− 0.0375938i
$681$	0	0
$682$	41771.9	0.0898081
$683$	− 330850.i	− 0.709234i	−0.935012	−	0.354617i	$-0.884611\pi$
$683$	− 330850.i	− 0.709234i	0.935012	−	0.354617i	$-0.115389\pi$
$684$	0	0
$685$	105311.	0.224436
$686$	− 170365.i	− 0.362020i
$687$	0	0
$688$	−170093.	−0.359343
$689$	− 97484.9i	− 0.205352i
$690$	0	0
$691$	−422232.	−0.884290	−0.442145	−	0.896944i	$-0.645782\pi$
$691$	−422232.	−0.884290	−0.442145	+	0.896944i	$0.645782\pi$
$692$	31299.9i	0.0653628i
$693$	0	0
$694$	−308085.	−0.639664
$695$	382795.i	0.792496i
$696$	0	0
$697$	186739.	0.384388
$698$	11094.6i	0.0227719i
$699$	0	0
$700$	−61875.2	−0.126276
$701$	390766.i	0.795208i	0.917557	+	0.397604i	$0.130158\pi$
$701$	390766.i	0.795208i	−0.917557	+	0.397604i	$0.869842\pi$
$702$	0	0
$703$	−486645.	−0.984696
$704$	− 5096.72i	− 0.0102836i
$705$	0	0
$706$	−419722.	−0.842078
$707$	183290.i	0.366691i
$708$	0	0
$709$	−56502.2	−0.112402	−0.0562009	−	0.998419i	$-0.517899\pi$
$709$	−56502.2	−0.112402	−0.0562009	+	0.998419i	$0.517899\pi$
$710$	204577.i	0.405826i
$711$	0	0
$712$	−146680.	−0.289342
$713$	1.13441e6i	2.23147i
$714$	0	0
$715$	21921.0	0.0428794
$716$	− 258770.i	− 0.504763i
$717$	0	0
$718$	−325620.	−0.631628
$719$	615632.i	1.19087i	0.803405	+	0.595434i	$0.203020\pi$
$719$	615632.i	1.19087i	−0.803405	+	0.595434i	$0.796980\pi$
$720$	0	0
$721$	−283673.	−0.545691
$722$	− 211556.i	− 0.405837i
$723$	0	0
$724$	−308588.	−0.588710
$725$	− 13007.4i	− 0.0247466i
$726$	0	0
$727$	479255.	0.906772	0.453386	−	0.891314i	$-0.350216\pi$
$727$	479255.	0.906772	0.453386	+	0.891314i	$0.350216\pi$
$728$	− 275763.i	− 0.520323i
$729$	0	0
$730$	139949.	0.262617
$731$	182621.i	0.341755i
$732$	0	0
$733$	−97739.4	−0.181912	−0.0909561	−	0.995855i	$-0.528992\pi$
$733$	−97739.4	−0.181912	−0.0909561	+	0.995855i	$0.528992\pi$
$734$	− 286231.i	− 0.531281i
$735$	0	0
$736$	138413.	0.255518
$737$	45535.0i	0.0838321i
$738$	0	0
$739$	131781.	0.241304	0.120652	−	0.992695i	$-0.461502\pi$
$739$	131781.	0.241304	0.120652	+	0.992695i	$0.461502\pi$
$740$	96107.3i	0.175506i
$741$	0	0
$742$	86619.2	0.157328
$743$	200175.i	0.362603i	0.983428	+	0.181301i	$0.0580310\pi$
$743$	200175.i	0.362603i	−0.983428	+	0.181301i	$0.941969\pi$
$744$	0	0
$745$	−352648.	−0.635373
$746$	124435.i	0.223596i
$747$	0	0
$748$	−5472.11	−0.00978028
$749$	− 150055.i	− 0.267478i
$750$	0	0
$751$	−697834.	−1.23729	−0.618646	−	0.785669i	$-0.712319\pi$
$751$	−697834.	−1.23729	−0.618646	+	0.785669i	$0.712319\pi$
$752$	− 56220.4i	− 0.0994165i
$753$	0	0
$754$	57971.1	0.101969
$755$	31843.4i	0.0558632i
$756$	0	0
$757$	26929.2	0.0469929	0.0234964	−	0.999724i	$-0.492520\pi$
$757$	26929.2	0.0469929	0.0234964	+	0.999724i	$0.492520\pi$
$758$	− 67878.4i	− 0.118139i
$759$	0	0
$760$	−114575.	−0.198365
$761$	− 281719.i	− 0.486460i	−0.969969	−	0.243230i	$-0.921793\pi$
$761$	− 281719.i	− 0.486460i	0.969969	−	0.243230i	$-0.0782069\pi$
$762$	0	0
$763$	−999690.	−1.71718
$764$	152729.i	0.261658i
$765$	0	0
$766$	517591.	0.882122
$767$	− 871051.i	− 1.48065i
$768$	0	0
$769$	59132.8	0.0999944	0.0499972	−	0.998749i	$-0.484079\pi$
$769$	59132.8	0.0999944	0.0499972	+	0.998749i	$0.484079\pi$
$770$	19477.7i	0.0328516i
$771$	0	0
$772$	250417.	0.420174
$773$	− 368583.i	− 0.616846i	−0.951249	−	0.308423i	$-0.900199\pi$
$773$	− 368583.i	− 0.616846i	0.951249	−	0.308423i	$-0.0998012\pi$
$774$	0	0
$775$	−185450.	−0.308762
$776$	210085.i	0.348877i
$777$	0	0
$778$	287640.	0.475215
$779$	− 1.23082e6i	− 2.02824i
$780$	0	0
$781$	64398.8	0.105579
$782$	− 148607.i	− 0.243011i
$783$	0	0
$784$	91362.3	0.148640
$785$	− 369642.i	− 0.599849i
$786$	0	0
$787$	−369902.	−0.597224	−0.298612	−	0.954375i	$-0.596524\pi$
$787$	−369902.	−0.597224	−0.298612	+	0.954375i	$0.596524\pi$
$788$	195559.i	0.314938i
$789$	0	0
$790$	210848.	0.337844
$791$	794904.i	1.27046i
$792$	0	0
$793$	975198.	1.55077
$794$	− 274395.i	− 0.435246i
$795$	0	0
$796$	227671.	0.359320
$797$	749997.i	1.18071i	0.807144	+	0.590355i	$0.201012\pi$
$797$	749997.i	1.18071i	−0.807144	+	0.590355i	$0.798988\pi$
$798$	0	0
$799$	−60361.2	−0.0945506
$800$	22627.4i	0.0353553i
$801$	0	0
$802$	−54365.9	−0.0845235
$803$	− 44054.5i	− 0.0683218i
$804$	0	0
$805$	−528960.	−0.816265
$806$	− 826509.i	− 1.27226i
$807$	0	0
$808$	67028.2	0.102668
$809$	− 95022.1i	− 0.145187i	−0.997362	−	0.0725935i	$-0.976872\pi$
$809$	− 95022.1i	− 0.145187i	0.997362	−	0.0725935i	$-0.0231276\pi$
$810$	0	0
$811$	859069.	1.30613	0.653065	−	0.757302i	$-0.273483\pi$
$811$	859069.	1.30613	0.653065	+	0.757302i	$0.273483\pi$
$812$	51509.6i	0.0781225i
$813$	0	0
$814$	30253.6	0.0456592
$815$	169890.i	0.255772i
$816$	0	0
$817$	1.20367e6	1.80328
$818$	− 698766.i	− 1.04430i
$819$	0	0
$820$	−243073.	−0.361501
$821$	− 884514.i	− 1.31225i	−0.754650	−	0.656127i	$-0.772193\pi$
$821$	− 884514.i	− 1.31225i	0.754650	−	0.656127i	$-0.227807\pi$
$822$	0	0
$823$	1.01633e6	1.50050	0.750250	−	0.661154i	$-0.229933\pi$
$823$	1.01633e6	1.50050	0.750250	+	0.661154i	$0.229933\pi$
$824$	103738.i	0.152785i
$825$	0	0
$826$	773964.	1.13438
$827$	− 750150.i	− 1.09682i	−0.836208	−	0.548412i	$-0.815232\pi$
$827$	− 750150.i	− 1.09682i	0.836208	−	0.548412i	$-0.184768\pi$
$828$	0	0
$829$	−1.15714e6	−1.68375	−0.841873	−	0.539676i	$-0.818547\pi$
$829$	−1.15714e6	−1.68375	−0.841873	+	0.539676i	$0.818547\pi$
$830$	238283.i	0.345889i
$831$	0	0
$832$	−100845.	−0.145683
$833$	− 98091.3i	− 0.141365i
$834$	0	0
$835$	−495617.	−0.710843
$836$	36067.2i	0.0516060i
$837$	0	0
$838$	233143.	0.331997
$839$	441690.i	0.627471i	0.949510	+	0.313736i	$0.101581\pi$
$839$	441690.i	0.627471i	−0.949510	+	0.313736i	$0.898419\pi$
$840$	0	0
$841$	696453.	0.984690
$842$	− 50827.3i	− 0.0716924i
$843$	0	0
$844$	−339276.	−0.476287
$845$	− 114413.i	− 0.160237i
$846$	0	0
$847$	−899783.	−1.25421
$848$	− 31676.2i	− 0.0440495i
$849$	0	0
$850$	24294.0	0.0336249
$851$	821604.i	1.13450i
$852$	0	0
$853$	802656.	1.10314	0.551571	−	0.834128i	$-0.314029\pi$
$853$	802656.	1.10314	0.551571	+	0.834128i	$0.314029\pi$
$854$	866502.i	1.18810i
$855$	0	0
$856$	−54874.4	−0.0748898
$857$	200409.i	0.272870i	0.990649	+	0.136435i	$0.0435644\pi$
$857$	200409.i	0.272870i	−0.990649	+	0.136435i	$0.956436\pi$
$858$	0	0
$859$	−1.06213e6	−1.43944	−0.719718	−	0.694267i	$-0.755729\pi$
$859$	−1.06213e6	−1.43944	−0.719718	+	0.694267i	$0.755729\pi$
$860$	− 237712.i	− 0.321406i
$861$	0	0
$862$	−541653.	−0.728965
$863$	− 1.39351e6i	− 1.87106i	−0.353248	−	0.935530i	$-0.614923\pi$
$863$	− 1.39351e6i	− 1.87106i	0.353248	−	0.935530i	$-0.385077\pi$
$864$	0	0
$865$	−43743.0	−0.0584623
$866$	− 602155.i	− 0.802921i
$867$	0	0
$868$	734386.	0.974731
$869$	− 66373.0i	− 0.0878926i
$870$	0	0
$871$	900967.	1.18761
$872$	365581.i	0.480785i
$873$	0	0
$874$	−979485.	−1.28226
$875$	− 86473.2i	− 0.112945i
$876$	0	0
$877$	−1.08681e6	−1.41304	−0.706522	−	0.707691i	$-0.749737\pi$
$877$	−1.08681e6	−1.41304	−0.706522	+	0.707691i	$0.749737\pi$
$878$	− 932721.i	− 1.20994i
$879$	0	0
$880$	7122.89	0.00919794
$881$	− 982562.i	− 1.26593i	−0.774182	−	0.632963i	$-0.781838\pi$
$881$	− 982562.i	− 1.26593i	0.774182	−	0.632963i	$-0.218162\pi$
$882$	0	0
$883$	615464.	0.789371	0.394686	−	0.918816i	$-0.370854\pi$
$883$	615464.	0.789371	0.394686	+	0.918816i	$0.370854\pi$
$884$	108272.i	0.138552i
$885$	0	0
$886$	−125414.	−0.159764
$887$	− 446766.i	− 0.567848i	−0.958847	−	0.283924i	$-0.908364\pi$
$887$	− 446766.i	− 0.567848i	0.958847	−	0.283924i	$-0.0916364\pi$
$888$	0	0
$889$	−851431.	−1.07732
$890$	− 204992.i	− 0.258796i
$891$	0	0
$892$	−434246.	−0.545765
$893$	397847.i	0.498899i
$894$	0	0
$895$	361642.	0.451474
$896$	− 89604.8i	− 0.111613i
$897$	0	0
$898$	1.04697e6	1.29832
$899$	154383.i	0.191021i
$900$	0	0
$901$	−34009.2	−0.0418935
$902$	76517.1i	0.0940471i
$903$	0	0
$904$	290692.	0.355710
$905$	− 431264.i	− 0.526558i
$906$	0	0
$907$	−66149.4	−0.0804102	−0.0402051	−	0.999191i	$-0.512801\pi$
$907$	−66149.4	−0.0804102	−0.0402051	+	0.999191i	$0.512801\pi$
$908$	704808.i	0.854868i
$909$	0	0
$910$	385391.	0.465391
$911$	837148.i	1.00871i	0.863497	+	0.504354i	$0.168269\pi$
$911$	837148.i	1.00871i	−0.863497	+	0.504354i	$0.831731\pi$
$912$	0	0
$913$	75009.2	0.0899856
$914$	− 228819.i	− 0.273905i
$915$	0	0
$916$	108792.	0.129659
$917$	1.58088e6i	1.88002i
$918$	0	0
$919$	−213911.	−0.253280	−0.126640	−	0.991949i	$-0.540419\pi$
$919$	−213911.	−0.253280	−0.126640	+	0.991949i	$0.540419\pi$
$920$	193438.i	0.228542i
$921$	0	0
$922$	39449.6	0.0464067
$923$	− 1.27421e6i	− 1.49568i
$924$	0	0
$925$	−134314.	−0.156978
$926$	124859.i	0.145612i
$927$	0	0
$928$	18836.8	0.0218731
$929$	− 581487.i	− 0.673765i	−0.941547	−	0.336883i	$-0.890627\pi$
$929$	− 581487.i	− 0.673765i	0.941547	−	0.336883i	$-0.109373\pi$
$930$	0	0
$931$	−646530.	−0.745915
$932$	− 75348.4i	− 0.0867446i
$933$	0	0
$934$	663126.	0.760155
$935$	− 7647.50i	− 0.00874775i
$936$	0	0
$937$	641743.	0.730940	0.365470	−	0.930823i	$-0.380908\pi$
$937$	641743.	0.730940	0.365470	+	0.930823i	$0.380908\pi$
$938$	800544.i	0.909871i
$939$	0	0
$940$	78570.5	0.0889208
$941$	− 1.10840e6i	− 1.25175i	−0.779925	−	0.625873i	$-0.784743\pi$
$941$	− 1.10840e6i	− 1.25175i	0.779925	−	0.625873i	$-0.215257\pi$
$942$	0	0
$943$	−2.07799e6	−2.33679
$944$	− 283034.i	− 0.317611i
$945$	0	0
$946$	−74829.5	−0.0836162
$947$	559210.i	0.623555i	0.950155	+	0.311778i	$0.100924\pi$
$947$	559210.i	0.623555i	−0.950155	+	0.311778i	$0.899076\pi$
$948$	0	0
$949$	−871674.	−0.967880
$950$	− 160124.i	− 0.177423i
$951$	0	0
$952$	−96204.4	−0.106150
$953$	− 1.38363e6i	− 1.52347i	−0.647887	−	0.761737i	$-0.724347\pi$
$953$	− 1.38363e6i	− 1.52347i	0.647887	−	0.761737i	$-0.275653\pi$
$954$	0	0
$955$	−213445.	−0.234034
$956$	− 558984.i	− 0.611623i
$957$	0	0
$958$	280303.	0.305420
$959$	− 582821.i	− 0.633721i
$960$	0	0
$961$	1.27756e6	1.38336
$962$	− 598606.i	− 0.646831i
$963$	0	0
$964$	774274.	0.833184
$965$	349968.i	0.375815i
$966$	0	0
$967$	−764463.	−0.817530	−0.408765	−	0.912640i	$-0.634040\pi$
$967$	−764463.	−0.817530	−0.408765	+	0.912640i	$0.634040\pi$
$968$	329046.i	0.351160i
$969$	0	0
$970$	−293603.	−0.312045
$971$	163479.i	0.173390i	0.996235	+	0.0866948i	$0.0276305\pi$
$971$	163479.i	0.173390i	−0.996235	+	0.0866948i	$0.972369\pi$
$972$	0	0
$973$	2.11850e6	2.23770
$974$	− 904976.i	− 0.953936i
$975$	0	0
$976$	316875.	0.332651
$977$	209157.i	0.219121i	0.993980	+	0.109560i	$0.0349443\pi$
$977$	209157.i	0.219121i	−0.993980	+	0.109560i	$0.965056\pi$
$978$	0	0
$979$	−64529.5	−0.0673275
$980$	127683.i	0.132947i
$981$	0	0
$982$	939850.	0.974620
$983$	− 1.46858e6i	− 1.51982i	−0.650030	−	0.759909i	$-0.725244\pi$
$983$	− 1.46858e6i	− 1.51982i	0.650030	−	0.759909i	$-0.274756\pi$
$984$	0	0
$985$	−273302.	−0.281689
$986$	− 20224.2i	− 0.0208025i
$987$	0	0
$988$	713635.	0.731075
$989$	− 2.03216e6i	− 2.07762i
$990$	0	0
$991$	−891859.	−0.908132	−0.454066	−	0.890968i	$-0.650027\pi$
$991$	−891859.	−0.908132	−0.454066	+	0.890968i	$0.650027\pi$
$992$	− 268561.i	− 0.272910i
$993$	0	0
$994$	1.13219e6	1.14590
$995$	318180.i	0.321385i
$996$	0	0
$997$	−1.86794e6	−1.87920	−0.939601	−	0.342272i	$-0.888804\pi$
$997$	−1.86794e6	−1.87920	−0.939601	+	0.342272i	$0.888804\pi$
$998$	527739.i	0.529856i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	810.5.d.c.161.15		32
3.2	odd	2	inner	810.5.d.c.161.16		32
9.2	odd	6		270.5.h.a.71.16		32
9.4	even	3		270.5.h.a.251.16		32
9.5	odd	6		90.5.h.a.11.1	✓	32
9.7	even	3		90.5.h.a.41.1	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
90.5.h.a.11.1	✓	32	9.5	odd	6
90.5.h.a.41.1	yes	32	9.7	even	3
270.5.h.a.71.16		32	9.2	odd	6
270.5.h.a.251.16		32	9.4	even	3
810.5.d.c.161.15		32	1.1	even	1	trivial
810.5.d.c.161.16		32	3.2	odd	2	inner

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:	\( N \)	\(=\)	\( 810 = 2 \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	810.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.82843i q^{2} -8.00000 q^{4} -11.1803i q^{5} -61.8752 q^{7} +22.6274i q^{8} +O(q^{10})\)	\(q-2.82843i q^{2} -8.00000 q^{4} -11.1803i q^{5} -61.8752 q^{7} +22.6274i q^{8} -31.6228 q^{10} +9.95454i q^{11} +196.963 q^{13} +175.009i q^{14} +64.0000 q^{16} -68.7137i q^{17} -452.899 q^{19} +89.4427i q^{20} +28.1557 q^{22} +764.630i q^{23} -125.000 q^{25} -557.095i q^{26} +495.001 q^{28} +104.060i q^{29} +1483.60 q^{31} -181.019i q^{32} -194.352 q^{34} +691.785i q^{35} +1074.51 q^{37} +1280.99i q^{38} +252.982 q^{40} +2717.64i q^{41} -2657.70 q^{43} -79.6363i q^{44} +2162.70 q^{46} -878.444i q^{47} +1427.54 q^{49} +353.553i q^{50} -1575.70 q^{52} -494.940i q^{53} +111.295 q^{55} -1400.08i q^{56} +294.325 q^{58} -4422.41i q^{59} +4951.17 q^{61} -4196.27i q^{62} -512.000 q^{64} -2202.11i q^{65} +4574.29 q^{67} +549.710i q^{68} +1956.66 q^{70} -6469.29i q^{71} -4425.57 q^{73} -3039.18i q^{74} +3623.19 q^{76} -615.939i q^{77} -6667.61 q^{79} -715.542i q^{80} +7686.65 q^{82} -7535.17i q^{83} -768.243 q^{85} +7517.12i q^{86} -225.246 q^{88} +6482.42i q^{89} -12187.1 q^{91} -6117.04i q^{92} -2484.62 q^{94} +5063.56i q^{95} +9284.54 q^{97} -4037.68i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 256 q^{4} - 104 q^{7}+O(q^{10}) \) 32 * q - 256 * q^4 - 104 * q^7	\( 32 q - 256 q^{4} - 104 q^{7} + 40 q^{13} + 2048 q^{16} - 200 q^{19} + 1344 q^{22} - 4000 q^{25} + 832 q^{28} - 1472 q^{31} - 384 q^{34} - 4136 q^{37} - 272 q^{43} - 2112 q^{46} + 22296 q^{49} - 320 q^{52} - 12344 q^{61} - 16384 q^{64} + 40936 q^{67} + 4800 q^{70} - 41432 q^{73} + 1600 q^{76} - 14048 q^{79} - 17664 q^{82} - 17400 q^{85} - 10752 q^{88} + 69392 q^{91} + 1344 q^{94} + 36664 q^{97}+O(q^{100}) \) 32 * q - 256 * q^4 - 104 * q^7 + 40 * q^13 + 2048 * q^16 - 200 * q^19 + 1344 * q^22 - 4000 * q^25 + 832 * q^28 - 1472 * q^31 - 384 * q^34 - 4136 * q^37 - 272 * q^43 - 2112 * q^46 + 22296 * q^49 - 320 * q^52 - 12344 * q^61 - 16384 * q^64 + 40936 * q^67 + 4800 * q^70 - 41432 * q^73 + 1600 * q^76 - 14048 * q^79 - 17664 * q^82 - 17400 * q^85 - 10752 * q^88 + 69392 * q^91 + 1344 * q^94 + 36664 * q^97

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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