LMFDB - Embedded newform 825.4.c.i.199.3

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:	$4$
Coefficient field:	$\Q(i, \sqrt{33})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 17x^{2} + 64 $ x^4 + 17*x^2 + 64
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 33)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.3
Root			$2.37228i$ of defining polynomial
Character	$\chi$	$=$	825.199
Dual form			825.4.c.i.199.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.37228i q^{2} +3.00000i q^{3} +2.37228 q^{4} -7.11684 q^{6} -6.74456i q^{7} +24.6060i q^{8} -9.00000 q^{9} +11.0000 q^{11} +7.11684i q^{12} -60.9783i q^{13} +16.0000 q^{14} -39.3940 q^{16} +99.1684i q^{17} -21.3505i q^{18} -24.7011 q^{19} +20.2337 q^{21} +26.0951i q^{22} +112.000i q^{23} -73.8179 q^{24} +144.658 q^{26} -27.0000i q^{27} -16.0000i q^{28} +21.1249 q^{29} -318.717 q^{31} +103.394i q^{32} +33.0000i q^{33} -235.255 q^{34} -21.3505 q^{36} +150.380i q^{37} -58.5979i q^{38} +182.935 q^{39} -252.745 q^{41} +48.0000i q^{42} +214.016i q^{43} +26.0951 q^{44} -265.696 q^{46} -105.870i q^{47} -118.182i q^{48} +297.511 q^{49} -297.505 q^{51} -144.658i q^{52} +325.652i q^{53} +64.0516 q^{54} +165.957 q^{56} -74.1032i q^{57} +50.1143i q^{58} -196.000 q^{59} -402.641 q^{61} -756.087i q^{62} +60.7011i q^{63} -560.432 q^{64} -78.2853 q^{66} -27.4132i q^{67} +235.255i q^{68} -336.000 q^{69} -300.500 q^{71} -221.454i q^{72} +427.815i q^{73} -356.745 q^{74} -58.5979 q^{76} -74.1902i q^{77} +433.973i q^{78} -97.5488 q^{79} +81.0000 q^{81} -599.581i q^{82} +1104.62i q^{83} +48.0000 q^{84} -507.707 q^{86} +63.3748i q^{87} +270.666i q^{88} -463.022 q^{89} -411.272 q^{91} +265.696i q^{92} -956.152i q^{93} +251.152 q^{94} -310.182 q^{96} +1798.36i q^{97} +705.779i q^{98} -99.0000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{4} + 6 q^{6} - 36 q^{9} + 44 q^{11} + 64 q^{14} - 238 q^{16} + 108 q^{19} + 12 q^{21} - 54 q^{24} + 188 q^{26} - 444 q^{29} - 80 q^{31} - 964 q^{34} + 18 q^{36} + 456 q^{39} - 988 q^{41} - 22 q^{44}+ \cdots - 396 q^{99}+O(q^{100}) $ 4 * q - 2 * q^4 + 6 * q^6 - 36 * q^9 + 44 * q^11 + 64 * q^14 - 238 * q^16 + 108 * q^19 + 12 * q^21 - 54 * q^24 + 188 * q^26 - 444 * q^29 - 80 * q^31 - 964 * q^34 + 18 * q^36 + 456 * q^39 - 988 * q^41 - 22 * q^44 + 224 * q^46 + 1236 * q^49 - 156 * q^51 - 54 * q^54 + 480 * q^56 - 784 * q^59 - 2208 * q^61 - 1426 * q^64 + 66 * q^66 - 1344 * q^69 + 912 * q^71 - 1404 * q^74 - 648 * q^76 + 460 * q^79 + 324 * q^81 + 192 * q^84 - 2904 * q^86 - 1944 * q^89 - 680 * q^91 + 1648 * q^94 - 1482 * q^96 - 396 * 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 $

$2$	2.37228i	0.838728i	0.907818	+	0.419364i	$0.137747\pi$
$2$	2.37228i	0.838728i	−0.907818	+	0.419364i	$0.862253\pi$
$3$	3.00000i	0.577350i
$3$	3.00000i	0.577350i
$4$	2.37228	0.296535
$5$	0	0
$5$	0	0
$6$	−7.11684	−0.484240
$7$	− 6.74456i	− 0.364172i	−0.983283	−	0.182086i	$-0.941715\pi$
$7$	− 6.74456i	− 0.364172i	0.983283	−	0.182086i	$-0.0582850\pi$
$8$	24.6060i	1.08744i
$9$	−9.00000	−0.333333
$10$	0	0
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	7.11684i	0.171205i
$13$	− 60.9783i	− 1.30095i	−0.759529	−	0.650474i	$-0.774570\pi$
$13$	− 60.9783i	− 1.30095i	0.759529	−	0.650474i	$-0.225430\pi$
$14$	16.0000	0.305441
$15$	0	0
$16$	−39.3940	−0.615532
$17$	99.1684i	1.41482i	0.706805	+	0.707408i	$0.250136\pi$
$17$	99.1684i	1.41482i	−0.706805	+	0.707408i	$0.749864\pi$
$18$	− 21.3505i	− 0.279576i
$19$	−24.7011	−0.298253	−0.149127	−	0.988818i	$-0.547646\pi$
$19$	−24.7011	−0.298253	−0.149127	+	0.988818i	$0.547646\pi$
$20$	0	0
$21$	20.2337	0.210255
$22$	26.0951i	0.252886i
$23$	112.000i	1.01537i	0.861541	+	0.507687i	$0.169499\pi$
$23$	112.000i	1.01537i	−0.861541	+	0.507687i	$0.830501\pi$
$24$	−73.8179	−0.627834
$25$	0	0
$26$	144.658	1.09114
$27$	− 27.0000i	− 0.192450i
$28$	− 16.0000i	− 0.107990i
$29$	21.1249	0.135269	0.0676345	−	0.997710i	$-0.478455\pi$
$29$	21.1249	0.135269	0.0676345	+	0.997710i	$0.478455\pi$
$30$	0	0
$31$	−318.717	−1.84656	−0.923279	−	0.384130i	$-0.874502\pi$
$31$	−318.717	−1.84656	−0.923279	+	0.384130i	$0.874502\pi$
$32$	103.394i	0.571177i
$33$	33.0000i	0.174078i
$34$	−235.255	−1.18665
$35$	0	0
$36$	−21.3505	−0.0988451
$37$	150.380i	0.668172i	0.942543	+	0.334086i	$0.108428\pi$
$37$	150.380i	0.668172i	−0.942543	+	0.334086i	$0.891572\pi$
$38$	− 58.5979i	− 0.250153i
$39$	182.935	0.751103
$40$	0	0
$41$	−252.745	−0.962733	−0.481367	−	0.876519i	$-0.659859\pi$
$41$	−252.745	−0.962733	−0.481367	+	0.876519i	$0.659859\pi$
$42$	48.0000i	0.176347i
$43$	214.016i	0.759004i	0.925191	+	0.379502i	$0.123905\pi$
$43$	214.016i	0.759004i	−0.925191	+	0.379502i	$0.876095\pi$
$44$	26.0951	0.0894087
$45$	0	0
$46$	−265.696	−0.851623
$47$	− 105.870i	− 0.328567i	−0.986413	−	0.164284i	$-0.947469\pi$
$47$	− 105.870i	− 0.328567i	0.986413	−	0.164284i	$-0.0525312\pi$
$48$	− 118.182i	− 0.355377i
$49$	297.511	0.867379
$50$	0	0
$51$	−297.505	−0.816845
$52$	− 144.658i	− 0.385777i
$53$	325.652i	0.843995i	0.906597	+	0.421998i	$0.138671\pi$
$53$	325.652i	0.843995i	−0.906597	+	0.421998i	$0.861329\pi$
$54$	64.0516	0.161413
$55$	0	0
$56$	165.957	0.396016
$57$	− 74.1032i	− 0.172197i
$58$	50.1143i	0.113454i
$59$	−196.000	−0.432492	−0.216246	−	0.976339i	$-0.569381\pi$
$59$	−196.000	−0.432492	−0.216246	+	0.976339i	$0.569381\pi$
$60$	0	0
$61$	−402.641	−0.845130	−0.422565	−	0.906333i	$-0.638870\pi$
$61$	−402.641	−0.845130	−0.422565	+	0.906333i	$0.638870\pi$
$62$	− 756.087i	− 1.54876i
$63$	60.7011i	0.121391i
$64$	−560.432	−1.09459
$65$	0	0
$66$	−78.2853	−0.146004
$67$	− 27.4132i	− 0.0499860i	−0.999688	−	0.0249930i	$-0.992044\pi$
$67$	− 27.4132i	− 0.0499860i	0.999688	−	0.0249930i	$-0.00795634\pi$
$68$	235.255i	0.419543i
$69$	−336.000	−0.586227
$70$	0	0
$71$	−300.500	−0.502292	−0.251146	−	0.967949i	$-0.580807\pi$
$71$	−300.500	−0.502292	−0.251146	+	0.967949i	$0.580807\pi$
$72$	− 221.454i	− 0.362480i
$73$	427.815i	0.685917i	0.939351	+	0.342959i	$0.111429\pi$
$73$	427.815i	0.685917i	−0.939351	+	0.342959i	$0.888571\pi$
$74$	−356.745	−0.560415
$75$	0	0
$76$	−58.5979	−0.0884426
$77$	− 74.1902i	− 0.109802i
$78$	433.973i	0.629971i
$79$	−97.5488	−0.138925	−0.0694627	−	0.997585i	$-0.522128\pi$
$79$	−97.5488	−0.138925	−0.0694627	+	0.997585i	$0.522128\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	− 599.581i	− 0.807472i
$83$	1104.62i	1.46082i	0.683011	+	0.730408i	$0.260670\pi$
$83$	1104.62i	1.46082i	−0.683011	+	0.730408i	$0.739330\pi$
$84$	48.0000	0.0623480
$85$	0	0
$86$	−507.707	−0.636598
$87$	63.3748i	0.0780976i
$88$	270.666i	0.327876i
$89$	−463.022	−0.551463	−0.275732	−	0.961235i	$-0.588920\pi$
$89$	−463.022	−0.551463	−0.275732	+	0.961235i	$0.588920\pi$
$90$	0	0
$91$	−411.272	−0.473769
$92$	265.696i	0.301094i
$93$	− 956.152i	− 1.06611i
$94$	251.152	0.275578
$95$	0	0
$96$	−310.182	−0.329769
$97$	1798.36i	1.88243i	0.337810	+	0.941214i	$0.390314\pi$
$97$	1798.36i	1.88243i	−0.337810	+	0.941214i	$0.609686\pi$
$98$	705.779i	0.727495i
$99$	−99.0000	−0.100504
$100$	0	0
$101$	741.918	0.730927	0.365463	−	0.930826i	$-0.380911\pi$
$101$	741.918	0.730927	0.365463	+	0.930826i	$0.380911\pi$
$102$	− 705.766i	− 0.685111i
$103$	389.837i	0.372930i	0.982462	+	0.186465i	$0.0597031\pi$
$103$	389.837i	0.372930i	−0.982462	+	0.186465i	$0.940297\pi$
$104$	1500.43	1.41470
$105$	0	0
$106$	−772.538	−0.707882
$107$	− 1538.42i	− 1.38995i	−0.719032	−	0.694977i	$-0.755415\pi$
$107$	− 1538.42i	− 1.38995i	0.719032	−	0.694977i	$-0.244585\pi$
$108$	− 64.0516i	− 0.0570682i
$109$	−779.783	−0.685226	−0.342613	−	0.939477i	$-0.611312\pi$
$109$	−779.783	−0.685226	−0.342613	+	0.939477i	$0.611312\pi$
$110$	0	0
$111$	−451.141	−0.385770
$112$	265.696i	0.224160i
$113$	− 1514.55i	− 1.26086i	−0.776246	−	0.630430i	$-0.782878\pi$
$113$	− 1514.55i	− 1.26086i	0.776246	−	0.630430i	$-0.217122\pi$
$114$	175.794	0.144426
$115$	0	0
$116$	50.1143	0.0401120
$117$	548.804i	0.433649i
$118$	− 464.967i	− 0.362743i
$119$	668.848	0.515237
$120$	0	0
$121$	121.000	0.0909091
$122$	− 955.179i	− 0.708834i
$123$	− 758.234i	− 0.555834i
$124$	−756.087	−0.547569
$125$	0	0
$126$	−144.000	−0.101814
$127$	− 2302.74i	− 1.60894i	−0.593992	−	0.804471i	$-0.702449\pi$
$127$	− 2302.74i	− 1.60894i	0.593992	−	0.804471i	$-0.297551\pi$
$128$	− 502.350i	− 0.346890i
$129$	−642.049	−0.438211
$130$	0	0
$131$	−2020.59	−1.34763	−0.673815	−	0.738900i	$-0.735345\pi$
$131$	−2020.59	−1.34763	−0.673815	+	0.738900i	$0.735345\pi$
$132$	78.2853i	0.0516201i
$133$	166.598i	0.108616i
$134$	65.0319	0.0419246
$135$	0	0
$136$	−2440.14	−1.53853
$137$	− 1475.40i	− 0.920088i	−0.887896	−	0.460044i	$-0.847834\pi$
$137$	− 1475.40i	− 0.920088i	0.887896	−	0.460044i	$-0.152166\pi$
$138$	− 797.087i	− 0.491685i
$139$	1623.71	0.990802	0.495401	−	0.868665i	$-0.335021\pi$
$139$	1623.71	0.990802	0.495401	+	0.868665i	$0.335021\pi$
$140$	0	0
$141$	317.609	0.189698
$142$	− 712.870i	− 0.421287i
$143$	− 670.761i	− 0.392251i
$144$	354.546	0.205177
$145$	0	0
$146$	−1014.90	−0.575298
$147$	892.533i	0.500781i
$148$	356.745i	0.198137i
$149$	1104.11	0.607064	0.303532	−	0.952821i	$-0.401834\pi$
$149$	1104.11	0.607064	0.303532	+	0.952821i	$0.401834\pi$
$150$	0	0
$151$	2980.07	1.60606	0.803029	−	0.595940i	$-0.203221\pi$
$151$	2980.07	1.60606	0.803029	+	0.595940i	$0.203221\pi$
$152$	− 607.794i	− 0.324333i
$153$	− 892.516i	− 0.471605i
$154$	176.000	0.0920941
$155$	0	0
$156$	433.973	0.222728
$157$	− 2844.67i	− 1.44605i	−0.690823	−	0.723024i	$-0.742751\pi$
$157$	− 2844.67i	− 1.44605i	0.690823	−	0.723024i	$-0.257249\pi$
$158$	− 231.413i	− 0.116521i
$159$	−976.956	−0.487281
$160$	0	0
$161$	755.391	0.369771
$162$	192.155i	0.0931920i
$163$	− 1528.51i	− 0.734492i	−0.930124	−	0.367246i	$-0.880301\pi$
$163$	− 1528.51i	− 0.734492i	0.930124	−	0.367246i	$-0.119699\pi$
$164$	−599.581	−0.285484
$165$	0	0
$166$	−2620.47	−1.22523
$167$	383.881i	0.177878i	0.996037	+	0.0889388i	$0.0283476\pi$
$167$	383.881i	0.177878i	−0.996037	+	0.0889388i	$0.971652\pi$
$168$	497.870i	0.228640i
$169$	−1521.35	−0.692466
$170$	0	0
$171$	222.310	0.0994178
$172$	507.707i	0.225071i
$173$	2702.85i	1.18783i	0.804529	+	0.593914i	$0.202418\pi$
$173$	2702.85i	1.18783i	−0.804529	+	0.593914i	$0.797582\pi$
$174$	−150.343	−0.0655027
$175$	0	0
$176$	−433.334	−0.185590
$177$	− 588.000i	− 0.249699i
$178$	− 1098.42i	− 0.462528i
$179$	2777.61	1.15982	0.579911	−	0.814680i	$-0.303087\pi$
$179$	2777.61	1.15982	0.579911	+	0.814680i	$0.303087\pi$
$180$	0	0
$181$	−3993.61	−1.64001	−0.820007	−	0.572354i	$-0.806030\pi$
$181$	−3993.61	−1.64001	−0.820007	+	0.572354i	$0.806030\pi$
$182$	− 975.652i	− 0.397363i
$183$	− 1207.92i	− 0.487936i
$184$	−2755.87	−1.10416
$185$	0	0
$186$	2268.26	0.894177
$187$	1090.85i	0.426583i
$188$	− 251.152i	− 0.0974317i
$189$	−182.103	−0.0700850
$190$	0	0
$191$	895.587	0.339280	0.169640	−	0.985506i	$-0.445740\pi$
$191$	895.587	0.339280	0.169640	+	0.985506i	$0.445740\pi$
$192$	− 1681.30i	− 0.631964i
$193$	1328.24i	0.495382i	0.968839	+	0.247691i	$0.0796718\pi$
$193$	1328.24i	0.495382i	−0.968839	+	0.247691i	$0.920328\pi$
$194$	−4266.21	−1.57885
$195$	0	0
$196$	705.779	0.257208
$197$	− 154.114i	− 0.0557370i	−0.999612	−	0.0278685i	$-0.991128\pi$
$197$	− 154.114i	− 0.0557370i	0.999612	−	0.0278685i	$-0.00887197\pi$
$198$	− 234.856i	− 0.0842953i
$199$	−1316.15	−0.468842	−0.234421	−	0.972135i	$-0.575319\pi$
$199$	−1316.15	−0.468842	−0.234421	+	0.972135i	$0.575319\pi$
$200$	0	0
$201$	82.2397	0.0288594
$202$	1760.04i	0.613049i
$203$	− 142.478i	− 0.0492612i
$204$	−705.766	−0.242223
$205$	0	0
$206$	−924.803	−0.312787
$207$	− 1008.00i	− 0.338458i
$208$	2402.18i	0.800775i
$209$	−271.712	−0.0899268
$210$	0	0
$211$	−1735.76	−0.566324	−0.283162	−	0.959072i	$-0.591383\pi$
$211$	−1735.76	−0.566324	−0.283162	+	0.959072i	$0.591383\pi$
$212$	772.538i	0.250274i
$213$	− 901.499i	− 0.289999i
$214$	3649.57	1.16579
$215$	0	0
$216$	664.361	0.209278
$217$	2149.61i	0.672465i
$218$	− 1849.86i	− 0.574718i
$219$	−1283.44	−0.396014
$220$	0	0
$221$	6047.12	1.84060
$222$	− 1070.23i	− 0.323556i
$223$	1663.71i	0.499597i	0.968298	+	0.249798i	$0.0803643\pi$
$223$	1663.71i	0.499597i	−0.968298	+	0.249798i	$0.919636\pi$
$224$	697.348	0.208007
$225$	0	0
$226$	3592.95	1.05752
$227$	3658.84i	1.06980i	0.844914	+	0.534902i	$0.179651\pi$
$227$	3658.84i	1.06980i	−0.844914	+	0.534902i	$0.820349\pi$
$228$	− 175.794i	− 0.0510624i
$229$	−1927.07	−0.556090	−0.278045	−	0.960568i	$-0.589686\pi$
$229$	−1927.07	−0.556090	−0.278045	+	0.960568i	$0.589686\pi$
$230$	0	0
$231$	222.571	0.0633942
$232$	519.800i	0.147097i
$233$	− 2784.27i	− 0.782847i	−0.920211	−	0.391423i	$-0.871983\pi$
$233$	− 2784.27i	− 0.782847i	0.920211	−	0.391423i	$-0.128017\pi$
$234$	−1301.92	−0.363714
$235$	0	0
$236$	−464.967	−0.128249
$237$	− 292.646i	− 0.0802086i
$238$	1586.70i	0.432144i
$239$	−1222.60	−0.330892	−0.165446	−	0.986219i	$-0.552906\pi$
$239$	−1222.60	−0.330892	−0.165446	+	0.986219i	$0.552906\pi$
$240$	0	0
$241$	−2013.01	−0.538047	−0.269024	−	0.963134i	$-0.586701\pi$
$241$	−2013.01	−0.538047	−0.269024	+	0.963134i	$0.586701\pi$
$242$	287.046i	0.0762480i
$243$	243.000i	0.0641500i
$244$	−955.179	−0.250611
$245$	0	0
$246$	1798.74	0.466194
$247$	1506.23i	0.388012i
$248$	− 7842.35i	− 2.00802i
$249$	−3313.86	−0.843402
$250$	0	0
$251$	2706.63	0.680641	0.340320	−	0.940310i	$-0.389464\pi$
$251$	2706.63	0.680641	0.340320	+	0.940310i	$0.389464\pi$
$252$	144.000i	0.0359966i
$253$	1232.00i	0.306147i
$254$	5462.76	1.34946
$255$	0	0
$256$	−3291.74	−0.803647
$257$	− 225.741i	− 0.0547912i	−0.999625	−	0.0273956i	$-0.991279\pi$
$257$	− 225.741i	− 0.0547912i	0.999625	−	0.0273956i	$-0.00872138\pi$
$258$	− 1523.12i	− 0.367540i
$259$	1014.25	0.243330
$260$	0	0
$261$	−190.124	−0.0450897
$262$	− 4793.40i	− 1.13029i
$263$	1953.42i	0.457997i	0.973427	+	0.228998i	$0.0735451\pi$
$263$	1953.42i	0.457997i	−0.973427	+	0.228998i	$0.926455\pi$
$264$	−811.997	−0.189299
$265$	0	0
$266$	−395.217	−0.0910989
$267$	− 1389.07i	− 0.318387i
$268$	− 65.0319i	− 0.0148226i
$269$	350.848	0.0795225	0.0397613	−	0.999209i	$-0.487340\pi$
$269$	350.848	0.0795225	0.0397613	+	0.999209i	$0.487340\pi$
$270$	0	0
$271$	254.779	0.0571096	0.0285548	−	0.999592i	$-0.490909\pi$
$271$	254.779	0.0571096	0.0285548	+	0.999592i	$0.490909\pi$
$272$	− 3906.64i	− 0.870864i
$273$	− 1233.81i	− 0.273531i
$274$	3500.07	0.771704
$275$	0	0
$276$	−797.087	−0.173837
$277$	− 116.478i	− 0.0252654i	−0.999920	−	0.0126327i	$-0.995979\pi$
$277$	− 116.478i	− 0.0252654i	0.999920	−	0.0126327i	$-0.00402122\pi$
$278$	3851.90i	0.831013i
$279$	2868.46	0.615519
$280$	0	0
$281$	8226.47	1.74644	0.873221	−	0.487325i	$-0.162027\pi$
$281$	8226.47	1.74644	0.873221	+	0.487325i	$0.162027\pi$
$282$	753.457i	0.159105i
$283$	1561.52i	0.327995i	0.986461	+	0.163997i	$0.0524389\pi$
$283$	1561.52i	0.327995i	−0.986461	+	0.163997i	$0.947561\pi$
$284$	−712.870	−0.148947
$285$	0	0
$286$	1591.23	0.328992
$287$	1704.65i	0.350601i
$288$	− 930.546i	− 0.190392i
$289$	−4921.38	−1.00171
$290$	0	0
$291$	−5395.07	−1.08682
$292$	1014.90i	0.203399i
$293$	9486.19i	1.89143i	0.324997	+	0.945715i	$0.394637\pi$
$293$	9486.19i	1.89143i	−0.324997	+	0.945715i	$0.605363\pi$
$294$	−2117.34	−0.420019
$295$	0	0
$296$	−3700.25	−0.726598
$297$	− 297.000i	− 0.0580259i
$298$	2619.27i	0.509161i
$299$	6829.56	1.32095
$300$	0	0
$301$	1443.45	0.276408
$302$	7069.56i	1.34705i
$303$	2225.75i	0.422001i
$304$	973.074	0.183584
$305$	0	0
$306$	2117.30	0.395549
$307$	8443.06i	1.56961i	0.619742	+	0.784806i	$0.287237\pi$
$307$	8443.06i	1.56961i	−0.619742	+	0.784806i	$0.712763\pi$
$308$	− 176.000i	− 0.0325602i
$309$	−1169.51	−0.215311
$310$	0	0
$311$	4447.17	0.810856	0.405428	−	0.914127i	$-0.367123\pi$
$311$	4447.17	0.810856	0.405428	+	0.914127i	$0.367123\pi$
$312$	4501.29i	0.816779i
$313$	6480.75i	1.17033i	0.810914	+	0.585165i	$0.198970\pi$
$313$	6480.75i	1.17033i	−0.810914	+	0.585165i	$0.801030\pi$
$314$	6748.36	1.21284
$315$	0	0
$316$	−231.413	−0.0411962
$317$	1252.75i	0.221960i	0.993823	+	0.110980i	$0.0353990\pi$
$317$	1252.75i	0.221960i	−0.993823	+	0.110980i	$0.964601\pi$
$318$	− 2317.61i	− 0.408696i
$319$	232.374	0.0407852
$320$	0	0
$321$	4615.27	0.802490
$322$	1792.00i	0.310137i
$323$	− 2449.57i	− 0.421974i
$324$	192.155	0.0329484
$325$	0	0
$326$	3626.06	0.616039
$327$	− 2339.35i	− 0.395615i
$328$	− 6219.02i	− 1.04692i
$329$	−714.043	−0.119655
$330$	0	0
$331$	3801.44	0.631258	0.315629	−	0.948883i	$-0.397785\pi$
$331$	3801.44	0.631258	0.315629	+	0.948883i	$0.397785\pi$
$332$	2620.47i	0.433183i
$333$	− 1353.42i	− 0.222724i
$334$	−910.673	−0.149191
$335$	0	0
$336$	−797.087	−0.129419
$337$	5456.53i	0.882007i	0.897506	+	0.441003i	$0.145377\pi$
$337$	5456.53i	0.882007i	−0.897506	+	0.441003i	$0.854623\pi$
$338$	− 3609.06i	− 0.580790i
$339$	4543.66	0.727958
$340$	0	0
$341$	−3505.89	−0.556758
$342$	527.381i	0.0833845i
$343$	− 4319.97i	− 0.680047i
$344$	−5266.08	−0.825371
$345$	0	0
$346$	−6411.93	−0.996264
$347$	4240.53i	0.656033i	0.944672	+	0.328017i	$0.106380\pi$
$347$	4240.53i	0.656033i	−0.944672	+	0.328017i	$0.893620\pi$
$348$	150.343i	0.0231587i
$349$	8471.53	1.29934	0.649672	−	0.760215i	$-0.274906\pi$
$349$	8471.53	1.29934	0.649672	+	0.760215i	$0.274906\pi$
$350$	0	0
$351$	−1646.41	−0.250368
$352$	1137.33i	0.172216i
$353$	981.282i	0.147956i	0.997260	+	0.0739778i	$0.0235694\pi$
$353$	981.282i	0.147956i	−0.997260	+	0.0739778i	$0.976431\pi$
$354$	1394.90	0.209430
$355$	0	0
$356$	−1098.42	−0.163528
$357$	2006.54i	0.297472i
$358$	6589.27i	0.972776i
$359$	3020.09	0.443995	0.221997	−	0.975047i	$-0.428742\pi$
$359$	3020.09	0.443995	0.221997	+	0.975047i	$0.428742\pi$
$360$	0	0
$361$	−6248.86	−0.911045
$362$	− 9473.96i	− 1.37553i
$363$	363.000i	0.0524864i
$364$	−975.652	−0.140489
$365$	0	0
$366$	2865.54	0.409246
$367$	− 8689.40i	− 1.23592i	−0.786209	−	0.617960i	$-0.787959\pi$
$367$	− 8689.40i	− 1.23592i	0.786209	−	0.617960i	$-0.212041\pi$
$368$	− 4412.13i	− 0.624995i
$369$	2274.70	0.320911
$370$	0	0
$371$	2196.38	0.307360
$372$	− 2268.26i	− 0.316139i
$373$	5340.53i	0.741346i	0.928763	+	0.370673i	$0.120873\pi$
$373$	5340.53i	0.741346i	−0.928763	+	0.370673i	$0.879127\pi$
$374$	−2587.81	−0.357787
$375$	0	0
$376$	2605.02	0.357297
$377$	− 1288.16i	− 0.175978i
$378$	− 432.000i	− 0.0587822i
$379$	1603.49	0.217324	0.108662	−	0.994079i	$-0.465343\pi$
$379$	1603.49	0.217324	0.108662	+	0.994079i	$0.465343\pi$
$380$	0	0
$381$	6908.23	0.928923
$382$	2124.58i	0.284563i
$383$	− 830.236i	− 0.110765i	−0.998465	−	0.0553826i	$-0.982362\pi$
$383$	− 830.236i	− 0.110765i	0.998465	−	0.0553826i	$-0.0176379\pi$
$384$	1507.05	0.200277
$385$	0	0
$386$	−3150.96	−0.415491
$387$	− 1926.15i	− 0.253001i
$388$	4266.21i	0.558206i
$389$	1746.12	0.227588	0.113794	−	0.993504i	$-0.463700\pi$
$389$	1746.12	0.227588	0.113794	+	0.993504i	$0.463700\pi$
$390$	0	0
$391$	−11106.9	−1.43657
$392$	7320.54i	0.943223i
$393$	− 6061.76i	− 0.778054i
$394$	365.602	0.0467482
$395$	0	0
$396$	−234.856	−0.0298029
$397$	10016.2i	1.26624i	0.774054	+	0.633119i	$0.218226\pi$
$397$	10016.2i	1.26624i	−0.774054	+	0.633119i	$0.781774\pi$
$398$	− 3122.28i	− 0.393231i
$399$	−499.794	−0.0627092
$400$	0	0
$401$	8228.38	1.02470	0.512351	−	0.858776i	$-0.328775\pi$
$401$	8228.38	1.02470	0.512351	+	0.858776i	$0.328775\pi$
$402$	195.096i	0.0242052i
$403$	19434.8i	2.40228i
$404$	1760.04	0.216746
$405$	0	0
$406$	337.999	0.0413168
$407$	1654.18i	0.201462i
$408$	− 7320.41i	− 0.888270i
$409$	12311.5	1.48843	0.744213	−	0.667943i	$-0.232825\pi$
$409$	12311.5	1.48843	0.744213	+	0.667943i	$0.232825\pi$
$410$	0	0
$411$	4426.21	0.531213
$412$	924.803i	0.110587i
$413$	1321.93i	0.157502i
$414$	2391.26	0.283874
$415$	0	0
$416$	6304.79	0.743071
$417$	4871.14i	0.572040i
$418$	− 644.577i	− 0.0754241i
$419$	−13260.4	−1.54610	−0.773048	−	0.634347i	$-0.781269\pi$
$419$	−13260.4	−1.54610	−0.773048	+	0.634347i	$0.781269\pi$
$420$	0	0
$421$	−6177.74	−0.715165	−0.357583	−	0.933881i	$-0.616399\pi$
$421$	−6177.74	−0.715165	−0.357583	+	0.933881i	$0.616399\pi$
$422$	− 4117.70i	− 0.474992i
$423$	952.826i	0.109522i
$424$	−8012.98	−0.917795
$425$	0	0
$426$	2138.61	0.243230
$427$	2715.64i	0.307773i
$428$	− 3649.57i	− 0.412170i
$429$	2012.28	0.226466
$430$	0	0
$431$	1668.05	0.186421	0.0932103	−	0.995646i	$-0.470287\pi$
$431$	1668.05	0.186421	0.0932103	+	0.995646i	$0.470287\pi$
$432$	1063.64i	0.118459i
$433$	− 731.748i	− 0.0812138i	−0.999175	−	0.0406069i	$-0.987071\pi$
$433$	− 731.748i	− 0.0812138i	0.999175	−	0.0406069i	$-0.0129291\pi$
$434$	−5099.48	−0.564015
$435$	0	0
$436$	−1849.86	−0.203194
$437$	− 2766.52i	− 0.302839i
$438$	− 3044.69i	− 0.332148i
$439$	14248.0	1.54902	0.774508	−	0.632564i	$-0.217998\pi$
$439$	14248.0	1.54902	0.774508	+	0.632564i	$0.217998\pi$
$440$	0	0
$441$	−2677.60	−0.289126
$442$	14345.5i	1.54377i
$443$	174.262i	0.0186895i	0.999956	+	0.00934473i	$0.00297456\pi$
$443$	174.262i	0.0186895i	−0.999956	+	0.00934473i	$0.997025\pi$
$444$	−1070.23	−0.114394
$445$	0	0
$446$	−3946.78	−0.419026
$447$	3312.34i	0.350488i
$448$	3779.87i	0.398621i
$449$	−7469.97	−0.785144	−0.392572	−	0.919721i	$-0.628415\pi$
$449$	−7469.97	−0.785144	−0.392572	+	0.919721i	$0.628415\pi$
$450$	0	0
$451$	−2780.19	−0.290275
$452$	− 3592.95i	− 0.373890i
$453$	8940.21i	0.927258i
$454$	−8679.79	−0.897275
$455$	0	0
$456$	1823.38	0.187254
$457$	− 8762.68i	− 0.896939i	−0.893798	−	0.448469i	$-0.851969\pi$
$457$	− 8762.68i	− 0.896939i	0.893798	−	0.448469i	$-0.148031\pi$
$458$	− 4571.56i	− 0.466409i
$459$	2677.55	0.272282
$460$	0	0
$461$	4339.67	0.438435	0.219218	−	0.975676i	$-0.429650\pi$
$461$	4339.67	0.438435	0.219218	+	0.975676i	$0.429650\pi$
$462$	528.000i	0.0531705i
$463$	− 3932.49i	− 0.394726i	−0.980330	−	0.197363i	$-0.936762\pi$
$463$	− 3932.49i	− 0.394726i	0.980330	−	0.197363i	$-0.0632378\pi$
$464$	−832.197	−0.0832624
$465$	0	0
$466$	6605.06	0.656596
$467$	8383.97i	0.830758i	0.909648	+	0.415379i	$0.136351\pi$
$467$	8383.97i	0.830758i	−0.909648	+	0.415379i	$0.863649\pi$
$468$	1301.92i	0.128592i
$469$	−184.890	−0.0182035
$470$	0	0
$471$	8534.02	0.834876
$472$	− 4822.77i	− 0.470309i
$473$	2354.18i	0.228848i
$474$	694.240	0.0672732
$475$	0	0
$476$	1586.70	0.152786
$477$	− 2930.87i	− 0.281332i
$478$	− 2900.35i	− 0.277529i
$479$	−9534.10	−0.909445	−0.454722	−	0.890633i	$-0.650261\pi$
$479$	−9534.10	−0.909445	−0.454722	+	0.890633i	$0.650261\pi$
$480$	0	0
$481$	9169.93	0.869258
$482$	− 4775.43i	− 0.451275i
$483$	2266.17i	0.213487i
$484$	287.046	0.0269577
$485$	0	0
$486$	−576.464	−0.0538044
$487$	− 4451.09i	− 0.414164i	−0.978324	−	0.207082i	$-0.933603\pi$
$487$	− 4451.09i	− 0.414164i	0.978324	−	0.207082i	$-0.0663967\pi$
$488$	− 9907.38i	− 0.919029i
$489$	4585.53	0.424059
$490$	0	0
$491$	−9757.40	−0.896833	−0.448417	−	0.893825i	$-0.648012\pi$
$491$	−9757.40	−0.896833	−0.448417	+	0.893825i	$0.648012\pi$
$492$	− 1798.74i	− 0.164824i
$493$	2094.93i	0.191381i
$494$	−3573.20	−0.325437
$495$	0	0
$496$	12555.6	1.13662
$497$	2026.74i	0.182921i
$498$	− 7861.40i	− 0.707385i
$499$	7173.69	0.643564	0.321782	−	0.946814i	$-0.395718\pi$
$499$	7173.69	0.643564	0.321782	+	0.946814i	$0.395718\pi$
$500$	0	0
$501$	−1151.64	−0.102698
$502$	6420.88i	0.570873i
$503$	− 15617.0i	− 1.38435i	−0.721728	−	0.692177i	$-0.756652\pi$
$503$	− 15617.0i	− 1.38435i	0.721728	−	0.692177i	$-0.243348\pi$
$504$	−1493.61	−0.132005
$505$	0	0
$506$	−2922.65	−0.256774
$507$	− 4564.04i	− 0.399795i
$508$	− 5462.76i	− 0.477108i
$509$	8789.23	0.765375	0.382688	−	0.923878i	$-0.374999\pi$
$509$	8789.23	0.765375	0.382688	+	0.923878i	$0.374999\pi$
$510$	0	0
$511$	2885.42	0.249792
$512$	− 11827.7i	− 1.02093i
$513$	666.929i	0.0573989i
$514$	535.521	0.0459549
$515$	0	0
$516$	−1523.12	−0.129945
$517$	− 1164.56i	− 0.0990667i
$518$	2406.09i	0.204088i
$519$	−8108.56	−0.685792
$520$	0	0
$521$	13099.3	1.10152	0.550760	−	0.834664i	$-0.314338\pi$
$521$	13099.3	1.10152	0.550760	+	0.834664i	$0.314338\pi$
$522$	− 451.029i	− 0.0378180i
$523$	16824.2i	1.40664i	0.710876	+	0.703318i	$0.248299\pi$
$523$	16824.2i	1.40664i	−0.710876	+	0.703318i	$0.751701\pi$
$524$	−4793.40	−0.399620
$525$	0	0
$526$	−4634.07	−0.384135
$527$	− 31606.7i	− 2.61254i
$528$	− 1300.00i	− 0.107150i
$529$	−377.000	−0.0309855
$530$	0	0
$531$	1764.00	0.144164
$532$	395.217i	0.0322083i
$533$	15411.9i	1.25247i
$534$	3295.25	0.267040
$535$	0	0
$536$	674.529	0.0543568
$537$	8332.83i	0.669624i
$538$	832.310i	0.0666978i
$539$	3272.62	0.261525
$540$	0	0
$541$	18863.1	1.49905	0.749527	−	0.661974i	$-0.230281\pi$
$541$	18863.1	1.49905	0.749527	+	0.661974i	$0.230281\pi$
$542$	604.407i	0.0478995i
$543$	− 11980.8i	− 0.946862i
$544$	−10253.4	−0.808110
$545$	0	0
$546$	2926.96	0.229418
$547$	12283.0i	0.960119i	0.877236	+	0.480059i	$0.159385\pi$
$547$	12283.0i	0.960119i	−0.877236	+	0.480059i	$0.840615\pi$
$548$	− 3500.07i	− 0.272839i
$549$	3623.77	0.281710
$550$	0	0
$551$	−521.809	−0.0403444
$552$	− 8267.61i	− 0.637487i
$553$	657.924i	0.0505927i
$554$	276.320	0.0211908
$555$	0	0
$556$	3851.90	0.293808
$557$	9752.05i	0.741845i	0.928664	+	0.370923i	$0.120958\pi$
$557$	9752.05i	0.741845i	−0.928664	+	0.370923i	$0.879042\pi$
$558$	6804.78i	0.516253i
$559$	13050.3	0.987424
$560$	0	0
$561$	−3272.56	−0.246288
$562$	19515.5i	1.46479i
$563$	− 3447.50i	− 0.258072i	−0.991640	−	0.129036i	$-0.958812\pi$
$563$	− 3447.50i	− 0.258072i	0.991640	−	0.129036i	$-0.0411884\pi$
$564$	753.457	0.0562522
$565$	0	0
$566$	−3704.36	−0.275098
$567$	− 546.310i	− 0.0404636i
$568$	− 7394.09i	− 0.546213i
$569$	3371.02	0.248366	0.124183	−	0.992259i	$-0.460369\pi$
$569$	3371.02	0.248366	0.124183	+	0.992259i	$0.460369\pi$
$570$	0	0
$571$	−15852.5	−1.16183	−0.580916	−	0.813964i	$-0.697305\pi$
$571$	−15852.5	−1.16183	−0.580916	+	0.813964i	$0.697305\pi$
$572$	− 1591.23i	− 0.116316i
$573$	2686.76i	0.195883i
$574$	−4043.91	−0.294059
$575$	0	0
$576$	5043.89	0.364865
$577$	− 1376.35i	− 0.0993036i	−0.998767	−	0.0496518i	$-0.984189\pi$
$577$	− 1376.35i	− 0.0993036i	0.998767	−	0.0496518i	$-0.0158112\pi$
$578$	− 11674.9i	− 0.840159i
$579$	−3984.72	−0.286009
$580$	0	0
$581$	7450.17	0.531988
$582$	− 12798.6i	− 0.911547i
$583$	3582.17i	0.254474i
$584$	−10526.8	−0.745894
$585$	0	0
$586$	−22503.9	−1.58640
$587$	23021.4i	1.61873i	0.587304	+	0.809366i	$0.300189\pi$
$587$	23021.4i	1.61873i	−0.587304	+	0.809366i	$0.699811\pi$
$588$	2117.34i	0.148499i
$589$	7872.66	0.550742
$590$	0	0
$591$	462.343	0.0321798
$592$	− 5924.09i	− 0.411281i
$593$	2818.69i	0.195194i	0.995226	+	0.0975968i	$0.0311156\pi$
$593$	2818.69i	0.195194i	−0.995226	+	0.0975968i	$0.968884\pi$
$594$	704.568	0.0486679
$595$	0	0
$596$	2619.27	0.180016
$597$	− 3948.46i	− 0.270686i
$598$	16201.6i	1.10792i
$599$	17691.4	1.20676	0.603380	−	0.797454i	$-0.293820\pi$
$599$	17691.4	1.20676	0.603380	+	0.797454i	$0.293820\pi$
$600$	0	0
$601$	−24516.1	−1.66394	−0.831972	−	0.554817i	$-0.812788\pi$
$601$	−24516.1	−1.66394	−0.831972	+	0.554817i	$0.812788\pi$
$602$	3424.26i	0.231831i
$603$	246.719i	0.0166620i
$604$	7069.56	0.476252
$605$	0	0
$606$	−5280.12	−0.353944
$607$	− 4288.59i	− 0.286768i	−0.989667	−	0.143384i	$-0.954202\pi$
$607$	− 4288.59i	− 0.286768i	0.989667	−	0.143384i	$-0.0457984\pi$
$608$	− 2553.94i	− 0.170355i
$609$	427.435	0.0284410
$610$	0	0
$611$	−6455.74	−0.427449
$612$	− 2117.30i	− 0.139848i
$613$	− 8124.07i	− 0.535283i	−0.963519	−	0.267641i	$-0.913756\pi$
$613$	− 8124.07i	− 0.535283i	0.963519	−	0.267641i	$-0.0862442\pi$
$614$	−20029.3	−1.31648
$615$	0	0
$616$	1825.52	0.119403
$617$	14923.1i	0.973714i	0.873482	+	0.486857i	$0.161857\pi$
$617$	14923.1i	0.973714i	−0.873482	+	0.486857i	$0.838143\pi$
$618$	− 2774.41i	− 0.180588i
$619$	−3627.68	−0.235556	−0.117778	−	0.993040i	$-0.537577\pi$
$619$	−3627.68	−0.235556	−0.117778	+	0.993040i	$0.537577\pi$
$620$	0	0
$621$	3024.00	0.195409
$622$	10549.9i	0.680087i
$623$	3122.88i	0.200827i
$624$	−7206.54	−0.462328
$625$	0	0
$626$	−15374.2	−0.981589
$627$	− 815.135i	− 0.0519192i
$628$	− 6748.36i	− 0.428804i
$629$	−14913.0	−0.945341
$630$	0	0
$631$	12576.5	0.793445	0.396723	−	0.917939i	$-0.370147\pi$
$631$	12576.5	0.793445	0.396723	+	0.917939i	$0.370147\pi$
$632$	− 2400.28i	− 0.151073i
$633$	− 5207.27i	− 0.326967i
$634$	−2971.87	−0.186164
$635$	0	0
$636$	−2317.61	−0.144496
$637$	− 18141.7i	− 1.12841i
$638$	551.257i	0.0342077i
$639$	2704.50	0.167431
$640$	0	0
$641$	−7292.77	−0.449371	−0.224686	−	0.974431i	$-0.572136\pi$
$641$	−7292.77	−0.449371	−0.224686	+	0.974431i	$0.572136\pi$
$642$	10948.7i	0.673071i
$643$	− 12946.3i	− 0.794016i	−0.917815	−	0.397008i	$-0.870049\pi$
$643$	− 12946.3i	− 0.794016i	0.917815	−	0.397008i	$-0.129951\pi$
$644$	1792.00	0.109650
$645$	0	0
$646$	5811.06	0.353921
$647$	− 21973.3i	− 1.33518i	−0.744530	−	0.667589i	$-0.767327\pi$
$647$	− 21973.3i	− 1.33518i	0.744530	−	0.667589i	$-0.232673\pi$
$648$	1993.08i	0.120827i
$649$	−2156.00	−0.130401
$650$	0	0
$651$	−6448.83	−0.388248
$652$	− 3626.06i	− 0.217803i
$653$	− 27495.6i	− 1.64776i	−0.566764	−	0.823880i	$-0.691805\pi$
$653$	− 27495.6i	− 1.64776i	0.566764	−	0.823880i	$-0.308195\pi$
$654$	5549.59	0.331814
$655$	0	0
$656$	9956.63	0.592593
$657$	− 3850.33i	− 0.228639i
$658$	− 1693.91i	− 0.100358i
$659$	−5156.28	−0.304796	−0.152398	−	0.988319i	$-0.548699\pi$
$659$	−5156.28	−0.304796	−0.152398	+	0.988319i	$0.548699\pi$
$660$	0	0
$661$	9328.59	0.548926	0.274463	−	0.961598i	$-0.411500\pi$
$661$	9328.59	0.548926	0.274463	+	0.961598i	$0.411500\pi$
$662$	9018.10i	0.529454i
$663$	18141.4i	1.06267i
$664$	−27180.2	−1.58855
$665$	0	0
$666$	3210.70	0.186805
$667$	2365.99i	0.137349i
$668$	910.673i	0.0527470i
$669$	−4991.12	−0.288442
$670$	0	0
$671$	−4429.06	−0.254816
$672$	2092.04i	0.120093i
$673$	22182.4i	1.27053i	0.772293	+	0.635267i	$0.219110\pi$
$673$	22182.4i	1.27053i	−0.772293	+	0.635267i	$0.780890\pi$
$674$	−12944.4	−0.739764
$675$	0	0
$676$	−3609.06	−0.205340
$677$	13507.1i	0.766797i	0.923583	+	0.383399i	$0.125246\pi$
$677$	13507.1i	0.766797i	−0.923583	+	0.383399i	$0.874754\pi$
$678$	10778.8i	0.610559i
$679$	12129.1	0.685528
$680$	0	0
$681$	−10976.5	−0.617652
$682$	− 8316.96i	− 0.466969i
$683$	− 19465.6i	− 1.09053i	−0.838264	−	0.545264i	$-0.816429\pi$
$683$	− 19465.6i	− 1.09053i	0.838264	−	0.545264i	$-0.183571\pi$
$684$	527.381	0.0294809
$685$	0	0
$686$	10248.2	0.570375
$687$	− 5781.22i	− 0.321059i
$688$	− 8430.96i	− 0.467191i
$689$	19857.7	1.09799
$690$	0	0
$691$	−19971.8	−1.09951	−0.549757	−	0.835324i	$-0.685280\pi$
$691$	−19971.8	−1.09951	−0.549757	+	0.835324i	$0.685280\pi$
$692$	6411.93i	0.352233i
$693$	667.712i	0.0366007i
$694$	−10059.7	−0.550234
$695$	0	0
$696$	−1559.40	−0.0849265
$697$	− 25064.3i	− 1.36209i
$698$	20096.9i	1.08980i
$699$	8352.80	0.451977
$700$	0	0
$701$	−14180.1	−0.764018	−0.382009	−	0.924159i	$-0.624768\pi$
$701$	−14180.1	−0.764018	−0.382009	+	0.924159i	$0.624768\pi$
$702$	− 3905.75i	− 0.209990i
$703$	− 3714.56i	− 0.199285i
$704$	−6164.75	−0.330032
$705$	0	0
$706$	−2327.88	−0.124095
$707$	− 5003.91i	− 0.266183i
$708$	− 1394.90i	− 0.0740446i
$709$	−15870.6	−0.840667	−0.420334	−	0.907370i	$-0.638087\pi$
$709$	−15870.6	−0.840667	−0.420334	+	0.907370i	$0.638087\pi$
$710$	0	0
$711$	877.939	0.0463084
$712$	− 11393.1i	− 0.599683i
$713$	− 35696.3i	− 1.87495i
$714$	−4760.09	−0.249498
$715$	0	0
$716$	6589.27	0.343928
$717$	− 3667.79i	− 0.191041i
$718$	7164.50i	0.372391i
$719$	6040.05	0.313290	0.156645	−	0.987655i	$-0.449932\pi$
$719$	6040.05	0.313290	0.156645	+	0.987655i	$0.449932\pi$
$720$	0	0
$721$	2629.28	0.135811
$722$	− 14824.0i	− 0.764119i
$723$	− 6039.03i	− 0.310642i
$724$	−9473.96	−0.486322
$725$	0	0
$726$	−861.138	−0.0440218
$727$	− 37252.3i	− 1.90043i	−0.311601	−	0.950213i	$-0.600865\pi$
$727$	− 37252.3i	− 1.90043i	0.311601	−	0.950213i	$-0.399135\pi$
$728$	− 10119.7i	− 0.515196i
$729$	−729.000	−0.0370370
$730$	0	0
$731$	−21223.7	−1.07385
$732$	− 2865.54i	− 0.144690i
$733$	6125.52i	0.308665i	0.988019	+	0.154332i	$0.0493227\pi$
$733$	6125.52i	0.308665i	−0.988019	+	0.154332i	$0.950677\pi$
$734$	20613.7	1.03660
$735$	0	0
$736$	−11580.1	−0.579958
$737$	− 301.546i	− 0.0150713i
$738$	5396.23i	0.269157i
$739$	5225.97	0.260136	0.130068	−	0.991505i	$-0.458480\pi$
$739$	5225.97	0.260136	0.130068	+	0.991505i	$0.458480\pi$
$740$	0	0
$741$	−4518.68	−0.224019
$742$	5210.43i	0.257791i
$743$	7062.96i	0.348741i	0.984680	+	0.174371i	$0.0557891\pi$
$743$	7062.96i	0.348741i	−0.984680	+	0.174371i	$0.944211\pi$
$744$	23527.0	1.15933
$745$	0	0
$746$	−12669.2	−0.621788
$747$	− 9941.57i	− 0.486939i
$748$	2587.81i	0.126497i
$749$	−10376.0	−0.506182
$750$	0	0
$751$	−20755.4	−1.00849	−0.504244	−	0.863561i	$-0.668229\pi$
$751$	−20755.4	−1.00849	−0.504244	+	0.863561i	$0.668229\pi$
$752$	4170.63i	0.202243i
$753$	8119.89i	0.392968i
$754$	3055.88	0.147598
$755$	0	0
$756$	−432.000	−0.0207827
$757$	− 31182.9i	− 1.49717i	−0.663037	−	0.748587i	$-0.730733\pi$
$757$	− 31182.9i	− 1.49717i	0.663037	−	0.748587i	$-0.269267\pi$
$758$	3803.93i	0.182276i
$759$	−3696.00	−0.176754
$760$	0	0
$761$	32047.8	1.52659	0.763293	−	0.646052i	$-0.223581\pi$
$761$	32047.8	1.52659	0.763293	+	0.646052i	$0.223581\pi$
$762$	16388.3i	0.779114i
$763$	5259.29i	0.249540i
$764$	2124.58	0.100608
$765$	0	0
$766$	1969.55	0.0929019
$767$	11951.7i	0.562650i
$768$	− 9875.22i	− 0.463986i
$769$	−2215.88	−0.103910	−0.0519548	−	0.998649i	$-0.516545\pi$
$769$	−2215.88	−0.103910	−0.0519548	+	0.998649i	$0.516545\pi$
$770$	0	0
$771$	677.223	0.0316337
$772$	3150.96i	0.146898i
$773$	5300.56i	0.246634i	0.992367	+	0.123317i	$0.0393532\pi$
$773$	5300.56i	0.246634i	−0.992367	+	0.123317i	$0.960647\pi$
$774$	4569.36	0.212199
$775$	0	0
$776$	−44250.3	−2.04703
$777$	3042.75i	0.140487i
$778$	4142.29i	0.190885i
$779$	6243.06	0.287138
$780$	0	0
$781$	−3305.50	−0.151447
$782$	− 26348.6i	− 1.20489i
$783$	− 570.373i	− 0.0260325i
$784$	−11720.2	−0.533899
$785$	0	0
$786$	14380.2	0.652576
$787$	− 34374.1i	− 1.55693i	−0.627687	−	0.778466i	$-0.715998\pi$
$787$	− 34374.1i	− 1.55693i	0.627687	−	0.778466i	$-0.284002\pi$
$788$	− 365.602i	− 0.0165280i
$789$	−5860.27	−0.264425
$790$	0	0
$791$	−10215.0	−0.459170
$792$	− 2435.99i	− 0.109292i
$793$	24552.4i	1.09947i
$794$	−23761.2	−1.06203
$795$	0	0
$796$	−3122.28	−0.139028
$797$	− 29867.1i	− 1.32741i	−0.747993	−	0.663707i	$-0.768982\pi$
$797$	− 29867.1i	− 1.32741i	0.747993	−	0.663707i	$-0.231018\pi$
$798$	− 1185.65i	− 0.0525960i
$799$	10498.9	0.464862
$800$	0	0
$801$	4167.20	0.183821
$802$	19520.0i	0.859447i
$803$	4705.96i	0.206812i
$804$	195.096	0.00855783
$805$	0	0
$806$	−46104.9	−2.01486
$807$	1052.54i	0.0459124i
$808$	18255.6i	0.794839i
$809$	15857.2	0.689133	0.344566	−	0.938762i	$-0.388026\pi$
$809$	15857.2	0.689133	0.344566	+	0.938762i	$0.388026\pi$
$810$	0	0
$811$	36122.6	1.56404	0.782020	−	0.623253i	$-0.214189\pi$
$811$	36122.6	1.56404	0.782020	+	0.623253i	$0.214189\pi$
$812$	− 337.999i	− 0.0146077i
$813$	764.337i	0.0329723i
$814$	−3924.19	−0.168971
$815$	0	0
$816$	11719.9	0.502794
$817$	− 5286.43i	− 0.226375i
$818$	29206.4i	1.24838i
$819$	3701.44	0.157923
$820$	0	0
$821$	25779.6	1.09588	0.547938	−	0.836519i	$-0.315413\pi$
$821$	25779.6	1.09588	0.547938	+	0.836519i	$0.315413\pi$
$822$	10500.2i	0.445544i
$823$	− 22130.2i	− 0.937315i	−0.883380	−	0.468658i	$-0.844738\pi$
$823$	− 22130.2i	− 0.937315i	0.883380	−	0.468658i	$-0.155262\pi$
$824$	−9592.32	−0.405539
$825$	0	0
$826$	−3136.00	−0.132101
$827$	− 18288.6i	− 0.768994i	−0.923126	−	0.384497i	$-0.874375\pi$
$827$	− 18288.6i	− 0.768994i	0.923126	−	0.384497i	$-0.125625\pi$
$828$	− 2391.26i	− 0.100365i
$829$	34956.6	1.46453	0.732263	−	0.681022i	$-0.238464\pi$
$829$	34956.6	1.46453	0.732263	+	0.681022i	$0.238464\pi$
$830$	0	0
$831$	349.435	0.0145870
$832$	34174.2i	1.42401i
$833$	29503.7i	1.22718i
$834$	−11555.7	−0.479786
$835$	0	0
$836$	−644.577	−0.0266664
$837$	8605.37i	0.355370i
$838$	− 31457.5i	− 1.29675i
$839$	14507.5	0.596965	0.298482	−	0.954415i	$-0.403520\pi$
$839$	14507.5	0.596965	0.298482	+	0.954415i	$0.403520\pi$
$840$	0	0
$841$	−23942.7	−0.981702
$842$	− 14655.3i	− 0.599829i
$843$	24679.4i	1.00831i
$844$	−4117.70	−0.167935
$845$	0	0
$846$	−2260.37	−0.0918595
$847$	− 816.092i	− 0.0331066i
$848$	− 12828.7i	− 0.519506i
$849$	−4684.55	−0.189368
$850$	0	0
$851$	−16842.6	−0.678445
$852$	− 2138.61i	− 0.0859948i
$853$	− 44146.9i	− 1.77205i	−0.463634	−	0.886027i	$-0.653455\pi$
$853$	− 44146.9i	− 1.77205i	0.463634	−	0.886027i	$-0.346545\pi$
$854$	−6442.26	−0.258138
$855$	0	0
$856$	37854.4	1.51149
$857$	47679.3i	1.90046i	0.311549	+	0.950230i	$0.399152\pi$
$857$	47679.3i	1.90046i	−0.311549	+	0.950230i	$0.600848\pi$
$858$	4773.70i	0.189943i
$859$	−7525.09	−0.298897	−0.149449	−	0.988769i	$-0.547750\pi$
$859$	−7525.09	−0.298897	−0.149449	+	0.988769i	$0.547750\pi$
$860$	0	0
$861$	−5113.95	−0.202419
$862$	3957.09i	0.156356i
$863$	45816.1i	1.80718i	0.428396	+	0.903591i	$0.359079\pi$
$863$	45816.1i	1.80718i	−0.428396	+	0.903591i	$0.640921\pi$
$864$	2791.64	0.109923
$865$	0	0
$866$	1735.91	0.0681163
$867$	− 14764.1i	− 0.578335i
$868$	5099.48i	0.199410i
$869$	−1073.04	−0.0418876
$870$	0	0
$871$	−1671.61	−0.0650292
$872$	− 19187.3i	− 0.745142i
$873$	− 16185.2i	− 0.627476i
$874$	6562.96	0.253999
$875$	0	0
$876$	−3044.69	−0.117432
$877$	34168.3i	1.31560i	0.753192	+	0.657801i	$0.228513\pi$
$877$	34168.3i	1.31560i	−0.753192	+	0.657801i	$0.771487\pi$
$878$	33800.2i	1.29920i
$879$	−28458.6	−1.09202
$880$	0	0
$881$	−100.796	−0.00385460	−0.00192730	−	0.999998i	$-0.500613\pi$
$881$	−100.796	−0.00385460	−0.00192730	+	0.999998i	$0.500613\pi$
$882$	− 6352.02i	− 0.242498i
$883$	− 12346.1i	− 0.470531i	−0.971931	−	0.235266i	$-0.924404\pi$
$883$	− 12346.1i	− 0.470531i	0.971931	−	0.235266i	$-0.0755960\pi$
$884$	14345.5	0.545803
$885$	0	0
$886$	−413.398	−0.0156754
$887$	37345.1i	1.41367i	0.707379	+	0.706834i	$0.249877\pi$
$887$	37345.1i	1.41367i	−0.707379	+	0.706834i	$0.750123\pi$
$888$	− 11100.8i	− 0.419501i
$889$	−15531.0	−0.585932
$890$	0	0
$891$	891.000	0.0335013
$892$	3946.78i	0.148148i
$893$	2615.09i	0.0979962i
$894$	−7857.80	−0.293964
$895$	0	0
$896$	−3388.13	−0.126328
$897$	20488.7i	0.762651i
$898$	− 17720.9i	− 0.658523i
$899$	−6732.88	−0.249782
$900$	0	0
$901$	−32294.4	−1.19410
$902$	− 6595.39i	− 0.243462i
$903$	4330.34i	0.159584i
$904$	37267.1	1.37111
$905$	0	0
$906$	−21208.7	−0.777717
$907$	− 10308.6i	− 0.377390i	−0.982036	−	0.188695i	$-0.939574\pi$
$907$	− 10308.6i	− 0.377390i	0.982036	−	0.188695i	$-0.0604258\pi$
$908$	8679.79i	0.317235i
$909$	−6677.26	−0.243642
$910$	0	0
$911$	−22590.8	−0.821589	−0.410795	−	0.911728i	$-0.634749\pi$
$911$	−22590.8	−0.821589	−0.410795	+	0.911728i	$0.634749\pi$
$912$	2919.22i	0.105992i
$913$	12150.8i	0.440453i
$914$	20787.5	0.752288
$915$	0	0
$916$	−4571.56	−0.164900
$917$	13628.0i	0.490769i
$918$	6351.90i	0.228370i
$919$	−47712.1	−1.71260	−0.856299	−	0.516481i	$-0.827242\pi$
$919$	−47712.1	−1.71260	−0.856299	+	0.516481i	$0.827242\pi$
$920$	0	0
$921$	−25329.2	−0.906216
$922$	10294.9i	0.367728i
$923$	18323.9i	0.653456i
$924$	528.000	0.0187986
$925$	0	0
$926$	9328.96	0.331068
$927$	− 3508.53i	− 0.124310i
$928$	2184.19i	0.0772625i
$929$	−10714.3	−0.378391	−0.189196	−	0.981939i	$-0.560588\pi$
$929$	−10714.3	−0.378391	−0.189196	+	0.981939i	$0.560588\pi$
$930$	0	0
$931$	−7348.84	−0.258699
$932$	− 6605.06i	− 0.232142i
$933$	13341.5i	0.468148i
$934$	−19889.1	−0.696780
$935$	0	0
$936$	−13503.9	−0.471568
$937$	14719.5i	0.513197i	0.966518	+	0.256599i	$0.0826018\pi$
$937$	14719.5i	0.513197i	−0.966518	+	0.256599i	$0.917398\pi$
$938$	− 438.612i	− 0.0152678i
$939$	−19442.2	−0.675691
$940$	0	0
$941$	3694.44	0.127987	0.0639933	−	0.997950i	$-0.479616\pi$
$941$	3694.44	0.127987	0.0639933	+	0.997950i	$0.479616\pi$
$942$	20245.1i	0.700234i
$943$	− 28307.4i	− 0.977535i
$944$	7721.23	0.266213
$945$	0	0
$946$	−5584.77	−0.191941
$947$	− 36416.7i	− 1.24961i	−0.780780	−	0.624806i	$-0.785178\pi$
$947$	− 36416.7i	− 1.24961i	0.780780	−	0.624806i	$-0.214822\pi$
$948$	− 694.240i	− 0.0237847i
$949$	26087.4	0.892342
$950$	0	0
$951$	−3758.25	−0.128149
$952$	16457.6i	0.560289i
$953$	20779.4i	0.706306i	0.935566	+	0.353153i	$0.114890\pi$
$953$	20779.4i	0.706306i	−0.935566	+	0.353153i	$0.885110\pi$
$954$	6952.84	0.235961
$955$	0	0
$956$	−2900.35	−0.0981212
$957$	697.123i	0.0235473i
$958$	− 22617.6i	− 0.762777i
$959$	−9950.94	−0.335071
$960$	0	0
$961$	71789.7	2.40978
$962$	21753.7i	0.729071i
$963$	13845.8i	0.463318i
$964$	−4775.43	−0.159550
$965$	0	0
$966$	−5376.00	−0.179058
$967$	− 56812.8i	− 1.88932i	−0.328044	−	0.944662i	$-0.606389\pi$
$967$	− 56812.8i	− 1.88932i	0.328044	−	0.944662i	$-0.393611\pi$
$968$	2977.32i	0.0988582i
$969$	7348.70	0.243627
$970$	0	0
$971$	−26459.7	−0.874493	−0.437247	−	0.899342i	$-0.644046\pi$
$971$	−26459.7	−0.874493	−0.437247	+	0.899342i	$0.644046\pi$
$972$	576.464i	0.0190227i
$973$	− 10951.2i	− 0.360822i
$974$	10559.2	0.347371
$975$	0	0
$976$	15861.7	0.520204
$977$	21009.2i	0.687967i	0.938976	+	0.343984i	$0.111776\pi$
$977$	21009.2i	0.687967i	−0.938976	+	0.343984i	$0.888224\pi$
$978$	10878.2i	0.355670i
$979$	−5093.24	−0.166272
$980$	0	0
$981$	7018.04	0.228409
$982$	− 23147.3i	− 0.752199i
$983$	− 9076.80i	− 0.294512i	−0.989098	−	0.147256i	$-0.952956\pi$
$983$	− 9076.80i	− 0.294512i	0.989098	−	0.147256i	$-0.0470441\pi$
$984$	18657.1	0.604437
$985$	0	0
$986$	−4969.76	−0.160517
$987$	− 2142.13i	− 0.0690828i
$988$	3573.20i	0.115059i
$989$	−23969.8	−0.770673
$990$	0	0
$991$	2629.24	0.0842791	0.0421395	−	0.999112i	$-0.486583\pi$
$991$	2629.24	0.0842791	0.0421395	+	0.999112i	$0.486583\pi$
$992$	− 32953.5i	− 1.05471i
$993$	11404.3i	0.364457i
$994$	−4808.00	−0.153421
$995$	0	0
$996$	−7861.40	−0.250098
$997$	50423.9i	1.60175i	0.598833	+	0.800874i	$0.295631\pi$
$997$	50423.9i	1.60175i	−0.598833	+	0.800874i	$0.704369\pi$
$998$	17018.0i	0.539775i
$999$	4060.27	0.128590

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.4.c.i.199.3		4
5.2	odd	4		33.4.a.d.1.1	✓	2
5.3	odd	4		825.4.a.k.1.2		2
5.4	even	2	inner	825.4.c.i.199.2		4
15.2	even	4		99.4.a.e.1.2		2
15.8	even	4		2475.4.a.o.1.1		2
20.7	even	4		528.4.a.o.1.2		2
35.27	even	4		1617.4.a.j.1.1		2
40.27	even	4		2112.4.a.bh.1.1		2
40.37	odd	4		2112.4.a.ba.1.1		2
55.32	even	4		363.4.a.j.1.2		2
60.47	odd	4		1584.4.a.x.1.1		2
165.32	odd	4		1089.4.a.t.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.4.a.d.1.1	✓	2	5.2	odd	4
99.4.a.e.1.2		2	15.2	even	4
363.4.a.j.1.2		2	55.32	even	4
528.4.a.o.1.2		2	20.7	even	4
825.4.a.k.1.2		2	5.3	odd	4
825.4.c.i.199.2		4	5.4	even	2	inner
825.4.c.i.199.3		4	1.1	even	1	trivial
1089.4.a.t.1.1		2	165.32	odd	4
1584.4.a.x.1.1		2	60.47	odd	4
1617.4.a.j.1.1		2	35.27	even	4
2112.4.a.ba.1.1		2	40.37	odd	4
2112.4.a.bh.1.1		2	40.27	even	4
2475.4.a.o.1.1		2	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)

\(f(q)\)	\(=\)	\(q+2.37228i q^{2} +3.00000i q^{3} +2.37228 q^{4} -7.11684 q^{6} -6.74456i q^{7} +24.6060i q^{8} -9.00000 q^{9} +11.0000 q^{11} +7.11684i q^{12} -60.9783i q^{13} +16.0000 q^{14} -39.3940 q^{16} +99.1684i q^{17} -21.3505i q^{18} -24.7011 q^{19} +20.2337 q^{21} +26.0951i q^{22} +112.000i q^{23} -73.8179 q^{24} +144.658 q^{26} -27.0000i q^{27} -16.0000i q^{28} +21.1249 q^{29} -318.717 q^{31} +103.394i q^{32} +33.0000i q^{33} -235.255 q^{34} -21.3505 q^{36} +150.380i q^{37} -58.5979i q^{38} +182.935 q^{39} -252.745 q^{41} +48.0000i q^{42} +214.016i q^{43} +26.0951 q^{44} -265.696 q^{46} -105.870i q^{47} -118.182i q^{48} +297.511 q^{49} -297.505 q^{51} -144.658i q^{52} +325.652i q^{53} +64.0516 q^{54} +165.957 q^{56} -74.1032i q^{57} +50.1143i q^{58} -196.000 q^{59} -402.641 q^{61} -756.087i q^{62} +60.7011i q^{63} -560.432 q^{64} -78.2853 q^{66} -27.4132i q^{67} +235.255i q^{68} -336.000 q^{69} -300.500 q^{71} -221.454i q^{72} +427.815i q^{73} -356.745 q^{74} -58.5979 q^{76} -74.1902i q^{77} +433.973i q^{78} -97.5488 q^{79} +81.0000 q^{81} -599.581i q^{82} +1104.62i q^{83} +48.0000 q^{84} -507.707 q^{86} +63.3748i q^{87} +270.666i q^{88} -463.022 q^{89} -411.272 q^{91} +265.696i q^{92} -956.152i q^{93} +251.152 q^{94} -310.182 q^{96} +1798.36i q^{97} +705.779i q^{98} -99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{4} + 6 q^{6} - 36 q^{9} + 44 q^{11} + 64 q^{14} - 238 q^{16} + 108 q^{19} + 12 q^{21} - 54 q^{24} + 188 q^{26} - 444 q^{29} - 80 q^{31} - 964 q^{34} + 18 q^{36} + 456 q^{39} - 988 q^{41} - 22 q^{44}+ \cdots - 396 q^{99}+O(q^{100}) \) 4 * q - 2 * q^4 + 6 * q^6 - 36 * q^9 + 44 * q^11 + 64 * q^14 - 238 * q^16 + 108 * q^19 + 12 * q^21 - 54 * q^24 + 188 * q^26 - 444 * q^29 - 80 * q^31 - 964 * q^34 + 18 * q^36 + 456 * q^39 - 988 * q^41 - 22 * q^44 + 224 * q^46 + 1236 * q^49 - 156 * q^51 - 54 * q^54 + 480 * q^56 - 784 * q^59 - 2208 * q^61 - 1426 * q^64 + 66 * q^66 - 1344 * q^69 + 912 * q^71 - 1404 * q^74 - 648 * q^76 + 460 * q^79 + 324 * q^81 + 192 * q^84 - 2904 * q^86 - 1944 * q^89 - 680 * q^91 + 1648 * q^94 - 1482 * q^96 - 396 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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