Properties

Label 349.2.k.a
Level 349
Weight 2
Character orbit 349.k
Analytic conductor 2.787
Analytic rank 0
Dimension 1624
CM No

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Newspace parameters

Level: \( N \) = \( 349 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 349.k (of order \(174\) and degree \(56\))

Newform invariants

Self dual: No
Analytic conductor: \(2.78677903054\)
Analytic rank: \(0\)
Dimension: \(1624\)
Relative dimension: \(29\) over \(\Q(\zeta_{174})\)
Sato-Tate group: $\mathrm{SU}(2)[C_{174}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 1624q - 55q^{2} - 58q^{3} - 85q^{4} - 60q^{5} - 58q^{6} - 58q^{7} - 145q^{8} - 29q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 1624q - 55q^{2} - 58q^{3} - 85q^{4} - 60q^{5} - 58q^{6} - 58q^{7} - 145q^{8} - 29q^{9} - 58q^{10} - 58q^{11} - 57q^{12} - 55q^{13} - 48q^{14} - 59q^{15} - 29q^{16} - 48q^{17} + 43q^{18} + 67q^{19} - 58q^{20} + 58q^{21} - 55q^{22} - 41q^{23} - 39q^{24} - 29q^{25} - 112q^{26} - 76q^{27} - 58q^{28} - 56q^{29} - 67q^{30} - 90q^{31} - 67q^{32} - 46q^{33} + 5q^{34} - 82q^{36} + 45q^{37} - 58q^{38} - 58q^{39} - 112q^{40} + 10q^{41} + 305q^{42} - 13q^{43} + 40q^{44} - 56q^{45} - 85q^{46} - 58q^{47} - 122q^{48} - 93q^{49} - 441q^{50} - 160q^{51} + 58q^{52} - 58q^{53} - 85q^{54} + 34q^{55} - 69q^{56} - 29q^{57} - 58q^{58} - 40q^{59} + 522q^{60} + 87q^{61} - 49q^{62} - 37q^{63} + 335q^{64} - 58q^{65} - 565q^{66} - 116q^{67} - 58q^{68} + 16q^{69} - 98q^{70} - 34q^{71} + 44q^{72} - 52q^{73} - 204q^{74} - 95q^{76} - 54q^{77} + 439q^{78} - 58q^{79} - 192q^{80} - 209q^{81} - 79q^{82} + 162q^{83} + 330q^{84} - 192q^{85} + 126q^{86} - 190q^{87} - 8q^{88} + 345q^{89} - 46q^{90} - 30q^{91} + 62q^{92} - 89q^{93} - 87q^{94} - 350q^{95} + 990q^{96} - 43q^{97} - 464q^{98} - 97q^{99} + O(q^{100}) \)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1 −2.69504 0.644448i 2.08866 0.381263i 5.06425 + 2.56886i −0.186969 1.47146i −5.87473 0.318518i 3.45075 + 0.0623105i −7.76892 6.59898i 1.41063 0.532742i −0.444394 + 4.08614i
3.2 −2.68985 0.643207i −1.01045 + 0.184447i 5.03791 + 2.55550i 0.223073 + 1.75561i 2.83660 + 0.153796i −3.93989 0.0711429i −7.69170 6.53339i −1.81953 + 0.687167i 0.529188 4.86580i
3.3 −2.36455 0.565422i −2.19178 + 0.400086i 3.48776 + 1.76918i −0.444524 3.49845i 5.40880 + 0.293257i 2.72845 + 0.0492679i −3.54070 3.00750i 1.83731 0.693883i −0.926999 + 8.52361i
3.4 −2.12865 0.509011i 1.03227 0.188430i 2.48839 + 1.26225i −0.245130 1.92920i −2.29326 0.124337i −3.23078 0.0583385i −1.31819 1.11968i −1.77644 + 0.670894i −0.460187 + 4.23135i
3.5 −2.03001 0.485425i −0.306098 + 0.0558749i 2.10166 + 1.06608i 0.440893 + 3.46988i 0.648506 + 0.0351610i 2.50769 + 0.0452815i −0.567264 0.481838i −2.71595 + 1.02571i 0.789345 7.25791i
3.6 −1.91236 0.457292i −3.10103 + 0.566059i 1.66436 + 0.844252i 0.130497 + 1.02702i 6.18914 + 0.335566i −0.693808 0.0125282i 0.200454 + 0.170267i 6.48944 2.45081i 0.220092 2.02371i
3.7 −1.84881 0.442095i 0.747190 0.136391i 1.43900 + 0.729936i −0.125119 0.984695i −1.44171 0.0781672i 2.06249 + 0.0372425i 0.559908 + 0.475590i −2.26683 + 0.856097i −0.204009 + 1.87583i
3.8 −1.46055 0.349254i 3.22574 0.588823i 0.227590 + 0.115446i −0.420184 3.30689i −4.91701 0.266593i 1.13449 + 0.0204856i 1.99704 + 1.69630i 7.25215 2.73886i −0.541243 + 4.97664i
3.9 −1.25687 0.300549i −1.37630 + 0.251229i −0.294249 0.149259i −0.00486642 0.0382992i 1.80534 + 0.0978827i −0.786455 0.0142011i 2.29487 + 1.94928i −0.975439 + 0.368386i −0.00539431 + 0.0495999i
3.10 −1.25278 0.299570i 1.86516 0.340464i −0.303938 0.154174i 0.0720620 + 0.567135i −2.43862 0.132218i −4.65222 0.0840055i 2.29806 + 1.95199i 0.556366 0.210119i 0.0796187 0.732082i
3.11 −0.795786 0.190292i 2.47576 0.451923i −1.18658 0.601899i 0.341620 + 2.68858i −2.05617 0.111482i 1.87800 + 0.0339112i 2.07697 + 1.76419i 3.11862 1.17778i 0.239759 2.20454i
3.12 −0.787498 0.188310i −1.01339 + 0.184983i −1.19896 0.608175i −0.327670 2.57880i 0.832873 + 0.0451571i 3.73113 + 0.0673734i 2.06390 + 1.75309i −1.81379 + 0.685000i −0.227573 + 2.09250i
3.13 −0.633957 0.151594i −2.19680 + 0.401003i −1.40473 0.712553i −0.512518 4.03357i 1.45347 + 0.0788048i −5.00988 0.0904639i 1.77612 + 1.50865i 1.85862 0.701932i −0.286552 + 2.63480i
3.14 −0.314630 0.0752356i 0.582766 0.106378i −1.69032 0.857419i 0.173657 + 1.36670i −0.191359 0.0103752i −1.44508 0.0260940i 0.960435 + 0.815801i −2.47822 + 0.935931i 0.0481867 0.443069i
3.15 −0.186100 0.0445010i −2.45768 + 0.448624i −1.75100 0.888199i 0.514582 + 4.04981i 0.477340 + 0.0258806i −0.327657 0.00591653i 0.578009 + 0.490966i 3.03242 1.14523i 0.0844570 0.776569i
3.16 0.380511 + 0.0909894i 1.76989 0.323074i −1.64714 0.835517i −0.175332 1.37988i 0.702859 + 0.0381079i 3.99930 + 0.0722157i −1.14711 0.974361i 0.221609 0.0836932i 0.0588387 0.541013i
3.17 0.512744 + 0.122610i −2.99844 + 0.547332i −1.53578 0.779027i −0.141464 1.11334i −1.60454 0.0869956i 1.44369 + 0.0260689i −1.49557 1.27035i 5.88452 2.22236i 0.0639708 0.588202i
3.18 0.716111 + 0.171240i 2.54499 0.464560i −1.30016 0.659509i −0.300536 2.36525i 1.90205 + 0.103126i −2.32067 0.0419046i −1.94048 1.64826i 3.45463 1.30468i 0.189807 1.74525i
3.19 0.815047 + 0.194897i −0.0632014 + 0.0115367i −1.15733 0.587061i −0.348458 2.74240i −0.0537606 0.00291482i −1.20872 0.0218260i −2.10629 1.78910i −2.80266 + 1.05846i 0.250477 2.30310i
3.20 0.830551 + 0.198605i −0.507815 + 0.0926962i −1.13328 0.574859i 0.160603 + 1.26396i −0.440176 0.0238657i −3.59787 0.0649670i −2.12880 1.80822i −2.55724 + 0.965773i −0.117640 + 1.08168i
See next 80 embeddings (of 1624 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 340.29
Significant digits:
Format:

Inner twists

This newform does not have CM; other inner twists have not been computed.

Hecke kernels

There are no other newforms in \(S_{2}^{\mathrm{new}}(349, [\chi])\).