LMFDB - Embedded newform 825.4.c.j.199.1

Newspace parameters

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

Self dual:	no
Analytic conductor:	$48.6765757547$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{17})$
Defining polynomial:	$ x^{4} + 9x^{2} + 16 $ x^4 + 9*x^2 + 16
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 165)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.1
Root			$-2.56155i$ of defining polynomial
Character	$\chi$	$=$	825.199
Dual form			825.4.c.j.199.4

$q$-expansion

$f(q)$	$=$	$q-2.56155i q^{2} -3.00000i q^{3} +1.43845 q^{4} -7.68466 q^{6} +6.24621i q^{7} -24.1771i q^{8} -9.00000 q^{9} +O(q^{10})$	\(q-2.56155i q^{2} -3.00000i q^{3} +1.43845 q^{4} -7.68466 q^{6} +6.24621i q^{7} -24.1771i q^{8} -9.00000 q^{9} -11.0000 q^{11} -4.31534i q^{12} +49.1231i q^{13} +16.0000 q^{14} -50.4233 q^{16} +82.7083i q^{17} +23.0540i q^{18} +130.354 q^{19} +18.7386 q^{21} +28.1771i q^{22} +185.693i q^{23} -72.5312 q^{24} +125.831 q^{26} +27.0000i q^{27} +8.98485i q^{28} +8.90720 q^{29} +5.26137 q^{31} -64.2547i q^{32} +33.0000i q^{33} +211.862 q^{34} -12.9460 q^{36} -416.894i q^{37} -333.909i q^{38} +147.369 q^{39} -298.479 q^{41} -48.0000i q^{42} +513.633i q^{43} -15.8229 q^{44} +475.663 q^{46} +557.295i q^{47} +151.270i q^{48} +303.985 q^{49} +248.125 q^{51} +70.6610i q^{52} +168.064i q^{53} +69.1619 q^{54} +151.015 q^{56} -391.062i q^{57} -22.8163i q^{58} -618.773 q^{59} +786.405 q^{61} -13.4773i q^{62} -56.2159i q^{63} -567.978 q^{64} +84.5312 q^{66} -339.015i q^{67} +118.972i q^{68} +557.080 q^{69} +1120.71 q^{71} +217.594i q^{72} +123.430i q^{73} -1067.90 q^{74} +187.508 q^{76} -68.7083i q^{77} -377.494i q^{78} +309.835 q^{79} +81.0000 q^{81} +764.570i q^{82} +1021.22i q^{83} +26.9545 q^{84} +1315.70 q^{86} -26.7216i q^{87} +265.948i q^{88} +141.879 q^{89} -306.833 q^{91} +267.110i q^{92} -15.7841i q^{93} +1427.54 q^{94} -192.764 q^{96} +798.345i q^{97} -778.673i q^{98} +99.0000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 14 q^{4} - 6 q^{6} - 36 q^{9}+O(q^{10}) $ 4 * q + 14 * q^4 - 6 * q^6 - 36 * q^9	$ 4 q + 14 q^{4} - 6 q^{6} - 36 q^{9} - 44 q^{11} + 64 q^{14} - 78 q^{16} + 340 q^{19} - 24 q^{21} - 18 q^{24} + 124 q^{26} + 316 q^{29} + 120 q^{31} + 732 q^{34} - 126 q^{36} + 540 q^{39} + 76 q^{41} - 154 q^{44} + 1144 q^{46} + 1084 q^{49} - 96 q^{51} + 54 q^{54} + 736 q^{56} - 496 q^{59} + 144 q^{61} - 1538 q^{64} + 66 q^{66} + 744 q^{69} + 4120 q^{71} - 2276 q^{74} + 816 q^{76} - 1284 q^{79} + 324 q^{81} - 288 q^{84} + 2624 q^{86} - 488 q^{89} + 224 q^{91} + 3896 q^{94} + 738 q^{96} + 396 q^{99}+O(q^{100}) $ 4 * q + 14 * q^4 - 6 * q^6 - 36 * q^9 - 44 * q^11 + 64 * q^14 - 78 * q^16 + 340 * q^19 - 24 * q^21 - 18 * q^24 + 124 * q^26 + 316 * q^29 + 120 * q^31 + 732 * q^34 - 126 * q^36 + 540 * q^39 + 76 * q^41 - 154 * q^44 + 1144 * q^46 + 1084 * q^49 - 96 * q^51 + 54 * q^54 + 736 * q^56 - 496 * q^59 + 144 * q^61 - 1538 * q^64 + 66 * q^66 + 744 * q^69 + 4120 * q^71 - 2276 * q^74 + 816 * q^76 - 1284 * q^79 + 324 * q^81 - 288 * q^84 + 2624 * q^86 - 488 * q^89 + 224 * q^91 + 3896 * q^94 + 738 * q^96 + 396 * q^99

Display 100 coefficients 10 coefficients

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 $

$2$	− 2.56155i	− 0.905646i	−0.891601	−	0.452823i	$-0.850417\pi$
$2$	− 2.56155i	− 0.905646i	0.891601	−	0.452823i	$-0.149583\pi$
$3$	− 3.00000i	− 0.577350i
$3$	− 3.00000i	− 0.577350i
$4$	1.43845	0.179806
$5$	0	0
$5$	0	0
$6$	−7.68466	−0.522875
$7$	6.24621i	0.337264i	0.985679	+	0.168632i	$0.0539349\pi$
$7$	6.24621i	0.337264i	−0.985679	+	0.168632i	$0.946065\pi$
$8$	− 24.1771i	− 1.06849i
$9$	−9.00000	−0.333333
$10$	0	0
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	− 4.31534i	− 0.103811i
$13$	49.1231i	1.04802i	0.851711	+	0.524011i	$0.175565\pi$
$13$	49.1231i	1.04802i	−0.851711	+	0.524011i	$0.824435\pi$
$14$	16.0000	0.305441
$15$	0	0
$16$	−50.4233	−0.787864
$17$	82.7083i	1.17998i	0.807409	+	0.589992i	$0.200869\pi$
$17$	82.7083i	1.17998i	−0.807409	+	0.589992i	$0.799131\pi$
$18$	23.0540i	0.301882i
$19$	130.354	1.57396	0.786981	−	0.616977i	$-0.211643\pi$
$19$	130.354	1.57396	0.786981	+	0.616977i	$0.211643\pi$
$20$	0	0
$21$	18.7386	0.194719
$22$	28.1771i	0.273062i
$23$	185.693i	1.68347i	0.539895	+	0.841733i	$0.318464\pi$
$23$	185.693i	1.68347i	−0.539895	+	0.841733i	$0.681536\pi$
$24$	−72.5312	−0.616891
$25$	0	0
$26$	125.831	0.949137
$27$	27.0000i	0.192450i
$28$	8.98485i	0.0606420i
$29$	8.90720	0.0570354	0.0285177	−	0.999593i	$-0.490921\pi$
$29$	8.90720	0.0570354	0.0285177	+	0.999593i	$0.490921\pi$
$30$	0	0
$31$	5.26137	0.0304829	0.0152414	−	0.999884i	$-0.495148\pi$
$31$	5.26137	0.0304829	0.0152414	+	0.999884i	$0.495148\pi$
$32$	− 64.2547i	− 0.354961i
$33$	33.0000i	0.174078i
$34$	211.862	1.06865
$35$	0	0
$36$	−12.9460	−0.0599353
$37$	− 416.894i	− 1.85235i	−0.377094	−	0.926175i	$-0.623077\pi$
$37$	− 416.894i	− 1.85235i	0.377094	−	0.926175i	$-0.376923\pi$
$38$	− 333.909i	− 1.42545i
$39$	147.369	0.605076
$40$	0	0
$41$	−298.479	−1.13694	−0.568471	−	0.822703i	$-0.692465\pi$
$41$	−298.479	−1.13694	−0.568471	+	0.822703i	$0.692465\pi$
$42$	− 48.0000i	− 0.176347i
$43$	513.633i	1.82159i	0.412863	+	0.910793i	$0.364529\pi$
$43$	513.633i	1.82159i	−0.412863	+	0.910793i	$0.635471\pi$
$44$	−15.8229	−0.0542135
$45$	0	0
$46$	475.663	1.52462
$47$	557.295i	1.72957i	0.502140	+	0.864786i	$0.332546\pi$
$47$	557.295i	1.72957i	−0.502140	+	0.864786i	$0.667454\pi$
$48$	151.270i	0.454873i
$49$	303.985	0.886253
$50$	0	0
$51$	248.125	0.681264
$52$	70.6610i	0.188441i
$53$	168.064i	0.435574i	0.975996	+	0.217787i	$0.0698838\pi$
$53$	168.064i	0.435574i	−0.975996	+	0.217787i	$0.930116\pi$
$54$	69.1619	0.174292
$55$	0	0
$56$	151.015	0.360362
$57$	− 391.062i	− 0.908728i
$58$	− 22.8163i	− 0.0516539i
$59$	−618.773	−1.36538	−0.682689	−	0.730709i	$-0.739190\pi$
$59$	−618.773	−1.36538	−0.682689	+	0.730709i	$0.739190\pi$
$60$	0	0
$61$	786.405	1.65064	0.825319	−	0.564667i	$-0.190996\pi$
$61$	786.405	1.65064	0.825319	+	0.564667i	$0.190996\pi$
$62$	− 13.4773i	− 0.0276067i
$63$	− 56.2159i	− 0.112421i
$64$	−567.978	−1.10933
$65$	0	0
$66$	84.5312	0.157653
$67$	− 339.015i	− 0.618169i	−0.951035	−	0.309084i	$-0.899977\pi$
$67$	− 339.015i	− 0.618169i	0.951035	−	0.309084i	$-0.100023\pi$
$68$	118.972i	0.212168i
$69$	557.080	0.971949
$70$	0	0
$71$	1120.71	1.87329	0.936645	−	0.350280i	$-0.113913\pi$
$71$	1120.71	1.87329	0.936645	+	0.350280i	$0.113913\pi$
$72$	217.594i	0.356162i
$73$	123.430i	0.197896i	0.995093	+	0.0989478i	$0.0315477\pi$
$73$	123.430i	0.197896i	−0.995093	+	0.0989478i	$0.968452\pi$
$74$	−1067.90	−1.67757
$75$	0	0
$76$	187.508	0.283008
$77$	− 68.7083i	− 0.101689i
$78$	− 377.494i	− 0.547985i
$79$	309.835	0.441255	0.220628	−	0.975358i	$-0.429189\pi$
$79$	309.835	0.441255	0.220628	+	0.975358i	$0.429189\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	764.570i	1.02967i
$83$	1021.22i	1.35053i	0.737577	+	0.675263i	$0.235970\pi$
$83$	1021.22i	1.35053i	−0.737577	+	0.675263i	$0.764030\pi$
$84$	26.9545	0.0350117
$85$	0	0
$86$	1315.70	1.64971
$87$	− 26.7216i	− 0.0329294i
$88$	265.948i	0.322161i
$89$	141.879	0.168979	0.0844894	−	0.996424i	$-0.473074\pi$
$89$	141.879	0.168979	0.0844894	+	0.996424i	$0.473074\pi$
$90$	0	0
$91$	−306.833	−0.353460
$92$	267.110i	0.302697i
$93$	− 15.7841i	− 0.0175993i
$94$	1427.54	1.56638
$95$	0	0
$96$	−192.764	−0.204937
$97$	798.345i	0.835666i	0.908524	+	0.417833i	$0.137210\pi$
$97$	798.345i	0.835666i	−0.908524	+	0.417833i	$0.862790\pi$
$98$	− 778.673i	− 0.802631i
$99$	99.0000	0.100504
$100$	0	0
$101$	241.400	0.237823	0.118912	−	0.992905i	$-0.462059\pi$
$101$	241.400	0.237823	0.118912	+	0.992905i	$0.462059\pi$
$102$	− 635.585i	− 0.616983i
$103$	− 1168.38i	− 1.11771i	−0.829267	−	0.558853i	$-0.811242\pi$
$103$	− 1168.38i	− 1.11771i	0.829267	−	0.558853i	$-0.188758\pi$
$104$	1187.65	1.11980
$105$	0	0
$106$	430.506	0.394476
$107$	− 2106.82i	− 1.90350i	−0.306882	−	0.951748i	$-0.599286\pi$
$107$	− 2106.82i	− 1.90350i	0.306882	−	0.951748i	$-0.400714\pi$
$108$	38.8381i	0.0346037i
$109$	−493.792	−0.433914	−0.216957	−	0.976181i	$-0.569613\pi$
$109$	−493.792	−0.433914	−0.216957	+	0.976181i	$0.569613\pi$
$110$	0	0
$111$	−1250.68	−1.06945
$112$	− 314.955i	− 0.265718i
$113$	− 170.000i	− 0.141524i	−0.997493	−	0.0707622i	$-0.977457\pi$
$113$	− 170.000i	− 0.141524i	0.997493	−	0.0707622i	$-0.0225431\pi$
$114$	−1001.73	−0.822986
$115$	0	0
$116$	12.8125	0.0102553
$117$	− 442.108i	− 0.349341i
$118$	1585.02i	1.23655i
$119$	−516.614	−0.397966
$120$	0	0
$121$	121.000	0.0909091
$122$	− 2014.42i	− 1.49489i
$123$	895.437i	0.656414i
$124$	7.56820	0.00548100
$125$	0	0
$126$	−144.000	−0.101814
$127$	− 948.182i	− 0.662500i	−0.943543	−	0.331250i	$-0.892530\pi$
$127$	− 948.182i	− 0.662500i	0.943543	−	0.331250i	$-0.107470\pi$
$128$	940.868i	0.649702i
$129$	1540.90	1.05169
$130$	0	0
$131$	−1484.84	−0.990312	−0.495156	−	0.868804i	$-0.664889\pi$
$131$	−1484.84	−0.990312	−0.495156	+	0.868804i	$0.664889\pi$
$132$	47.4688i	0.0313002i
$133$	814.220i	0.530841i
$134$	−868.405	−0.559842
$135$	0	0
$136$	1999.65	1.26080
$137$	684.928i	0.427134i	0.976928	+	0.213567i	$0.0685082\pi$
$137$	684.928i	0.427134i	−0.976928	+	0.213567i	$0.931492\pi$
$138$	− 1426.99i	− 0.880242i
$139$	−830.483	−0.506767	−0.253384	−	0.967366i	$-0.581543\pi$
$139$	−830.483	−0.506767	−0.253384	+	0.967366i	$0.581543\pi$
$140$	0	0
$141$	1671.89	0.998569
$142$	− 2870.75i	− 1.69654i
$143$	− 540.354i	− 0.315991i
$144$	453.810	0.262621
$145$	0	0
$146$	316.172	0.179223
$147$	− 911.955i	− 0.511679i
$148$	− 599.680i	− 0.333063i
$149$	1213.64	0.667285	0.333642	−	0.942700i	$-0.391722\pi$
$149$	1213.64	0.667285	0.333642	+	0.942700i	$0.391722\pi$
$150$	0	0
$151$	30.8466	0.0166242	0.00831212	−	0.999965i	$-0.497354\pi$
$151$	30.8466	0.0166242	0.00831212	+	0.999965i	$0.497354\pi$
$152$	− 3151.58i	− 1.68176i
$153$	− 744.375i	− 0.393328i
$154$	−176.000	−0.0920941
$155$	0	0
$156$	211.983	0.108796
$157$	345.239i	0.175497i	0.996143	+	0.0877485i	$0.0279672\pi$
$157$	345.239i	0.175497i	−0.996143	+	0.0877485i	$0.972033\pi$
$158$	− 793.659i	− 0.399621i
$159$	504.193	0.251479
$160$	0	0
$161$	−1159.88	−0.567772
$162$	− 207.486i	− 0.100627i
$163$	− 1921.49i	− 0.923331i	−0.887054	−	0.461665i	$-0.847252\pi$
$163$	− 1921.49i	− 0.923331i	0.887054	−	0.461665i	$-0.152748\pi$
$164$	−429.346	−0.204429
$165$	0	0
$166$	2615.91	1.22310
$167$	172.297i	0.0798369i	0.999203	+	0.0399185i	$0.0127098\pi$
$167$	172.297i	0.0798369i	−0.999203	+	0.0399185i	$0.987290\pi$
$168$	− 453.045i	− 0.208055i
$169$	−216.080	−0.0983521
$170$	0	0
$171$	−1173.19	−0.524654
$172$	738.833i	0.327532i
$173$	1025.29i	0.450587i	0.974291	+	0.225293i	$0.0723340\pi$
$173$	1025.29i	0.450587i	−0.974291	+	0.225293i	$0.927666\pi$
$174$	−68.4488	−0.0298224
$175$	0	0
$176$	554.656	0.237550
$177$	1856.32i	0.788302i
$178$	− 363.430i	− 0.153035i
$179$	−1658.29	−0.692437	−0.346219	−	0.938154i	$-0.612534\pi$
$179$	−1658.29	−0.692437	−0.346219	+	0.938154i	$0.612534\pi$
$180$	0	0
$181$	−2021.81	−0.830275	−0.415137	−	0.909759i	$-0.636266\pi$
$181$	−2021.81	−0.830275	−0.415137	+	0.909759i	$0.636266\pi$
$182$	785.970i	0.320110i
$183$	− 2359.22i	− 0.952996i
$184$	4489.52	1.79876
$185$	0	0
$186$	−40.4318	−0.0159387
$187$	− 909.792i	− 0.355778i
$188$	801.640i	0.310987i
$189$	−168.648	−0.0649064
$190$	0	0
$191$	1440.22	0.545607	0.272803	−	0.962070i	$-0.412049\pi$
$191$	1440.22	0.545607	0.272803	+	0.962070i	$0.412049\pi$
$192$	1703.93i	0.640473i
$193$	2798.05i	1.04356i	0.853079	+	0.521782i	$0.174733\pi$
$193$	2798.05i	1.04356i	−0.853079	+	0.521782i	$0.825267\pi$
$194$	2045.00	0.756817
$195$	0	0
$196$	437.266	0.159354
$197$	458.943i	0.165982i	0.996550	+	0.0829908i	$0.0264472\pi$
$197$	458.943i	0.165982i	−0.996550	+	0.0829908i	$0.973553\pi$
$198$	− 253.594i	− 0.0910208i
$199$	2371.04	0.844615	0.422308	−	0.906453i	$-0.361220\pi$
$199$	2371.04	0.844615	0.422308	+	0.906453i	$0.361220\pi$
$200$	0	0
$201$	−1017.05	−0.356900
$202$	− 618.358i	− 0.215384i
$203$	55.6363i	0.0192360i
$204$	356.915	0.122495
$205$	0	0
$206$	−2992.86	−1.01225
$207$	− 1671.24i	− 0.561155i
$208$	− 2476.95i	− 0.825699i
$209$	−1433.90	−0.474568
$210$	0	0
$211$	4319.87	1.40944	0.704721	−	0.709484i	$-0.251072\pi$
$211$	4319.87	1.40944	0.704721	+	0.709484i	$0.251072\pi$
$212$	241.752i	0.0783187i
$213$	− 3362.12i	− 1.08154i
$214$	−5396.73	−1.72389
$215$	0	0
$216$	652.781	0.205630
$217$	32.8636i	0.0102808i
$218$	1264.87i	0.392973i
$219$	370.290	0.114255
$220$	0	0
$221$	−4062.89	−1.23665
$222$	3203.69i	0.968547i
$223$	3837.73i	1.15244i	0.817295	+	0.576219i	$0.195473\pi$
$223$	3837.73i	1.15244i	−0.817295	+	0.576219i	$0.804527\pi$
$224$	401.349	0.119715
$225$	0	0
$226$	−435.464	−0.128171
$227$	− 5003.71i	− 1.46303i	−0.681825	−	0.731515i	$-0.738813\pi$
$227$	− 5003.71i	− 1.46303i	0.681825	−	0.731515i	$-0.261187\pi$
$228$	− 562.523i	− 0.163395i
$229$	277.375	0.0800412	0.0400206	−	0.999199i	$-0.487258\pi$
$229$	277.375	0.0800412	0.0400206	+	0.999199i	$0.487258\pi$
$230$	0	0
$231$	−206.125	−0.0587101
$232$	− 215.350i	− 0.0609415i
$233$	− 2269.91i	− 0.638225i	−0.947717	−	0.319113i	$-0.896615\pi$
$233$	− 2269.91i	− 0.638225i	0.947717	−	0.319113i	$-0.103385\pi$
$234$	−1132.48	−0.316379
$235$	0	0
$236$	−890.072	−0.245503
$237$	− 929.505i	− 0.254759i
$238$	1323.33i	0.360416i
$239$	1617.11	0.437665	0.218832	−	0.975762i	$-0.429775\pi$
$239$	1617.11	0.437665	0.218832	+	0.975762i	$0.429775\pi$
$240$	0	0
$241$	5646.63	1.50926	0.754629	−	0.656151i	$-0.227817\pi$
$241$	5646.63	1.50926	0.754629	+	0.656151i	$0.227817\pi$
$242$	− 309.948i	− 0.0823314i
$243$	− 243.000i	− 0.0641500i
$244$	1131.20	0.296794
$245$	0	0
$246$	2293.71	0.594478
$247$	6403.40i	1.64955i
$248$	− 127.204i	− 0.0325705i
$249$	3063.66	0.779726
$250$	0	0
$251$	−6217.61	−1.56355	−0.781777	−	0.623558i	$-0.785687\pi$
$251$	−6217.61	−1.56355	−0.781777	+	0.623558i	$0.785687\pi$
$252$	− 80.8636i	− 0.0202140i
$253$	− 2042.62i	− 0.507584i
$254$	−2428.82	−0.599991
$255$	0	0
$256$	−2133.74	−0.520933
$257$	7712.75i	1.87202i	0.351980	+	0.936008i	$0.385509\pi$
$257$	7712.75i	1.87202i	−0.351980	+	0.936008i	$0.614491\pi$
$258$	− 3947.09i	− 0.952462i
$259$	2604.01	0.624730
$260$	0	0
$261$	−80.1648	−0.0190118
$262$	3803.49i	0.896871i
$263$	− 206.347i	− 0.0483798i	−0.999707	−	0.0241899i	$-0.992299\pi$
$263$	− 206.347i	− 0.0483798i	0.999707	−	0.0241899i	$-0.00770063\pi$
$264$	797.844	0.186000
$265$	0	0
$266$	2085.67	0.480753
$267$	− 425.636i	− 0.0975600i
$268$	− 487.655i	− 0.111150i
$269$	−1712.47	−0.388146	−0.194073	−	0.980987i	$-0.562170\pi$
$269$	−1712.47	−0.388146	−0.194073	+	0.980987i	$0.562170\pi$
$270$	0	0
$271$	−477.081	−0.106940	−0.0534698	−	0.998569i	$-0.517028\pi$
$271$	−477.081	−0.106940	−0.0534698	+	0.998569i	$0.517028\pi$
$272$	− 4170.43i	− 0.929666i
$273$	920.500i	0.204070i
$274$	1754.48	0.386832
$275$	0	0
$276$	801.329	0.174762
$277$	4283.48i	0.929130i	0.885539	+	0.464565i	$0.153789\pi$
$277$	4283.48i	0.929130i	−0.885539	+	0.464565i	$0.846211\pi$
$278$	2127.33i	0.458952i
$279$	−47.3523	−0.0101610
$280$	0	0
$281$	3477.79	0.738319	0.369160	−	0.929366i	$-0.379646\pi$
$281$	3477.79	0.738319	0.369160	+	0.929366i	$0.379646\pi$
$282$	− 4282.62i	− 0.904350i
$283$	6568.27i	1.37966i	0.723973	+	0.689829i	$0.242314\pi$
$283$	6568.27i	1.37966i	−0.723973	+	0.689829i	$0.757686\pi$
$284$	1612.08	0.336829
$285$	0	0
$286$	−1384.15	−0.286176
$287$	− 1864.36i	− 0.383449i
$288$	578.292i	0.118320i
$289$	−1927.67	−0.392360
$290$	0	0
$291$	2395.03	0.482472
$292$	177.547i	0.0355828i
$293$	− 8352.29i	− 1.66534i	−0.553766	−	0.832672i	$-0.686810\pi$
$293$	− 8352.29i	− 1.66534i	0.553766	−	0.832672i	$-0.313190\pi$
$294$	−2336.02	−0.463399
$295$	0	0
$296$	−10079.3	−1.97921
$297$	− 297.000i	− 0.0580259i
$298$	− 3108.81i	− 0.604324i
$299$	−9121.83	−1.76431
$300$	0	0
$301$	−3208.26	−0.614355
$302$	− 79.0152i	− 0.0150557i
$303$	− 724.199i	− 0.137307i
$304$	−6572.89	−1.24007
$305$	0	0
$306$	−1906.76	−0.356216
$307$	5383.89i	1.00090i	0.865767	+	0.500448i	$0.166831\pi$
$307$	5383.89i	1.00090i	−0.865767	+	0.500448i	$0.833169\pi$
$308$	− 98.8333i	− 0.0182843i
$309$	−3505.14	−0.645308
$310$	0	0
$311$	−1790.41	−0.326447	−0.163223	−	0.986589i	$-0.552189\pi$
$311$	−1790.41	−0.326447	−0.163223	+	0.986589i	$0.552189\pi$
$312$	− 3562.96i	− 0.646516i
$313$	809.076i	0.146108i	0.997328	+	0.0730538i	$0.0232745\pi$
$313$	809.076i	0.146108i	−0.997328	+	0.0730538i	$0.976726\pi$
$314$	884.347	0.158938
$315$	0	0
$316$	445.682	0.0793403
$317$	10744.5i	1.90370i	0.306567	+	0.951849i	$0.400820\pi$
$317$	10744.5i	1.90370i	−0.306567	+	0.951849i	$0.599180\pi$
$318$	− 1291.52i	− 0.227751i
$319$	−97.9792	−0.0171968
$320$	0	0
$321$	−6320.46	−1.09898
$322$	2971.09i	0.514200i
$323$	10781.4i	1.85725i
$324$	116.514	0.0199784
$325$	0	0
$326$	−4922.00	−0.836210
$327$	1481.37i	0.250521i
$328$	7216.35i	1.21481i
$329$	−3480.98	−0.583322
$330$	0	0
$331$	3399.12	0.564449	0.282224	−	0.959348i	$-0.408928\pi$
$331$	3399.12	0.564449	0.282224	+	0.959348i	$0.408928\pi$
$332$	1468.97i	0.242832i
$333$	3752.05i	0.617450i
$334$	441.349	0.0723039
$335$	0	0
$336$	−944.864	−0.153412
$337$	− 11840.0i	− 1.91384i	−0.290356	−	0.956919i	$-0.593774\pi$
$337$	− 11840.0i	− 1.91384i	0.290356	−	0.956919i	$-0.406226\pi$
$338$	553.499i	0.0890721i
$339$	−510.000	−0.0817091
$340$	0	0
$341$	−57.8750	−0.00919093
$342$	3005.18i	0.475151i
$343$	4041.20i	0.636165i
$344$	12418.1	1.94634
$345$	0	0
$346$	2626.34	0.408072
$347$	− 2076.67i	− 0.321272i	−0.987014	−	0.160636i	$-0.948645\pi$
$347$	− 2076.67i	− 0.321272i	0.987014	−	0.160636i	$-0.0513545\pi$
$348$	− 38.4376i	− 0.00592090i
$349$	−5837.37	−0.895322	−0.447661	−	0.894203i	$-0.647743\pi$
$349$	−5837.37	−0.895322	−0.447661	+	0.894203i	$0.647743\pi$
$350$	0	0
$351$	−1326.32	−0.201692
$352$	706.802i	0.107025i
$353$	2423.64i	0.365431i	0.983166	+	0.182715i	$0.0584887\pi$
$353$	2423.64i	0.365431i	−0.983166	+	0.182715i	$0.941511\pi$
$354$	4755.06	0.713922
$355$	0	0
$356$	204.085	0.0303834
$357$	1549.84i	0.229765i
$358$	4247.79i	0.627103i
$359$	3882.22	0.570740	0.285370	−	0.958417i	$-0.407884\pi$
$359$	3882.22	0.570740	0.285370	+	0.958417i	$0.407884\pi$
$360$	0	0
$361$	10133.2	1.47736
$362$	5178.96i	0.751935i
$363$	− 363.000i	− 0.0524864i
$364$	−441.363	−0.0635542
$365$	0	0
$366$	−6043.26	−0.863077
$367$	5666.65i	0.805986i	0.915203	+	0.402993i	$0.132030\pi$
$367$	5666.65i	0.805986i	−0.915203	+	0.402993i	$0.867970\pi$
$368$	− 9363.26i	− 1.32634i
$369$	2686.31	0.378981
$370$	0	0
$371$	−1049.77	−0.146903
$372$	− 22.7046i	− 0.00316446i
$373$	− 174.771i	− 0.0242608i	−0.999926	−	0.0121304i	$-0.996139\pi$
$373$	− 174.771i	− 0.0242608i	0.999926	−	0.0121304i	$-0.00386132\pi$
$374$	−2330.48	−0.322209
$375$	0	0
$376$	13473.8	1.84802
$377$	437.550i	0.0597744i
$378$	432.000i	0.0587822i
$379$	−252.686	−0.0342470	−0.0171235	−	0.999853i	$-0.505451\pi$
$379$	−252.686	−0.0342470	−0.0171235	+	0.999853i	$0.505451\pi$
$380$	0	0
$381$	−2844.55	−0.382495
$382$	− 3689.21i	− 0.494126i
$383$	11014.5i	1.46950i	0.678340	+	0.734748i	$0.262700\pi$
$383$	11014.5i	1.46950i	−0.678340	+	0.734748i	$0.737300\pi$
$384$	2822.61	0.375105
$385$	0	0
$386$	7167.35	0.945099
$387$	− 4622.69i	− 0.607196i
$388$	1148.38i	0.150258i
$389$	−8099.40	−1.05567	−0.527835	−	0.849347i	$-0.676996\pi$
$389$	−8099.40	−1.05567	−0.527835	+	0.849347i	$0.676996\pi$
$390$	0	0
$391$	−15358.4	−1.98646
$392$	− 7349.47i	− 0.946949i
$393$	4454.51i	0.571757i
$394$	1175.61	0.150320
$395$	0	0
$396$	142.406	0.0180712
$397$	− 424.353i	− 0.0536465i	−0.999640	−	0.0268232i	$-0.991461\pi$
$397$	− 424.353i	− 0.0536465i	0.999640	−	0.0268232i	$-0.00853912\pi$
$398$	− 6073.54i	− 0.764922i
$399$	2442.66	0.306481
$400$	0	0
$401$	−5904.18	−0.735263	−0.367632	−	0.929972i	$-0.619831\pi$
$401$	−5904.18	−0.735263	−0.367632	+	0.929972i	$0.619831\pi$
$402$	2605.22i	0.323225i
$403$	258.455i	0.0319468i
$404$	347.241	0.0427620
$405$	0	0
$406$	142.515	0.0174210
$407$	4585.83i	0.558504i
$408$	− 5998.94i	− 0.727921i
$409$	−1370.47	−0.165686	−0.0828430	−	0.996563i	$-0.526400\pi$
$409$	−1370.47	−0.165686	−0.0828430	+	0.996563i	$0.526400\pi$
$410$	0	0
$411$	2054.78	0.246606
$412$	− 1680.65i	− 0.200970i
$413$	− 3864.98i	− 0.460493i
$414$	−4280.97	−0.508208
$415$	0	0
$416$	3156.39	0.372007
$417$	2491.45i	0.292582i
$418$	3673.00i	0.429790i
$419$	−1268.73	−0.147927	−0.0739635	−	0.997261i	$-0.523565\pi$
$419$	−1268.73	−0.147927	−0.0739635	+	0.997261i	$0.523565\pi$
$420$	0	0
$421$	−12241.9	−1.41719	−0.708594	−	0.705617i	$-0.750670\pi$
$421$	−12241.9	−1.41719	−0.708594	+	0.705617i	$0.750670\pi$
$422$	− 11065.6i	− 1.27646i
$423$	− 5015.66i	− 0.576524i
$424$	4063.31	0.465405
$425$	0	0
$426$	−8612.26	−0.979496
$427$	4912.05i	0.556700i
$428$	− 3030.55i	− 0.342260i
$429$	−1621.06	−0.182437
$430$	0	0
$431$	8050.11	0.899675	0.449838	−	0.893110i	$-0.351482\pi$
$431$	8050.11	0.899675	0.449838	+	0.893110i	$0.351482\pi$
$432$	− 1361.43i	− 0.151624i
$433$	16565.7i	1.83856i	0.393600	+	0.919282i	$0.371230\pi$
$433$	16565.7i	1.83856i	−0.393600	+	0.919282i	$0.628770\pi$
$434$	84.1819	0.00931073
$435$	0	0
$436$	−710.293	−0.0780203
$437$	24205.9i	2.64971i
$438$	− 948.517i	− 0.103475i
$439$	−4705.80	−0.511607	−0.255804	−	0.966729i	$-0.582340\pi$
$439$	−4705.80	−0.511607	−0.255804	+	0.966729i	$0.582340\pi$
$440$	0	0
$441$	−2735.86	−0.295418
$442$	10407.3i	1.11997i
$443$	− 15094.0i	− 1.61882i	−0.587246	−	0.809408i	$-0.699788\pi$
$443$	− 15094.0i	− 1.61882i	0.587246	−	0.809408i	$-0.300212\pi$
$444$	−1799.04	−0.192294
$445$	0	0
$446$	9830.56	1.04370
$447$	− 3640.93i	− 0.385257i
$448$	− 3547.71i	− 0.374138i
$449$	−973.478	−0.102319	−0.0511595	−	0.998690i	$-0.516292\pi$
$449$	−973.478	−0.102319	−0.0511595	+	0.998690i	$0.516292\pi$
$450$	0	0
$451$	3283.27	0.342801
$452$	− 244.536i	− 0.0254469i
$453$	− 92.5398i	− 0.00959801i
$454$	−12817.3	−1.32499
$455$	0	0
$456$	−9454.75	−0.970963
$457$	62.6577i	0.00641358i	0.999995	+	0.00320679i	$0.00102075\pi$
$457$	62.6577i	0.00641358i	−0.999995	+	0.00320679i	$0.998979\pi$
$458$	− 710.510i	− 0.0724890i
$459$	−2233.12	−0.227088
$460$	0	0
$461$	−11866.2	−1.19884	−0.599419	−	0.800436i	$-0.704602\pi$
$461$	−11866.2	−1.19884	−0.599419	+	0.800436i	$0.704602\pi$
$462$	528.000i	0.0531705i
$463$	13144.8i	1.31942i	0.751519	+	0.659711i	$0.229321\pi$
$463$	13144.8i	1.31942i	−0.751519	+	0.659711i	$0.770679\pi$
$464$	−449.131	−0.0449361
$465$	0	0
$466$	−5814.48	−0.578006
$467$	− 10176.8i	− 1.00840i	−0.863586	−	0.504201i	$-0.831787\pi$
$467$	− 10176.8i	− 1.00840i	0.863586	−	0.504201i	$-0.168213\pi$
$468$	− 635.949i	− 0.0628136i
$469$	2117.56	0.208486
$470$	0	0
$471$	1035.72	0.101323
$472$	14960.1i	1.45889i
$473$	− 5649.96i	− 0.549229i
$474$	−2380.98	−0.230721
$475$	0	0
$476$	−743.121	−0.0715565
$477$	− 1512.58i	− 0.145191i
$478$	− 4142.30i	− 0.396369i
$479$	−3431.25	−0.327302	−0.163651	−	0.986518i	$-0.552327\pi$
$479$	−3431.25	−0.327302	−0.163651	+	0.986518i	$0.552327\pi$
$480$	0	0
$481$	20479.1	1.94130
$482$	− 14464.1i	− 1.36685i
$483$	3479.64i	0.327803i
$484$	174.052	0.0163460
$485$	0	0
$486$	−622.457	−0.0580972
$487$	− 2833.20i	− 0.263624i	−0.991275	−	0.131812i	$-0.957921\pi$
$487$	− 2833.20i	− 0.263624i	0.991275	−	0.131812i	$-0.0420795\pi$
$488$	− 19013.0i	− 1.76368i
$489$	−5764.48	−0.533085
$490$	0	0
$491$	−2667.29	−0.245159	−0.122580	−	0.992459i	$-0.539117\pi$
$491$	−2667.29	−0.245159	−0.122580	+	0.992459i	$0.539117\pi$
$492$	1288.04i	0.118027i
$493$	736.700i	0.0673008i
$494$	16402.7	1.49391
$495$	0	0
$496$	−265.295	−0.0240164
$497$	7000.18i	0.631793i
$498$	− 7847.74i	− 0.706156i
$499$	11137.0	0.999120	0.499560	−	0.866279i	$-0.333495\pi$
$499$	11137.0	0.999120	0.499560	+	0.866279i	$0.333495\pi$
$500$	0	0
$501$	516.892	0.0460939
$502$	15926.7i	1.41603i
$503$	− 8780.30i	− 0.778319i	−0.921170	−	0.389159i	$-0.872766\pi$
$503$	− 8780.30i	− 0.778319i	0.921170	−	0.389159i	$-0.127234\pi$
$504$	−1359.14	−0.120121
$505$	0	0
$506$	−5232.29	−0.459691
$507$	648.239i	0.0567836i
$508$	− 1363.91i	− 0.119121i
$509$	−13597.4	−1.18408	−0.592039	−	0.805910i	$-0.701677\pi$
$509$	−13597.4	−1.18408	−0.592039	+	0.805910i	$0.701677\pi$
$510$	0	0
$511$	−770.969	−0.0667430
$512$	12992.6i	1.12148i
$513$	3519.56i	0.302909i
$514$	19756.6	1.69538
$515$	0	0
$516$	2216.50	0.189101
$517$	− 6130.25i	− 0.521486i
$518$	− 6670.30i	− 0.565784i
$519$	3075.88	0.260146
$520$	0	0
$521$	14001.3	1.17736	0.588682	−	0.808364i	$-0.299647\pi$
$521$	14001.3	1.17736	0.588682	+	0.808364i	$0.299647\pi$
$522$	205.346i	0.0172180i
$523$	14749.8i	1.23320i	0.787275	+	0.616602i	$0.211491\pi$
$523$	14749.8i	1.23320i	−0.787275	+	0.616602i	$0.788509\pi$
$524$	−2135.86	−0.178064
$525$	0	0
$526$	−528.568	−0.0438149
$527$	435.159i	0.0359693i
$528$	− 1663.97i	− 0.137150i
$529$	−22315.0	−1.83406
$530$	0	0
$531$	5568.95	0.455126
$532$	1171.21i	0.0954483i
$533$	− 14662.2i	− 1.19154i
$534$	−1090.29	−0.0883548
$535$	0	0
$536$	−8196.40	−0.660505
$537$	4974.86i	0.399779i
$538$	4386.59i	0.351523i
$539$	−3343.83	−0.267215
$540$	0	0
$541$	1484.06	0.117939	0.0589694	−	0.998260i	$-0.481219\pi$
$541$	1484.06	0.117939	0.0589694	+	0.998260i	$0.481219\pi$
$542$	1222.07i	0.0968493i
$543$	6065.42i	0.479359i
$544$	5314.40	0.418847
$545$	0	0
$546$	2357.91	0.184815
$547$	− 16562.2i	− 1.29460i	−0.762234	−	0.647302i	$-0.775897\pi$
$547$	− 16562.2i	− 1.29460i	0.762234	−	0.647302i	$-0.224103\pi$
$548$	985.233i	0.0768012i
$549$	−7077.65	−0.550212
$550$	0	0
$551$	1161.09	0.0897716
$552$	− 13468.6i	− 1.03851i
$553$	1935.30i	0.148819i
$554$	10972.3	0.841463
$555$	0	0
$556$	−1194.61	−0.0911197
$557$	− 8821.52i	− 0.671059i	−0.942030	−	0.335529i	$-0.891085\pi$
$557$	− 8821.52i	− 0.671059i	0.942030	−	0.335529i	$-0.108915\pi$
$558$	121.295i	0.00920223i
$559$	−25231.2	−1.90906
$560$	0	0
$561$	−2729.37	−0.205409
$562$	− 8908.55i	− 0.668656i
$563$	5985.53i	0.448064i	0.974582	+	0.224032i	$0.0719220\pi$
$563$	5985.53i	0.448064i	−0.974582	+	0.224032i	$0.928078\pi$
$564$	2404.92	0.179549
$565$	0	0
$566$	16825.0	1.24948
$567$	505.943i	0.0374737i
$568$	− 27095.5i	− 2.00158i
$569$	3453.08	0.254413	0.127206	−	0.991876i	$-0.459399\pi$
$569$	3453.08	0.254413	0.127206	+	0.991876i	$0.459399\pi$
$570$	0	0
$571$	−21484.5	−1.57460	−0.787302	−	0.616568i	$-0.788523\pi$
$571$	−21484.5	−1.57460	−0.787302	+	0.616568i	$0.788523\pi$
$572$	− 777.271i	− 0.0568170i
$573$	− 4320.67i	− 0.315006i
$574$	−4775.67	−0.347269
$575$	0	0
$576$	5111.80	0.369777
$577$	− 13294.4i	− 0.959189i	−0.877490	−	0.479594i	$-0.840784\pi$
$577$	− 13294.4i	− 0.959189i	0.877490	−	0.479594i	$-0.159216\pi$
$578$	4937.82i	0.355340i
$579$	8394.14	0.602502
$580$	0	0
$581$	−6378.77	−0.455483
$582$	− 6135.01i	− 0.436949i
$583$	− 1848.71i	− 0.131330i
$584$	2984.18	0.211449
$585$	0	0
$586$	−21394.8	−1.50821
$587$	6695.73i	0.470805i	0.971898	+	0.235402i	$0.0756408\pi$
$587$	6695.73i	0.470805i	−0.971898	+	0.235402i	$0.924359\pi$
$588$	− 1311.80i	− 0.0920028i
$589$	685.841	0.0479789
$590$	0	0
$591$	1376.83	0.0958295
$592$	21021.2i	1.45940i
$593$	10239.6i	0.709088i	0.935039	+	0.354544i	$0.115364\pi$
$593$	10239.6i	0.709088i	−0.935039	+	0.354544i	$0.884636\pi$
$594$	−760.781	−0.0525509
$595$	0	0
$596$	1745.76	0.119982
$597$	− 7113.11i	− 0.487639i
$598$	23366.0i	1.59784i
$599$	23890.8	1.62963	0.814817	−	0.579719i	$-0.196838\pi$
$599$	23890.8	1.62963	0.814817	+	0.579719i	$0.196838\pi$
$600$	0	0
$601$	−11343.8	−0.769920	−0.384960	−	0.922933i	$-0.625785\pi$
$601$	−11343.8	−0.769920	−0.384960	+	0.922933i	$0.625785\pi$
$602$	8218.12i	0.556388i
$603$	3051.14i	0.206056i
$604$	44.3712	0.00298914
$605$	0	0
$606$	−1855.07	−0.124352
$607$	− 26032.5i	− 1.74074i	−0.492399	−	0.870369i	$-0.663880\pi$
$607$	− 26032.5i	− 1.74074i	0.492399	−	0.870369i	$-0.336120\pi$
$608$	− 8375.87i	− 0.558695i
$609$	166.909	0.0111059
$610$	0	0
$611$	−27376.1	−1.81263
$612$	− 1070.74i	− 0.0707226i
$613$	4568.13i	0.300987i	0.988611	+	0.150493i	$0.0480863\pi$
$613$	4568.13i	0.300987i	−0.988611	+	0.150493i	$0.951914\pi$
$614$	13791.1	0.906457
$615$	0	0
$616$	−1661.17	−0.108653
$617$	− 12755.9i	− 0.832308i	−0.909294	−	0.416154i	$-0.863378\pi$
$617$	− 12755.9i	− 0.832308i	0.909294	−	0.416154i	$-0.136622\pi$
$618$	8978.59i	0.584421i
$619$	1138.94	0.0739545	0.0369772	−	0.999316i	$-0.488227\pi$
$619$	1138.94	0.0739545	0.0369772	+	0.999316i	$0.488227\pi$
$620$	0	0
$621$	−5013.72	−0.323983
$622$	4586.24i	0.295645i
$623$	886.205i	0.0569904i
$624$	−7430.85	−0.476718
$625$	0	0
$626$	2072.49	0.132322
$627$	4301.69i	0.273992i
$628$	496.607i	0.0315554i
$629$	34480.6	2.18574
$630$	0	0
$631$	7997.36	0.504548	0.252274	−	0.967656i	$-0.418822\pi$
$631$	7997.36	0.504548	0.252274	+	0.967656i	$0.418822\pi$
$632$	− 7490.91i	− 0.471475i
$633$	− 12959.6i	− 0.813742i
$634$	27522.7	1.72408
$635$	0	0
$636$	725.255	0.0452173
$637$	14932.7i	0.928814i
$638$	250.979i	0.0155742i
$639$	−10086.4	−0.624430
$640$	0	0
$641$	−573.115	−0.0353146	−0.0176573	−	0.999844i	$-0.505621\pi$
$641$	−573.115	−0.0353146	−0.0176573	+	0.999844i	$0.505621\pi$
$642$	16190.2i	0.995290i
$643$	16027.8i	0.983009i	0.870875	+	0.491504i	$0.163553\pi$
$643$	16027.8i	0.983009i	−0.870875	+	0.491504i	$0.836447\pi$
$644$	−1668.42	−0.102089
$645$	0	0
$646$	27617.1	1.68201
$647$	− 2622.74i	− 0.159367i	−0.996820	−	0.0796837i	$-0.974609\pi$
$647$	− 2622.74i	− 0.159367i	0.996820	−	0.0796837i	$-0.0253910\pi$
$648$	− 1958.34i	− 0.118721i
$649$	6806.50	0.411677
$650$	0	0
$651$	98.5908	0.00593560
$652$	− 2763.97i	− 0.166020i
$653$	− 3102.00i	− 0.185897i	−0.995671	−	0.0929484i	$-0.970371\pi$
$653$	− 3102.00i	− 0.185897i	0.995671	−	0.0929484i	$-0.0296292\pi$
$654$	3794.62	0.226883
$655$	0	0
$656$	15050.3	0.895755
$657$	− 1110.87i	− 0.0659652i
$658$	8916.73i	0.528283i
$659$	20840.2	1.23190	0.615948	−	0.787787i	$-0.288773\pi$
$659$	20840.2	1.23190	0.615948	+	0.787787i	$0.288773\pi$
$660$	0	0
$661$	18242.9	1.07348	0.536738	−	0.843749i	$-0.319657\pi$
$661$	18242.9	1.07348	0.536738	+	0.843749i	$0.319657\pi$
$662$	− 8707.03i	− 0.511191i
$663$	12188.7i	0.713980i
$664$	24690.2	1.44302
$665$	0	0
$666$	9611.06	0.559191
$667$	1654.01i	0.0960171i
$668$	247.841i	0.0143551i
$669$	11513.2	0.665361
$670$	0	0
$671$	−8650.46	−0.497686
$672$	− 1204.05i	− 0.0691177i
$673$	12746.6i	0.730084i	0.930991	+	0.365042i	$0.118945\pi$
$673$	12746.6i	0.730084i	−0.930991	+	0.365042i	$0.881055\pi$
$674$	−30328.7	−1.73326
$675$	0	0
$676$	−310.819	−0.0176843
$677$	− 7683.11i	− 0.436168i	−0.975930	−	0.218084i	$-0.930019\pi$
$677$	− 7683.11i	− 0.436168i	0.975930	−	0.218084i	$-0.0699807\pi$
$678$	1306.39i	0.0739995i
$679$	−4986.63	−0.281840
$680$	0	0
$681$	−15011.1	−0.844681
$682$	148.250i	0.00832373i
$683$	21397.1i	1.19874i	0.800473	+	0.599368i	$0.204582\pi$
$683$	21397.1i	1.19874i	−0.800473	+	0.599368i	$0.795418\pi$
$684$	−1687.57	−0.0943359
$685$	0	0
$686$	10351.8	0.576140
$687$	− 832.124i	− 0.0462118i
$688$	− 25899.0i	− 1.43516i
$689$	−8255.84	−0.456491
$690$	0	0
$691$	26137.5	1.43895	0.719477	−	0.694516i	$-0.244381\pi$
$691$	26137.5	1.43895	0.719477	+	0.694516i	$0.244381\pi$
$692$	1474.83i	0.0810181i
$693$	618.375i	0.0338963i
$694$	−5319.50	−0.290959
$695$	0	0
$696$	−646.051	−0.0351846
$697$	− 24686.7i	− 1.34157i
$698$	14952.7i	0.810845i
$699$	−6809.72	−0.368479
$700$	0	0
$701$	13382.4	0.721036	0.360518	−	0.932752i	$-0.382600\pi$
$701$	13382.4	0.721036	0.360518	+	0.932752i	$0.382600\pi$
$702$	3397.45i	0.182662i
$703$	− 54343.9i	− 2.91553i
$704$	6247.76	0.334476
$705$	0	0
$706$	6208.27	0.330951
$707$	1507.83i	0.0802092i
$708$	2670.22i	0.141741i
$709$	−18164.6	−0.962179	−0.481090	−	0.876671i	$-0.659759\pi$
$709$	−18164.6	−0.962179	−0.481090	+	0.876671i	$0.659759\pi$
$710$	0	0
$711$	−2788.52	−0.147085
$712$	− 3430.21i	− 0.180552i
$713$	977.000i	0.0513169i
$714$	3970.00	0.208086
$715$	0	0
$716$	−2385.36	−0.124504
$717$	− 4851.32i	− 0.252686i
$718$	− 9944.50i	− 0.516888i
$719$	−9665.62	−0.501344	−0.250672	−	0.968072i	$-0.580652\pi$
$719$	−9665.62	−0.501344	−0.250672	+	0.968072i	$0.580652\pi$
$720$	0	0
$721$	7297.94	0.376962
$722$	− 25956.7i	− 1.33796i
$723$	− 16939.9i	− 0.871371i
$724$	−2908.26	−0.149288
$725$	0	0
$726$	−929.844	−0.0475341
$727$	− 29779.6i	− 1.51921i	−0.650385	−	0.759605i	$-0.725392\pi$
$727$	− 29779.6i	− 1.51921i	0.650385	−	0.759605i	$-0.274608\pi$
$728$	7418.33i	0.377667i
$729$	−729.000	−0.0370370
$730$	0	0
$731$	−42481.7	−2.14944
$732$	− 3393.61i	− 0.171354i
$733$	− 35029.5i	− 1.76513i	−0.470187	−	0.882567i	$-0.655814\pi$
$733$	− 35029.5i	− 1.76513i	0.470187	−	0.882567i	$-0.344186\pi$
$734$	14515.4	0.729937
$735$	0	0
$736$	11931.7	0.597564
$737$	3729.17i	0.186385i
$738$	− 6881.13i	− 0.343222i
$739$	−23297.3	−1.15968	−0.579842	−	0.814729i	$-0.696886\pi$
$739$	−23297.3	−1.15968	−0.579842	+	0.814729i	$0.696886\pi$
$740$	0	0
$741$	19210.2	0.952368
$742$	2689.03i	0.133042i
$743$	− 21570.4i	− 1.06506i	−0.846411	−	0.532530i	$-0.821241\pi$
$743$	− 21570.4i	− 1.06506i	0.846411	−	0.532530i	$-0.178759\pi$
$744$	−381.613	−0.0188046
$745$	0	0
$746$	−447.684	−0.0219717
$747$	− 9190.99i	− 0.450175i
$748$	− 1308.69i	− 0.0639710i
$749$	13159.6	0.641980
$750$	0	0
$751$	28554.8	1.38746	0.693729	−	0.720236i	$-0.255966\pi$
$751$	28554.8	1.38746	0.693729	+	0.720236i	$0.255966\pi$
$752$	− 28100.7i	− 1.36267i
$753$	18652.8i	0.902719i
$754$	1120.81	0.0541344
$755$	0	0
$756$	−242.591	−0.0116706
$757$	7812.81i	0.375114i	0.982254	+	0.187557i	$0.0600570\pi$
$757$	7812.81i	0.375114i	−0.982254	+	0.187557i	$0.939943\pi$
$758$	647.268i	0.0310156i
$759$	−6127.87	−0.293054
$760$	0	0
$761$	2875.13	0.136956	0.0684778	−	0.997653i	$-0.478186\pi$
$761$	2875.13	0.136956	0.0684778	+	0.997653i	$0.478186\pi$
$762$	7286.45i	0.346405i
$763$	− 3084.33i	− 0.146344i
$764$	2071.69	0.0981033
$765$	0	0
$766$	28214.3	1.33084
$767$	− 30396.0i	− 1.43095i
$768$	6401.22i	0.300761i
$769$	27657.7	1.29696	0.648479	−	0.761233i	$-0.275406\pi$
$769$	27657.7	1.29696	0.648479	+	0.761233i	$0.275406\pi$
$770$	0	0
$771$	23138.2	1.08081
$772$	4024.84i	0.187639i
$773$	3929.35i	0.182832i	0.995813	+	0.0914160i	$0.0291393\pi$
$773$	3929.35i	0.182832i	−0.995813	+	0.0914160i	$0.970861\pi$
$774$	−11841.3	−0.549904
$775$	0	0
$776$	19301.6	0.892898
$777$	− 7812.02i	− 0.360688i
$778$	20747.0i	0.956064i
$779$	−38908.0	−1.78950
$780$	0	0
$781$	−12327.8	−0.564818
$782$	39341.3i	1.79903i
$783$	240.495i	0.0109765i
$784$	−15327.9	−0.698247
$785$	0	0
$786$	11410.5	0.517809
$787$	− 21125.7i	− 0.956860i	−0.878126	−	0.478430i	$-0.841206\pi$
$787$	− 21125.7i	− 0.956860i	0.878126	−	0.478430i	$-0.158794\pi$
$788$	660.166i	0.0298445i
$789$	−619.040	−0.0279321
$790$	0	0
$791$	1061.86	0.0477310
$792$	− 2393.53i	− 0.107387i
$793$	38630.7i	1.72991i
$794$	−1087.00	−0.0485847
$795$	0	0
$796$	3410.61	0.151867
$797$	11696.3i	0.519828i	0.965632	+	0.259914i	$0.0836942\pi$
$797$	11696.3i	0.519828i	−0.965632	+	0.259914i	$0.916306\pi$
$798$	− 6257.00i	− 0.277563i
$799$	−46093.0	−2.04087
$800$	0	0
$801$	−1276.91	−0.0563263
$802$	15123.9i	0.665888i
$803$	− 1357.73i	− 0.0596678i
$804$	−1462.97	−0.0641727
$805$	0	0
$806$	662.045	0.0289324
$807$	5137.42i	0.224096i
$808$	− 5836.34i	− 0.254111i
$809$	−14310.2	−0.621902	−0.310951	−	0.950426i	$-0.600648\pi$
$809$	−14310.2	−0.621902	−0.310951	+	0.950426i	$0.600648\pi$
$810$	0	0
$811$	21697.9	0.939477	0.469739	−	0.882806i	$-0.344348\pi$
$811$	21697.9	0.939477	0.469739	+	0.882806i	$0.344348\pi$
$812$	80.0299i	0.00345874i
$813$	1431.24i	0.0617416i
$814$	11746.9	0.505807
$815$	0	0
$816$	−12511.3	−0.536743
$817$	66954.1i	2.86711i
$818$	3510.54i	0.150053i
$819$	2761.50	0.117820
$820$	0	0
$821$	3613.00	0.153587	0.0767934	−	0.997047i	$-0.475532\pi$
$821$	3613.00	0.153587	0.0767934	+	0.997047i	$0.475532\pi$
$822$	− 5263.44i	− 0.223338i
$823$	4763.98i	0.201776i	0.994898	+	0.100888i	$0.0321684\pi$
$823$	4763.98i	0.201776i	−0.994898	+	0.100888i	$0.967832\pi$
$824$	−28248.0	−1.19425
$825$	0	0
$826$	−9900.36	−0.417043
$827$	− 33571.7i	− 1.41161i	−0.708405	−	0.705806i	$-0.750585\pi$
$827$	− 33571.7i	− 1.41161i	0.708405	−	0.705806i	$-0.249415\pi$
$828$	− 2403.99i	− 0.100899i
$829$	−17980.5	−0.753303	−0.376652	−	0.926355i	$-0.622925\pi$
$829$	−17980.5	−0.753303	−0.376652	+	0.926355i	$0.622925\pi$
$830$	0	0
$831$	12850.4	0.536434
$832$	− 27900.9i	− 1.16261i
$833$	25142.1i	1.04576i
$834$	6381.98	0.264976
$835$	0	0
$836$	−2062.58	−0.0853301
$837$	142.057i	0.00586643i
$838$	3249.91i	0.133969i
$839$	40139.5	1.65169	0.825847	−	0.563895i	$-0.190698\pi$
$839$	40139.5	1.65169	0.825847	+	0.563895i	$0.190698\pi$
$840$	0	0
$841$	−24309.7	−0.996747
$842$	31358.4i	1.28347i
$843$	− 10433.4i	− 0.426269i
$844$	6213.91	0.253426
$845$	0	0
$846$	−12847.9	−0.522127
$847$	755.792i	0.0306603i
$848$	− 8474.36i	− 0.343173i
$849$	19704.8	0.796546
$850$	0	0
$851$	77414.4	3.11837
$852$	− 4836.24i	− 0.194468i
$853$	15369.2i	0.616919i	0.951237	+	0.308459i	$0.0998134\pi$
$853$	15369.2i	0.616919i	−0.951237	+	0.308459i	$0.900187\pi$
$854$	12582.5	0.504173
$855$	0	0
$856$	−50936.8	−2.03386
$857$	10324.9i	0.411541i	0.978600	+	0.205770i	$0.0659700\pi$
$857$	10324.9i	0.411541i	−0.978600	+	0.205770i	$0.934030\pi$
$858$	4152.44i	0.165224i
$859$	27112.5	1.07691	0.538455	−	0.842655i	$-0.319008\pi$
$859$	27112.5	1.07691	0.538455	+	0.842655i	$0.319008\pi$
$860$	0	0
$861$	−5593.09	−0.221384
$862$	− 20620.8i	− 0.814787i
$863$	− 30463.6i	− 1.20161i	−0.799394	−	0.600807i	$-0.794846\pi$
$863$	− 30463.6i	− 1.20161i	0.799394	−	0.600807i	$-0.205154\pi$
$864$	1734.88	0.0683122
$865$	0	0
$866$	42434.0	1.66509
$867$	5783.00i	0.226529i
$868$	47.2726i	0.00184854i
$869$	−3408.19	−0.133044
$870$	0	0
$871$	16653.5	0.647855
$872$	11938.4i	0.463631i
$873$	− 7185.10i	− 0.278555i
$874$	62004.6	2.39970
$875$	0	0
$876$	532.642	0.0205437
$877$	− 5086.12i	− 0.195833i	−0.995195	−	0.0979167i	$-0.968782\pi$
$877$	− 5086.12i	− 0.195833i	0.995195	−	0.0979167i	$-0.0312179\pi$
$878$	12054.2i	0.463335i
$879$	−25056.9	−0.961487
$880$	0	0
$881$	−10625.5	−0.406338	−0.203169	−	0.979144i	$-0.565124\pi$
$881$	−10625.5	−0.406338	−0.203169	+	0.979144i	$0.565124\pi$
$882$	7008.06i	0.267544i
$883$	− 13112.2i	− 0.499728i	−0.968281	−	0.249864i	$-0.919614\pi$
$883$	− 13112.2i	− 0.499728i	0.968281	−	0.249864i	$-0.0803859\pi$
$884$	−5844.25	−0.222357
$885$	0	0
$886$	−38664.0	−1.46607
$887$	− 14442.8i	− 0.546719i	−0.961912	−	0.273360i	$-0.911865\pi$
$887$	− 14442.8i	− 0.546719i	0.961912	−	0.273360i	$-0.0881349\pi$
$888$	30237.8i	1.14270i
$889$	5922.54	0.223437
$890$	0	0
$891$	−891.000	−0.0335013
$892$	5520.38i	0.207215i
$893$	72645.8i	2.72228i
$894$	−9326.43	−0.348906
$895$	0	0
$896$	−5876.86	−0.219121
$897$	27365.5i	1.01863i
$898$	2493.61i	0.0926648i
$899$	46.8641	0.00173860
$900$	0	0
$901$	−13900.3	−0.513970
$902$	− 8410.27i	− 0.310456i
$903$	9624.77i	0.354698i
$904$	−4110.10	−0.151217
$905$	0	0
$906$	−237.045	−0.00869239
$907$	− 44981.9i	− 1.64675i	−0.567499	−	0.823374i	$-0.692089\pi$
$907$	− 44981.9i	− 1.64675i	0.567499	−	0.823374i	$-0.307911\pi$
$908$	− 7197.57i	− 0.263062i
$909$	−2172.60	−0.0792745
$910$	0	0
$911$	6841.96	0.248830	0.124415	−	0.992230i	$-0.460295\pi$
$911$	6841.96	0.248830	0.124415	+	0.992230i	$0.460295\pi$
$912$	19718.7i	0.715954i
$913$	− 11233.4i	− 0.407199i
$914$	160.501	0.00580843
$915$	0	0
$916$	398.989	0.0143919
$917$	− 9274.61i	− 0.333996i
$918$	5720.27i	0.205661i
$919$	4753.54	0.170625	0.0853127	−	0.996354i	$-0.472811\pi$
$919$	4753.54	0.170625	0.0853127	+	0.996354i	$0.472811\pi$
$920$	0	0
$921$	16151.7	0.577868
$922$	30395.9i	1.08572i
$923$	55052.7i	1.96325i
$924$	−296.500	−0.0105564
$925$	0	0
$926$	33671.2	1.19493
$927$	10515.4i	0.372569i
$928$	− 572.330i	− 0.0202453i
$929$	7507.93	0.265153	0.132576	−	0.991173i	$-0.457675\pi$
$929$	7507.93	0.265153	0.132576	+	0.991173i	$0.457675\pi$
$930$	0	0
$931$	39625.7	1.39493
$932$	− 3265.14i	− 0.114757i
$933$	5371.24i	0.188474i
$934$	−26068.3	−0.913256
$935$	0	0
$936$	−10688.9	−0.373266
$937$	8540.47i	0.297764i	0.988855	+	0.148882i	$0.0475675\pi$
$937$	8540.47i	0.297764i	−0.988855	+	0.148882i	$0.952433\pi$
$938$	− 5424.24i	− 0.188814i
$939$	2427.23	0.0843552
$940$	0	0
$941$	9101.13	0.315290	0.157645	−	0.987496i	$-0.449610\pi$
$941$	9101.13	0.315290	0.157645	+	0.987496i	$0.449610\pi$
$942$	− 2653.04i	− 0.0917630i
$943$	− 55425.5i	− 1.91400i
$944$	31200.6	1.07573
$945$	0	0
$946$	−14472.7	−0.497407
$947$	47540.0i	1.63130i	0.578546	+	0.815650i	$0.303620\pi$
$947$	47540.0i	1.63130i	−0.578546	+	0.815650i	$0.696380\pi$
$948$	− 1337.04i	− 0.0458072i
$949$	−6063.26	−0.207399
$950$	0	0
$951$	32233.6	1.09910
$952$	12490.2i	0.425221i
$953$	− 47370.7i	− 1.61016i	−0.593164	−	0.805082i	$-0.702121\pi$
$953$	− 47370.7i	− 1.61016i	0.593164	−	0.805082i	$-0.297879\pi$
$954$	−3874.55	−0.131492
$955$	0	0
$956$	2326.12	0.0786947
$957$	293.938i	0.00992859i
$958$	8789.33i	0.296420i
$959$	−4278.20	−0.144057
$960$	0	0
$961$	−29763.3	−0.999071
$962$	− 52458.4i	− 1.75813i
$963$	18961.4i	0.634498i
$964$	8122.38	0.271374
$965$	0	0
$966$	8913.27	0.296874
$967$	− 36171.6i	− 1.20290i	−0.798912	−	0.601448i	$-0.794591\pi$
$967$	− 36171.6i	− 1.20290i	0.798912	−	0.601448i	$-0.205409\pi$
$968$	− 2925.43i	− 0.0971351i
$969$	32344.1	1.07228
$970$	0	0
$971$	−31713.2	−1.04812	−0.524060	−	0.851681i	$-0.675583\pi$
$971$	−31713.2	−1.04812	−0.524060	+	0.851681i	$0.675583\pi$
$972$	− 349.543i	− 0.0115346i
$973$	− 5187.37i	− 0.170914i
$974$	−7257.40	−0.238750
$975$	0	0
$976$	−39653.1	−1.30048
$977$	22800.5i	0.746626i	0.927706	+	0.373313i	$0.121778\pi$
$977$	22800.5i	0.746626i	−0.927706	+	0.373313i	$0.878222\pi$
$978$	14766.0i	0.482786i
$979$	−1560.67	−0.0509490
$980$	0	0
$981$	4444.12	0.144638
$982$	6832.41i	0.222027i
$983$	44597.4i	1.44704i	0.690305	+	0.723518i	$0.257476\pi$
$983$	44597.4i	1.44704i	−0.690305	+	0.723518i	$0.742524\pi$
$984$	21649.1	0.701369
$985$	0	0
$986$	1887.10	0.0609507
$987$	10443.0i	0.336781i
$988$	9210.95i	0.296599i
$989$	−95378.1	−3.06658
$990$	0	0
$991$	−34788.1	−1.11512	−0.557558	−	0.830138i	$-0.688261\pi$
$991$	−34788.1	−1.11512	−0.557558	+	0.830138i	$0.688261\pi$
$992$	− 338.068i	− 0.0108202i
$993$	− 10197.4i	− 0.325885i
$994$	17931.3	0.572180
$995$	0	0
$996$	4406.92	0.140199
$997$	− 20360.5i	− 0.646765i	−0.946268	−	0.323383i	$-0.895180\pi$
$997$	− 20360.5i	− 0.646765i	0.946268	−	0.323383i	$-0.104820\pi$
$998$	− 28528.0i	− 0.904849i
$999$	11256.1	0.356485

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.4.c.j.199.1		4
5.2	odd	4		825.4.a.m.1.2		2
5.3	odd	4		165.4.a.c.1.1	✓	2
5.4	even	2	inner	825.4.c.j.199.4		4
15.2	even	4		2475.4.a.n.1.1		2
15.8	even	4		495.4.a.d.1.2		2
55.43	even	4		1815.4.a.n.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.4.a.c.1.1	✓	2	5.3	odd	4
495.4.a.d.1.2		2	15.8	even	4
825.4.a.m.1.2		2	5.2	odd	4
825.4.c.j.199.1		4	1.1	even	1	trivial
825.4.c.j.199.4		4	5.4	even	2	inner
1815.4.a.n.1.2		2	55.43	even	4
2475.4.a.n.1.1		2	15.2	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)

\(f(q)\)	\(=\)	\(q-2.56155i q^{2} -3.00000i q^{3} +1.43845 q^{4} -7.68466 q^{6} +6.24621i q^{7} -24.1771i q^{8} -9.00000 q^{9} +O(q^{10})\)	\(q-2.56155i q^{2} -3.00000i q^{3} +1.43845 q^{4} -7.68466 q^{6} +6.24621i q^{7} -24.1771i q^{8} -9.00000 q^{9} -11.0000 q^{11} -4.31534i q^{12} +49.1231i q^{13} +16.0000 q^{14} -50.4233 q^{16} +82.7083i q^{17} +23.0540i q^{18} +130.354 q^{19} +18.7386 q^{21} +28.1771i q^{22} +185.693i q^{23} -72.5312 q^{24} +125.831 q^{26} +27.0000i q^{27} +8.98485i q^{28} +8.90720 q^{29} +5.26137 q^{31} -64.2547i q^{32} +33.0000i q^{33} +211.862 q^{34} -12.9460 q^{36} -416.894i q^{37} -333.909i q^{38} +147.369 q^{39} -298.479 q^{41} -48.0000i q^{42} +513.633i q^{43} -15.8229 q^{44} +475.663 q^{46} +557.295i q^{47} +151.270i q^{48} +303.985 q^{49} +248.125 q^{51} +70.6610i q^{52} +168.064i q^{53} +69.1619 q^{54} +151.015 q^{56} -391.062i q^{57} -22.8163i q^{58} -618.773 q^{59} +786.405 q^{61} -13.4773i q^{62} -56.2159i q^{63} -567.978 q^{64} +84.5312 q^{66} -339.015i q^{67} +118.972i q^{68} +557.080 q^{69} +1120.71 q^{71} +217.594i q^{72} +123.430i q^{73} -1067.90 q^{74} +187.508 q^{76} -68.7083i q^{77} -377.494i q^{78} +309.835 q^{79} +81.0000 q^{81} +764.570i q^{82} +1021.22i q^{83} +26.9545 q^{84} +1315.70 q^{86} -26.7216i q^{87} +265.948i q^{88} +141.879 q^{89} -306.833 q^{91} +267.110i q^{92} -15.7841i q^{93} +1427.54 q^{94} -192.764 q^{96} +798.345i q^{97} -778.673i q^{98} +99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 14 q^{4} - 6 q^{6} - 36 q^{9}+O(q^{10}) \) 4 * q + 14 * q^4 - 6 * q^6 - 36 * q^9	\( 4 q + 14 q^{4} - 6 q^{6} - 36 q^{9} - 44 q^{11} + 64 q^{14} - 78 q^{16} + 340 q^{19} - 24 q^{21} - 18 q^{24} + 124 q^{26} + 316 q^{29} + 120 q^{31} + 732 q^{34} - 126 q^{36} + 540 q^{39} + 76 q^{41} - 154 q^{44} + 1144 q^{46} + 1084 q^{49} - 96 q^{51} + 54 q^{54} + 736 q^{56} - 496 q^{59} + 144 q^{61} - 1538 q^{64} + 66 q^{66} + 744 q^{69} + 4120 q^{71} - 2276 q^{74} + 816 q^{76} - 1284 q^{79} + 324 q^{81} - 288 q^{84} + 2624 q^{86} - 488 q^{89} + 224 q^{91} + 3896 q^{94} + 738 q^{96} + 396 q^{99}+O(q^{100}) \) 4 * q + 14 * q^4 - 6 * q^6 - 36 * q^9 - 44 * q^11 + 64 * q^14 - 78 * q^16 + 340 * q^19 - 24 * q^21 - 18 * q^24 + 124 * q^26 + 316 * q^29 + 120 * q^31 + 732 * q^34 - 126 * q^36 + 540 * q^39 + 76 * q^41 - 154 * q^44 + 1144 * q^46 + 1084 * q^49 - 96 * q^51 + 54 * q^54 + 736 * q^56 - 496 * q^59 + 144 * q^61 - 1538 * q^64 + 66 * q^66 + 744 * q^69 + 4120 * q^71 - 2276 * q^74 + 816 * q^76 - 1284 * q^79 + 324 * q^81 - 288 * q^84 + 2624 * q^86 - 488 * q^89 + 224 * q^91 + 3896 * q^94 + 738 * q^96 + 396 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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