LMFDB - Embedded newform 825.4.c.l.199.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [825,4,Mod(199,825)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(825, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("825.199");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 825 = 3 \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	825.c (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:	$48.6765757547$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.2230106176.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 41x^{4} + 452x^{2} + 676 $ x^6 + 41x^4 + 452x^2 + 676
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 165)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.6
Root			$4.26150i$ of defining polynomial
Character	$\chi$	$=$	825.199
Dual form			825.4.c.l.199.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+5.26150i q^{2} -3.00000i q^{3} -19.6833 q^{4} +15.7845 q^{6} +10.3207i q^{7} -61.4719i q^{8} -9.00000 q^{9} +11.0000 q^{11} +59.0500i q^{12} +63.9817i q^{13} -54.3024 q^{14} +165.967 q^{16} -17.1461i q^{17} -47.3535i q^{18} -90.2104 q^{19} +30.9621 q^{21} +57.8765i q^{22} -212.605i q^{23} -184.416 q^{24} -336.639 q^{26} +27.0000i q^{27} -203.146i q^{28} -57.5461 q^{29} -141.704 q^{31} +381.462i q^{32} -33.0000i q^{33} +90.2140 q^{34} +177.150 q^{36} +257.963i q^{37} -474.642i q^{38} +191.945 q^{39} -225.914 q^{41} +162.907i q^{42} -347.445i q^{43} -216.517 q^{44} +1118.62 q^{46} -404.364i q^{47} -497.902i q^{48} +236.483 q^{49} -51.4382 q^{51} -1259.37i q^{52} +259.568i q^{53} -142.060 q^{54} +634.433 q^{56} +270.631i q^{57} -302.779i q^{58} +853.067 q^{59} -203.699 q^{61} -745.573i q^{62} -92.8864i q^{63} -679.320 q^{64} +173.629 q^{66} -266.890i q^{67} +337.492i q^{68} -637.814 q^{69} +92.4460 q^{71} +553.247i q^{72} -242.026i q^{73} -1357.27 q^{74} +1775.64 q^{76} +113.528i q^{77} +1009.92i q^{78} +1021.60 q^{79} +81.0000 q^{81} -1188.65i q^{82} +706.415i q^{83} -609.438 q^{84} +1828.08 q^{86} +172.638i q^{87} -676.191i q^{88} +440.218 q^{89} -660.336 q^{91} +4184.77i q^{92} +425.111i q^{93} +2127.56 q^{94} +1144.38 q^{96} +197.761i q^{97} +1244.25i q^{98} -99.0000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 44 q^{4} + 24 q^{6} - 54 q^{9} + 66 q^{11} + 112 q^{14} + 100 q^{16} - 292 q^{19} + 24 q^{21} - 288 q^{24} - 1016 q^{26} - 136 q^{29} - 136 q^{31} + 352 q^{34} + 396 q^{36} - 392 q^{41} - 484 q^{44}+ \cdots - 594 q^{99}+O(q^{100}) $ 6 * q - 44 * q^4 + 24 * q^6 - 54 * q^9 + 66 * q^11 + 112 * q^14 + 100 * q^16 - 292 * q^19 + 24 * q^21 - 288 * q^24 - 1016 * q^26 - 136 * q^29 - 136 * q^31 + 352 * q^34 + 396 * q^36 - 392 * q^41 - 484 * q^44 + 2320 * q^46 + 314 * q^49 + 1308 * q^51 - 216 * q^54 + 2736 * q^56 + 2088 * q^59 + 1284 * q^61 - 2332 * q^64 + 264 * q^66 - 1200 * q^69 - 1088 * q^71 - 3072 * q^74 + 3992 * q^76 + 3172 * q^79 + 486 * q^81 - 2040 * q^84 + 7136 * q^86 + 4244 * q^89 - 16 * q^91 + 4304 * q^94 + 4128 * q^96 - 594 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/825\mathbb{Z}\right)^\times$.

$n$	$376$	$551$	$727$
$\chi(n)$	$1$	$1$	$-1$

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 $

$2$	5.26150i	1.86022i	0.367281	+	0.930110i	$0.380289\pi$
$2$	5.26150i	1.86022i	−0.367281	+	0.930110i	$0.619711\pi$
$3$	− 3.00000i	− 0.577350i
$3$	− 3.00000i	− 0.577350i
$4$	−19.6833	−2.46042
$5$	0	0
$5$	0	0
$6$	15.7845	1.07400
$7$	10.3207i	0.557266i	0.960398	+	0.278633i	$0.0898813\pi$
$7$	10.3207i	0.557266i	−0.960398	+	0.278633i	$0.910119\pi$
$8$	− 61.4719i	− 2.71670i
$9$	−9.00000	−0.333333
$10$	0	0
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	59.0500i	1.42052i
$13$	63.9817i	1.36502i	0.730874	+	0.682512i	$0.239113\pi$
$13$	63.9817i	1.36502i	−0.730874	+	0.682512i	$0.760887\pi$
$14$	−54.3024	−1.03664
$15$	0	0
$16$	165.967	2.59324
$17$	− 17.1461i	− 0.244620i	−0.992492	−	0.122310i	$-0.960970\pi$
$17$	− 17.1461i	− 0.244620i	0.992492	−	0.122310i	$-0.0390301\pi$
$18$	− 47.3535i	− 0.620073i
$19$	−90.2104	−1.08925	−0.544623	−	0.838681i	$-0.683327\pi$
$19$	−90.2104	−1.08925	−0.544623	+	0.838681i	$0.683327\pi$
$20$	0	0
$21$	30.9621	0.321738
$22$	57.8765i	0.560877i
$23$	− 212.605i	− 1.92744i	−0.266913	−	0.963721i	$-0.586004\pi$
$23$	− 212.605i	− 1.92744i	0.266913	−	0.963721i	$-0.413996\pi$
$24$	−184.416	−1.56849
$25$	0	0
$26$	−336.639	−2.53925
$27$	27.0000i	0.192450i
$28$	− 203.146i	− 1.37111i
$29$	−57.5461	−0.368484	−0.184242	−	0.982881i	$-0.558983\pi$
$29$	−57.5461	−0.368484	−0.184242	+	0.982881i	$0.558983\pi$
$30$	0	0
$31$	−141.704	−0.820991	−0.410496	−	0.911863i	$-0.634644\pi$
$31$	−141.704	−0.820991	−0.410496	+	0.911863i	$0.634644\pi$
$32$	381.462i	2.10730i
$33$	− 33.0000i	− 0.174078i
$34$	90.2140	0.455046
$35$	0	0
$36$	177.150	0.820139
$37$	257.963i	1.14619i	0.819490	+	0.573093i	$0.194257\pi$
$37$	257.963i	1.14619i	−0.819490	+	0.573093i	$0.805743\pi$
$38$	− 474.642i	− 2.02624i
$39$	191.945	0.788097
$40$	0	0
$41$	−225.914	−0.860533	−0.430266	−	0.902702i	$-0.641580\pi$
$41$	−225.914	−0.860533	−0.430266	+	0.902702i	$0.641580\pi$
$42$	162.907i	0.598503i
$43$	− 347.445i	− 1.23221i	−0.787666	−	0.616103i	$-0.788711\pi$
$43$	− 347.445i	− 1.23221i	0.787666	−	0.616103i	$-0.211289\pi$
$44$	−216.517	−0.741844
$45$	0	0
$46$	1118.62	3.58547
$47$	− 404.364i	− 1.25495i	−0.778638	−	0.627473i	$-0.784089\pi$
$47$	− 404.364i	− 1.25495i	0.778638	−	0.627473i	$-0.215911\pi$
$48$	− 497.902i	− 1.49721i
$49$	236.483	0.689455
$50$	0	0
$51$	−51.4382	−0.141231
$52$	− 1259.37i	− 3.35853i
$53$	259.568i	0.672726i	0.941732	+	0.336363i	$0.109197\pi$
$53$	259.568i	0.672726i	−0.941732	+	0.336363i	$0.890803\pi$
$54$	−142.060	−0.358000
$55$	0	0
$56$	634.433	1.51392
$57$	270.631i	0.628877i
$58$	− 302.779i	− 0.685462i
$59$	853.067	1.88237	0.941185	−	0.337891i	$-0.109713\pi$
$59$	853.067	1.88237	0.941185	+	0.337891i	$0.109713\pi$
$60$	0	0
$61$	−203.699	−0.427558	−0.213779	−	0.976882i	$-0.568577\pi$
$61$	−203.699	−0.427558	−0.213779	+	0.976882i	$0.568577\pi$
$62$	− 745.573i	− 1.52722i
$63$	− 92.8864i	− 0.185755i
$64$	−679.320	−1.32680
$65$	0	0
$66$	173.629	0.323823
$67$	− 266.890i	− 0.486653i	−0.969944	−	0.243327i	$-0.921761\pi$
$67$	− 266.890i	− 0.486653i	0.969944	−	0.243327i	$-0.0782387\pi$
$68$	337.492i	0.601866i
$69$	−637.814	−1.11281
$70$	0	0
$71$	92.4460	0.154526	0.0772629	−	0.997011i	$-0.475382\pi$
$71$	92.4460	0.154526	0.0772629	+	0.997011i	$0.475382\pi$
$72$	553.247i	0.905566i
$73$	− 242.026i	− 0.388040i	−0.980998	−	0.194020i	$-0.937847\pi$
$73$	− 242.026i	− 0.388040i	0.980998	−	0.194020i	$-0.0621527\pi$
$74$	−1357.27	−2.13216
$75$	0	0
$76$	1775.64	2.68000
$77$	113.528i	0.168022i
$78$	1009.92i	1.46603i
$79$	1021.60	1.45492	0.727460	−	0.686150i	$-0.240701\pi$
$79$	1021.60	1.45492	0.727460	+	0.686150i	$0.240701\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	− 1188.65i	− 1.60078i
$83$	706.415i	0.934206i	0.884203	+	0.467103i	$0.154702\pi$
$83$	706.415i	0.934206i	−0.884203	+	0.467103i	$0.845298\pi$
$84$	−609.438	−0.791609
$85$	0	0
$86$	1828.08	2.29217
$87$	172.638i	0.212745i
$88$	− 676.191i	− 0.819116i
$89$	440.218	0.524304	0.262152	−	0.965027i	$-0.415568\pi$
$89$	440.218	0.524304	0.262152	+	0.965027i	$0.415568\pi$
$90$	0	0
$91$	−660.336	−0.760682
$92$	4184.77i	4.74231i
$93$	425.111i	0.473999i
$94$	2127.56	2.33448
$95$	0	0
$96$	1144.38	1.21665
$97$	197.761i	0.207006i	0.994629	+	0.103503i	$0.0330051\pi$
$97$	197.761i	0.207006i	−0.994629	+	0.103503i	$0.966995\pi$
$98$	1244.25i	1.28254i
$99$	−99.0000	−0.100504
$100$	0	0
$101$	1400.62	1.37987	0.689937	−	0.723870i	$-0.257638\pi$
$101$	1400.62	1.37987	0.689937	+	0.723870i	$0.257638\pi$
$102$	− 270.642i	− 0.262721i
$103$	− 1345.70i	− 1.28734i	−0.765304	−	0.643669i	$-0.777411\pi$
$103$	− 1345.70i	− 1.28734i	0.765304	−	0.643669i	$-0.222589\pi$
$104$	3933.07	3.70836
$105$	0	0
$106$	−1365.72	−1.25142
$107$	889.178i	0.803366i	0.915779	+	0.401683i	$0.131575\pi$
$107$	889.178i	0.803366i	−0.915779	+	0.401683i	$0.868425\pi$
$108$	− 531.450i	− 0.473508i
$109$	−1256.29	−1.10395	−0.551974	−	0.833861i	$-0.686125\pi$
$109$	−1256.29	−1.10395	−0.551974	+	0.833861i	$0.686125\pi$
$110$	0	0
$111$	773.890	0.661751
$112$	1712.90i	1.44512i
$113$	− 2394.01i	− 1.99301i	−0.0835448	−	0.996504i	$-0.526624\pi$
$113$	− 2394.01i	− 1.99301i	0.0835448	−	0.996504i	$-0.473376\pi$
$114$	−1423.93	−1.16985
$115$	0	0
$116$	1132.70	0.906626
$117$	− 575.835i	− 0.455008i
$118$	4488.41i	3.50162i
$119$	176.960	0.136318
$120$	0	0
$121$	121.000	0.0909091
$122$	− 1071.76i	− 0.795352i
$123$	677.742i	0.496829i
$124$	2789.20	2.01998
$125$	0	0
$126$	488.721	0.345546
$127$	− 2065.57i	− 1.44322i	−0.692298	−	0.721612i	$-0.743402\pi$
$127$	− 2065.57i	− 1.44322i	0.692298	−	0.721612i	$-0.256598\pi$
$128$	− 522.548i	− 0.360837i
$129$	−1042.33	−0.711414
$130$	0	0
$131$	785.526	0.523907	0.261953	−	0.965081i	$-0.415633\pi$
$131$	785.526	0.523907	0.261953	+	0.965081i	$0.415633\pi$
$132$	649.550i	0.428304i
$133$	− 931.035i	− 0.607000i
$134$	1404.24	0.905282
$135$	0	0
$136$	−1054.00	−0.664558
$137$	− 1276.24i	− 0.795885i	−0.917410	−	0.397942i	$-0.869724\pi$
$137$	− 1276.24i	− 0.795885i	0.917410	−	0.397942i	$-0.130276\pi$
$138$	− 3355.86i	− 2.07007i
$139$	2703.21	1.64952	0.824760	−	0.565482i	$-0.191310\pi$
$139$	2703.21	1.64952	0.824760	+	0.565482i	$0.191310\pi$
$140$	0	0
$141$	−1213.09	−0.724544
$142$	486.405i	0.287452i
$143$	703.798i	0.411570i
$144$	−1493.71	−0.864413
$145$	0	0
$146$	1273.42	0.721840
$147$	− 709.449i	− 0.398057i
$148$	− 5077.58i	− 2.82010i
$149$	2400.99	1.32011	0.660056	−	0.751217i	$-0.270533\pi$
$149$	2400.99	1.32011	0.660056	+	0.751217i	$0.270533\pi$
$150$	0	0
$151$	−2517.30	−1.35665	−0.678326	−	0.734761i	$-0.737294\pi$
$151$	−2517.30	−1.35665	−0.678326	+	0.734761i	$0.737294\pi$
$152$	5545.40i	2.95916i
$153$	154.315i	0.0815399i
$154$	−597.326	−0.312558
$155$	0	0
$156$	−3778.12	−1.93905
$157$	− 1391.42i	− 0.707310i	−0.935376	−	0.353655i	$-0.884939\pi$
$157$	− 1391.42i	− 0.707310i	0.935376	−	0.353655i	$-0.115061\pi$
$158$	5375.13i	2.70647i
$159$	778.705	0.388398
$160$	0	0
$161$	2194.23	1.07410
$162$	426.181i	0.206691i
$163$	− 2720.53i	− 1.30729i	−0.756802	−	0.653644i	$-0.773239\pi$
$163$	− 2720.53i	− 1.30729i	0.756802	−	0.653644i	$-0.226761\pi$
$164$	4446.74	2.11727
$165$	0	0
$166$	−3716.80	−1.73783
$167$	− 2950.25i	− 1.36705i	−0.729927	−	0.683525i	$-0.760446\pi$
$167$	− 2950.25i	− 1.36705i	0.729927	−	0.683525i	$-0.239554\pi$
$168$	− 1903.30i	− 0.874064i
$169$	−1896.65	−0.863292
$170$	0	0
$171$	811.894	0.363082
$172$	6838.88i	3.03174i
$173$	537.049i	0.236018i	0.993013	+	0.118009i	$0.0376511\pi$
$173$	537.049i	0.236018i	−0.993013	+	0.118009i	$0.962349\pi$
$174$	−908.336	−0.395752
$175$	0	0
$176$	1825.64	0.781891
$177$	− 2559.20i	− 1.08679i
$178$	2316.21i	0.975320i
$179$	−2891.25	−1.20728	−0.603638	−	0.797259i	$-0.706283\pi$
$179$	−2891.25	−1.20728	−0.603638	+	0.797259i	$0.706283\pi$
$180$	0	0
$181$	435.209	0.178723	0.0893615	−	0.995999i	$-0.471517\pi$
$181$	435.209	0.178723	0.0893615	+	0.995999i	$0.471517\pi$
$182$	− 3474.36i	− 1.41504i
$183$	611.098i	0.246851i
$184$	−13069.2	−5.23628
$185$	0	0
$186$	−2236.72	−0.881743
$187$	− 188.607i	− 0.0737556i
$188$	7959.23i	3.08769i
$189$	−278.659	−0.107246
$190$	0	0
$191$	−3779.49	−1.43180	−0.715901	−	0.698202i	$-0.753984\pi$
$191$	−3779.49	−1.43180	−0.715901	+	0.698202i	$0.753984\pi$
$192$	2037.96i	0.766027i
$193$	3751.91i	1.39932i	0.714476	+	0.699660i	$0.246665\pi$
$193$	3751.91i	1.39932i	−0.714476	+	0.699660i	$0.753335\pi$
$194$	−1040.52	−0.385076
$195$	0	0
$196$	−4654.78	−1.69635
$197$	3920.73i	1.41797i	0.705223	+	0.708986i	$0.250847\pi$
$197$	3920.73i	1.41797i	−0.705223	+	0.708986i	$0.749153\pi$
$198$	− 520.888i	− 0.186959i
$199$	597.084	0.212694	0.106347	−	0.994329i	$-0.466085\pi$
$199$	597.084	0.212694	0.106347	+	0.994329i	$0.466085\pi$
$200$	0	0
$201$	−800.669	−0.280969
$202$	7369.37i	2.56687i
$203$	− 593.917i	− 0.205344i
$204$	1012.48	0.347488
$205$	0	0
$206$	7080.40	2.39473
$207$	1913.44i	0.642480i
$208$	10618.9i	3.53984i
$209$	−992.314	−0.328420
$210$	0	0
$211$	−4384.55	−1.43054	−0.715272	−	0.698846i	$-0.753697\pi$
$211$	−4384.55	−1.43054	−0.715272	+	0.698846i	$0.753697\pi$
$212$	− 5109.17i	− 1.65519i
$213$	− 277.338i	− 0.0892155i
$214$	−4678.41	−1.49444
$215$	0	0
$216$	1659.74	0.522829
$217$	− 1462.48i	− 0.457510i
$218$	− 6609.95i	− 2.05359i
$219$	−726.077	−0.224035
$220$	0	0
$221$	1097.03	0.333912
$222$	4071.82i	1.23100i
$223$	− 2333.03i	− 0.700587i	−0.936640	−	0.350294i	$-0.886082\pi$
$223$	− 2333.03i	− 0.700587i	0.936640	−	0.350294i	$-0.113918\pi$
$224$	−3936.95	−1.17433
$225$	0	0
$226$	12596.1	3.70743
$227$	2120.00i	0.619864i	0.950759	+	0.309932i	$0.100306\pi$
$227$	2120.00i	0.619864i	−0.950759	+	0.309932i	$0.899694\pi$
$228$	− 5326.93i	− 1.54730i
$229$	−2347.12	−0.677301	−0.338651	−	0.940912i	$-0.609970\pi$
$229$	−2347.12	−0.677301	−0.338651	+	0.940912i	$0.609970\pi$
$230$	0	0
$231$	340.583	0.0970075
$232$	3537.47i	1.00106i
$233$	− 375.499i	− 0.105578i	−0.998606	−	0.0527891i	$-0.983189\pi$
$233$	− 375.499i	− 0.105578i	0.998606	−	0.0527891i	$-0.0168111\pi$
$234$	3029.75	0.846415
$235$	0	0
$236$	−16791.2	−4.63142
$237$	− 3064.79i	− 0.839998i
$238$	931.072i	0.253582i
$239$	1428.15	0.386524	0.193262	−	0.981147i	$-0.438093\pi$
$239$	1428.15	0.386524	0.193262	+	0.981147i	$0.438093\pi$
$240$	0	0
$241$	190.819	0.0510032	0.0255016	−	0.999675i	$-0.491882\pi$
$241$	190.819	0.0510032	0.0255016	+	0.999675i	$0.491882\pi$
$242$	636.641i	0.169111i
$243$	− 243.000i	− 0.0641500i
$244$	4009.49	1.05197
$245$	0	0
$246$	−3565.94	−0.924211
$247$	− 5771.81i	− 1.48685i
$248$	8710.79i	2.23039i
$249$	2119.24	0.539364
$250$	0	0
$251$	−6294.80	−1.58297	−0.791483	−	0.611191i	$-0.790691\pi$
$251$	−6294.80	−1.58297	−0.791483	+	0.611191i	$0.790691\pi$
$252$	1828.31i	0.457036i
$253$	− 2338.65i	− 0.581145i
$254$	10868.0	2.68471
$255$	0	0
$256$	−2685.18	−0.655561
$257$	− 4459.44i	− 1.08238i	−0.840900	−	0.541191i	$-0.817974\pi$
$257$	− 4459.44i	− 1.08238i	0.840900	−	0.541191i	$-0.182026\pi$
$258$	− 5484.24i	− 1.32339i
$259$	−2662.36	−0.638731
$260$	0	0
$261$	517.915	0.122828
$262$	4133.04i	0.974581i
$263$	− 4416.65i	− 1.03552i	−0.855525	−	0.517761i	$-0.826766\pi$
$263$	− 4416.65i	− 1.03552i	0.855525	−	0.517761i	$-0.173234\pi$
$264$	−2028.57	−0.472917
$265$	0	0
$266$	4898.64	1.12915
$267$	− 1320.65i	− 0.302707i
$268$	5253.28i	1.19737i
$269$	−1914.86	−0.434020	−0.217010	−	0.976169i	$-0.569630\pi$
$269$	−1914.86	−0.434020	−0.217010	+	0.976169i	$0.569630\pi$
$270$	0	0
$271$	6088.34	1.36472	0.682362	−	0.731014i	$-0.260953\pi$
$271$	6088.34	1.36472	0.682362	+	0.731014i	$0.260953\pi$
$272$	− 2845.69i	− 0.634357i
$273$	1981.01i	0.439180i
$274$	6714.91	1.48052
$275$	0	0
$276$	12554.3	2.73798
$277$	832.321i	0.180539i	0.995917	+	0.0902696i	$0.0287729\pi$
$277$	832.321i	0.180539i	−0.995917	+	0.0902696i	$0.971227\pi$
$278$	14222.9i	3.06847i
$279$	1275.33	0.273664
$280$	0	0
$281$	−2545.32	−0.540360	−0.270180	−	0.962810i	$-0.587083\pi$
$281$	−2545.32	−0.540360	−0.270180	+	0.962810i	$0.587083\pi$
$282$	− 6382.67i	− 1.34781i
$283$	− 5911.71i	− 1.24175i	−0.783911	−	0.620874i	$-0.786778\pi$
$283$	− 5911.71i	− 1.24175i	0.783911	−	0.620874i	$-0.213222\pi$
$284$	−1819.65	−0.380198
$285$	0	0
$286$	−3703.03	−0.765611
$287$	− 2331.59i	− 0.479545i
$288$	− 3433.15i	− 0.702433i
$289$	4619.01	0.940161
$290$	0	0
$291$	593.282	0.119515
$292$	4763.87i	0.954742i
$293$	− 6871.03i	− 1.37000i	−0.728543	−	0.685000i	$-0.759802\pi$
$293$	− 6871.03i	− 1.37000i	0.728543	−	0.685000i	$-0.240198\pi$
$294$	3732.76	0.740473
$295$	0	0
$296$	15857.5	3.11384
$297$	297.000i	0.0580259i
$298$	12632.8i	2.45570i
$299$	13602.8	2.63100
$300$	0	0
$301$	3585.88	0.686666
$302$	− 13244.7i	− 2.52367i
$303$	− 4201.87i	− 0.796670i
$304$	−14972.0	−2.82468
$305$	0	0
$306$	−811.926	−0.151682
$307$	− 200.179i	− 0.0372144i	−0.999827	−	0.0186072i	$-0.994077\pi$
$307$	− 200.179i	− 0.0372144i	0.999827	−	0.0186072i	$-0.00592320\pi$
$308$	− 2234.61i	− 0.413404i
$309$	−4037.10	−0.743245
$310$	0	0
$311$	5734.93	1.04565	0.522827	−	0.852439i	$-0.324878\pi$
$311$	5734.93	1.04565	0.522827	+	0.852439i	$0.324878\pi$
$312$	− 11799.2i	− 2.14102i
$313$	− 3077.36i	− 0.555727i	−0.960621	−	0.277864i	$-0.910374\pi$
$313$	− 3077.36i	− 0.555727i	0.960621	−	0.277864i	$-0.0896264\pi$
$314$	7320.97	1.31575
$315$	0	0
$316$	−20108.5	−3.57971
$317$	− 2142.38i	− 0.379584i	−0.981824	−	0.189792i	$-0.939219\pi$
$317$	− 2142.38i	− 0.379584i	0.981824	−	0.189792i	$-0.0607813\pi$
$318$	4097.15i	0.722506i
$319$	−633.007	−0.111102
$320$	0	0
$321$	2667.54	0.463823
$322$	11544.9i	1.99806i
$323$	1546.75i	0.266451i
$324$	−1594.35	−0.273380
$325$	0	0
$326$	14314.0	2.43184
$327$	3768.86i	0.637365i
$328$	13887.4i	2.33781i
$329$	4173.32	0.699339
$330$	0	0
$331$	−1618.23	−0.268719	−0.134359	−	0.990933i	$-0.542898\pi$
$331$	−1618.23	−0.268719	−0.134359	+	0.990933i	$0.542898\pi$
$332$	− 13904.6i	− 2.29854i
$333$	− 2321.67i	− 0.382062i
$334$	15522.7	2.54301
$335$	0	0
$336$	5138.70	0.834343
$337$	− 2406.47i	− 0.388988i	−0.980904	−	0.194494i	$-0.937694\pi$
$337$	− 2406.47i	− 0.388988i	0.980904	−	0.194494i	$-0.0623064\pi$
$338$	− 9979.23i	− 1.60591i
$339$	−7182.04	−1.15066
$340$	0	0
$341$	−1558.74	−0.247538
$342$	4271.78i	0.675413i
$343$	5980.67i	0.941475i
$344$	−21358.1	−3.34753
$345$	0	0
$346$	−2825.68	−0.439045
$347$	6612.89i	1.02305i	0.859268	+	0.511525i	$0.170919\pi$
$347$	6612.89i	1.02305i	−0.859268	+	0.511525i	$0.829081\pi$
$348$	− 3398.10i	− 0.523441i
$349$	−349.871	−0.0536623	−0.0268311	−	0.999640i	$-0.508542\pi$
$349$	−349.871	−0.0536623	−0.0268311	+	0.999640i	$0.508542\pi$
$350$	0	0
$351$	−1727.50	−0.262699
$352$	4196.08i	0.635374i
$353$	1723.29i	0.259835i	0.991525	+	0.129917i	$0.0414712\pi$
$353$	1723.29i	0.259835i	−0.991525	+	0.129917i	$0.958529\pi$
$354$	13465.2	2.02166
$355$	0	0
$356$	−8664.97	−1.29001
$357$	− 530.879i	− 0.0787033i
$358$	− 15212.3i	− 2.24580i
$359$	−5875.74	−0.863816	−0.431908	−	0.901918i	$-0.642159\pi$
$359$	−5875.74	−0.863816	−0.431908	+	0.901918i	$0.642159\pi$
$360$	0	0
$361$	1278.92	0.186458
$362$	2289.85i	0.332464i
$363$	− 363.000i	− 0.0524864i
$364$	12997.6	1.87159
$365$	0	0
$366$	−3215.29	−0.459197
$367$	5368.28i	0.763548i	0.924256	+	0.381774i	$0.124687\pi$
$367$	5368.28i	0.763548i	−0.924256	+	0.381774i	$0.875313\pi$
$368$	− 35285.5i	− 4.99832i
$369$	2033.23	0.286844
$370$	0	0
$371$	−2678.93	−0.374887
$372$	− 8367.60i	− 1.16624i
$373$	10393.9i	1.44282i	0.692506	+	0.721412i	$0.256507\pi$
$373$	10393.9i	1.44282i	−0.692506	+	0.721412i	$0.743493\pi$
$374$	992.354	0.137202
$375$	0	0
$376$	−24857.0	−3.40931
$377$	− 3681.90i	− 0.502990i
$378$	− 1466.16i	− 0.199501i
$379$	−10918.9	−1.47986	−0.739928	−	0.672686i	$-0.765140\pi$
$379$	−10918.9	−1.47986	−0.739928	+	0.672686i	$0.765140\pi$
$380$	0	0
$381$	−6196.70	−0.833246
$382$	− 19885.8i	− 2.66347i
$383$	− 11663.6i	− 1.55609i	−0.628210	−	0.778044i	$-0.716212\pi$
$383$	− 11663.6i	− 1.55609i	0.628210	−	0.778044i	$-0.283788\pi$
$384$	−1567.64	−0.208329
$385$	0	0
$386$	−19740.7	−2.60304
$387$	3127.00i	0.410735i
$388$	− 3892.59i	− 0.509321i
$389$	5827.00	0.759487	0.379744	−	0.925092i	$-0.376012\pi$
$389$	5827.00	0.759487	0.379744	+	0.925092i	$0.376012\pi$
$390$	0	0
$391$	−3645.34	−0.471490
$392$	− 14537.1i	− 1.87304i
$393$	− 2356.58i	− 0.302478i
$394$	−20628.9	−2.63774
$395$	0	0
$396$	1948.65	0.247281
$397$	− 7366.99i	− 0.931332i	−0.884961	−	0.465666i	$-0.845815\pi$
$397$	− 7366.99i	− 0.931332i	0.884961	−	0.465666i	$-0.154185\pi$
$398$	3141.55i	0.395658i
$399$	−2793.11	−0.350452
$400$	0	0
$401$	14604.2	1.81870	0.909349	−	0.416035i	$-0.136581\pi$
$401$	14604.2	1.81870	0.909349	+	0.416035i	$0.136581\pi$
$402$	− 4212.72i	− 0.522665i
$403$	− 9066.43i	− 1.12067i
$404$	−27569.0	−3.39507
$405$	0	0
$406$	3124.89	0.381985
$407$	2837.60i	0.345588i
$408$	3162.00i	0.383683i
$409$	−12581.2	−1.52103	−0.760515	−	0.649320i	$-0.775054\pi$
$409$	−12581.2	−1.52103	−0.760515	+	0.649320i	$0.775054\pi$
$410$	0	0
$411$	−3828.71	−0.459504
$412$	26487.9i	3.16739i
$413$	8804.26i	1.04898i
$414$	−10067.6	−1.19516
$415$	0	0
$416$	−24406.6	−2.87651
$417$	− 8109.63i	− 0.952351i
$418$	− 5221.06i	− 0.610934i
$419$	3776.01	0.440263	0.220131	−	0.975470i	$-0.429351\pi$
$419$	3776.01	0.440263	0.220131	+	0.975470i	$0.429351\pi$
$420$	0	0
$421$	12683.4	1.46830	0.734148	−	0.678989i	$-0.237582\pi$
$421$	12683.4	1.46830	0.734148	+	0.678989i	$0.237582\pi$
$422$	− 23069.3i	− 2.66113i
$423$	3639.27i	0.418316i
$424$	15956.2	1.82759
$425$	0	0
$426$	1459.21	0.165960
$427$	− 2102.32i	− 0.238263i
$428$	− 17502.0i	− 1.97662i
$429$	2111.39	0.237620
$430$	0	0
$431$	14152.6	1.58168	0.790841	−	0.612022i	$-0.209644\pi$
$431$	14152.6	1.58168	0.790841	+	0.612022i	$0.209644\pi$
$432$	4481.12i	0.499069i
$433$	− 10950.2i	− 1.21532i	−0.794197	−	0.607661i	$-0.792108\pi$
$433$	− 10950.2i	− 1.21532i	0.794197	−	0.607661i	$-0.207892\pi$
$434$	7694.84	0.851070
$435$	0	0
$436$	24727.9	2.71618
$437$	19179.2i	2.09946i
$438$	− 3820.25i	− 0.416755i
$439$	11221.0	1.21993	0.609964	−	0.792429i	$-0.291184\pi$
$439$	11221.0	1.21993	0.609964	+	0.792429i	$0.291184\pi$
$440$	0	0
$441$	−2128.35	−0.229818
$442$	5772.04i	0.621149i
$443$	9647.11i	1.03465i	0.855790	+	0.517323i	$0.173071\pi$
$443$	9647.11i	1.03465i	−0.855790	+	0.517323i	$0.826929\pi$
$444$	−15232.7	−1.62818
$445$	0	0
$446$	12275.2	1.30325
$447$	− 7202.96i	− 0.762167i
$448$	− 7011.07i	− 0.739379i
$449$	−6482.03	−0.681305	−0.340652	−	0.940189i	$-0.610648\pi$
$449$	−6482.03	−0.681305	−0.340652	+	0.940189i	$0.610648\pi$
$450$	0	0
$451$	−2485.05	−0.259460
$452$	47122.2i	4.90363i
$453$	7551.89i	0.783264i
$454$	−11154.4	−1.15308
$455$	0	0
$456$	16636.2	1.70847
$457$	− 11319.8i	− 1.15868i	−0.815085	−	0.579342i	$-0.803310\pi$
$457$	− 11319.8i	− 1.15868i	0.815085	−	0.579342i	$-0.196690\pi$
$458$	− 12349.4i	− 1.25993i
$459$	462.944	0.0470771
$460$	0	0
$461$	−8406.73	−0.849329	−0.424664	−	0.905351i	$-0.639608\pi$
$461$	−8406.73	−0.849329	−0.424664	+	0.905351i	$0.639608\pi$
$462$	1791.98i	0.180455i
$463$	9758.56i	0.979523i	0.871857	+	0.489761i	$0.162916\pi$
$463$	9758.56i	0.979523i	−0.871857	+	0.489761i	$0.837084\pi$
$464$	−9550.78	−0.955568
$465$	0	0
$466$	1975.68	0.196399
$467$	16388.7i	1.62394i	0.583701	+	0.811969i	$0.301604\pi$
$467$	16388.7i	1.62394i	−0.583701	+	0.811969i	$0.698396\pi$
$468$	11334.4i	1.11951i
$469$	2754.49	0.271195
$470$	0	0
$471$	−4174.27	−0.408365
$472$	− 52439.6i	− 5.11384i
$473$	− 3821.89i	− 0.371524i
$474$	16125.4	1.56258
$475$	0	0
$476$	−3483.16	−0.335400
$477$	− 2336.11i	− 0.224242i
$478$	7514.20i	0.719020i
$479$	−13829.0	−1.31913	−0.659567	−	0.751646i	$-0.729260\pi$
$479$	−13829.0	−1.31913	−0.659567	+	0.751646i	$0.729260\pi$
$480$	0	0
$481$	−16504.9	−1.56457
$482$	1004.00i	0.0948771i
$483$	− 6582.69i	− 0.620130i
$484$	−2381.68	−0.223674
$485$	0	0
$486$	1278.54	0.119333
$487$	− 13264.4i	− 1.23423i	−0.786875	−	0.617113i	$-0.788302\pi$
$487$	− 13264.4i	− 1.23423i	0.786875	−	0.617113i	$-0.211698\pi$
$488$	12521.8i	1.16155i
$489$	−8161.58	−0.754763
$490$	0	0
$491$	−7468.22	−0.686428	−0.343214	−	0.939257i	$-0.611516\pi$
$491$	−7468.22	−0.686428	−0.343214	+	0.939257i	$0.611516\pi$
$492$	− 13340.2i	− 1.22241i
$493$	986.690i	0.0901385i
$494$	30368.4	2.76586
$495$	0	0
$496$	−23518.2	−2.12903
$497$	954.109i	0.0861119i
$498$	11150.4i	1.00334i
$499$	5276.64	0.473377	0.236688	−	0.971586i	$-0.423938\pi$
$499$	5276.64	0.473377	0.236688	+	0.971586i	$0.423938\pi$
$500$	0	0
$501$	−8850.76	−0.789267
$502$	− 33120.1i	− 2.94466i
$503$	− 10956.8i	− 0.971253i	−0.874166	−	0.485626i	$-0.838592\pi$
$503$	− 10956.8i	− 0.971253i	0.874166	−	0.485626i	$-0.161408\pi$
$504$	−5709.90	−0.504641
$505$	0	0
$506$	12304.8	1.08106
$507$	5689.96i	0.498422i
$508$	40657.3i	3.55093i
$509$	−12734.4	−1.10892	−0.554462	−	0.832209i	$-0.687076\pi$
$509$	−12734.4	−1.10892	−0.554462	+	0.832209i	$0.687076\pi$
$510$	0	0
$511$	2497.87	0.216242
$512$	− 18308.4i	− 1.58032i
$513$	− 2435.68i	− 0.209626i
$514$	23463.3	2.01347
$515$	0	0
$516$	20516.6	1.75038
$517$	− 4448.00i	− 0.378381i
$518$	− 14008.0i	− 1.18818i
$519$	1611.15	0.136265
$520$	0	0
$521$	−5650.70	−0.475166	−0.237583	−	0.971367i	$-0.576355\pi$
$521$	−5650.70	−0.475166	−0.237583	+	0.971367i	$0.576355\pi$
$522$	2725.01i	0.228487i
$523$	14103.2i	1.17914i	0.807716	+	0.589572i	$0.200703\pi$
$523$	14103.2i	1.17914i	−0.807716	+	0.589572i	$0.799297\pi$
$524$	−15461.8	−1.28903
$525$	0	0
$526$	23238.2	1.92630
$527$	2429.66i	0.200830i
$528$	− 5476.92i	− 0.451425i
$529$	−33033.8	−2.71503
$530$	0	0
$531$	−7677.60	−0.627457
$532$	18325.9i	1.49347i
$533$	− 14454.4i	− 1.17465i
$534$	6948.62	0.563102
$535$	0	0
$536$	−16406.2	−1.32209
$537$	8673.75i	0.697021i
$538$	− 10075.0i	− 0.807372i
$539$	2601.31	0.207878
$540$	0	0
$541$	−6391.90	−0.507965	−0.253983	−	0.967209i	$-0.581741\pi$
$541$	−6391.90	−0.507965	−0.253983	+	0.967209i	$0.581741\pi$
$542$	32033.8i	2.53869i
$543$	− 1305.63i	− 0.103186i
$544$	6540.57	0.515486
$545$	0	0
$546$	−10423.1	−0.816971
$547$	− 20786.7i	− 1.62482i	−0.583088	−	0.812409i	$-0.698156\pi$
$547$	− 20786.7i	− 1.62482i	0.583088	−	0.812409i	$-0.301844\pi$
$548$	25120.6i	1.95821i
$549$	1833.29	0.142519
$550$	0	0
$551$	5191.26	0.401370
$552$	39207.6i	3.02317i
$553$	10543.6i	0.810777i
$554$	−4379.26	−0.335843
$555$	0	0
$556$	−53208.2	−4.05851
$557$	15125.9i	1.15064i	0.817928	+	0.575320i	$0.195123\pi$
$557$	15125.9i	1.15064i	−0.817928	+	0.575320i	$0.804877\pi$
$558$	6710.16i	0.509075i
$559$	22230.1	1.68199
$560$	0	0
$561$	−565.820	−0.0425828
$562$	− 13392.2i	− 1.00519i
$563$	8706.42i	0.651744i	0.945414	+	0.325872i	$0.105658\pi$
$563$	8706.42i	0.651744i	−0.945414	+	0.325872i	$0.894342\pi$
$564$	23877.7	1.78268
$565$	0	0
$566$	31104.4	2.30992
$567$	835.977i	0.0619184i
$568$	− 5682.83i	− 0.419800i
$569$	−7067.76	−0.520731	−0.260366	−	0.965510i	$-0.583843\pi$
$569$	−7067.76	−0.520731	−0.260366	+	0.965510i	$0.583843\pi$
$570$	0	0
$571$	−3326.42	−0.243794	−0.121897	−	0.992543i	$-0.538898\pi$
$571$	−3326.42	−0.243794	−0.121897	+	0.992543i	$0.538898\pi$
$572$	− 13853.1i	− 1.01264i
$573$	11338.5i	0.826651i
$574$	12267.7	0.892060
$575$	0	0
$576$	6113.88	0.442266
$577$	3308.06i	0.238676i	0.992854	+	0.119338i	$0.0380773\pi$
$577$	3308.06i	0.238676i	−0.992854	+	0.119338i	$0.961923\pi$
$578$	24302.9i	1.74891i
$579$	11255.7	0.807897
$580$	0	0
$581$	−7290.70	−0.520601
$582$	3121.55i	0.222324i
$583$	2855.25i	0.202834i
$584$	−14877.8	−1.05419
$585$	0	0
$586$	36151.9	2.54850
$587$	5694.88i	0.400431i	0.979752	+	0.200215i	$0.0641642\pi$
$587$	5694.88i	0.400431i	−0.979752	+	0.200215i	$0.935836\pi$
$588$	13964.3i	0.979387i
$589$	12783.1	0.894262
$590$	0	0
$591$	11762.2	0.818666
$592$	42813.5i	2.97234i
$593$	− 3907.69i	− 0.270606i	−0.990804	−	0.135303i	$-0.956799\pi$
$593$	− 3907.69i	− 0.270606i	0.990804	−	0.135303i	$-0.0432008\pi$
$594$	−1562.66	−0.107941
$595$	0	0
$596$	−47259.5	−3.24803
$597$	− 1791.25i	− 0.122799i
$598$	71571.1i	4.89425i
$599$	−10727.2	−0.731720	−0.365860	−	0.930670i	$-0.619225\pi$
$599$	−10727.2	−0.731720	−0.365860	+	0.930670i	$0.619225\pi$
$600$	0	0
$601$	3348.98	0.227301	0.113650	−	0.993521i	$-0.463746\pi$
$601$	3348.98	0.227301	0.113650	+	0.993521i	$0.463746\pi$
$602$	18867.1i	1.27735i
$603$	2402.01i	0.162218i
$604$	49548.8	3.33793
$605$	0	0
$606$	22108.1	1.48198
$607$	21539.9i	1.44032i	0.693806	+	0.720162i	$0.255932\pi$
$607$	21539.9i	1.44032i	−0.693806	+	0.720162i	$0.744068\pi$
$608$	− 34411.8i	− 2.29537i
$609$	−1781.75	−0.118555
$610$	0	0
$611$	25871.9	1.71303
$612$	− 3037.43i	− 0.200622i
$613$	− 9284.33i	− 0.611730i	−0.952075	−	0.305865i	$-0.901054\pi$
$613$	− 9284.33i	− 0.611730i	0.952075	−	0.305865i	$-0.0989457\pi$
$614$	1053.24	0.0692270
$615$	0	0
$616$	6978.77	0.456465
$617$	20711.3i	1.35139i	0.737181	+	0.675695i	$0.236156\pi$
$617$	20711.3i	1.35139i	−0.737181	+	0.675695i	$0.763844\pi$
$618$	− 21241.2i	− 1.38260i
$619$	13282.9	0.862496	0.431248	−	0.902233i	$-0.358073\pi$
$619$	13282.9	0.862496	0.431248	+	0.902233i	$0.358073\pi$
$620$	0	0
$621$	5740.33	0.370936
$622$	30174.3i	1.94515i
$623$	4543.36i	0.292177i
$624$	31856.6	2.04373
$625$	0	0
$626$	16191.5	1.03378
$627$	2976.94i	0.189613i
$628$	27387.9i	1.74028i
$629$	4423.06	0.280380
$630$	0	0
$631$	−14789.9	−0.933086	−0.466543	−	0.884498i	$-0.654501\pi$
$631$	−14789.9	−0.933086	−0.466543	+	0.884498i	$0.654501\pi$
$632$	− 62799.5i	− 3.95258i
$633$	13153.6i	0.825925i
$634$	11272.1	0.706109
$635$	0	0
$636$	−15327.5	−0.955622
$637$	15130.6i	0.941123i
$638$	− 3330.57i	− 0.206675i
$639$	−832.014	−0.0515086
$640$	0	0
$641$	−5808.52	−0.357914	−0.178957	−	0.983857i	$-0.557272\pi$
$641$	−5808.52	−0.357914	−0.178957	+	0.983857i	$0.557272\pi$
$642$	14035.2i	0.862813i
$643$	− 18891.2i	− 1.15862i	−0.815106	−	0.579311i	$-0.803322\pi$
$643$	− 18891.2i	− 1.15862i	0.815106	−	0.579311i	$-0.196678\pi$
$644$	−43189.8	−2.64273
$645$	0	0
$646$	−8138.24	−0.495657
$647$	243.046i	0.0147684i	0.999973	+	0.00738418i	$0.00235048\pi$
$647$	243.046i	0.0147684i	−0.999973	+	0.00738418i	$0.997650\pi$
$648$	− 4979.22i	− 0.301855i
$649$	9383.74	0.567556
$650$	0	0
$651$	−4387.44	−0.264144
$652$	53549.0i	3.21648i
$653$	− 15920.4i	− 0.954081i	−0.878881	−	0.477041i	$-0.841709\pi$
$653$	− 15920.4i	− 0.954081i	0.878881	−	0.477041i	$-0.158291\pi$
$654$	−19829.8	−1.18564
$655$	0	0
$656$	−37494.4	−2.23157
$657$	2178.23i	0.129347i
$658$	21957.9i	1.30092i
$659$	7476.38	0.441940	0.220970	−	0.975281i	$-0.429078\pi$
$659$	7476.38	0.441940	0.220970	+	0.975281i	$0.429078\pi$
$660$	0	0
$661$	−14920.1	−0.877948	−0.438974	−	0.898500i	$-0.644658\pi$
$661$	−14920.1	−0.877948	−0.438974	+	0.898500i	$0.644658\pi$
$662$	− 8514.30i	− 0.499876i
$663$	− 3291.10i	− 0.192784i
$664$	43424.7	2.53796
$665$	0	0
$666$	12215.5	0.710720
$667$	12234.6i	0.710232i
$668$	58070.8i	3.36351i
$669$	−6999.08	−0.404484
$670$	0	0
$671$	−2240.69	−0.128914
$672$	11810.9i	0.677997i
$673$	563.692i	0.0322864i	0.999870	+	0.0161432i	$0.00513876\pi$
$673$	563.692i	0.0322864i	−0.999870	+	0.0161432i	$0.994861\pi$
$674$	12661.6	0.723602
$675$	0	0
$676$	37332.5	2.12406
$677$	− 13280.2i	− 0.753914i	−0.926231	−	0.376957i	$-0.876970\pi$
$677$	− 13280.2i	− 0.753914i	0.926231	−	0.376957i	$-0.123030\pi$
$678$	− 37788.3i	− 2.14049i
$679$	−2041.03	−0.115357
$680$	0	0
$681$	6359.99	0.357879
$682$	− 8201.30i	− 0.460475i
$683$	6856.80i	0.384141i	0.981381	+	0.192070i	$0.0615202\pi$
$683$	6856.80i	0.384141i	−0.981381	+	0.192070i	$0.938480\pi$
$684$	−15980.8	−0.893334
$685$	0	0
$686$	−31467.3	−1.75135
$687$	7041.36i	0.391040i
$688$	− 57664.5i	− 3.19541i
$689$	−16607.6	−0.918287
$690$	0	0
$691$	−28374.6	−1.56211	−0.781057	−	0.624460i	$-0.785319\pi$
$691$	−28374.6	−1.56211	−0.781057	+	0.624460i	$0.785319\pi$
$692$	− 10570.9i	− 0.580702i
$693$	− 1021.75i	− 0.0560073i
$694$	−34793.7	−1.90310
$695$	0	0
$696$	10612.4	0.577963
$697$	3873.54i	0.210503i
$698$	− 1840.84i	− 0.0998237i
$699$	−1126.50	−0.0609556
$700$	0	0
$701$	667.753	0.0359781	0.0179891	−	0.999838i	$-0.494274\pi$
$701$	667.753	0.0359781	0.0179891	+	0.999838i	$0.494274\pi$
$702$	− 9089.26i	− 0.488678i
$703$	− 23271.0i	− 1.24848i
$704$	−7472.52	−0.400044
$705$	0	0
$706$	−9067.10	−0.483349
$707$	14455.4i	0.768956i
$708$	50373.6i	2.67395i
$709$	−23667.8	−1.25368	−0.626842	−	0.779147i	$-0.715653\pi$
$709$	−23667.8	−1.25368	−0.626842	+	0.779147i	$0.715653\pi$
$710$	0	0
$711$	−9194.37	−0.484973
$712$	− 27061.0i	− 1.42438i
$713$	30126.9i	1.58241i
$714$	2793.22	0.146405
$715$	0	0
$716$	56909.5	2.97040
$717$	− 4284.45i	− 0.223160i
$718$	− 30915.2i	− 1.60689i
$719$	−11835.5	−0.613896	−0.306948	−	0.951726i	$-0.599308\pi$
$719$	−11835.5	−0.613896	−0.306948	+	0.951726i	$0.599308\pi$
$720$	0	0
$721$	13888.6	0.717390
$722$	6729.01i	0.346853i
$723$	− 572.458i	− 0.0294467i
$724$	−8566.38	−0.439733
$725$	0	0
$726$	1909.92	0.0976362
$727$	− 15633.2i	− 0.797530i	−0.917053	−	0.398765i	$-0.869439\pi$
$727$	− 15633.2i	− 0.797530i	0.917053	−	0.398765i	$-0.130561\pi$
$728$	40592.1i	2.06654i
$729$	−729.000	−0.0370370
$730$	0	0
$731$	−5957.31	−0.301422
$732$	− 12028.5i	− 0.607356i
$733$	− 14870.7i	− 0.749335i	−0.927159	−	0.374668i	$-0.877757\pi$
$733$	− 14870.7i	− 0.749335i	0.927159	−	0.374668i	$-0.122243\pi$
$734$	−28245.2	−1.42037
$735$	0	0
$736$	81100.6	4.06169
$737$	− 2935.79i	− 0.146731i
$738$	10697.8i	0.533593i
$739$	−20850.3	−1.03788	−0.518939	−	0.854812i	$-0.673673\pi$
$739$	−20850.3	−1.03788	−0.518939	+	0.854812i	$0.673673\pi$
$740$	0	0
$741$	−17315.4	−0.858432
$742$	− 14095.2i	− 0.697372i
$743$	− 29254.7i	− 1.44448i	−0.691641	−	0.722242i	$-0.743112\pi$
$743$	− 29254.7i	− 1.44448i	0.691641	−	0.722242i	$-0.256888\pi$
$744$	26132.4	1.28771
$745$	0	0
$746$	−54687.2	−2.68397
$747$	− 6357.73i	− 0.311402i
$748$	3712.41i	0.181470i
$749$	−9176.95	−0.447688
$750$	0	0
$751$	10936.8	0.531411	0.265705	−	0.964054i	$-0.414395\pi$
$751$	10936.8	0.531411	0.265705	+	0.964054i	$0.414395\pi$
$752$	− 67111.2i	− 3.25438i
$753$	18884.4i	0.913926i
$754$	19372.3	0.935672
$755$	0	0
$756$	5484.94	0.263870
$757$	8476.34i	0.406972i	0.979078	+	0.203486i	$0.0652272\pi$
$757$	8476.34i	0.406972i	−0.979078	+	0.203486i	$0.934773\pi$
$758$	− 57449.6i	− 2.75286i
$759$	−7015.96	−0.335524
$760$	0	0
$761$	913.964	0.0435364	0.0217682	−	0.999763i	$-0.493070\pi$
$761$	913.964	0.0435364	0.0217682	+	0.999763i	$0.493070\pi$
$762$	− 32603.9i	− 1.55002i
$763$	− 12965.8i	− 0.615193i
$764$	74393.0	3.52283
$765$	0	0
$766$	61367.9	2.89466
$767$	54580.6i	2.56948i
$768$	8055.53i	0.378488i
$769$	−32215.2	−1.51067	−0.755337	−	0.655337i	$-0.772527\pi$
$769$	−32215.2	−1.51067	−0.755337	+	0.655337i	$0.772527\pi$
$770$	0	0
$771$	−13378.3	−0.624914
$772$	− 73850.2i	− 3.44291i
$773$	72.6900i	0.00338225i	0.999999	+	0.00169112i	$0.000538302\pi$
$773$	72.6900i	0.00338225i	−0.999999	+	0.00169112i	$0.999462\pi$
$774$	−16452.7	−0.764058
$775$	0	0
$776$	12156.7	0.562373
$777$	7987.09i	0.368771i
$778$	30658.7i	1.41281i
$779$	20379.8	0.937332
$780$	0	0
$781$	1016.91	0.0465913
$782$	− 19179.9i	− 0.877075i
$783$	− 1553.75i	− 0.0709148i
$784$	39248.5	1.78792
$785$	0	0
$786$	12399.1	0.562675
$787$	− 487.318i	− 0.0220724i	−0.999939	−	0.0110362i	$-0.996487\pi$
$787$	− 487.318i	− 0.0220724i	0.999939	−	0.0110362i	$-0.00351301\pi$
$788$	− 77173.1i	− 3.48880i
$789$	−13249.9	−0.597859
$790$	0	0
$791$	24707.9	1.11064
$792$	6085.72i	0.273039i
$793$	− 13033.0i	− 0.583627i
$794$	38761.4	1.73248
$795$	0	0
$796$	−11752.6	−0.523317
$797$	− 31379.9i	− 1.39465i	−0.716756	−	0.697324i	$-0.754374\pi$
$797$	− 31379.9i	− 1.39465i	0.716756	−	0.697324i	$-0.245626\pi$
$798$	− 14695.9i	− 0.651917i
$799$	−6933.25	−0.306985
$800$	0	0
$801$	−3961.96	−0.174768
$802$	76839.8i	3.38318i
$803$	− 2662.28i	− 0.116999i
$804$	15759.8	0.691302
$805$	0	0
$806$	47703.0	2.08470
$807$	5744.59i	0.250581i
$808$	− 86099.0i	− 3.74870i
$809$	−1824.26	−0.0792802	−0.0396401	−	0.999214i	$-0.512621\pi$
$809$	−1824.26	−0.0792802	−0.0396401	+	0.999214i	$0.512621\pi$
$810$	0	0
$811$	−4364.52	−0.188976	−0.0944878	−	0.995526i	$-0.530121\pi$
$811$	−4364.52	−0.188976	−0.0944878	+	0.995526i	$0.530121\pi$
$812$	11690.3i	0.505232i
$813$	− 18265.0i	− 0.787924i
$814$	−14930.0	−0.642870
$815$	0	0
$816$	−8537.06	−0.366246
$817$	31343.1i	1.34218i
$818$	− 66196.1i	− 2.82945i
$819$	5943.02	0.253561
$820$	0	0
$821$	3306.16	0.140543	0.0702714	−	0.997528i	$-0.477613\pi$
$821$	3306.16	0.140543	0.0702714	+	0.997528i	$0.477613\pi$
$822$	− 20144.7i	− 0.854779i
$823$	− 19183.7i	− 0.812519i	−0.913758	−	0.406259i	$-0.866833\pi$
$823$	− 19183.7i	− 0.812519i	0.913758	−	0.406259i	$-0.133167\pi$
$824$	−82722.8	−3.49731
$825$	0	0
$826$	−46323.6	−1.95134
$827$	− 20646.4i	− 0.868131i	−0.900881	−	0.434066i	$-0.857079\pi$
$827$	− 20646.4i	− 0.868131i	0.900881	−	0.434066i	$-0.142921\pi$
$828$	− 37663.0i	− 1.58077i
$829$	−5345.44	−0.223950	−0.111975	−	0.993711i	$-0.535718\pi$
$829$	−5345.44	−0.223950	−0.111975	+	0.993711i	$0.535718\pi$
$830$	0	0
$831$	2496.96	0.104234
$832$	− 43464.0i	− 1.81111i
$833$	− 4054.75i	− 0.168654i
$834$	42668.8	1.77158
$835$	0	0
$836$	19532.1	0.808051
$837$	− 3826.00i	− 0.158000i
$838$	19867.5i	0.818985i
$839$	−29284.9	−1.20504	−0.602519	−	0.798105i	$-0.705836\pi$
$839$	−29284.9	−1.20504	−0.602519	+	0.798105i	$0.705836\pi$
$840$	0	0
$841$	−21077.4	−0.864219
$842$	66733.8i	2.73135i
$843$	7635.97i	0.311977i
$844$	86302.6	3.51974
$845$	0	0
$846$	−19148.0	−0.778159
$847$	1248.81i	0.0506605i
$848$	43079.9i	1.74454i
$849$	−17735.1	−0.716923
$850$	0	0
$851$	54844.2	2.20921
$852$	5458.94i	0.219507i
$853$	− 8070.62i	− 0.323954i	−0.986795	−	0.161977i	$-0.948213\pi$
$853$	− 8070.62i	− 0.323954i	0.986795	−	0.161977i	$-0.0517870\pi$
$854$	11061.4	0.443222
$855$	0	0
$856$	54659.5	2.18250
$857$	11344.2i	0.452169i	0.974108	+	0.226085i	$0.0725926\pi$
$857$	11344.2i	0.452169i	−0.974108	+	0.226085i	$0.927407\pi$
$858$	11109.1i	0.442026i
$859$	−25470.6	−1.01170	−0.505848	−	0.862623i	$-0.668820\pi$
$859$	−25470.6	−1.01170	−0.505848	+	0.862623i	$0.668820\pi$
$860$	0	0
$861$	−6994.78	−0.276866
$862$	74463.6i	2.94228i
$863$	− 14558.4i	− 0.574243i	−0.957894	−	0.287122i	$-0.907302\pi$
$863$	− 14558.4i	− 0.574243i	0.957894	−	0.287122i	$-0.0926984\pi$
$864$	−10299.5	−0.405550
$865$	0	0
$866$	57614.6	2.26077
$867$	− 13857.0i	− 0.542802i
$868$	28786.5i	1.12567i
$869$	11237.6	0.438675
$870$	0	0
$871$	17076.0	0.664294
$872$	77226.3i	2.99910i
$873$	− 1779.85i	− 0.0690019i
$874$	−100911.	−3.90546
$875$	0	0
$876$	14291.6	0.551220
$877$	− 185.528i	− 0.00714350i	−0.999994	−	0.00357175i	$-0.998863\pi$
$877$	− 185.528i	− 0.00714350i	0.999994	−	0.00357175i	$-0.00113693\pi$
$878$	59039.1i	2.26933i
$879$	−20613.1	−0.790969
$880$	0	0
$881$	18950.1	0.724681	0.362340	−	0.932046i	$-0.381978\pi$
$881$	18950.1	0.724681	0.362340	+	0.932046i	$0.381978\pi$
$882$	− 11198.3i	− 0.427513i
$883$	21258.3i	0.810189i	0.914275	+	0.405095i	$0.132761\pi$
$883$	21258.3i	0.810189i	−0.914275	+	0.405095i	$0.867239\pi$
$884$	−21593.3	−0.821563
$885$	0	0
$886$	−50758.2	−1.92467
$887$	− 35707.4i	− 1.35168i	−0.737049	−	0.675839i	$-0.763781\pi$
$887$	− 35707.4i	− 1.35168i	0.737049	−	0.675839i	$-0.236219\pi$
$888$	− 47572.5i	− 1.79778i
$889$	21318.1	0.804259
$890$	0	0
$891$	891.000	0.0335013
$892$	45921.8i	1.72374i
$893$	36477.8i	1.36695i
$894$	37898.4	1.41780
$895$	0	0
$896$	5393.06	0.201082
$897$	− 40808.4i	− 1.51901i
$898$	− 34105.2i	− 1.26738i
$899$	8154.49	0.302522
$900$	0	0
$901$	4450.58	0.164562
$902$	− 13075.1i	− 0.482653i
$903$	− 10757.6i	− 0.396447i
$904$	−147165.	−5.41440
$905$	0	0
$906$	−39734.2	−1.45704
$907$	6542.52i	0.239516i	0.992803	+	0.119758i	$0.0382118\pi$
$907$	6542.52i	0.239516i	−0.992803	+	0.119758i	$0.961788\pi$
$908$	− 41728.6i	− 1.52512i
$909$	−12605.6	−0.459958
$910$	0	0
$911$	31171.2	1.13364	0.566821	−	0.823841i	$-0.308173\pi$
$911$	31171.2	1.13364	0.566821	+	0.823841i	$0.308173\pi$
$912$	44915.9i	1.63083i
$913$	7770.56i	0.281674i
$914$	59559.2	2.15541
$915$	0	0
$916$	46199.2	1.66644
$917$	8107.19i	0.291955i
$918$	2435.78i	0.0875737i
$919$	−12031.9	−0.431877	−0.215938	−	0.976407i	$-0.569281\pi$
$919$	−12031.9	−0.431877	−0.215938	+	0.976407i	$0.569281\pi$
$920$	0	0
$921$	−600.537	−0.0214857
$922$	− 44232.0i	− 1.57994i
$923$	5914.85i	0.210931i
$924$	−6703.82	−0.238679
$925$	0	0
$926$	−51344.7	−1.82213
$927$	12111.3i	0.429113i
$928$	− 21951.6i	− 0.776506i
$929$	12546.2	0.443085	0.221542	−	0.975151i	$-0.428891\pi$
$929$	12546.2	0.443085	0.221542	+	0.975151i	$0.428891\pi$
$930$	0	0
$931$	−21333.2	−0.750986
$932$	7391.07i	0.259767i
$933$	− 17204.8i	− 0.603708i
$934$	−86229.1	−3.02088
$935$	0	0
$936$	−35397.7	−1.23612
$937$	− 17909.8i	− 0.624427i	−0.950012	−	0.312214i	$-0.898930\pi$
$937$	− 17909.8i	− 0.624427i	0.950012	−	0.312214i	$-0.101070\pi$
$938$	14492.7i	0.504483i
$939$	−9232.08	−0.320849
$940$	0	0
$941$	829.893	0.0287500	0.0143750	−	0.999897i	$-0.495424\pi$
$941$	829.893	0.0287500	0.0143750	+	0.999897i	$0.495424\pi$
$942$	− 21962.9i	− 0.759650i
$943$	48030.4i	1.65863i
$944$	141581.	4.88144
$945$	0	0
$946$	20108.9	0.691116
$947$	17654.9i	0.605814i	0.953020	+	0.302907i	$0.0979572\pi$
$947$	17654.9i	0.605814i	−0.953020	+	0.302907i	$0.902043\pi$
$948$	60325.4i	2.06675i
$949$	15485.2	0.529685
$950$	0	0
$951$	−6427.14	−0.219153
$952$	− 10878.0i	− 0.370335i
$953$	30736.6i	1.04476i	0.852713	+	0.522380i	$0.174956\pi$
$953$	30736.6i	1.04476i	−0.852713	+	0.522380i	$0.825044\pi$
$954$	12291.5	0.417139
$955$	0	0
$956$	−28110.8	−0.951012
$957$	1899.02i	0.0641449i
$958$	− 72761.5i	− 2.45388i
$959$	13171.7	0.443519
$960$	0	0
$961$	−9711.08	−0.325974
$962$	− 86840.6i	− 2.91045i
$963$	− 8002.61i	− 0.267789i
$964$	−3755.97	−0.125489
$965$	0	0
$966$	34634.8	1.15358
$967$	− 23645.2i	− 0.786327i	−0.919469	−	0.393163i	$-0.871381\pi$
$967$	− 23645.2i	− 0.786327i	0.919469	−	0.393163i	$-0.128619\pi$
$968$	− 7438.10i	− 0.246973i
$969$	4640.26	0.153836
$970$	0	0
$971$	27402.0	0.905635	0.452818	−	0.891603i	$-0.350419\pi$
$971$	27402.0	0.905635	0.452818	+	0.891603i	$0.350419\pi$
$972$	4783.05i	0.157836i
$973$	27899.1i	0.919222i
$974$	69790.6	2.29593
$975$	0	0
$976$	−33807.5	−1.10876
$977$	49118.7i	1.60844i	0.594332	+	0.804220i	$0.297416\pi$
$977$	49118.7i	1.60844i	−0.594332	+	0.804220i	$0.702584\pi$
$978$	− 42942.1i	− 1.40403i
$979$	4842.40	0.158084
$980$	0	0
$981$	11306.6	0.367983
$982$	− 39294.0i	− 1.27691i
$983$	52630.5i	1.70768i	0.520533	+	0.853842i	$0.325733\pi$
$983$	52630.5i	1.70768i	−0.520533	+	0.853842i	$0.674267\pi$
$984$	41662.1	1.34973
$985$	0	0
$986$	−5191.47	−0.167677
$987$	− 12520.0i	− 0.403764i
$988$	113609.i	3.65827i
$989$	−73868.4	−2.37500
$990$	0	0
$991$	45472.7	1.45761	0.728803	−	0.684724i	$-0.240077\pi$
$991$	45472.7	1.45761	0.728803	+	0.684724i	$0.240077\pi$
$992$	− 54054.5i	− 1.73007i
$993$	4854.68i	0.155145i
$994$	−5020.04	−0.160187
$995$	0	0
$996$	−41713.8	−1.32706
$997$	− 55068.1i	− 1.74927i	−0.484779	−	0.874637i	$-0.661100\pi$
$997$	− 55068.1i	− 1.74927i	0.484779	−	0.874637i	$-0.338900\pi$
$998$	27763.0i	0.880585i
$999$	−6965.01	−0.220584

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.4.c.l.199.6		6
5.2	odd	4		165.4.a.d.1.1	✓	3
5.3	odd	4		825.4.a.s.1.3		3
5.4	even	2	inner	825.4.c.l.199.1		6
15.2	even	4		495.4.a.l.1.3		3
15.8	even	4		2475.4.a.s.1.1		3
55.32	even	4		1815.4.a.s.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.4.a.d.1.1	✓	3	5.2	odd	4
495.4.a.l.1.3		3	15.2	even	4
825.4.a.s.1.3		3	5.3	odd	4
825.4.c.l.199.1		6	5.4	even	2	inner
825.4.c.l.199.6		6	1.1	even	1	trivial
1815.4.a.s.1.3		3	55.32	even	4
2475.4.a.s.1.1		3	15.8	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	825.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+5.26150i q^{2} -3.00000i q^{3} -19.6833 q^{4} +15.7845 q^{6} +10.3207i q^{7} -61.4719i q^{8} -9.00000 q^{9} +11.0000 q^{11} +59.0500i q^{12} +63.9817i q^{13} -54.3024 q^{14} +165.967 q^{16} -17.1461i q^{17} -47.3535i q^{18} -90.2104 q^{19} +30.9621 q^{21} +57.8765i q^{22} -212.605i q^{23} -184.416 q^{24} -336.639 q^{26} +27.0000i q^{27} -203.146i q^{28} -57.5461 q^{29} -141.704 q^{31} +381.462i q^{32} -33.0000i q^{33} +90.2140 q^{34} +177.150 q^{36} +257.963i q^{37} -474.642i q^{38} +191.945 q^{39} -225.914 q^{41} +162.907i q^{42} -347.445i q^{43} -216.517 q^{44} +1118.62 q^{46} -404.364i q^{47} -497.902i q^{48} +236.483 q^{49} -51.4382 q^{51} -1259.37i q^{52} +259.568i q^{53} -142.060 q^{54} +634.433 q^{56} +270.631i q^{57} -302.779i q^{58} +853.067 q^{59} -203.699 q^{61} -745.573i q^{62} -92.8864i q^{63} -679.320 q^{64} +173.629 q^{66} -266.890i q^{67} +337.492i q^{68} -637.814 q^{69} +92.4460 q^{71} +553.247i q^{72} -242.026i q^{73} -1357.27 q^{74} +1775.64 q^{76} +113.528i q^{77} +1009.92i q^{78} +1021.60 q^{79} +81.0000 q^{81} -1188.65i q^{82} +706.415i q^{83} -609.438 q^{84} +1828.08 q^{86} +172.638i q^{87} -676.191i q^{88} +440.218 q^{89} -660.336 q^{91} +4184.77i q^{92} +425.111i q^{93} +2127.56 q^{94} +1144.38 q^{96} +197.761i q^{97} +1244.25i q^{98} -99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 44 q^{4} + 24 q^{6} - 54 q^{9} + 66 q^{11} + 112 q^{14} + 100 q^{16} - 292 q^{19} + 24 q^{21} - 288 q^{24} - 1016 q^{26} - 136 q^{29} - 136 q^{31} + 352 q^{34} + 396 q^{36} - 392 q^{41} - 484 q^{44}+ \cdots - 594 q^{99}+O(q^{100}) \) 6 * q - 44 * q^4 + 24 * q^6 - 54 * q^9 + 66 * q^11 + 112 * q^14 + 100 * q^16 - 292 * q^19 + 24 * q^21 - 288 * q^24 - 1016 * q^26 - 136 * q^29 - 136 * q^31 + 352 * q^34 + 396 * q^36 - 392 * q^41 - 484 * q^44 + 2320 * q^46 + 314 * q^49 + 1308 * q^51 - 216 * q^54 + 2736 * q^56 + 2088 * q^59 + 1284 * q^61 - 2332 * q^64 + 264 * q^66 - 1200 * q^69 - 1088 * q^71 - 3072 * q^74 + 3992 * q^76 + 3172 * q^79 + 486 * q^81 - 2040 * q^84 + 7136 * q^86 + 4244 * q^89 - 16 * q^91 + 4304 * q^94 + 4128 * q^96 - 594 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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